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The value of $$\,{}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}} {C_3}$$ is
Options:
[{"identifier": "A", "content": "$${}^{55}{C_4}$$ "}, {"identifier": "B", "content": "$${}^{55}{C_3}$$"}, {"identifier": "C", "content": "$${}^{56}{C_3}$$"}, {"identifier": "D", "content": "$${}^{56}{C_4}$$"}] | ["D"]
Explanation:
Given, $${}^{50}{C_4} + \sum\limits_{n = 1}^6 {{}^{56 - r}{C_3}} $$
<br><br>$$ \Rightarrow $$ $${}^{50}{C_4}$$ + $${}^{55}{C_3}$$ + $${}^{54}{C_3}$$ + $${}^{53}{C_3}$$ + $${}^{52}{C_3}$$ + $${}^{51}{C_3}$$ + $${}^{50}{C_3}$$
<br><br>Arrange those this way
<br><br>$$ \Rightarrow $$ $${}^{50}{C_4}$$ +... |
For natural numbers $$m$$ , $$n$$, if $${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\, = 1 + {a_1}y + {a_2}{y^2} + ..........$$ and $${a_1} = {a_2} = 10,$$ then $$\left( {m,\,n} \right)$$ is
Options:
[{"identifier": "A", "content": "$$\\left( {20,\\,45} \\right)$$ "}, {"identifier": "B", "content": "$$\\le... | ["D"]
Explanation:
$${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\,$$
<br><br>= $$\left( {{}^m{C_0} - {}^m{C_1}y + {}^m{C_2}{y^2} + ....} \right)$$ -
<br> $$\left( {{}^n{C_0} + {}^n{C_1}y + {}^n{C_2}{y^2} + ....} ... |
The sum of the series $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .....\, - \,.....\, + {}^{20}{C_{10}}$$ is
Options:
[{"identifier": "A", "content": "$$0$$"}, {"identifier": "B", "content": "$${}^{20}{C_{10}}$$ "}, {"identifier": "C", "content": "$$ - {}^{20}{C_{10}}$$ "}, {"identifier": "D", "cont... | ["D"]
Explanation:
We know
<br><br>$${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... + {}^{20}{C_{10}} - {}^{20}{C_{11}}+ ...... + {}^{20}{C_{20}} = 0$$
<br><br>$$ \Rightarrow $$ $$({}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .... - {}^{20} {C_{9}})$$ $$+{}^{20}{C_{10}}$$ $$(-{}^{20}{C... |
<b>Statement - 1 :</b> $$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.} $$
<br/><b>Statement - 2 :</b> $$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r} = {{\left( {1 + x} \right)}^n} + nx{{\left( {1 + x} \right)}^{n - 1}}.} $$
Options:
[{"identifier": "A"... | ["B"]
Explanation:
<b>Check Statement - 1</b>
<br><br>$$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}}$$
<br><br>= $$\sum\limits_{r = 0}^n {r.{}^n{C_r}} $$ + $$\sum\limits_{r = 0}^n {{}^n{C_r}} $$
<br><br>= $$\sum\limits_{r = 1}^n {r.{n \over r}{}^{n - 1}{C_{r - 1}}} $$ $$ + {2^n}$$
<br><br>= $$n\sum\limits... |
Let $${s_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right){}^{10}} {C_j}$$,
<br/><br/>$${{s_2} = \sum\limits_{j = 1}^{10} {} } j.{}^{10}{C_j}$$ and <br/><br/>$${{s_3} = \sum\limits_{j = 1}^{10} {{j^2}.{}^{10}{C_j}.} }$$
<p><b>Statement-1 :</b> $${{S_3} = 55 \times {2^9}}$$.
<br/><b>Statement-2 :</b> $${{S_1} = 90... | ["B"]
Explanation:
<b>Note :</b>
<br><br>$$\sum\limits_{r = 0}^n {r.{}^n{C_r}} $$ = $$ = n{.2^{n - 1}}$$
<br><br>$$\sum\limits_{r = 0}^n {{r^2}.{}^n{C_r}} = n\left( {n + 1} \right){2^{n - 2}}$$
<br><br>Given that,
<br><br>$${s_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right){}^{10}} {C_j}$$
<br><br>=$$\sum\limi... |
The coefficient of $${x^7}$$ in the expansion of $${\left( {1 - x - {x^2} + {x^3}} \right)^6}$$ is
Options:
[{"identifier": "A", "content": "$$-132$$ "}, {"identifier": "B", "content": "$$-144$$ "}, {"identifier": "C", "content": "$$132$$ "}, {"identifier": "D", "content": "$$144$$"}] | ["B"]
Explanation:
Given,
<br>$${\left( {1 - x - {x^2} + {x^3}} \right)^6}$$
<br><br>= $${\left[ {\left( {1 - x} \right) - {x^2}\left( {1 - x} \right)} \right]^6}$$
<br><br>= $${\left( {1 - x} \right)^6}{\left( {1 - {x^2}} \right)^6}$$
<br><br>= $$\left( {1 + {}^6{C_1}( - x) + {}^6{C_2}{{( - x)}^2} + {}^6{C_3}{{( - x... |
If the coefficints of $${x^3}$$ and $${x^4}$$ in the expansion of $$\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}$$ in powers of $$x$$ are both zero, then $$\left( {a,\,b} \right)$$ is equal to:
