Answer: -9

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 12x + (-13) = -121.

Solution: Isolate the variable. Subtract -13 from both sides: 12x = -108. Divide by 12: x = -9.


Answer: -2

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 6x + (-5) = -17.

Solution: Isolate the variable. Subtract -5 from both sides: 6x = -12. Divide by 6: x = -2.


Answer: 8

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 4x + (-14) = 18.

Solution: Isolate the variable. Subtract -14 from both sides: 4x = 32. Divide by 4: x = 8.


Answer: 4

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 3x + (17) = 29.

Solution: Isolate the variable. Subtract 17 from both sides: 3x = 12. Divide by 3: x = 4.


Answer: -7

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 2x + (-19) = -33.

Solution: Isolate the variable. Subtract -19 from both sides: 2x = -14. Divide by 2: x = -7.


Answer: 7

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 5x + (-6) = 29.

Solution: Isolate the variable. Subtract -6 from both sides: 5x = 35. Divide by 5: x = 7.


Answer: 8

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 11x + (-19) = 69.

Solution: Isolate the variable. Subtract -19 from both sides: 11x = 88. Divide by 11: x = 8.


Answer: 4

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 5x + (14) = 34.

Solution: Isolate the variable. Subtract 14 from both sides: 5x = 20. Divide by 5: x = 4.


Answer: 9

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 5x + (8) = 53.

Solution: Isolate the variable. Subtract 8 from both sides: 5x = 45. Divide by 5: x = 9.


Answer: -4

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 6x + (-20) = -44.

Solution: Isolate the variable. Subtract -20 from both sides: 6x = -24. Divide by 6: x = -4.


Answer: -1

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 8x + (1) = -7.

Solution: Isolate the variable. Subtract 1 from both sides: 8x = -8. Divide by 8: x = -1.


Answer: 1

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 4x + (-7) = -3.

Solution: Isolate the variable. Subtract -7 from both sides: 4x = 4. Divide by 4: x = 1.


Answer: 3

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 3x + (-15) = -6.

Solution: Isolate the variable. Subtract -15 from both sides: 3x = 9. Divide by 3: x = 3.


Answer: 2

Exact integer or rational solution
Single real unknown

Move constants
Divide by the coefficient of x
Simplify

Prompt: Solve for x: 3x + (2) = 8.

Solution: Isolate the variable. Subtract 2 from both sides: 3x = 6. Divide by 3: x = 2.


Answer: -2, 3

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 - x - 6 = 0.

Solution: The monic factorisation is proportional to (x - (3))(x - (-2)). The roots are 3 and -2.


Answer: 1, 5

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 - 6*x + 5 = 0.

Solution: The monic factorisation is proportional to (x - (5))(x - (1)). The roots are 5 and 1.


Answer: -5, 0

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 + 5*x = 0.

Solution: The monic factorisation is proportional to (x - (0))(x - (-5)). The roots are 0 and -5.


Answer: 3, 4

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 - 7*x + 12 = 0.

Solution: The monic factorisation is proportional to (x - (4))(x - (3)). The roots are 4 and 3.


Answer: -3, 3

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 - 9 = 0.

Solution: The monic factorisation is proportional to (x - (3))(x - (-3)). The roots are 3 and -3.


Answer: -6, 4

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 + 2*x - 24 = 0.

Solution: The monic factorisation is proportional to (x - (-6))(x - (4)). The roots are -6 and 4.


Answer: -2, 6

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 - 4*x - 12 = 0.

Solution: The monic factorisation is proportional to (x - (6))(x - (-2)). The roots are 6 and -2.


Answer: -5, -3

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of 2*x**2 + 16*x + 30 = 0.

Solution: The monic factorisation is proportional to (x - (-3))(x - (-5)). The roots are -3 and -5.


Answer: -2, 1

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 + x - 2 = 0.

Solution: The monic factorisation is proportional to (x - (-2))(x - (1)). The roots are -2 and 1.


Answer: -4, -1

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 + 5*x + 4 = 0.

Solution: The monic factorisation is proportional to (x - (-4))(x - (-1)). The roots are -4 and -1.


Answer: -3, 4

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 - x - 12 = 0.

Solution: The monic factorisation is proportional to (x - (-3))(x - (4)). The roots are -3 and 4.


Answer: 4, 5

Work over the rationals
Report every root

Identify a quadratic
Factor or use the quadratic formula
Check by substitution

Prompt: Find all roots of x**2 - 9*x + 20 = 0.

Solution: The monic factorisation is proportional to (x - (5))(x - (4)). The roots are 5 and 4.


Answer: x=4, y=5

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 2x + 5y = 33; 2x + 2y = 18.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = 4, y = 5.


Answer: x=2, y=1

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 3x + 5y = 11; 2x + 3y = 7.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = 2, y = 1.


Answer: x=-5, y=-2

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 1x + 3y = -11; 4x + 3y = -26.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = -5, y = -2.


Answer: x=-4, y=-2

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 5x + 3y = -26; 2x + 4y = -16.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = -4, y = -2.


Answer: x=1, y=5

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 4x + 2y = 14; 3x + 2y = 13.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = 1, y = 5.


Answer: x=-2, y=3

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 5x + 3y = -1; 5x + 4y = 2.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = -2, y = 3.


Answer: x=4, y=1

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 3x + 2y = 14; 2x + 5y = 13.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = 4, y = 1.


Answer: x=2, y=-4

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 1x + 1y = -2; 2x + 3y = -8.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = 2, y = -4.


Answer: x=5, y=1

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 5x + 1y = 26; 4x + 4y = 24.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = 5, y = 1.


Answer: x=4, y=2

Unique real solution
Integer coefficients

Write the coefficient matrix
Invert or eliminate
Back-substitute

Prompt: Solve the system: 5x + 3y = 26; 5x + 1y = 22.

Solution: Use elimination or Cramer's rule. The unique solution of this invertible 2×2 system is x = 4, y = 2.


Answer: -5/3

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 2/6 - 6/3 as a simplified rational number.

Solution: Convert to rationals and simplify: 1/3 - 2 = -5/3.


Answer: 7

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 7/4 * 8/2 as a simplified rational number.

Solution: Convert to rationals and simplify: 7/4 * 4 = 7.


Answer: 15/4

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 5/4 * 9/3 as a simplified rational number.

Solution: Convert to rationals and simplify: 5/4 * 3 = 15/4.


Answer: 10/7

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 5/5 + 3/7 as a simplified rational number.

Solution: Convert to rationals and simplify: 1 + 3/7 = 10/7.


Answer: 31/6

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 9/2 + 6/9 as a simplified rational number.

Solution: Convert to rationals and simplify: 9/2 + 2/3 = 31/6.


Answer: 9/7

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 2/7 + 5/5 as a simplified rational number.

Solution: Convert to rationals and simplify: 2/7 + 1 = 9/7.


Answer: 14/9

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 4/3 + 2/9 as a simplified rational number.

Solution: Convert to rationals and simplify: 4/3 + 2/9 = 14/9.


Answer: 3/4

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 9/4 * 3/9 as a simplified rational number.

Solution: Convert to rationals and simplify: 9/4 * 1/3 = 3/4.


Answer: 13/8

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 3/6 + 9/8 as a simplified rational number.

Solution: Convert to rationals and simplify: 1/2 + 9/8 = 13/8.


Answer: 9/8

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 9/5 * 5/8 as a simplified rational number.

Solution: Convert to rationals and simplify: 9/5 * 5/8 = 9/8.


Answer: 5/3

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 6/9 + 9/9 as a simplified rational number.

Solution: Convert to rationals and simplify: 2/3 + 1 = 5/3.


Answer: 38/35

Write each term as a Rational
Apply the operator
Reduce

Prompt: Compute 4/5 + 2/7 as a simplified rational number.

Solution: Convert to rationals and simplify: 4/5 + 2/7 = 38/35.


Answer: 1

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(76, 35).

Solution: gcd(76, 35) = 1, so lcm(76, 35) = 76×35/gcd = 2660. Requested value: 1.


Answer: 3

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(6, 15).

Solution: gcd(6, 15) = 3, so lcm(6, 15) = 6×15/gcd = 30. Requested value: 3.


Answer: 7

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(35, 14).

Solution: gcd(35, 14) = 7, so lcm(35, 14) = 35×14/gcd = 70. Requested value: 7.


Answer: 3

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(48, 15).

Solution: gcd(48, 15) = 3, so lcm(48, 15) = 48×15/gcd = 240. Requested value: 3.


Answer: 1

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(41, 68).

Solution: gcd(41, 68) = 1, so lcm(41, 68) = 41×68/gcd = 2788. Requested value: 1.


Answer: 1650

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute LCM(75, 22).

Solution: gcd(75, 22) = 1, so lcm(75, 22) = 75×22/gcd = 1650. Requested value: 1650.


Answer: 2442

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute LCM(37, 66).

Solution: gcd(37, 66) = 1, so lcm(37, 66) = 37×66/gcd = 2442. Requested value: 2442.


Answer: 6

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(30, 18).

Solution: gcd(30, 18) = 6, so lcm(30, 18) = 30×18/gcd = 90. Requested value: 6.


Answer: 3111

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute LCM(61, 51).

Solution: gcd(61, 51) = 1, so lcm(61, 51) = 61×51/gcd = 3111. Requested value: 3111.


Answer: 1

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(58, 65).

Solution: gcd(58, 65) = 1, so lcm(58, 65) = 58×65/gcd = 3770. Requested value: 1.


Answer: 234

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute LCM(18, 13).

Solution: gcd(18, 13) = 1, so lcm(18, 13) = 18×13/gcd = 234. Requested value: 234.


Answer: 1

Euclidean algorithm for gcd
Use lcm = |ab|/gcd if needed

Prompt: Compute GCD(49, 19).

Solution: gcd(49, 19) = 1, so lcm(49, 19) = 49×19/gcd = 931. Requested value: 1.


Answer: 1

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 59^6 mod 6.

Solution: Use modular exponentiation: 59^6 ≡ 1 (mod 6).


Answer: 4

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 124^5 mod 5.

Solution: Use modular exponentiation: 124^5 ≡ 4 (mod 5).


Answer: 0

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 56^5 mod 7.

Solution: Use modular exponentiation: 56^5 ≡ 0 (mod 7).


Answer: 1

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 73^2 mod 16.

Solution: Use modular exponentiation: 73^2 ≡ 1 (mod 16).


Answer: 6

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 123^8 mod 15.

Solution: Use modular exponentiation: 123^8 ≡ 6 (mod 15).


Answer: 0

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 150^2 mod 4.

Solution: Use modular exponentiation: 150^2 ≡ 0 (mod 4).


Answer: 0

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 176^8 mod 11.

Solution: Use modular exponentiation: 176^8 ≡ 0 (mod 11).


Answer: 1

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 13^8 mod 4.

Solution: Use modular exponentiation: 13^8 ≡ 1 (mod 4).


Answer: 0

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 70^5 mod 5.

Solution: Use modular exponentiation: 70^5 ≡ 0 (mod 5).


Answer: 4

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 134^3 mod 10.

Solution: Use modular exponentiation: 134^3 ≡ 4 (mod 10).


Answer: 15

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 112^2 mod 17.

Solution: Use modular exponentiation: 112^2 ≡ 15 (mod 17).


Answer: 4

Reduce the base modulo m
Square-and-multiply

Prompt: Compute 52^2 mod 9.

Solution: Use modular exponentiation: 52^2 ≡ 4 (mod 9).


Answer: 12*x**2 - 1

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 4*x**3 - x + 6.

Solution: Power rule: d/dx[4x^3] = 12x^2, plus -1. Derivative: 12*x**2 - 1.


Answer: 12*x + 3

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 6*x**2 + 3*x + 7.

Solution: Power rule: d/dx[6x^2] = 12x^1, plus 3. Derivative: 12*x + 3.


Answer: 12*x**3 - 2

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 3*x**4 - 2*x - 7.

Solution: Power rule: d/dx[3x^4] = 12x^3, plus -2. Derivative: 12*x**3 - 2.


Answer: 12*x

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 6*x**2 - 7.

Solution: Power rule: d/dx[6x^2] = 12x^1, plus 0. Derivative: 12*x.


Answer: 10*x + 3

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 5*x**2 + 3*x - 3.

Solution: Power rule: d/dx[5x^2] = 10x^1, plus 3. Derivative: 10*x + 3.


Answer: 8*x**3 - 4

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 2*x**4 - 4*x - 6.

Solution: Power rule: d/dx[2x^4] = 8x^3, plus -4. Derivative: 8*x**3 - 4.


Answer: 16*x**3 - 4

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 4*x**4 - 4*x - 1.

Solution: Power rule: d/dx[4x^4] = 16x^3, plus -4. Derivative: 16*x**3 - 4.


Answer: 20*x**3 - 4

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 5*x**4 - 4*x + 5.

Solution: Power rule: d/dx[5x^4] = 20x^3, plus -4. Derivative: 20*x**3 - 4.


Answer: 6*x - 2

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 3*x**2 - 2*x + 2.

Solution: Power rule: d/dx[3x^2] = 6x^1, plus -2. Derivative: 6*x - 2.


Answer: 12*x**2 - 3

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: 4*x**3 - 3*x + 1.

Solution: Power rule: d/dx[4x^3] = 12x^2, plus -3. Derivative: 12*x**2 - 3.


Answer: 4*x**3 - 5

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: x**4 - 5*x + 6.

Solution: Power rule: d/dx[1x^4] = 4x^3, plus -5. Derivative: 4*x**3 - 5.


Answer: 4*x**3 + 3

Symbolic derivative
Do not expand unnecessarily

Apply the power rule termwise
Differentiate the linear term
Constant vanishes

Prompt: Differentiate with respect to x: x**4 + 3*x - 2.

Solution: Power rule: d/dx[1x^4] = 4x^3, plus 3. Derivative: 4*x**3 + 3.


Answer: 16/3

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^2 2*x**2 dx.

Solution: Antiderivative is 2x^3/3. Evaluate from 0 to 2: 16/3.


Answer: 128/3

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^4 2*x**2 dx.

Solution: Antiderivative is 2x^3/3. Evaluate from 0 to 4: 128/3.


Answer: 20

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^2 5*x**3 dx.

Solution: Antiderivative is 5x^4/4. Evaluate from 0 to 2: 20.


Answer: 40/3

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^2 5*x**2 dx.

Solution: Antiderivative is 5x^3/3. Evaluate from 0 to 2: 40/3.


Answer: 81/4

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^3 x**3 dx.

Solution: Antiderivative is 1x^4/4. Evaluate from 0 to 3: 81/4.


Answer: 27

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^3 3*x**2 dx.

Solution: Antiderivative is 3x^3/3. Evaluate from 0 to 3: 27.


Answer: 24

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^4 3*x dx.

Solution: Antiderivative is 3x^2/2. Evaluate from 0 to 4: 24.


Answer: 320/3

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^4 5*x**2 dx.

Solution: Antiderivative is 5x^3/3. Evaluate from 0 to 4: 320/3.


Answer: 25/2

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^5 x dx.

Solution: Antiderivative is 1x^2/2. Evaluate from 0 to 5: 25/2.


Answer: 6

Find an antiderivative
Apply the fundamental theorem

Prompt: Evaluate the definite integral ∫_0^2 3*x dx.

Solution: Antiderivative is 3x^2/2. Evaluate from 0 to 2: 6.


Answer: -1

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of cos(pi).

Solution: This is a standard unit-circle value: cos(pi) = -1.


Answer: 1/2

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of sin(pi/6).

Solution: This is a standard unit-circle value: sin(pi/6) = 1/2.


Answer: sqrt(3)/3

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of tan(pi/6).

Solution: This is a standard unit-circle value: tan(pi/6) = sqrt(3)/3.


Answer: 0

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of sin(0).

Solution: This is a standard unit-circle value: sin(0) = 0.


Answer: sqrt(3)/2

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of sin(pi/3).

Solution: This is a standard unit-circle value: sin(pi/3) = sqrt(3)/2.


Answer: -1/2

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of cos(2*pi/3).

Solution: This is a standard unit-circle value: cos(2*pi/3) = -1/2.


Answer: 1/2

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of cos(pi/3).

Solution: This is a standard unit-circle value: cos(pi/3) = 1/2.


Answer: sqrt(3)/2

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of cos(pi/6).

Solution: This is a standard unit-circle value: cos(pi/6) = sqrt(3)/2.


Answer: 1

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of tan(pi/4).

Solution: This is a standard unit-circle value: tan(pi/4) = 1.


Answer: 1

Exact radical or rational form
No decimal approximation

Recall the unit circle
Reduce the angle if needed
Simplify

Prompt: Give the exact value of sin(pi/2).

Solution: This is a standard unit-circle value: sin(pi/2) = 1.


Answer: 20

Identify the formula
Substitute
Simplify

Prompt: A rectangle has width 5 and height 4. What is its area?

Solution: Area = width × height = 5×4 = 20.


Answer: 16*pi

Identify the formula
Substitute
Simplify

Prompt: A circle has radius 4. Give its area in terms of pi.

Solution: Area = πr2 = π·42 = 16*pi.


Answer: sqrt(137)

Identify the formula
Substitute
Simplify

Prompt: A right triangle has legs 11 and 4. What is the hypotenuse?

Solution: Pythagoras: sqrt(112 + 42) = sqrt(137).


Answer: 7*sqrt(39)/4

Identify the formula
Substitute
Simplify

Prompt: A triangle has side lengths 7, 10, and 4. Give its area using Heron's formula.

Solution: s = 21/2. Area = sqrt(s(s-a)(s-b)(s-c)) = 7*sqrt(39)/4.


Answer: 90

Identify the formula
Substitute
Simplify

Prompt: A rectangle has width 15 and height 6. What is its area?

Solution: Area = width × height = 15×6 = 90.


Answer: 4*pi

Identify the formula
Substitute
Simplify

Prompt: A circle has radius 2. Give its area in terms of pi.

Solution: Area = πr2 = π·22 = 4*pi.


Answer: sqrt(130)

Identify the formula
Substitute
Simplify

Prompt: A right triangle has legs 7 and 9. What is the hypotenuse?

Solution: Pythagoras: sqrt(72 + 92) = sqrt(130).


Answer: 45

Identify the formula
Substitute
Simplify

Prompt: A rectangle has width 9 and height 5. What is its area?

Solution: Area = width × height = 9×5 = 45.


Answer: 49*pi

Identify the formula
Substitute
Simplify

Prompt: A circle has radius 7. Give its area in terms of pi.

Solution: Area = πr2 = π·72 = 49*pi.


Answer: 3*sqrt(41)

Identify the formula
Substitute
Simplify

Prompt: A right triangle has legs 15 and 12. What is the hypotenuse?

Solution: Pythagoras: sqrt(152 + 122) = 3*sqrt(41).


Answer: 3*sqrt(119)/4

Identify the formula
Substitute
Simplify

Prompt: A triangle has side lengths 8, 4, and 5. Give its area using Heron's formula.

Solution: s = 17/2. Area = sqrt(s(s-a)(s-b)(s-c)) = 3*sqrt(119)/4.


Answer: 49

Identify the formula
Substitute
Simplify

Prompt: A rectangle has width 7 and height 7. What is its area?

Solution: Area = width × height = 7×7 = 49.


Answer: 64*pi

Identify the formula
Substitute
Simplify

Prompt: A circle has radius 8. Give its area in terms of pi.

Solution: Area = πr2 = π·82 = 64*pi.


Answer: 3*sqrt(5)

Identify the formula
Substitute
Simplify

Prompt: A right triangle has legs 3 and 6. What is the hypotenuse?

Solution: Pythagoras: sqrt(32 + 62) = 3*sqrt(5).


Answer: 10*sqrt(3)

Identify the formula
Substitute
Simplify

Prompt: A triangle has side lengths 7, 8, and 5. Give its area using Heron's formula.

Solution: s = 10. Area = sqrt(s(s-a)(s-b)(s-c)) = 10*sqrt(3).


Answer: 84

Prompt: Compute C(9, 3) = n choose k.

Solution: C(n,k) = n! / (k!(n-k)!) = 84.


Answer: 110

Prompt: Compute P(11, 2) = n permute k.

Solution: P(n,k) = n! / (n-k)! = 110.


Answer: 336

Prompt: Compute P(8, 3) = n permute k.

Solution: P(n,k) = n! / (n-k)! = 336.


Answer: 210

Prompt: Compute C(10, 4) = n choose k.

Solution: C(n,k) = n! / (k!(n-k)!) = 210.


Answer: 28

Prompt: Compute C(8, 2) = n choose k.

Solution: C(n,k) = n! / (k!(n-k)!) = 28.


Answer: 7920

Prompt: Compute P(11, 4) = n permute k.

Solution: P(n,k) = n! / (n-k)! = 7920.


Answer: 360

Prompt: Compute P(6, 4) = n permute k.

Solution: P(n,k) = n! / (n-k)! = 360.


Answer: 330

Prompt: Compute C(11, 4) = n choose k.

Solution: C(n,k) = n! / (k!(n-k)!) = 330.


Answer: 15

Prompt: Compute C(6, 4) = n choose k.

Solution: C(n,k) = n! / (k!(n-k)!) = 15.


Answer: 36

Prompt: Compute C(9, 2) = n choose k.

Solution: C(n,k) = n! / (k!(n-k)!) = 36.


Answer: 256/2187

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 1/3. In 9 independent trials, what is P(exactly 1 successes)?

Solution: Binomial pmf: C(9,1) 1/3^1 (1-1/3)^8 = 256/2187.


Answer: 1280/6561

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 1/3. In 10 independent trials, what is P(exactly 2 successes)?

Solution: Binomial pmf: C(10,2) 1/3^2 (1-1/3)^8 = 1280/6561.


Answer: 5103/16384

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 1/4. In 7 independent trials, what is P(exactly 2 successes)?

Solution: Binomial pmf: C(7,2) 1/4^2 (1-1/4)^5 = 5103/16384.


Answer: 512/2187

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 1/3. In 9 independent trials, what is P(exactly 2 successes)?

Solution: Binomial pmf: C(9,2) 1/3^2 (1-1/3)^7 = 512/2187.


Answer: 405/1048576

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 3/4. In 10 independent trials, what is P(exactly 2 successes)?

Solution: Binomial pmf: C(10,2) 3/4^2 (1-3/4)^8 = 405/1048576.


Answer: 236196/1953125

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 2/5. In 10 independent trials, what is P(exactly 2 successes)?

Solution: Binomial pmf: C(10,2) 2/5^2 (1-2/5)^8 = 236196/1953125.


Answer: 432/3125

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 3/5. In 6 independent trials, what is P(exactly 2 successes)?

Solution: Binomial pmf: C(6,2) 3/5^2 (1-3/5)^4 = 432/3125.


Answer: 32805/131072

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 1/4. In 10 independent trials, what is P(exactly 3 successes)?

Solution: Binomial pmf: C(10,3) 1/4^3 (1-1/4)^7 = 32805/131072.