Options:
[{"identifier": "A", "content": "$$\\left( {14,{{272} \\over 3}} \\right)$$ "}, {"identifier": "B"... | ["B"]
Explanation:
$$\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}$$
<br><br>= $${\left( {1 - 2x} \right)^{18}} + ax{\left( {1 - 2x} \right)^{18}} + b{x^2}{\left( {1 - 2x} \right)^{18}}$$
<br><br>= $$\left( {1 + ax + b{x^2}} \right)\left[ {{}^{18}{C_0} - {}^{18}{C_1}\left( {2x} \right) + {}^{18}{C_2}{... |
The sum of coefficients of integral power of $$x$$ in the binomial expansion $${\left( {1 - 2\sqrt x } \right)^{50}}$$ is :
Options:
[{"identifier": "A", "content": "$${1 \\over 2}\\left( {{3^{50}} - 1} \\right)$$ "}, {"identifier": "B", "content": "$${1 \\over 2}\\left( {{2^{50}} + 1} \\right)$$ "}, {"identifier": "... | ["C"]
Explanation:
$${\left( {1 - 2\sqrt x } \right)^{50}}$$
<br><br>= $${}^{50}{C_0} + {}^{50}{C_1}.\left( { - 2\sqrt x } \right) + {}^{50}{C_2}.{\left( { - 2\sqrt x } \right)^2} + ....$$
<br><br>Now we need to find out those coefficient where degree of x is integer and you can see at odd terms power of x is integer.... |
If the number of terms in the expansion of $${\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n},\,x \ne 0,$$ is 28, then the sum of the coefficients of all the terms in this expansion, is :
Options:
[{"identifier": "A", "content": "243 "}, {"identifier": "B", "content": "729 "}, {"identifier": "C", "content": "6... | ["B"]
Explanation:
Total no of terms in $${\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n}$$ = $${}^{n + 2}{C_2}$$ = 28
<br><br>(n+2)(n+1) = 56
<br><br>$$ \Rightarrow n = 6$$
<br><br>Sum of coefficient = (1 - 2 + 4)<sup>6</sup> = 3<sup>6</sup> = 729 |
The value of $$\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} - {}^{10}{C_3}} \right)$$
<br/>$$\left( {{}^{21}{C_4} - {}^{10}{C_4}} \right)$$$$ + .... + \left( {{}^{21}{C_{10}} - {}^{10}{C_{10}}} \right)$$ is
Options:
[{"identifier": "A", "content": "... | ["C"]
Explanation:
<img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265679/exam_images/bx4nbx1k931n5ip0ploo.webp" loading="lazy" alt="JEE Main 2017 (Offline) Mathematics - Binomial Theorem Question 184 English Explanation"> |
The sum of the co-efficients of all odd degree terms in the expansion of
<br/><br/>$${\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5}$$, $$\left( {x > 1} \right)$$ is
Options:
[{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "-1"}, {"identifier": "C", "co... | ["A"]
Explanation:
<img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266144/exam_images/sxwjinh9ykk33t08ossz.webp" loading="lazy" alt="JEE Main 2018 (Offline) Mathematics - Binomial Theorem Question 185 English Explanation"> |
If n is the degree of the polynomial,
<br/><br/>$${\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8} + $$ $${\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}$$
<br/><br/>and m is the coefficient of x<sup>n</sup> in it, then the ordered pair (n, m) is equal to :
Optio... | ["C"]
Explanation:
Given, <br>
$${\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8}$$ + $${\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}$$<br><br>
= $${\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }} \times {{\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} } ... |
The coefficien of x<sup>10</sup> in the expansion of (1 + x)<sup>2</sup>(1 + x<sup>2</sup>)<sup>3</sup>(1 + x<sup>3</sup>)<sup>4</sup> is equal to :
Options:
[{"identifier": "A", "content": "52"}, {"identifier": "B", "content": "56"}, {"identifier": "C", "content": "50"}, {"identifier": "D", "content": "44"}] | ["A"]
Explanation:
$$ \because $$$$\,\,\,$$ (1 + x)<sup>2</sup> = 1 + 2x + x<sup>2</sup>,
<br><br>(1 + x<sup>2</sup>)<sup>3</sup> = 1 + 3x<sup>2</sup> + 3x<sup>4</sup> + x<sup>6</sup>