Answer: 1701/8192

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 1/4. In 8 independent trials, what is P(exactly 3 successes)?

Solution: Binomial pmf: C(8,3) 1/4^3 (1-1/4)^5 = 1701/8192.


Answer: 7/128

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 1/2. In 7 independent trials, what is P(exactly 1 successes)?

Solution: Binomial pmf: C(7,1) 1/2^1 (1-1/2)^6 = 7/128.


Answer: 112/6561

Independent trials
Exact rational probability

Identify n, k, p
Apply the binomial formula

Prompt: A Bernoulli trial succeeds with probability 2/3. In 8 independent trials, what is P(exactly 2 successes)?

Solution: Binomial pmf: C(8,2) 2/3^2 (1-2/3)^6 = 112/6561.


Answer: 10

Compute the mean
If variance, average squared deviations

Prompt: For the sample [6, 3, 10, 17, 20, 11, 3], compute the population mean (divide by n, not n-1).

Solution: mean = 10; population variance = 264/7. Requested: 10.


Answer: 281/36

Compute the mean
If variance, average squared deviations

Prompt: For the sample [8, 7, 5, 1, 2, 8], compute the population variance (divide by n, not n-1).

Solution: mean = 31/6; population variance = 281/36. Requested: 281/36.


Answer: 299/16

Compute the mean
If variance, average squared deviations

Prompt: For the sample [15, 14, 19, 7], compute the population variance (divide by n, not n-1).

Solution: mean = 55/4; population variance = 299/16. Requested: 299/16.


Answer: 53/7

Compute the mean
If variance, average squared deviations

Prompt: For the sample [13, 8, 5, 1, 4, 14, 8], compute the population mean (divide by n, not n-1).

Solution: mean = 53/7; population variance = 936/49. Requested: 53/7.


Answer: 1692/49

Compute the mean
If variance, average squared deviations

Prompt: For the sample [2, 18, 8, 4, 15, 5, 15], compute the population variance (divide by n, not n-1).

Solution: mean = 67/7; population variance = 1692/49. Requested: 1692/49.


Answer: 846/49

Compute the mean
If variance, average squared deviations

Prompt: For the sample [20, 17, 14, 18, 15, 6, 16], compute the population variance (divide by n, not n-1).

Solution: mean = 106/7; population variance = 846/49. Requested: 846/49.


Answer: 521/36

Compute the mean
If variance, average squared deviations

Prompt: For the sample [8, 9, 17, 16, 8, 9], compute the population variance (divide by n, not n-1).

Solution: mean = 67/6; population variance = 521/36. Requested: 521/36.


Answer: 5/4

Compute the mean
If variance, average squared deviations

Prompt: For the sample [10, 8, 9, 11], compute the population variance (divide by n, not n-1).

Solution: mean = 19/2; population variance = 5/4. Requested: 5/4.


Answer: 31/4

Compute the mean
If variance, average squared deviations

Prompt: For the sample [5, 5, 8, 13], compute the population mean (divide by n, not n-1).

Solution: mean = 31/4; population variance = 171/16. Requested: 31/4.


Answer: 126/5

Compute the mean
If variance, average squared deviations

Prompt: For the sample [3, 14, 14, 11, 18], compute the population variance (divide by n, not n-1).

Solution: mean = 12; population variance = 126/5. Requested: 126/5.


Answer: 2

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[1, -5], [-2, 1]]. Compute tr(A).

Solution: det = ad-bc = 1·1 - -5·-2 = -9; trace = 1+1 = 2.


Answer: 5

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[4, -5], [4, 1]]. Compute tr(A).

Solution: det = ad-bc = 4·1 - -5·4 = 24; trace = 4+1 = 5.


Answer: -4

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[-5, 0], [-1, 1]]. Compute tr(A).

Solution: det = ad-bc = -5·1 - 0·-1 = -5; trace = -5+1 = -4.


Answer: 1

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[3, 3], [4, -2]]. Compute tr(A).

Solution: det = ad-bc = 3·-2 - 3·4 = -18; trace = 3+-2 = 1.


Answer: -3

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[-2, -1], [1, 2]]. Compute det(A).

Solution: det = ad-bc = -2·2 - -1·1 = -3; trace = -2+2 = 0.


Answer: 6

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[1, 0], [5, 5]]. Compute tr(A).

Solution: det = ad-bc = 1·5 - 0·5 = 5; trace = 1+5 = 6.


Answer: -6

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[-3, 2], [-3, 4]]. Compute det(A).

Solution: det = ad-bc = -3·4 - 2·-3 = -6; trace = -3+4 = 1.


Answer: -11

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[1, 4], [4, 5]]. Compute det(A).

Solution: det = ad-bc = 1·5 - 4·4 = -11; trace = 1+5 = 6.


Answer: -7

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[-4, 5], [1, -3]]. Compute tr(A).

Solution: det = ad-bc = -4·-3 - 5·1 = 7; trace = -4+-3 = -7.


Answer: -2

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[-3, -5], [-1, 1]]. Compute tr(A).

Solution: det = ad-bc = -3·1 - -5·-1 = -8; trace = -3+1 = -2.


Answer: -2

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[-2, 2], [0, 0]]. Compute tr(A).

Solution: det = ad-bc = -2·0 - 2·0 = 0; trace = -2+0 = -2.


Answer: -5

Write the matrix
Apply det = ad-bc or tr = a+d

Prompt: Let A = [[-1, 1], [-1, -4]]. Compute tr(A).

Solution: det = ad-bc = -1·-4 - 1·-1 = 5; trace = -1+-4 = -5.


Answer: 49/4

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -x**2 - 7*x on the real line.

Solution: f'(x) = -2*x - 7 = 0 at x = -7/2. f''(x) = -2 < 0, so this is a maximum, f = 49/4.


Answer: 19/2

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -2*x**2 - 6*x + 5 on the real line.

Solution: f'(x) = -4*x - 6 = 0 at x = -3/2. f''(x) = -4 < 0, so this is a maximum, f = 19/2.


Answer: 14

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -x**2 - 8*x - 2 on the real line.

Solution: f'(x) = -2*x - 8 = 0 at x = -4. f''(x) = -2 < 0, so this is a maximum, f = 14.


Answer: 12

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -2*x**2 - 8*x + 4 on the real line.

Solution: f'(x) = -4*x - 8 = 0 at x = -2. f''(x) = -4 < 0, so this is a maximum, f = 12.


Answer: -23/8

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -2*x**2 - x - 3 on the real line.

Solution: f'(x) = -4*x - 1 = 0 at x = -1/4. f''(x) = -4 < 0, so this is a maximum, f = -23/8.


Answer: 89/16

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -4*x**2 - 5*x + 4 on the real line.

Solution: f'(x) = -8*x - 5 = 0 at x = -5/8. f''(x) = -8 < 0, so this is a maximum, f = 89/16.


Answer: 7/2

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -2*x**2 + 6*x - 1 on the real line.

Solution: f'(x) = 6 - 4*x = 0 at x = 3/2. f''(x) = -4 < 0, so this is a maximum, f = 7/2.


Answer: 19/4

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -3*x**2 - 3*x + 4 on the real line.

Solution: f'(x) = -6*x - 3 = 0 at x = -1/2. f''(x) = -6 < 0, so this is a maximum, f = 19/4.


Answer: 5/4

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -x**2 - 3*x - 1 on the real line.

Solution: f'(x) = -2*x - 3 = 0 at x = -3/2. f''(x) = -2 < 0, so this is a maximum, f = 5/4.


Answer: 15

Differentiate
Solve f'=0
Confirm with the second-derivative test

Prompt: Find the maximum value of f(x) = -x**2 - 8*x - 1 on the real line.

Solution: f'(x) = -2*x - 8 = 0 at x = -4. f''(x) = -2 < 0, so this is a maximum, f = 15.


Answer: 57

Prompt: An arithmetic sequence has first term 7 and common difference 5. What is term 11?

Solution: a_k = a1 + (k-1)d = 7 + (11-1)·5 = 57.


Answer: 62

Prompt: An arithmetic sequence has first term 2 and common difference 6. What is term 11?

Solution: a_k = a1 + (k-1)d = 2 + (11-1)·6 = 62.


Answer: 4096

Prompt: A geometric sequence has first term 2 and common ratio 2. What is term 12?

Solution: a_k = a1 · r^(k-1) = 2·2^11 = 4096.


Answer: 19683

Prompt: A geometric sequence has first term 1 and common ratio 3. What is term 10?

Solution: a_k = a1 · r^(k-1) = 1·3^9 = 19683.


Answer: 2048

Prompt: A geometric sequence has first term 2 and common ratio 2. What is term 11?

Solution: a_k = a1 · r^(k-1) = 2·2^10 = 2048.


Answer: 1701

Prompt: A geometric sequence has first term 7 and common ratio 3. What is term 6?

Solution: a_k = a1 · r^(k-1) = 7·3^5 = 1701.


Answer: 512

Prompt: A geometric sequence has first term 8 and common ratio 2. What is term 7?

Solution: a_k = a1 · r^(k-1) = 8·2^6 = 512.


Answer: 885735

Prompt: A geometric sequence has first term 5 and common ratio 3. What is term 12?

Solution: a_k = a1 · r^(k-1) = 5·3^11 = 885735.


Answer: 137781

Prompt: A geometric sequence has first term 7 and common ratio 3. What is term 10?

Solution: a_k = a1 · r^(k-1) = 7·3^9 = 137781.


Answer: 512

Prompt: A geometric sequence has first term 4 and common ratio 2. What is term 8?

Solution: a_k = a1 · r^(k-1) = 4·2^7 = 512.


Answer: 157464

Prompt: A geometric sequence has first term 8 and common ratio 3. What is term 10?

Solution: a_k = a1 · r^(k-1) = 8·3^9 = 157464.


Answer: 2048

Prompt: A geometric sequence has first term 1 and common ratio 2. What is term 12?

Solution: a_k = a1 · r^(k-1) = 1·2^11 = 2048.


Answer: 3

Write the argument as a power of the base

Prompt: Evaluate log_5(125).

Solution: 5^3 = 125, so the logarithm is 3.


Answer: 4

Write the argument as a power of the base

Prompt: Evaluate log_4(256).

Solution: 4^4 = 256, so the logarithm is 4.


Answer: 4

Write the argument as a power of the base

Prompt: Evaluate log_10(10000).

Solution: 10^4 = 10000, so the logarithm is 4.


Answer: 2

Write the argument as a power of the base

Prompt: Evaluate log_10(100).

Solution: 10^2 = 100, so the logarithm is 2.


Answer: 3

Write the argument as a power of the base

Prompt: Evaluate log_10(1000).

Solution: 10^3 = 1000, so the logarithm is 3.


Answer: 3

Write the argument as a power of the base

Prompt: Evaluate log_2(8).

Solution: 2^3 = 8, so the logarithm is 3.


Answer: 5

Write the argument as a power of the base

Prompt: Evaluate log_5(3125).

Solution: 5^5 = 3125, so the logarithm is 5.


Answer: 1

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 9x + (-12) > -3. Give the threshold x must exceed.

Solution: Subtract -12: 9x > 9. Divide by positive 9: x > 1.


Answer: 2

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 6x + (-5) > 7. Give the threshold x must exceed.

Solution: Subtract -5: 6x > 12. Divide by positive 6: x > 2.


Answer: 9/5

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 5x + (-3) > 6. Give the threshold x must exceed.

Solution: Subtract -3: 5x > 9. Divide by positive 5: x > 9/5.


Answer: 1/9

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 9x + (5) > 6. Give the threshold x must exceed.