<br><br>and (1 + x<sup>3</sup>)<sup>4</sup> = 1 + 4x<sup>3</sup> + 6x<sup>6</sup> + 4x<sup>9</sup> + x<sup>12</sup>
<br><br>So, the pos... |
The coefficient of x<sup>2</sup> in the expansion of the product
<br/>(2$$-$$x<sup>2</sup>) .((1 + 2x + 3x<sup>2</sup>)<sup>6</sup> + (1 $$-$$ 4x<sup>2</sup>)<sup>6</sup>) is :
Options:
[{"identifier": "A", "content": "107"}, {"identifier": "B", "content": "106"}, {"identifier": "C", "content": "108"}, {"identifier... | ["B"]
Explanation:
Given,
<br><br>(2 $$-$$ x<sup>2</sup>) . (1 + 2x + 3x<sup>2</sup>) <sup>6</sup> + (1 $$-$$ 4x<sup>2</sup>)<sup>6</sup>)
<br><br>Let, a = ((1 + 2x + 3x<sup>2</sup>)<sup>6</sup> + (1 $$-$$ 4x<sup>2</sup>)<sup>6</sup>)
<br><br>$$\therefore\,\,\,\,$$ Given statement becomes,
<br><br>(2 $$-$$ x<sup>2</... |
If <sup>20</sup>C<sub>1</sub> + (2<sup>2</sup>) <sup>20</sup>C<sub>2</sub> + (3<sup>2</sup>) <sup>20</sup>C<sub>3</sub> + ..... + (20<sup>2</sup>
)
<sup>20</sup>C<sub>20</sub> = A(2<sup>$$\beta $$</sup>), then the ordered pair (A, $$\beta $$) is equal to :
Options:
[{"identifier": "A", "content": "(420, 19)"}, {"ident... | ["B"]
Explanation:
S = $${1^2}\,{}^{20}{C_1} + {2^2}\,{}^{20}{C_2} + {3^2}\,{}^{20}{C_3} + .......... + {20^2}\,{}^{20}{C_{20}}$$<br><br>
$$ \Rightarrow $$ $$\sum\limits_{r = 1}^{20} {{r^2}\,{}^{20}{C_r}} $$ <br><br>
$$ \Rightarrow $$ $$\sum\limits_{r = 1}^{20} {r\,.\left( {r.{}^{20}{C_r}} \right)} $$<br><br>
$$ \Righ... |
If the coefficients of x<sup>2</sup>
and x<sup>3</sup>
are both zero, in the expansion of the expression (1 + ax + bx<sup>2</sup>
) (1 – 3x)<sup>15</sup> in
powers of x, then the ordered pair (a,b) is equal to :
Options:
[{"identifier": "A", "content": "(28, 861)"}, {"identifier": "B", "content": "(28, 315)"}, {"id... | ["B"]
Explanation:
(1 + ax + bx<sup>2</sup>)(1 – 3x)<sup>15</sup><br><br>
Co-eff. of x<sup>2</sup> = 1.<sup>15</sup>C<sub>2</sub>(–3)<sup>2</sup> + a.<sup>15</sup>C<sub>1</sub>(–3) + b.<sup>15</sup>C<sub>0</sub><br><br>
$$ = {{15 \times 14} \over 2} \times 9 - 15 \times 3a + b = 0$$ (given)<br><br>
$$ \Rightarrow $$ 9... |
If some three consecutive in the binomial
expansion of (x + 1)<sup>n</sup> is powers of x are in the ratio
2 : 15 : 70, then the average of these three
coefficient is :-
Options:
[{"identifier": "A", "content": "625"}, {"identifier": "B", "content": "227"}, {"identifier": "C", "content": "964"}, {"identifier": "D", "c... | ["D"]
Explanation:
Given $${}^n{C_{r - 1}}:{}^n{C_r}:{}^n{C_{r + 1}} = 2:15:70$$<br><br>
$${{{}^n{C_{r - 1}}} \over {{}^n{C_r}}} = {2 \over {15}}$$<br><br>
$$ \Rightarrow {r \over {n - r + 1}} = {2 \over {15}}$$<br><br>
$$ \Rightarrow 15r = 2n - 2r + 2$$<br><br>
$$ \Rightarrow 17r = 2n + 2$$ .... (i)<br><br>
Now $${{... |
The sum of the co-efficients of all even
degree terms in x in the expansion of<br/>
$${\left( {x + \sqrt {{x^3} - 1} } \right)^6}$$ + $${\left( {x - \sqrt {{x^3} - 1} } \right)^6}$$, (x > 1) is equal to:
Options:
[{"identifier": "A", "content": "32"}, {"identifier": "B", "content": "26"}, {"identifier": "C", "conte... | ["D"]
Explanation:
Let $${\left( {a + x} \right)^n}$$ = Odd trems(A) + Even terms(B)
<br><br>So $${\left( {a - x} \right)^n}$$ = Odd terms(A) - Even terms(B)
<br><br>$$\therefore$$ $${\left( {a + x} \right)^n} - {\left( {a - x} \right)^n}$$
<br><br>= (A + B) + (A - B)
<br><br>= 2A
<br><br>= 2[odd terms]
<br><br>= 2[ ... |
Let S<sub>n</sub> = 1 + q + q<sup>2</sup> + . . . . . + q<sup>n</sup> and T<sub>n</sub> = 1 + $$\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}$$ + . . . . . .+ $${\left( {{{q + 1} \over 2}} \right)^n}$$ where q is a real number and q $$ \ne $$ 1. If <sup>101</sup>C<sub>1</sub> + <su... | ["C"]
Explanation:
<sup>101</sup>C<sub>1</sub> + <sup>101</sup>C<sub>2</sub>S<sub>1</sub> + . . . . . . . + <sup>101</sup>C<sub>101</sub>S<sub>100</sub>
<br><br>$$=$$ $$\alpha $$T<sub>100</sub>