Solution: Subtract 5: 9x > 1. Divide by positive 9: x > 1/9.


Answer: -3/4

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 8x + (11) > 5. Give the threshold x must exceed.

Solution: Subtract 11: 8x > -6. Divide by positive 8: x > -3/4.


Answer: -1/7

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 7x + (10) > 9. Give the threshold x must exceed.

Solution: Subtract 10: 7x > -1. Divide by positive 7: x > -1/7.


Answer: 1

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 6x + (-3) > 3. Give the threshold x must exceed.

Solution: Subtract -3: 6x > 6. Divide by positive 6: x > 1.


Answer: 2

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 5x + (-9) > 1. Give the threshold x must exceed.

Solution: Subtract -9: 5x > 10. Divide by positive 5: x > 2.


Answer: 9/7

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 7x + (-9) > 0. Give the threshold x must exceed.

Solution: Subtract -9: 7x > 9. Divide by positive 7: x > 9/7.


Answer: 16/5

a > 0 so the inequality direction is preserved

Isolate x
Divide by the positive coefficient

Prompt: Solve the inequality 5x + (-6) > 10. Give the threshold x must exceed.

Solution: Subtract -6: 5x > 16. Divide by positive 5: x > 16/5.


Answer: 2.25

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-5=0 starting at x0=2. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 2 - (2^2-5)/(2·2) = 2.25.


Answer: 1.416666667

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-2=0 starting at x0=1.5. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 1.5 - (1.5^2-2)/(2·1.5) = 1.416666667.


Answer: 2.416666667

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-5=0 starting at x0=1.5. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 1.5 - (1.5^2-5)/(2·1.5) = 2.416666667.


Answer: 3

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-5=0 starting at x0=1. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 1 - (1^2-5)/(2·1) = 3.


Answer: 4.083333333

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-10=0 starting at x0=1.5. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 1.5 - (1.5^2-10)/(2·1.5) = 4.083333333.


Answer: 1.5

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-2=0 starting at x0=2. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 2 - (2^2-2)/(2·2) = 1.5.


Answer: 2

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-3=0 starting at x0=3. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 3 - (3^2-3)/(2·3) = 2.


Answer: 1.5

One iteration only
Use 64-bit floating point

Evaluate f and f' at x0
Apply the Newton update

Prompt: Perform one Newton iteration for f(x)=x^2-2=0 starting at x0=1. Report x1.

Solution: f'(x)=2x. x1 = x0 - f(x0)/f'(x0) = 1 - (1^2-2)/(2·1) = 1.5.


Answer: 240

Proof over the integers
Include the requested numeric check

Write an algebraic factorisation or induction

Prompt: Prove that for every integer n, n(n+1) is even. Then give the value of n(n+1) for n = 15 as a check.

Solution: If n is even, n=2k and n(n+1)=2k(2k+1) is even. If n is odd, n+1 is even, so the product is even. Computational check: 15×16 = 240.


Answer: 81

Proof over the integers
Include the requested numeric check

Write an algebraic factorisation or induction

Prompt: Prove that the sum of the first m positive odd numbers is m^2. Then report the sum of the first 9 positive odd numbers.

Solution: The k-th odd number is 2k-1. Sum_{k=1..m}(2k-1) = 2·m(m+1)/2 - m = m^2. For m=9, the sum is 81.


Answer: 504

Proof over the integers
Include the requested numeric check

Write an algebraic factorisation or induction

Prompt: Show that 3 divides n^3-n for every integer n. Report n^3-n for n=8.

Solution: n^3-n = n(n-1)(n+1), three consecutive integers, hence divisible by 3. For n=8: 512-8=504.


Answer: 1456

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 4, ratio 3, and 6 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 4(3^6-1)/(3-1) = 1456.


Answer: 200

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 5, ratio 3, and 4 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 5(3^4-1)/(3-1) = 200.


Answer: 728

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 2, ratio 3, and 6 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 2(3^6-1)/(3-1) = 728.


Answer: 126

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 2, ratio 2, and 6 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 2(2^6-1)/(2-1) = 126.


Answer: 35

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 5, ratio 2, and 3 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 5(2^3-1)/(2-1) = 35.


Answer: 93

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 3, ratio 2, and 5 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 3(2^5-1)/(2-1) = 93.


Answer: 14

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 2, ratio 2, and 3 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 2(2^3-1)/(2-1) = 14.


Answer: 315

Write S=a(r^n-1)/(r-1)
Substitute
Simplify

Prompt: A geometric series has first term 5, ratio 2, and 6 terms. What is the sum?

Solution: S = a(r^n-1)/(r-1) = 5(2^6-1)/(2-1) = 315.


Answer: 175

Divide by the base
Collect remainders
Reverse

Prompt: Write 125 in octal (no 0o prefix).

Solution: Repeated division by 8 yields 175, and int('175', 8) = 125.


Answer: 56

Divide by the base
Collect remainders
Reverse

Prompt: Write 86 in hexadecimal using lowercase letters (no 0x prefix).

Solution: Repeated division by 16 yields 56, and int('56', 16) = 86.


Answer: 167

Divide by the base
Collect remainders
Reverse

Prompt: Write 119 in octal (no 0o prefix).

Solution: Repeated division by 8 yields 167, and int('167', 8) = 119.


Answer: 9b

Divide by the base
Collect remainders
Reverse

Prompt: Write 155 in hexadecimal using lowercase letters (no 0x prefix).

Solution: Repeated division by 16 yields 9b, and int('9b', 16) = 155.


Answer: 19

Divide by the base
Collect remainders
Reverse

Prompt: Write 25 in hexadecimal using lowercase letters (no 0x prefix).

Solution: Repeated division by 16 yields 19, and int('19', 16) = 25.


Answer: 11000111

Divide by the base
Collect remainders
Reverse

Prompt: Write 199 in binary (no 0b prefix).

Solution: Repeated division by 2 yields 11000111, and int('11000111', 2) = 199.


Answer: 3f

Divide by the base
Collect remainders
Reverse

Prompt: Write 63 in hexadecimal using lowercase letters (no 0x prefix).

Solution: Repeated division by 16 yields 3f, and int('3f', 16) = 63.


Answer: 100

Divide by the base
Collect remainders
Reverse

Prompt: Write 64 in octal (no 0o prefix).

Solution: Repeated division by 8 yields 100, and int('100', 8) = 64.


Answer: 8

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when x**2 - 3*x - 2 is divided by (x - (-2))?

Solution: The remainder is p(-2) = 8.


Answer: -5

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when x**2 - 3*x - 5 is divided by (x - (0))?

Solution: The remainder is p(0) = -5.


Answer: 2

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when 4*x**2 + 4*x + 2 is divided by (x - (-1))?

Solution: The remainder is p(-1) = 2.


Answer: -1

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when x**2 - 2*x - 1 is divided by (x - (2))?

Solution: The remainder is p(2) = -1.


Answer: 12

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when 3*x**2 + 2*x - 4 is divided by (x - (2))?

Solution: The remainder is p(2) = 12.


Answer: 6

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when 2*x**2 - x + 5 is divided by (x - (1))?

Solution: The remainder is p(1) = 6.


Answer: -1

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when 2*x**2 + x - 4 is divided by (x - (1))?

Solution: The remainder is p(1) = -1.


Answer: -6

Polynomial over the integers
Exact remainder

Evaluate p at the root of the divisor
Report that value

Prompt: By the remainder theorem, what is the remainder when 2*x**2 + 5*x - 3 is divided by (x - (-1))?

Solution: The remainder is p(-1) = -6.


Answer: 77

Prompt: What is the squared Euclidean norm of the vector (-4, -5, -6)?

Solution: ||v||^2 = -4^2+-5^2+-6^2 = 77.


Answer: 56

Prompt: What is the squared Euclidean norm of the vector (-4, 6, -2)?

Solution: ||v||^2 = -4^2+6^2+-2^2 = 56.


Answer: 43

Prompt: What is the squared Euclidean norm of the vector (3, 5, 3)?

Solution: ||v||^2 = 3^2+5^2+3^2 = 43.


Answer: 30

Prompt: What is the squared Euclidean norm of the vector (-2, 1, -5)?

Solution: ||v||^2 = -2^2+1^2+-5^2 = 30.


Answer: 30

Prompt: What is the squared Euclidean norm of the vector (1, 5, -2)?

Solution: ||v||^2 = 1^2+5^2+-2^2 = 30.


Answer: 29

Prompt: What is the squared Euclidean norm of the vector (5, 0, -2)?

Solution: ||v||^2 = 5^2+0^2+-2^2 = 29.


Answer: 9

Prompt: What is the squared Euclidean norm of the vector (2, 2, 1)?

Solution: ||v||^2 = 2^2+2^2+1^2 = 9.


Answer: 35

Prompt: What is the squared Euclidean norm of the vector (1, -5, 3)?

Solution: ||v||^2 = 1^2+-5^2+3^2 = 35.


Answer: 10

Prompt: |A|=5, |B|=8, |A ∩ B|=3. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 5+8-3 = 10.


Answer: 11

Prompt: |A|=7, |B|=5, |A ∩ B|=1. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 7+5-1 = 11.


Answer: 10

Prompt: |A|=7, |B|=8, |A ∩ B|=5. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 7+8-5 = 10.


Answer: 11

Prompt: |A|=9, |B|=5, |A ∩ B|=3. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 9+5-3 = 11.


Answer: 10

Prompt: |A|=8, |B|=4, |A ∩ B|=2. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 8+4-2 = 10.


Answer: 15

Prompt: |A|=9, |B|=10, |A ∩ B|=4. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 9+10-4 = 15.


Answer: 12

Prompt: |A|=9, |B|=8, |A ∩ B|=5. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 9+8-5 = 12.


Answer: 16

Prompt: |A|=9, |B|=11, |A ∩ B|=4. What is |A ∪ B|?

Solution: |A ∪ B| = |A|+|B|-|A ∩ B| = 9+11-4 = 16.


Answer: 26

Prompt: Let z = -5 + 1i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = -5^2+1^2 = 26.


Answer: 1

Prompt: Let z = -1 + 0i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = -1^2+0^2 = 1.


Answer: 2

Prompt: Let z = -1 + -1i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = -1^2+-1^2 = 2.


Answer: 41

Prompt: Let z = 4 + -5i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = 4^2+-5^2 = 41.


Answer: 17

Prompt: Let z = -4 + -1i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = -4^2+-1^2 = 17.


Answer: 25

Prompt: Let z = 0 + 5i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = 0^2+5^2 = 25.


Answer: 5

Prompt: Let z = 1 + -2i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = 1^2+-2^2 = 5.


Answer: 16

Prompt: Let z = 4 + 0i. What is |z|^2 (the square of the modulus)?

Solution: |z|^2 = a^2+b^2 = 4^2+0^2 = 16.


Answer: 36

Exact integer
Combinations, not permutations

Prompt: What is C(9,2), the number of ways to choose 2 items from 9 distinct items without regard to order?

Solution: C(n,k)=n!/(k!(n-k)!) = 36.


Answer: 28

Exact integer
Combinations, not permutations

Prompt: What is C(8,2), the number of ways to choose 2 items from 8 distinct items without regard to order?

Solution: C(n,k)=n!/(k!(n-k)!) = 28.


Answer: 35

Exact integer
Combinations, not permutations

Prompt: What is C(7,3), the number of ways to choose 3 items from 7 distinct items without regard to order?

Solution: C(n,k)=n!/(k!(n-k)!) = 35.


Answer: 21

Exact integer
Combinations, not permutations

Prompt: What is C(7,2), the number of ways to choose 2 items from 7 distinct items without regard to order?