<br><br><sup>101</sup>C<sub>1</sub> + <sup>101</sup>C<sub>2</sub>(1 + q) + <sup>101</sup>C<sub>3</sub>(1 + q + q<sup>2</sup>... |
Let (x + 10)<sup>50</sup> + (x $$-$$ 10)<sup>50</sup> = a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub>x<sup>2</sup> + . . . . + a<sub>50</sub>x<sup>50</sup>, for all x $$ \in $$ R; then $${{{a_2}} \over {{a_0}}}$$ is equal to
Options:
[{"identifier": "A", "content": "12.25 "}, {"identifier": "B", "content": "12.75"},... | ["A"]
Explanation:
(10 + x)<sup>50</sup> + (10 $$-$$ x)<sup>50</sup>
<br><br>$$ \Rightarrow $$ a<sub>2</sub> = 2.<sup>50</sup>C<sub>2</sub> 10<sup>48</sup>, a<sub>0</sub> = 2.10<sup>50</sup>
<br><br>$${{{a_2}} \over {{a_0}}} = {{^{50}{C_2}} \over {{{10}^2}}} = 12.25$$ |
The value of r for which <sup>20</sup>C<sub>r</sub> <sup>20</sup>C<sub>0</sub> + <sup>20</sup>C<sub>r$$-$$1</sub> <sup>20</sup>C<sub>1</sub> + <sup>20</sup>C<sub>r$$-$$2</sub> <sup>20</sup>C<sub>2</sub> + . . . . .+ <sup>20</sup>C<sub>0</sub> <sup>20</sup>C<sub>r</sub> is maximum, is
Options:
[{"identifier": "A", "c... | ["A"]
Explanation:
Given sum = coefficient of x<sup>r</sup> in the expansion of
<br><br>(1 + x)<sup>20</sup>(1 + x)<sup>20</sup>,
<br><br>Which is equal to <sup>40</sup>C<sub>r</sub>
<br><br>It is maximum when r = 20 |
If $${\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}$$ then k is equal to
Options:
[{"identifier": "A", "content": "100"}, {"identifier": "B", "content": "200"}, {"identifier": "C", "content": "50"}, {"identifier": "D", "content": "4... | ["A"]
Explanation:
$${\sum\limits_{i = 1}^{20} {\left( {{{^{20}{C_{i - 1}}} \over {^{20}{C_i}{ + ^{20}}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}$$
<br><br>$$ \Rightarrow \,\,\sum\limits_{i = 1}^{20} {{{\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{21}{C_i}}}} \right)}^3}} = {k \over {21}}$$
<br><br>$$ \Rightarrow \,\,\... |
The coefficient of t<sup>4</sup> in the expansion of $${\left( {{{1 - {t^6}} \over {1 - t}}} \right)^3}$$ is :
Options:
[{"identifier": "A", "content": "14"}, {"identifier": "B", "content": "15"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "12"}] | ["B"]
Explanation:
$${\left( {{{1 - {t^6}} \over {1 - t}}} \right)^3}$$
<br><br>= (1 $$-$$ t<sup>6</sup>)<sup>3</sup> (1 $$-$$ t)<sup>$$-$$3</sup>
<br><br>= (1 $$-$$ <sup>3</sup>C<sub>1</sub>t<sup>6</sup> + <sup>3</sup>C<sub>2</sub>t<sup>12</sup> $$-$$ <sup>3</sup>C<sub>3</sub>t<sup>18</sup>) $$ \times $$ (1 $$-$$ t... |
The sum of the series
<br/><br/>2.<sup>20</sup>C<sub>0</sub>
+ 5.<sup>20</sup>C<sub>1</sub> + 8.<sup>20</sup>C<sub>2</sub> + 11.<sup>20</sup>C<sub>3</sub> + ... +62.<sup>20</sup>C<sub>20</sub> is equal to :
Options:
[{"identifier": "A", "content": "2<sup>25</sup>"}, {"identifier": "B", "content": "2<sup>24</sup>"},... | ["A"]
Explanation:
Here general term = (3r + 2)<sup>20</sup>C<sub>r</sub>
<br><br>$$ \therefore $$ Sum of the series = $$\sum\limits_{r = 0}^{20} {\left( {3r + 2} \right)} {}^{20}{C_r}$$
<br><br>= $$3\sum\limits_{r = 0}^{20} {r.} {}^{20}{C_r} + 2\sum\limits_{r = 0}^{20} {{}^{20}{C_r}} $$
<br><br>= 3 $$ \times $$ 20$$ ... |
If for some positive integer n, the coefficients<br/> of three consecutive terms in the binomial<br/> expansion of (1 + x)<sup>n + 5</sup> are in the ratio<br/> 5 : 10 : 14, then the largest coefficient in this expansion is :
Options:
[{"identifier": "A", "content": "330"}, {"identifier": "B", "content": "792 "}, {"id... | ["D"]
Explanation:
Consider the three consecutive coefficients as <br><br>$$^{n + 5}{C_r},{\,^{n + 5}}{C_{r + 1}},{\,^{n + 5}}{C_{r + 2}}$$<br><br>$$ \because $$ $${{^{n + 5}{C_r}} \over {^{n + 5}{C_{r + 1}}}} = {1 \over 2}$$<br><br>$$ \Rightarrow {{r + 1} \over {n + 5 - r}} = {1 \over 2} \Rightarrow 3r = n + 3$$ ...(... |
The coefficient of x<sup>4</sup>
in the expansion of
<br/>(1 + x + x<sup>2</sup>
+ x<sup>3</sup>)<sup>6</sup>
in powers of x, is ______.
Options:
[] | 120
Explanation:
(1 + x + x<sup>2</sup>
+ x<sup>3</sup>)<sup>6</sup>
<br><br>= ((1 + x) (1 + x<sup>2</sup>))<sup>6</sup>
<br><br>= (1 + x)<sup>6</sup>(1 + x<sup>2</sup>)<sup>6</sup>
<br><br>= $$\sum\limits_{r = 0}^6 {{}^6{C_r}.{x^r}} $$ $$\sum\limits_{r = 0}^6 {{}^6{C_r}.{x^{2r}}} $$
<br><br>Coefficient of x<sup>4</s... |
The value of $$\sum\limits_{r = 0}^{20} {{}^{50 - r}{C_6}} $$ is equal to:
Options:
[{"identifier": "A", "content": "$${}^{50}{C_6} - {}^{30}{C_6}$$"}, {"identifier": "B", "content": "$${}^{51}{C_7} - {}^{30}{C_7}$$"}, {"identifier": "C", "content": "$${}^{50}{C_7} - {}^{30}{C_7}$$"}, {"identifier": "D", "content": "$... | ["B"]
Explanation:
$$\sum\limits_{r = 0}^{20} {} {}^{50 - r}{C_6} = {}^{50}{C_6} + {}^{49}{C_6} + {}^{48}{C_6} + .... + {}^{30}{C_6}$$<br><br>$$ = {}^{50}{C_6} + {}^{49}{C_6} + .... + {}^{31}{C_6} + ({}^{30}{C_6} + {}^{30}{C_7}) - {}^{30}{C_7}$$<br><br>$$ = {}^{50}{C_6} + {}^{49}{C_6} + .... + ({}^{31}{C_6} + {}^{31}{... |
For a positive integer n,
<br/>$${\left( {1 + {1 \over x}} \right)^n}$$ is expanded
<br/>in increasing powers of x. If three consecutive
<br/>coefficients in this expansion are in the ratio,
<br/>2 : 5 : 12, then n is equal to________.
Options:
[] | 118
Explanation:
Let, three consecutive coefficients are<br><br>
$${}^n{C_{r - 1}},{}^n{C_r},{}^n{C_{r + 1}}$$<br><br>
$${}^n{C_{r - 1}}:{}^n{C_r}:{}^n{C_{r + 1}} = 2:5:12$$<br><br>
Now, $${{{}^n{C_{r - 1}}} \over {{}^n{C_r}}} = {2 \over 5}$$<br><br>
$$ \Rightarrow 7r = 2n + 2$$ ...(i)<br><br>
$... |
If C<sub>r</sub> $$ \equiv $$ <sup>25</sup>C<sub>r</sub> and<br/>
C<sub>0</sub> + 5.C<sub>1</sub> + 9.C<sub>2</sub> + .... + (101).C<sub>25</sub> = 2<sup>25</sup>.k, then k is equal to _____.
Options:
[] | 51
Explanation:
S = 1.<sup>25</sup>C<sub>0</sub> + 5.<sup>25</sup>C<sub>1</sub> + 9.<sup>25</sup>C<sub>2</sub> + .... + (101)<sup>25</sup>C<sub>25</sub>
<br><br>S = (101).<sup>25</sup>C<sub>25</sub> + (97).<sup>25</sup>C<sub>24</sub> + .......... + (1).<sup>25</sup>C<sub>0</sub>
<br>___________________________________... |
If the sum of the coefficients of all even powers of x in the product<br/> (1 + x + x<sup>2</sup>
+ ....+ x<sup>2n</sup>)(1 - x + x<sup>2</sup> - x<sup>3</sup> + ...... + x<sup>2n</sup>) is 61, then n is equal to _______.
Options:
[] | 30
Explanation:
(1 + x + x<sup>2</sup>
+ ....+ x<sup>2n</sup>)(1 - x + x<sup>2</sup> - x<sup>3</sup> + ...... + x<sup>2n</sup>)
<br><br>= a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub>x<sup>2</sup>
+ …..
<br><br>put x = 1
<br><br> (2n + 1)$$ \times $$1 = a<sub>0</sub> + a<sub>1</sub> + a<sub>2</sub> + …… (1)
<br><b... |
The value of
<br/>-<sup>15</sup>C<sub>1</sub> + 2.<sup>15</sup>C<sub>2</sub> – 3.<sup>15</sup>C<sub>3</sub> + ... - 15.<sup>15</sup>C<sub>15</sub> + <sup>14</sup>C<sub>1</sub> + <sup>14</sup>C<sub>3</sub> + <sup>14</sup>C<sub>5</sub> + ...+ <sup>14</sup>C<sub>11</sub> is :
Options:
[{"identifier": "A", "content": "2<s... | ["D"]
Explanation:
$$ - {}^{15}{C_1} + 2.{}^{15}{C_2} - 3.{}^{15}{C_3} + ....\,. - 15.{}^{15}{C_{15}}$$<br><br>$$ = \sum\limits_{r = 1}^{15} {{{( - 1)}^r}.r.{}^{15}{C_r}} $$<br><br>$$ = \sum\limits_{r = 1}^{15} {{{( - 1)}^2}.r.{{15} \over r}} .{}^{14}{C_{r - 1}}$$<br><br>$$ = 15\sum\limits_{r = 1}^{15} {{{( - 1)}^2}.{... |
If $$n \ge 2$$ is a positive integer, then the sum of the series $${}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}^4{C_2} + ... + {}^n{C_2}} \right)$$ is :
Options:
[{"identifier": "A", "content": "$${{n(2n + 1)(3n + 1)} \\over 6}$$"}, {"identifier": "B", "content": "$${{n(n + 1)(2n + 1)} \\over 6}$$"}, {"identi... | ["B"]
Explanation:
$${}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}^4{C_2} + ........ + {}^n{C_2}} \right)$$<br><br>$${}^{n + 1}{C_2} + 2\left( {{}^3{C_2} + {}^3{C_2} + {}^4{C_2} + ........ + {}^n{C_2}} \right)$$<br><br>use $$\left\{ {{}^n{C_{r + 1}} + {}^n{C_r} = {}^{n + 1}{C_r}} \right\}$$<br><br>$$ = {}^{n +... |
For integers n and r, let $$\left( {\matrix{
n \cr
r \cr
} } \right) = \left\{ {\matrix{
{{}^n{C_r},} & {if\,n \ge r \ge 0} \cr
{0,} & {otherwise} \cr
} } \right.$$ The maximum value of k for which the sum $$\sum\limits_{i = 0}^k {\left( {\matrix{
{10} \cr
i \cr
} } \right)\l... | 12
Explanation:
As k is unbounded so maximum value is not defined.