Solution: C(n,k)=n!/(k!(n-k)!) = 21.


Answer: 70

Exact integer
Combinations, not permutations

Prompt: What is C(8,4), the number of ways to choose 4 items from 8 distinct items without regard to order?

Solution: C(n,k)=n!/(k!(n-k)!) = 70.


Answer: 5

Exact integer
Combinations, not permutations

Prompt: What is C(5,4), the number of ways to choose 4 items from 5 distinct items without regard to order?

Solution: C(n,k)=n!/(k!(n-k)!) = 5.


Answer: 84

Exact integer
Combinations, not permutations

Prompt: What is C(9,3), the number of ways to choose 3 items from 9 distinct items without regard to order?

Solution: C(n,k)=n!/(k!(n-k)!) = 84.


Answer: -7

Prompt: What is det([[-1, 4], [1, 3]]) for a 2×2 matrix?

Solution: ad-bc = -1·3 - 4·1 = -7.


Answer: -4

Prompt: What is det([[3, -4], [-1, 0]]) for a 2×2 matrix?

Solution: ad-bc = 3·0 - -4·-1 = -4.


Answer: 12

Prompt: What is det([[4, -2], [0, 3]]) for a 2×2 matrix?

Solution: ad-bc = 4·3 - -2·0 = 12.


Answer: -15

Prompt: What is det([[3, -3], [-4, -1]]) for a 2×2 matrix?

Solution: ad-bc = 3·-1 - -3·-4 = -15.


Answer: 8

Prompt: What is det([[-2, 0], [4, -4]]) for a 2×2 matrix?

Solution: ad-bc = -2·-4 - 0·4 = 8.


Answer: 2

Prompt: What is det([[4, 2], [-3, -1]]) for a 2×2 matrix?

Solution: ad-bc = 4·-1 - 2·-3 = 2.


Answer: 15

Prompt: What is det([[-3, 3], [-3, -2]]) for a 2×2 matrix?

Solution: ad-bc = -3·-2 - 3·-3 = 15.


Answer: 0

Prompt: What is det([[3, 0], [4, 0]]) for a 2×2 matrix?

Solution: ad-bc = 3·0 - 0·4 = 0.


Answer: 8

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 15 modulo 17 in 1..16.

Solution: 15·8 ≡ 1 (mod 17).


Answer: 2

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 9 modulo 17 in 1..16.

Solution: 9·2 ≡ 1 (mod 17).


Answer: 5

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 9 modulo 11 in 1..10.

Solution: 9·5 ≡ 1 (mod 11).


Answer: 7

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 8 modulo 11 in 1..10.

Solution: 8·7 ≡ 1 (mod 11).


Answer: 10

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 10 modulo 11 in 1..10.

Solution: 10·10 ≡ 1 (mod 11).


Answer: 4

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 3 modulo 11 in 1..10.

Solution: 3·4 ≡ 1 (mod 11).


Answer: 5

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 8 modulo 13 in 1..12.

Solution: 8·5 ≡ 1 (mod 13).


Answer: 4

Coprime to the modulus
Least positive residue

Extended Euclid or trial
Check a·inv ≡ 1

Prompt: Find the inverse of 10 modulo 13 in 1..12.

Solution: 10·4 ≡ 1 (mod 13).


Answer: 12

Prompt: Let f(x)=4x and g(x)=1x+1. What is (f ∘ g)(2)?

Solution: g(2)=3, then f of that is 12.


Answer: 20

Prompt: Let f(x)=4x and g(x)=4x+1. What is (f ∘ g)(1)?

Solution: g(1)=5, then f of that is 20.


Answer: 45

Prompt: Let f(x)=5x and g(x)=4x+1. What is (f ∘ g)(2)?

Solution: g(2)=9, then f of that is 45.


Answer: 52

Prompt: Let f(x)=4x and g(x)=4x+1. What is (f ∘ g)(3)?

Solution: g(3)=13, then f of that is 52.


Answer: 12

Prompt: Let f(x)=4x and g(x)=2x+1. What is (f ∘ g)(1)?

Solution: g(1)=3, then f of that is 12.


Answer: 35

Prompt: Let f(x)=5x and g(x)=2x+1. What is (f ∘ g)(3)?

Solution: g(3)=7, then f of that is 35.


Answer: 20

Prompt: Let f(x)=2x and g(x)=3x+1. What is (f ∘ g)(3)?

Solution: g(3)=10, then f of that is 20.


Answer: 25

Prompt: Let f(x)=5x and g(x)=4x+1. What is (f ∘ g)(1)?

Solution: g(1)=5, then f of that is 25.


Answer: -1

Prompt: Decompose 1/((x-5)(x-6)) into partial fractions A/(x-5)+B/(x-6). Report A as a simplified rational.

Solution: A=1/(5-6)=-1. Check: B=1/(6-5)=1.


Answer: -1/3

Prompt: Decompose 1/((x-3)(x-6)) into partial fractions A/(x-3)+B/(x-6). Report A as a simplified rational.

Solution: A=1/(3-6)=-1/3. Check: B=1/(6-3)=1/3.


Answer: -1/4

Prompt: Decompose 1/((x-2)(x-6)) into partial fractions A/(x-2)+B/(x-6). Report A as a simplified rational.

Solution: A=1/(2-6)=-1/4. Check: B=1/(6-2)=1/4.


Answer: -1/2

Prompt: Decompose 1/((x-2)(x-4)) into partial fractions A/(x-2)+B/(x-4). Report A as a simplified rational.

Solution: A=1/(2-4)=-1/2. Check: B=1/(4-2)=1/2.


Answer: -1/2

Prompt: Decompose 1/((x-5)(x-7)) into partial fractions A/(x-5)+B/(x-7). Report A as a simplified rational.

Solution: A=1/(5-7)=-1/2. Check: B=1/(7-5)=1/2.


Answer: 784

Prompt: What is sum_{k=1}^{7} k^3?

Solution: (n(n+1)/2)^2 = 784.


Answer: 8281

Prompt: What is sum_{k=1}^{13} k^3?

Solution: (n(n+1)/2)^2 = 8281.


Answer: 289

Prompt: What is the sum of the first 17 positive odd numbers?

Solution: The sum is n^2 = 289.


Answer: 23409

Prompt: What is sum_{k=1}^{17} k^3?

Solution: (n(n+1)/2)^2 = 23409.


Answer: 400

Prompt: What is the sum of the first 20 positive odd numbers?

Solution: The sum is n^2 = 400.


Answer: 36

Prompt: What is the sum of the first 6 positive odd numbers?

Solution: The sum is n^2 = 36.


Answer: 140

Prompt: What is sum_{k=1}^{7} k^2?

Solution: n(n+1)(2n+1)/6 = 140.


Answer: 0,5

Evaluate cos and sin at the axis angle
Scale by r

Prompt: Convert polar (r=5, θ=90°) to rectangular (x, y) with integer coordinates. Report as x,y.

Solution: x=r cos θ, y=r sin θ at a right angle: (0, 5).


Answer: 0,-2

Evaluate cos and sin at the axis angle
Scale by r

Prompt: Convert polar (r=2, θ=270°) to rectangular (x, y) with integer coordinates. Report as x,y.

Solution: x=r cos θ, y=r sin θ at a right angle: (0, -2).


Answer: 2,0

Evaluate cos and sin at the axis angle
Scale by r

Prompt: Convert polar (r=2, θ=0°) to rectangular (x, y) with integer coordinates. Report as x,y.

Solution: x=r cos θ, y=r sin θ at a right angle: (2, 0).


Answer: 0,10

Evaluate cos and sin at the axis angle
Scale by r

Prompt: Convert polar (r=10, θ=90°) to rectangular (x, y) with integer coordinates. Report as x,y.

Solution: x=r cos θ, y=r sin θ at a right angle: (0, 10).


Answer: 5,0

Evaluate cos and sin at the axis angle
Scale by r

Prompt: Convert polar (r=5, θ=0°) to rectangular (x, y) with integer coordinates. Report as x,y.

Solution: x=r cos θ, y=r sin θ at a right angle: (5, 0).


Answer: -2,0

Evaluate cos and sin at the axis angle
Scale by r

Prompt: Convert polar (r=2, θ=180°) to rectangular (x, y) with integer coordinates. Report as x,y.

Solution: x=r cos θ, y=r sin θ at a right angle: (-2, 0).


Answer: -5,0

Evaluate cos and sin at the axis angle
Scale by r

Prompt: Convert polar (r=5, θ=180°) to rectangular (x, y) with integer coordinates. Report as x,y.

Solution: x=r cos θ, y=r sin θ at a right angle: (-5, 0).


Answer: -3

Prompt: The diagonal matrix diag(1, -3) has eigenvalues 1 and -3. What is the product of the eigenvalues?

Solution: Product of eigenvalues equals det = 1·-3 = -3.


Answer: -6

Prompt: The diagonal matrix diag(-2, 3) has eigenvalues -2 and 3. What is the product of the eigenvalues?

Solution: Product of eigenvalues equals det = -2·3 = -6.


Answer: 4

Prompt: The diagonal matrix diag(1, 4) has eigenvalues 1 and 4. What is the product of the eigenvalues?

Solution: Product of eigenvalues equals det = 1·4 = 4.


Answer: -15

Prompt: The diagonal matrix diag(5, -3) has eigenvalues 5 and -3. What is the product of the eigenvalues?

Solution: Product of eigenvalues equals det = 5·-3 = -15.


Answer: 0

Prompt: The diagonal matrix diag(-3, 0) has eigenvalues -3 and 0. What is the product of the eigenvalues?

Solution: Product of eigenvalues equals det = -3·0 = 0.


Answer: -16

Prompt: The diagonal matrix diag(-4, 4) has eigenvalues -4 and 4. What is the product of the eigenvalues?

Solution: Product of eigenvalues equals det = -4·4 = -16.


Answer: 5

Prompt: What is floor(37/7)?

Solution: floor(37/7) = 5.


Answer: 6

Prompt: What is floor(25/4)?

Solution: floor(25/4) = 6.


Answer: 4

Prompt: What is ceil(24/6)?

Solution: ceil(24/6) = 4.


Answer: 6

Prompt: What is ceil(18/3)?

Solution: ceil(18/3) = 6.


Answer: 6

Prompt: What is floor(38/6)?

Solution: floor(38/6) = 6.


Answer: 2

Prompt: What is ceil(12/7)?

Solution: ceil(12/7) = 2.


Answer: 5

Prompt: What is ceil(28/6)?

Solution: ceil(28/6) = 5.


Answer: 6

Prompt: What is ceil(32/6)?

Solution: ceil(32/6) = 6.


Answer: 42

Prompt: How many injective functions are there from a set of 2 elements to a set of 7 elements? Equivalently P(7,2).

Solution: P(n,k)=n!/(n-k)! = 42.


Answer: 336

Prompt: How many injective functions are there from a set of 3 elements to a set of 8 elements? Equivalently P(8,3).

Solution: P(n,k)=n!/(n-k)! = 336.


Answer: 20

Prompt: How many injective functions are there from a set of 2 elements to a set of 5 elements? Equivalently P(5,2).

Solution: P(n,k)=n!/(n-k)! = 20.


Answer: 1680

Prompt: How many injective functions are there from a set of 4 elements to a set of 8 elements? Equivalently P(8,4).

Solution: P(n,k)=n!/(n-k)! = 1680.


Answer: 840

Prompt: How many injective functions are there from a set of 4 elements to a set of 7 elements? Equivalently P(7,4).

Solution: P(n,k)=n!/(n-k)! = 840.


Answer: 1

Prompt: What is log2(2) as an integer?