<br><br>Question will be <b>BONUS</b>. |
Let m, n$$\in$$N and gcd (2, n) = 1. If $$30\left( {\matrix{
{30} \cr
0 \cr
} } \right) + 29\left( {\matrix{
{30} \cr
1 \cr
} } \right) + ...... + 2\left( {\matrix{
{30} \cr
{28} \cr
} } \right) + 1\left( {\matrix{
{30} \cr
{29} \cr
} } \right) = n{.2^m}$$, then n + m is ... | 45
Explanation:
$$30({}^{30}{C_0}) + 29({}^{30}{C_1}) + .... + 2({}^{30}{C_{28}}) + 1({}^{30}{C_{29}})$$<br><br>$$ = 30({}^{30}{C_{30}}) + 29({}^{30}{C_{29}}) + ...... + 2({}^{30}{C_2}) + 1({}^{30}{C_1})$$<br><br>$$ = \sum\limits_{r = 1}^{30} {r({}^{30}{C_r})} $$<br><br>$$ = \sum\limits_{r = 1}^{30} {r\left( {{{30} \o... |
Let [ x ] denote greatest integer less than or equal to x. If for n$$\in$$N, <br/><br/>$${(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}} $$, <br/><br/>then $$\sum\limits_{j = 0}^{\left[ {{{3n} \over 2}} \right]} {{a_{2j}} + 4} \sum\limits_{j = 0}^{\left[ {{{3n - 1} \over 2}} \right]} {{a_{2j}} + 1} $$ is eq... | ["C"]
Explanation:
$${(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}} $$<br><br>$$(1 - x + {x^3}) = {a_0} + {a_1}x + {a_2}{x^2} + ...... + {a_{3n}}{x^{3n}}$$<br><br>Put x = 1<br><br>$$1 = {a_0} + {a_1} + {a_2} + {a_3} + {a_4} + ........ + {a_{3n}}$$ ...... (1)<br><br>Put x = $$-$$1<br><br>$$1 = {a_0} - {a_1... |
Let n be a positive integer. Let <br/><br/>$$A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^k} + {{\left( {{{31} \over {32}}} \right)}^k}} \right]} $$. If <br/><b... | 6
Explanation:
$$A = \sum {{{( - 1)}^k}{}^n{C_k}{{\left( {{1 \over 2}} \right)}^k}} + \sum {{{( - 1)}^k}{}^n{C_k}{{\left( {{3 \over 4}} \right)}^k}} + .....$$<br><br>$$ = {\left( {1 - {1 \over 2}} \right)^n} + {\left( {1 - {3 \over 4}} \right)^n} + ..... + {\left( {1 - {{31} \over {32}}} \right)^n}$$<br><br>$$ = {\l... |
The value of $$\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)} $$ is equal to :
Options:
[{"identifier": "A", "content": "924"}, {"identifier": "B", "content": "1024"}, {"identifier": "C", "content": "1124"}, {"identifier": "D", "content": "1324"}] | ["A"]
Explanation:
Given,<br><br>$$\sum\limits_{r = 0}^6 {{}^6{C_r}{}^6{C_{6 - r}}} $$
<br><br>= $${}^6{C_0}.{}^6{C_6} + {}^6{C_1}.{}^6{C_5} + ... + {}^6{C_6}.{}^6{C_0}$$
<br><br>Now,
<br><br>$$\eqalign{
& = \left( {{}^6{C_0} + {}^6{C_1}x + {}^6{C_2}{x^2} + ... + {}^6{C_6}{x^6}} \right) \cr
& \left( {{}... |
Let (1 + x + 2x<sup>2</sup>)<sup>20</sup> = a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub>x<sup>2</sup> + .... + a<sub>40</sub>x<sup>40</sup>. Then a<sub>1</sub> + a<sub>3</sub> + a<sub>5</sub> + ..... + a<sub>37</sub> is equal to
Options:
[{"identifier": "A", "content": "2<sup>20</sup>(2<sup>20</sup> $$-$$ 21)"}, {"... | ["B"]
Explanation:
$${(1 + x + 2{x^2})^{20}} = {a_0} + {a_1}x + {a_2}{x^2} + .... + {a_{40}}{x^{40}}$$<br><br>Put x = 1<br><br>$$ \Rightarrow {4^{20}} = {a_0} + {a_1} + ....... + {a_{40}}$$ ..... (i)<br><br>Put x = $$-$$1<br><br>$$ \Rightarrow {2^{20}} = {a_0} - {a_1} + ....... + - {a_{39}} + {a_{40}}$$ ..... (ii)<br... |
Let $${}^n{C_r}$$ denote the binomial coefficient of x<sup>r</sup> in the expansion of (1 + x)<sup>n</sup>.
If $$\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R$$, then $$\alpha$$ + $$\beta$$ is equal to ___________.
Options:
[] | 19
Explanation:
$$\sum\limits_{k = 0}^{10} {({2^2} + 3k){}^{10}{C_k}} $$<br><br>$$ = 4\sum\limits_{k = 0}^{10} {{}^{10}{C_k}} + 3\sum\limits_{k = 0}^{10} {k.{}^{10}{C_k}} $$<br><br>$$ = 4({2^{10}}) + 3\sum\limits_{k = 0}^{10} {k.{{10} \over k}.{}^9{C_{k - 1}}} $$<br><br>= $$4({2^{10}}) + 3.10({2^9})$$<br><br>$$ = 4({... |
The coefficient of x<sup>256</sup> in the expansion of <br/><br/>(1 $$-$$ x)<sup>101</sup> (x<sup>2</sup> + x + 1)<sup>100</sup> is :
Options:
[{"identifier": "A", "content": "$${}^{100}{C_{16}}$$"}, {"identifier": "B", "content": "$${}^{100}{C_{15}}$$"}, {"identifier": "C", "content": "$$-$$ $${}^{100}{C_{16}}$$"}, {... | ["B"]
Explanation:
$${(1 - x)^{101}}{({x^2} + x + 1)^{100}}$$<br/><br/>Coefficient of $${x^{256}} = {[(1 - x)(1 + x + {x^2})]^{100}}(1 - x) = {(1 - {x^3})^{100}}(1 - x)$$<br/><br/>$$ \Rightarrow ({}^{100}{C_0} - {}^{100}{C_1}{x^3} + {}^{100}{C_2}{x^6} - {}^{100}{C_3}{x^9}...)(1 - x)$$<br/><br/>$$\sum {{{( - 1)}^r}{}^{... |
For the natural numbers m, n, if $${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$$ and $${a_1} = {a_2} = 10$$, then the value of (m + n) is equal to :
Options:
[{"identifier": "A", "content": "88"}, {"identifier": "B", "content": "64"}, {"identifier": "C", "content": "100"}, {"identi... | ["D"]
Explanation:
$${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$$<br><br>Given, ($${a_1} = {a_2} = 10$$)$$(1 - my + {}^m{C_2}{y^2} + .....)(1 + ny + {}^n{C_2}{y^2} + .....) = 1 + {a_1}y + {a_2}{y^2} + ....$$<br><br>$$ \Rightarrow n - m = 10$$ ..... (i)<br><br>$$ \Rightarrow {}^m{C... |
The number of elements in the set {n $$\in$$ {1, 2, 3, ......., 100} | (11)<sup>n</sup> > (10)<sup>n</sup> + (9)<sup>n</sup>} is ______________.
Options:
[] | 96
Explanation:
$${11^n} > {10^n} + {9^n}$$<br><br>$$ \Rightarrow {11^n} - {9^n} > {10^n}$$<br><br>$$ \Rightarrow {(10 + 1)^n} - {(10 - 1)^n} > {10^n}$$<br><br>$$ \Rightarrow 2\{ {}^n{C_1}{.10^{n - 1}} + {}^n{C_3}{10^{n - 10}} + {}^n{C_5}{10^{n - 5}} + .....\} > {10^n}$$
<br><br>$$ \Rightarrow $$ $${1 \ov... |
Let n$$\in$$N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms $${}^n{C_0},3.{}^n{C_1},5.{}^n{C_2},7.{}^n{C_3},.....$$ is equal to 2<sup>100</sup> . 101, then $$2\left[ {{{n - 1} \over 2}} \right]$$ is equal to _______________.
Options:
[] | 98
Explanation:
1. $${}^n{C_0} + 3.{}^n{C_1} + 5.{}^n{C_2} + ... + (2n + 1).{}^n{C_n}$$<br><br>$${T_r} = (2r + 1){}^n{C_r}$$<br><br>$$S = \sum {{T_r}} $$<br><br>$$S = \sum {(2r + 1){}^n{C_r}} = \sum {2r{}^n{C_r} + \sum {{}^n{C_r}} } $$<br><br>$$S = 2(n{.2^{n - 1}}) + {2^n} = {2^n}(n + 1)$$<br><br>$${2^n}(n + 1) = {2^... |
If $${{}^{20}{C_r}}$$ is the co-efficient of x<sup>r</sup> in the expansion of (1 + x)<sup>20</sup>, then the value of $$\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}} $$ is equal to :
Options:
[{"identifier": "A", "content": "$$420 \\times {2^{19}}$$"}, {"identifier": "B", "content": "$$380 \\times {2^{19}}$$"}, {"ide... | ["D"]
Explanation:
$$\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}} $$<br><br>$$\sum {(4(r - 1) + r).{}^{20}{C_r}} $$<br><br>$$\sum {r(r - 1).{{20 \times 19} \over {r(r - 1)}}} .{}^{18}{C_r} + r.{{20} \over r}.\sum {{}^{19}{C_{r - 1}}} $$<br><br>$$ \Rightarrow 20 \times {19.2^{18}} + {20.2^{19}}$$<br><br>$$ \Rightarrow... |
Let $$\left( {\matrix{
n \cr
k \cr
} } \right)$$ denotes $${}^n{C_k}$$ and $$\left[ {\matrix{
n \cr
k \cr
} } \right] = \left\{ {\matrix{
{\left( {\matrix{
n \cr
k \cr
} } \right),} & {if\,0 \le k \le n} \cr
{0,} & {otherwise} \cr
} } \right.$$<br/><br/>If $${A_k}... | 49
Explanation:
$${A_k} = \sum\limits_{i = 0}^9 {{}^9{C_i}} {}^{12}{C_{k - i}} + \sum\limits_{i = 0}^8 {{}^8{C_i}} {}^{13}{C_{k - i}}$$<br><br>$${A_k} = {}^{21}{C_k} + {}^{21}{C_k} = 2.{}^{21}{C_k}$$<br><br>$${A_4} - {A_3} = 2\left( {{}^{21}{C_4} - {}^{21}{C_3}} \right) = 2(5985 - 1330)$$<br><br>$$190p = 2(5985 - 1330... |
$$\sum\limits_{k = 0}^{20} {{{\left( {{}^{20}{C_k}} \right)}^2}} $$ is equal to :
Options:
[{"identifier": "A", "content": "$${}^{40}{C_{21}}$$"}, {"identifier": "B", "content": "$${}^{40}{C_{19}}$$"}, {"identifier": "C", "content": "$${}^{40}{C_{20}}$$"}, {"identifier": "D", "content": "$${}^{41}{C_{20}}$$"}] | ["C"]
Explanation:
$$\sum\limits_{k = 0}^{20} {{{\left( {{}^{20}{C_k}} \right)}^2}} $$
<br><br>= $${\left( {{}^{20}{C_0}} \right)^2} + {\left( {{}^{20}{C_1}} \right)^2} + {\left( {{}^{20}{C_2}} \right)^2} + .... + {\left( {{}^{20}{C_{20}}} \right)^2}$$
<br><br>= <sup>40</sup>C<sub>20</sub>
<br><br><b>Using the formul... |
If the sum of the coefficients in the expansion of (x + y)<sup>n</sup> is 4096, then the greatest coefficient in the expansion is _____________.