Solution: 2^1 = 2.


Answer: 3

Prompt: What is log3(27) as an integer?

Solution: 3^3 = 27.


Answer: 3

Prompt: What is log2(8) as an integer?

Solution: 2^3 = 8.


Answer: 1

Prompt: What is log5(5) as an integer?

Solution: 5^1 = 5.


Answer: 2

Prompt: What is log3(9) as an integer?

Solution: 3^2 = 9.


Answer: 9

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 5x^2 + (0)x + (4) is divided by (x - -1)?

Solution: Remainder = p(-1) = 5(-1)^2+(0)(-1)+(4) = 9.


Answer: -2

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (-4)x + (-2) is divided by (x - 4)?

Solution: Remainder = p(4) = 1(4)^2+(-4)(4)+(-2) = -2.


Answer: 5

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (1)x + (5) is divided by (x - -1)?

Solution: Remainder = p(-1) = 1(-1)^2+(1)(-1)+(5) = 5.


Answer: 8

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (0)x + (-1) is divided by (x - 3)?

Solution: Remainder = p(3) = 1(3)^2+(0)(3)+(-1) = 8.


Answer: 2

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 2x^2 + (-1)x + (-4) is divided by (x - 2)?

Solution: Remainder = p(2) = 2(2)^2+(-1)(2)+(-4) = 2.


Answer: -1

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 5x^2 + (4)x + (-2) is divided by (x - -1)?

Solution: Remainder = p(-1) = 5(-1)^2+(4)(-1)+(-2) = -1.


Answer: 12

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 3x^2 + (3)x + (6) is divided by (x - 1)?

Solution: Remainder = p(1) = 3(1)^2+(3)(1)+(6) = 12.


Answer: 19

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 3x^2 + (-3)x + (1) is divided by (x - -2)?

Solution: Remainder = p(-2) = 3(-2)^2+(-3)(-2)+(1) = 19.


Answer: 19

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 2x^2 + (-1)x + (4) is divided by (x - 3)?

Solution: Remainder = p(3) = 2(3)^2+(-1)(3)+(4) = 19.


Answer: 43

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 5x^2 + (1)x + (-5) is divided by (x - 3)?

Solution: Remainder = p(3) = 5(3)^2+(1)(3)+(-5) = 43.


Answer: 6

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (0)x + (2) is divided by (x - -2)?

Solution: Remainder = p(-2) = 1(-2)^2+(0)(-2)+(2) = 6.


Answer: 9

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 4x^2 + (1)x + (4) is divided by (x - 1)?

Solution: Remainder = p(1) = 4(1)^2+(1)(1)+(4) = 9.


Answer: 28

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 5x^2 + (2)x + (4) is divided by (x - 2)?

Solution: Remainder = p(2) = 5(2)^2+(2)(2)+(4) = 28.


Answer: 13

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (-1)x + (1) is divided by (x - -3)?

Solution: Remainder = p(-3) = 1(-3)^2+(-1)(-3)+(1) = 13.


Answer: -1

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 5x^2 + (4)x + (-1) is divided by (x - 0)?

Solution: Remainder = p(0) = 5(0)^2+(4)(0)+(-1) = -1.


Answer: 9

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (3)x + (5) is divided by (x - 1)?

Solution: Remainder = p(1) = 1(1)^2+(3)(1)+(5) = 9.


Answer: 41

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 4x^2 + (-3)x + (-4) is divided by (x - -3)?

Solution: Remainder = p(-3) = 4(-3)^2+(-3)(-3)+(-4) = 41.


Answer: 5

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (0)x + (1) is divided by (x - -2)?

Solution: Remainder = p(-2) = 1(-2)^2+(0)(-2)+(1) = 5.


Answer: 4

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 1x^2 + (-1)x + (2) is divided by (x - -1)?

Solution: Remainder = p(-1) = 1(-1)^2+(-1)(-1)+(2) = 4.


Answer: 44

Polynomial division over the integers
Exact remainder

Evaluate the polynomial at the root of the divisor

Prompt: By the remainder theorem, what is the remainder when 4x^2 + (3)x + (-1) is divided by (x - 3)?

Solution: Remainder = p(3) = 4(3)^2+(3)(3)+(-1) = 44.


Answer: -13

Prompt: An arithmetic sequence has first term 9 and common difference -2. What is the 12th term?

Solution: a_n = a_1 + (n-1)d = 9 + (12-1)·-2 = -13.


Answer: -25

Prompt: An arithmetic sequence has first term 11 and common difference -3. What is the 13th term?

Solution: a_n = a_1 + (n-1)d = 11 + (13-1)·-3 = -25.


Answer: 55

Prompt: An arithmetic sequence has first term 10 and common difference 5. What is the 10th term?

Solution: a_n = a_1 + (n-1)d = 10 + (10-1)·5 = 55.


Answer: 24

Prompt: An arithmetic sequence has first term -8 and common difference 4. What is the 9th term?

Solution: a_n = a_1 + (n-1)d = -8 + (9-1)·4 = 24.


Answer: 61

Prompt: An arithmetic sequence has first term 5 and common difference 7. What is the 9th term?

Solution: a_n = a_1 + (n-1)d = 5 + (9-1)·7 = 61.


Answer: -43

Prompt: An arithmetic sequence has first term 2 and common difference -3. What is the 16th term?

Solution: a_n = a_1 + (n-1)d = 2 + (16-1)·-3 = -43.


Answer: 22

Prompt: An arithmetic sequence has first term 2 and common difference 4. What is the 6th term?

Solution: a_n = a_1 + (n-1)d = 2 + (6-1)·4 = 22.


Answer: 27

Prompt: An arithmetic sequence has first term 3 and common difference 3. What is the 9th term?

Solution: a_n = a_1 + (n-1)d = 3 + (9-1)·3 = 27.


Answer: 36

Prompt: An arithmetic sequence has first term -6 and common difference 7. What is the 7th term?

Solution: a_n = a_1 + (n-1)d = -6 + (7-1)·7 = 36.


Answer: 42

Prompt: An arithmetic sequence has first term 12 and common difference 2. What is the 16th term?

Solution: a_n = a_1 + (n-1)d = 12 + (16-1)·2 = 42.


Answer: -14

Prompt: An arithmetic sequence has first term 11 and common difference -5. What is the 6th term?

Solution: a_n = a_1 + (n-1)d = 11 + (6-1)·-5 = -14.


Answer: 15

Prompt: An arithmetic sequence has first term 1 and common difference 2. What is the 8th term?

Solution: a_n = a_1 + (n-1)d = 1 + (8-1)·2 = 15.


Answer: 21

Prompt: An arithmetic sequence has first term 9 and common difference 2. What is the 7th term?

Solution: a_n = a_1 + (n-1)d = 9 + (7-1)·2 = 21.


Answer: 36

Prompt: An arithmetic sequence has first term 8 and common difference 2. What is the 15th term?

Solution: a_n = a_1 + (n-1)d = 8 + (15-1)·2 = 36.


Answer: 17

Prompt: An arithmetic sequence has first term -3 and common difference 2. What is the 11th term?

Solution: a_n = a_1 + (n-1)d = -3 + (11-1)·2 = 17.


Answer: -50

Prompt: An arithmetic sequence has first term -5 and common difference -5. What is the 10th term?

Solution: a_n = a_1 + (n-1)d = -5 + (10-1)·-5 = -50.


Answer: -31

Prompt: An arithmetic sequence has first term -5 and common difference -2. What is the 14th term?

Solution: a_n = a_1 + (n-1)d = -5 + (14-1)·-2 = -31.


Answer: -13

Prompt: An arithmetic sequence has first term -1 and common difference -2. What is the 7th term?

Solution: a_n = a_1 + (n-1)d = -1 + (7-1)·-2 = -13.


Answer: -23

Prompt: An arithmetic sequence has first term 12 and common difference -5. What is the 8th term?

Solution: a_n = a_1 + (n-1)d = 12 + (8-1)·-5 = -23.


Answer: 59

Prompt: An arithmetic sequence has first term -9 and common difference 4. What is the 18th term?

Solution: a_n = a_1 + (n-1)d = -9 + (18-1)·4 = 59.


Answer: -42

Prompt: Find the sum of the first 6 terms of the geometric series with first term 2 and common ratio -2.

Solution: S_n = a(r^n-1)/(r-1) = 2(-2^6-1)/(-2-1) = -42.


Answer: 63

Prompt: Find the sum of the first 6 terms of the geometric series with first term 1 and common ratio 2.

Solution: S_n = a(r^n-1)/(r-1) = 1(2^6-1)/(2-1) = 63.


Answer: 127

Prompt: Find the sum of the first 7 terms of the geometric series with first term 1 and common ratio 2.

Solution: S_n = a(r^n-1)/(r-1) = 1(2^7-1)/(2-1) = 127.


Answer: -5

Prompt: Find the sum of the first 4 terms of the geometric series with first term 1 and common ratio -2.

Solution: S_n = a(r^n-1)/(r-1) = 1(-2^4-1)/(-2-1) = -5.


Answer: 1820

Prompt: Find the sum of the first 6 terms of the geometric series with first term 5 and common ratio 3.

Solution: S_n = a(r^n-1)/(r-1) = 5(3^6-1)/(3-1) = 1820.


Answer: 121

Prompt: Find the sum of the first 5 terms of the geometric series with first term 1 and common ratio 3.

Solution: S_n = a(r^n-1)/(r-1) = 1(3^5-1)/(3-1) = 121.


Answer: -21

Prompt: Find the sum of the first 6 terms of the geometric series with first term 1 and common ratio -2.

Solution: S_n = a(r^n-1)/(r-1) = 1(-2^6-1)/(-2-1) = -21.


Answer: -25

Prompt: Find the sum of the first 4 terms of the geometric series with first term 5 and common ratio -2.

Solution: S_n = a(r^n-1)/(r-1) = 5(-2^4-1)/(-2-1) = -25.


Answer: 1093

Prompt: Find the sum of the first 7 terms of the geometric series with first term 1 and common ratio 3.

Solution: S_n = a(r^n-1)/(r-1) = 1(3^7-1)/(3-1) = 1093.


Answer: 215

Prompt: Find the sum of the first 7 terms of the geometric series with first term 5 and common ratio -2.

Solution: S_n = a(r^n-1)/(r-1) = 5(-2^7-1)/(-2-1) = 215.


Answer: 62

Prompt: Find the sum of the first 5 terms of the geometric series with first term 2 and common ratio 2.

Solution: S_n = a(r^n-1)/(r-1) = 2(2^5-1)/(2-1) = 62.


Answer: -63

Prompt: Find the sum of the first 6 terms of the geometric series with first term 3 and common ratio -2.

Solution: S_n = a(r^n-1)/(r-1) = 3(-2^6-1)/(-2-1) = -63.


Answer: -105

Prompt: Find the sum of the first 6 terms of the geometric series with first term 5 and common ratio -2.

Solution: S_n = a(r^n-1)/(r-1) = 5(-2^6-1)/(-2-1) = -105.


Answer: 2186

Prompt: Find the sum of the first 7 terms of the geometric series with first term 2 and common ratio 3.

Solution: S_n = a(r^n-1)/(r-1) = 2(3^7-1)/(3-1) = 2186.


Answer: 2

Prompt: The parabola y = -1(x - (1))^2 + (2) opens down. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(1,2).


Answer: 5

Prompt: The parabola y = 2(x - (1))^2 + (5) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(1,5).


Answer: 5

Prompt: The parabola y = -2(x - (0))^2 + (5) opens down. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(0,5).


Answer: 0

Prompt: The parabola y = 1(x - (5))^2 + (0) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(5,0).


Answer: 7

Prompt: The parabola y = 1(x - (-3))^2 + (7) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(-3,7).


Answer: -6

Prompt: The parabola y = -2(x - (-4))^2 + (-6) opens down. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(-4,-6).


Answer: -5

Prompt: The parabola y = -2(x - (1))^2 + (-5) opens down. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(1,-5).


Answer: 2

Prompt: The parabola y = 3(x - (3))^2 + (2) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(3,2).


Answer: 3

Prompt: The parabola y = -2(x - (1))^2 + (3) opens down. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(1,3).


Answer: 1

Prompt: The parabola y = 2(x - (-5))^2 + (1) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(-5,1).


Answer: 3

Prompt: The parabola y = 3(x - (-1))^2 + (3) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(-1,3).


Answer: 8

Prompt: The parabola y = -2(x - (4))^2 + (8) opens down. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(4,8).


Answer: 7

Prompt: The parabola y = -1(x - (-3))^2 + (7) opens down. What is the y-coordinate of the vertex?


Answer: 3

Prompt: The parabola y = -1(x - (-4))^2 + (3) opens down. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(-4,3).


Answer: -5

Prompt: The parabola y = 3(x - (0))^2 + (-5) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(0,-5).


Answer: -1

Prompt: The parabola y = 1(x - (4))^2 + (-1) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(4,-1).


Answer: -8

Prompt: The parabola y = 2(x - (3))^2 + (-8) opens up. What is the y-coordinate of the vertex?

Solution: Vertex form y=a(x-h)^2+k has vertex (h,k)=(3,-8).


Answer: 65

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (5,0,-4) and (-2,0,0) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 65.


Answer: 99

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (1,1,-4) and (-2,-2,5) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 99.


Answer: 72

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (2,-3,-2) and (-4,-3,4) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 72.


Answer: 33

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (-1,2,2) and (3,1,-2) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 33.


Answer: 57

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (1,0,1) and (5,5,-3) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 57.


Answer: 86

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (-4,-2,-2) and (5,-4,-3) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 86.


Answer: 14

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (0,3,2) and (3,5,3) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 14.


Answer: 17

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (2,0,-1) and (4,-3,1) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 17.


Answer: 61

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (2,-3,0) and (5,3,4) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 61.


Answer: 94

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (0,-4,-1) and (2,5,-4) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 94.


Answer: 10

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (-4,-1,0) and (-1,-2,0) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 10.


Answer: 45

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (0,1,-3) and (-4,3,2) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 45.


Answer: 41

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (-2,-2,2) and (4,-1,4) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 41.


Answer: 54

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (4,1,-3) and (2,-4,2) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 54.


Answer: 54

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (-4,3,-3) and (1,5,2) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 54.


Answer: 50

Squared distance, not the root
Exact integer

Prompt: What is the squared Euclidean distance between (5,2,2) and (0,-3,2) in R^3?

Solution: d^2 = Δx^2+Δy^2+Δz^2 = 50.


Answer: 12

Prompt: A triangle has side lengths 5, 5, and 6. Using Heron's formula, what is its area?

Solution: s=8.0; area=sqrt(s(s-a)(s-b)(s-c))=12.


Answer: 9.16515139

Prompt: A triangle has side lengths 5, 5, and 4. Using Heron's formula, what is its area?

Solution: s=7.0; area=sqrt(s(s-a)(s-b)(s-c))=9.16515139.


Answer: 84

Prompt: A triangle has side lengths 13, 14, and 15. Using Heron's formula, what is its area?

Solution: s=21.0; area=sqrt(s(s-a)(s-b)(s-c))=84.


Answer: 42

Prompt: A triangle has side lengths 7, 15, and 20. Using Heron's formula, what is its area?

Solution: s=21.0; area=sqrt(s(s-a)(s-b)(s-c))=42.


Answer: 60

Prompt: A triangle has side lengths 6, 25, and 29. Using Heron's formula, what is its area?

Solution: s=30.0; area=sqrt(s(s-a)(s-b)(s-c))=60.


Answer: 36

Prompt: A triangle has side lengths 9, 10, and 17. Using Heron's formula, what is its area?

Solution: s=18.0; area=sqrt(s(s-a)(s-b)(s-c))=36.


Answer: -5

Prompt: What is the trace of the matrix [[-1, 5], [5, -4]]?

Solution: tr(A)=sum of diagonal entries=-5.


Answer: 11

Prompt: What is the trace of the matrix [[7, -3, 0], [0, 5, -3], [-3, 0, -1]]?

Solution: tr(A)=sum of diagonal entries=11.


Answer: -1

Prompt: What is the trace of the matrix [[7, 2, 0], [0, -5, 2], [2, 0, -3]]?

Solution: tr(A)=sum of diagonal entries=-1.


Answer: -3

Prompt: What is the trace of the matrix [[2, 0, 0], [0, -5, 0], [0, 0, 0]]?

Solution: tr(A)=sum of diagonal entries=-3.


Answer: 2

Prompt: What is the trace of the matrix [[-3, -2, 0], [0, -1, -2], [-2, 0, 6]]?

Solution: tr(A)=sum of diagonal entries=2.


Answer: 6

Prompt: What is the trace of the matrix [[2, -3], [-3, 4]]?

Solution: tr(A)=sum of diagonal entries=6.


Answer: 5

Prompt: What is the trace of the matrix [[6, 1], [1, -1]]?

Solution: tr(A)=sum of diagonal entries=5.


Answer: -8

Prompt: What is the trace of the matrix [[-3, -2], [-2, -5]]?

Solution: tr(A)=sum of diagonal entries=-8.


Answer: -4

Prompt: What is the trace of the matrix [[-3, -2, 0], [0, -4, -2], [-2, 0, 3]]?

Solution: tr(A)=sum of diagonal entries=-4.


Answer: 2

Prompt: What is the trace of the matrix [[-4, 3], [3, 6]]?

Solution: tr(A)=sum of diagonal entries=2.


Answer: 15

Prompt: What is the trace of the matrix [[6, 5, 0], [0, 3, 5], [5, 0, 6]]?

Solution: tr(A)=sum of diagonal entries=15.


Answer: 11

Prompt: What is the trace of the matrix [[4, 5, 0], [0, 3, 5], [5, 0, 4]]?

Solution: tr(A)=sum of diagonal entries=11.


Answer: 10

Prompt: What is the trace of the matrix [[7, -2, 0], [0, 4, -2], [-2, 0, -1]]?

Solution: tr(A)=sum of diagonal entries=10.


Answer: -3

Prompt: What is the trace of the matrix [[-5, 0, 0], [0, 1, 0], [0, 0, 1]]?

Solution: tr(A)=sum of diagonal entries=-3.


Answer: -4

Prompt: What is the trace of the matrix [[3, 4, 0], [0, -6, 4], [4, 0, -1]]?

Solution: tr(A)=sum of diagonal entries=-4.


Answer: -1

Prompt: What is the trace of the matrix [[-2, 3], [3, 1]]?

Solution: tr(A)=sum of diagonal entries=-1.


Answer: 86

Prompt: Let u=(-3,-3,1) and v=(1,-2,0). What is ||u × v||^2?

Solution: u×v=(2,1,9); squared length=86.


Answer: 676

Prompt: Let u=(3,4,2) and v=(-3,4,0). What is ||u × v||^2?

Solution: u×v=(-8,-6,24); squared length=676.


Answer: 120

Prompt: Let u=(2,4,2) and v=(-1,-3,1). What is ||u × v||^2?

Solution: u×v=(10,-4,-2); squared length=120.


Answer: 56

Prompt: Let u=(-2,2,2) and v=(-2,-1,0). What is ||u × v||^2?

Solution: u×v=(2,-4,6); squared length=56.


Answer: 21

Prompt: Let u=(0,4,1) and v=(-1,2,1). What is ||u × v||^2?

Solution: u×v=(2,-1,4); squared length=21.


Answer: 118

Prompt: Let u=(1,3,1) and v=(2,-3,1). What is ||u × v||^2?

Solution: u×v=(6,1,-9); squared length=118.


Answer: 72

Prompt: Let u=(-2,2,0) and v=(3,1,1). What is ||u × v||^2?

Solution: u×v=(2,2,-8); squared length=72.


Answer: 5

Prompt: Let u=(-1,2,0) and v=(-1,2,1). What is ||u × v||^2?

Solution: u×v=(2,1,0); squared length=5.


Answer: 59

Prompt: Let u=(3,-1,2) and v=(-2,1,1). What is ||u × v||^2?

Solution: u×v=(-3,-7,1); squared length=59.


Answer: 101

Prompt: Let u=(2,-1,2) and v=(-2,3,1). What is ||u × v||^2?

Solution: u×v=(-7,-6,4); squared length=101.


Answer: 12

Prompt: Let u=(-3,2,1) and v=(-1,0,1). What is ||u × v||^2?

Solution: u×v=(2,2,2); squared length=12.


Answer: 16

Prompt: Let u=(4,0,0) and v=(-1,-1,0). What is ||u × v||^2?

Solution: u×v=(0,0,-4); squared length=16.


Answer: 68

Prompt: Let u=(1,-2,2) and v=(-3,2,0). What is ||u × v||^2?

Solution: u×v=(-4,-6,-4); squared length=68.


Answer: 26

Prompt: Let u=(3,-1,0) and v=(-1,-1,1). What is ||u × v||^2?

Solution: u×v=(-1,-3,-4); squared length=26.


Answer: 165

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 00:30?

Solution: Hour hand at 0.5*(60h+m)=15.0°, minute at 6m=180°, smaller difference=165.


Answer: 102.5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 08:25?

Solution: Hour hand at 0.5*(60h+m)=252.5°, minute at 6m=150°, smaller difference=102.5.


Answer: 75

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 03:30?

Solution: Hour hand at 0.5*(60h+m)=105.0°, minute at 6m=180°, smaller difference=75.


Answer: 107.5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 01:25?

Solution: Hour hand at 0.5*(60h+m)=42.5°, minute at 6m=150°, smaller difference=107.5.


Answer: 90

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 03:00?

Solution: Hour hand at 0.5*(60h+m)=90.0°, minute at 6m=0°, smaller difference=90.


Answer: 122.5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 05:05?

Solution: Hour hand at 0.5*(60h+m)=152.5°, minute at 6m=30°, smaller difference=122.5.


Answer: 167.5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 11:25?

Solution: Hour hand at 0.5*(60h+m)=342.5°, minute at 6m=150°, smaller difference=167.5.


Answer: 5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 02:10?

Solution: Hour hand at 0.5*(60h+m)=65.0°, minute at 6m=60°, smaller difference=5.


Answer: 110

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 00:20?

Solution: Hour hand at 0.5*(60h+m)=10.0°, minute at 6m=120°, smaller difference=110.


Answer: 155

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 07:10?

Solution: Hour hand at 0.5*(60h+m)=215.0°, minute at 6m=60°, smaller difference=155.


Answer: 110

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 11:40?

Solution: Hour hand at 0.5*(60h+m)=350.0°, minute at 6m=240°, smaller difference=110.


Answer: 150

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 07:00?

Solution: Hour hand at 0.5*(60h+m)=210.0°, minute at 6m=0°, smaller difference=150.


Answer: 30

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 01:00?

Solution: Hour hand at 0.5*(60h+m)=30.0°, minute at 6m=0°, smaller difference=30.


Answer: 37.5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 04:15?

Solution: Hour hand at 0.5*(60h+m)=127.5°, minute at 6m=90°, smaller difference=37.5.


Answer: 172.5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 02:45?

Solution: Hour hand at 0.5*(60h+m)=82.5°, minute at 6m=270°, smaller difference=172.5.


Answer: 82.5

Prompt: On a 12-hour analog clock, what is the smaller angle in degrees between the hour and minute hands at 11:45?

Solution: Hour hand at 0.5*(60h+m)=352.5°, minute at 6m=270°, smaller difference=82.5.


Answer: 14

Prompt: What is the sum of the decimal digits of 7034?

Solution: Digits of 7034 sum to 14.


Answer: 13

Prompt: What is the sum of the decimal digits of 1921?

Solution: Digits of 1921 sum to 13.


Answer: 14

Prompt: What is the sum of the decimal digits of 4820?

Solution: Digits of 4820 sum to 14.


Answer: 24

Prompt: What is the sum of the decimal digits of 3993?

Solution: Digits of 3993 sum to 24.


Answer: 12

Prompt: What is the sum of the decimal digits of 5034?

Solution: Digits of 5034 sum to 12.


Answer: 17

Prompt: What is the sum of the decimal digits of 2096?

Solution: Digits of 2096 sum to 17.


Answer: 18

Prompt: What is the sum of the decimal digits of 882?

Solution: Digits of 882 sum to 18.


Answer: 10

Prompt: What is the sum of the decimal digits of 4006?

Solution: Digits of 4006 sum to 10.


Answer: 30

Prompt: What is the sum of the decimal digits of 6978?

Solution: Digits of 6978 sum to 30.


Answer: 29

Prompt: What is the sum of the decimal digits of 7589?

Solution: Digits of 7589 sum to 29.


Answer: 13

Prompt: What is the sum of the decimal digits of 1129?

Solution: Digits of 1129 sum to 13.


Answer: 17

Prompt: What is the sum of the decimal digits of 1916?

Solution: Digits of 1916 sum to 17.


Answer: 27

Prompt: What is the sum of the decimal digits of 8289?

Solution: Digits of 8289 sum to 27.


Answer: 30

Prompt: What is the sum of the decimal digits of 9876?

Solution: Digits of 9876 sum to 30.


Answer: 26

Prompt: What is the sum of the decimal digits of 8882?

Solution: Digits of 8882 sum to 26.


Answer: 18

Prompt: What is the sum of the decimal digits of 369?

Solution: Digits of 369 sum to 18.


Answer: 820

Prompt: What is the 40th triangular number T_n = n(n+1)/2?


Answer: 276

Prompt: What is the 23th triangular number T_n = n(n+1)/2?


Answer: 153

Prompt: What is the 17th triangular number T_n = n(n+1)/2?


Answer: 351

Prompt: What is the 26th triangular number T_n = n(n+1)/2?


Answer: 630

Prompt: What is the 35th triangular number T_n = n(n+1)/2?


Answer: 36

Prompt: What is the 8th triangular number T_n = n(n+1)/2?


Answer: 465

Prompt: What is the 30th triangular number T_n = n(n+1)/2?


Answer: 595

Prompt: What is the 34th triangular number T_n = n(n+1)/2?


Answer: 190

Prompt: What is the 19th triangular number T_n = n(n+1)/2?


Answer: 91

Prompt: What is the 13th triangular number T_n = n(n+1)/2?


Answer: 496

Prompt: What is the 31th triangular number T_n = n(n+1)/2?


Answer: 78

Prompt: What is the 12th triangular number T_n = n(n+1)/2?


Answer: 45

Prompt: What is the 9th triangular number T_n = n(n+1)/2?


Answer: 163

Prompt: Write the positive integer 115 in base 8 (no prefix, lowercase if hex).

Solution: 115 in base 8 is 163 (check: int('163', 8)=115).


Answer: 1b

Prompt: Write the positive integer 27 in base 16 (no prefix, lowercase if hex).

Solution: 27 in base 16 is 1b (check: int('1b', 16)=27).


Answer: 50

Prompt: Write the positive integer 80 in base 16 (no prefix, lowercase if hex).

Solution: 80 in base 16 is 50 (check: int('50', 16)=80).


Answer: 24

Prompt: Write the positive integer 20 in base 8 (no prefix, lowercase if hex).

Solution: 20 in base 8 is 24 (check: int('24', 8)=20).


Answer: 41

Prompt: Write the positive integer 33 in base 8 (no prefix, lowercase if hex).

Solution: 33 in base 8 is 41 (check: int('41', 8)=33).


Answer: 4d

Prompt: Write the positive integer 77 in base 16 (no prefix, lowercase if hex).

Solution: 77 in base 16 is 4d (check: int('4d', 16)=77).


Answer: b8

Prompt: Write the positive integer 184 in base 16 (no prefix, lowercase if hex).

Solution: 184 in base 16 is b8 (check: int('b8', 16)=184).


Answer: 2a

Prompt: Write the positive integer 42 in base 16 (no prefix, lowercase if hex).

Solution: 42 in base 16 is 2a (check: int('2a', 16)=42).


Answer: 1100101

Prompt: Write the positive integer 101 in base 2 (no prefix, lowercase if hex).

Solution: 101 in base 2 is 1100101 (check: int('1100101', 2)=101).


Answer: 33

Prompt: Write the positive integer 27 in base 8 (no prefix, lowercase if hex).

Solution: 27 in base 8 is 33 (check: int('33', 8)=27).


Answer: 233

Prompt: Write the positive integer 155 in base 8 (no prefix, lowercase if hex).

Solution: 155 in base 8 is 233 (check: int('233', 8)=155).


Answer: 10110100

Prompt: Write the positive integer 180 in base 2 (no prefix, lowercase if hex).

Solution: 180 in base 2 is 10110100 (check: int('10110100', 2)=180).


Answer: 277

Prompt: Write the positive integer 191 in base 8 (no prefix, lowercase if hex).

Solution: 191 in base 8 is 277 (check: int('277', 8)=191).


Answer: 302

Prompt: Write the positive integer 194 in base 8 (no prefix, lowercase if hex).

Solution: 194 in base 8 is 302 (check: int('302', 8)=194).


Answer: 10110001

Prompt: Write the positive integer 177 in base 2 (no prefix, lowercase if hex).

Solution: 177 in base 2 is 10110001 (check: int('10110001', 2)=177).


Answer: -8

Prompt: Evaluate p(x)=-1x^3 + (-2)x^2 + (4)x + (-3) at x=-1.


Answer: -8

Prompt: Evaluate p(x)=2x^3 + (-3)x^2 + (0)x + (-3) at x=-1.


Answer: -11

Prompt: Evaluate p(x)=1x^3 + (3)x^2 + (4)x + (1) at x=-3.


Answer: 66

Prompt: Evaluate p(x)=2x^3 + (1)x^2 + (2)x + (-3) at x=3.


Answer: 2

Prompt: Evaluate p(x)=2x^3 + (1)x^2 + (-2)x + (2) at x=0.


Answer: -12

Prompt: Evaluate p(x)=3x^3 + (4)x^2 + (3)x + (2) at x=-2.


Answer: 151

Prompt: Evaluate p(x)=4x^3 + (4)x^2 + (2)x + (1) at x=3.


Answer: 2

Prompt: Evaluate p(x)=-2x^3 + (3)x^2 + (-2)x + (3) at x=1.


Answer: 28

Prompt: Evaluate p(x)=-1x^3 + (1)x^2 + (2)x + (-2) at x=-3.


Answer: 8

Prompt: Evaluate p(x)=2x^3 + (1)x^2 + (1)x + (4) at x=1.


Answer: 2

Prompt: Evaluate p(x)=3x^3 + (-1)x^2 + (4)x + (2) at x=0.


Answer: -9

Prompt: Evaluate p(x)=-3x^3 + (2)x^2 + (3)x + (1) at x=2.


Answer: 0

Prompt: Evaluate p(x)=-3x^3 + (-2)x^2 + (3)x + (2) at x=1.


Answer: 7

Prompt: Evaluate p(x)=-1x^3 + (-3)x^2 + (-1)x + (4) at x=-3.


Answer: 1

Prompt: Evaluate p(x)=-1x^3 + (-2)x^2 + (0)x + (2) at x=-1.


Answer: -1

Prompt: Evaluate p(x)=3x^3 + (-3)x^2 + (-1)x + (4) at x=-1.


Answer: 42

Prompt: y is inversely proportional to x. When x=7, y=18. What is y when x=3?

Solution: xy constant=126; y=126/3=42.


Answer: 10

Prompt: y is inversely proportional to x. When x=4, y=20. What is y when x=8?

Solution: xy constant=80; y=80/8=10.


Answer: 6

Prompt: y is inversely proportional to x. When x=3, y=8. What is y when x=4?

Solution: xy constant=24; y=24/4=6.


Answer: 3.125

Prompt: A circular sector has radius 5 and central angle 45 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (45/360)·5^2 = 3.125.


Answer: 8.333333333

Prompt: A circular sector has radius 10 and central angle 30 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (30/360)·10^2 = 8.333333333333334.


Answer: 21.33333333

Prompt: A circular sector has radius 8 and central angle 120 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (120/360)·8^2 = 21.333333333333332.


Answer: 24

Prompt: A circular sector has radius 12 and central angle 60 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (60/360)·12^2 = 24.0.


Answer: 33.33333333

Prompt: A circular sector has radius 10 and central angle 120 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (120/360)·10^2 = 33.333333333333336.


Answer: 8.333333333

Prompt: A circular sector has radius 5 and central angle 120 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (120/360)·5^2 = 8.333333333333334.


Answer: 1.333333333

Prompt: A circular sector has radius 4 and central angle 30 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (30/360)·4^2 = 1.3333333333333333.


Answer: 12.25

Prompt: A circular sector has radius 7 and central angle 90 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (90/360)·7^2 = 12.25.


Answer: 20.25

Prompt: A circular sector has radius 9 and central angle 90 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (90/360)·9^2 = 20.25.


Answer: 12

Prompt: A circular sector has radius 12 and central angle 30 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (30/360)·12^2 = 12.0.


Answer: 4.166666667

Prompt: A circular sector has radius 5 and central angle 60 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (60/360)·5^2 = 4.166666666666667.


Answer: 4

Prompt: A circular sector has radius 4 and central angle 90 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (90/360)·4^2 = 4.0.


Answer: 50

Prompt: A circular sector has radius 10 and central angle 180 degrees. What is its area divided by π (exact decimal or integer)?

Solution: A/π = (θ/360)r^2 = (180/360)·10^2 = 50.0.


Answer: 140

Prompt: What is lcm(10, 7, 4)?


Answer: 660

Prompt: What is lcm(10, 12, 11)?


Answer: 8

Prompt: What is lcm(4, 4, 8)?


Answer: 30

Prompt: What is lcm(10, 2, 3)?


Answer: 60

Prompt: What is lcm(10, 4, 6)?


Answer: 28

Prompt: What is lcm(4, 4, 7)?


Answer: 70

Prompt: What is lcm(5, 7, 10)?


Answer: 6

Prompt: What is lcm(6, 3, 6)?


Answer: 60

Prompt: What is lcm(5, 12, 10)?


Answer: 12

Prompt: What is lcm(6, 4, 12)?


Answer: 330

Prompt: What is lcm(6, 11, 10)?


Answer: 60

Prompt: What is lcm(3, 10, 12)?


Answer: 44

Prompt: What is lcm(4, 11, 11)?


Answer: 12

Prompt: What is lcm(4, 4, 12)?


Answer: 0

Prompt: Compute 7^7 mod 7.


Answer: 1

Prompt: Compute 2^3 mod 7.


Answer: 4

Prompt: Compute 6^8 mod 11.