Options:
[] | 924
Explanation:
(x + y)<sup>n</sup> $$\Rightarrow$$ 2<sup>n</sup> = 4096<br><br>2<sup>10</sup> = 1024 $$\times$$ 2<br><br>$$\Rightarrow$$ 2<sup>n</sup> = 2<sup>12</sup><br><br>2<sup>11</sup> = 2048<br><br>n = 12<br><br>2<sup>12</sup> = 4096<br><br>$${}^{12}{C_6}={{12 \times 11 \times 10 \times 9 \times 8 \times 7} \o... |
<p>Let n $$\ge$$ 5 be an integer. If 9<sup>n</sup> $$-$$ 8n $$-$$ 1 = 64$$\alpha$$ and 6<sup>n</sup> $$-$$ 5n $$-$$ 1 = 25$$\beta$$, then $$\alpha$$ $$-$$ $$\beta$$ is equal to</p>
Options:
[{"identifier": "A", "content": "1 + <sup>n</sup>C<sub>2</sub> (8 $$-$$ 5) + <sup>n</sup>C<sub>3</sub> (8<sup>2</sup> $$-$$ 5<sup... | ["C"]
Explanation:
<p>Given,</p>
<p>$${9^n} - 8n - 1 = 64\alpha $$</p>
<p>$$ \Rightarrow \alpha = {{{{(1 + 8)}^n} - 8n - 1} \over {64}}$$</p>
<p>$$ = {{\left( {{}^n{C_0}\,.\,1 + {}^n{C_1}\,.\,{8^1} + {}^n{C_2}\,.\,{8^2}\,\, + \,\,.....\,\, + \,\,{}^n{C_n}\,.\,{8^n}} \right) - 8n - 1} \over {{8^2}}}$$</p>
<p>$$ = {{1 ... |
<p>If <br/><br/>$$\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} } $$, <br/><br/>where $$\alpha$$ $$\in$$ R, then the value of 16$$\alpha$$ is e... | ["A"]
Explanation:
<p>Given,</p>
<p>$$\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} } $$</p>
<p>Now,</p>
<p>$$\sum\limits_{k = 1}^{31} {\left(... |
<p>If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of $${\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}$$ is 939, then the sum of all the possible integral values of n is _________.</p>
Options:
[] | 57
Explanation:
<p>Given, Binomial expression is</p>
<p>$$ = {\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}$$</p>
<p>$$\therefore$$ General term</p>
<p>$${T_{r + 1}} = {}^7{C_r}\,.\,{({x^n})^{7 - r}}\,.\,{\left( {{2 \over {{x^5}}}} \right)^r}$$</p>
<p>$$ = {}^7{C_r}\,.\,{x^{7n - nr - 5r}}\,.\,{2^r}$$</p>
<p>For positi... |
If $$\left( {{}^{40}{C_0}} \right) + \left( {{}^{41}{C_1}} \right) + \left( {{}^{42}{C_2}} \right) + \,\,.....\,\, + \,\,\left( {{}^{60}{C_{20}}} \right) = {m \over n}{}^{60}{C_{20}}$$ m and n are coprime, then m + n is equal to ___________.
Options:
[] | 102
Explanation:
<p>Here property used is</p>
<p>$${}^n{C_r} + {}^n{C_{r + 1}} = {}^{n + 1}{C_{r + 1}}$$</p>
<p>Given, $${}^{40}{C_0} + {}^{41}{C_1} + {}^{42}{C_2} + \,\,....\,\, + \,\,{}^{60}{C_{20}} = {m \over n}{}^{60}{C_{20}}$$</p>
<p>As $${}^{40}{C_0} = {}^{41}{C_0} = 1$$</p>
<p>So, we replace $${}^{40}{C_0}$$ wi... |
<p>Let C<sub>r</sub> denote the binomial coefficient of x<sup>r</sup> in the expansion of $${(1 + x)^{10}}$$. If for $$\alpha$$, $$\beta$$ $$\in$$ R, $${C_1} + 3.2{C_2} + 5.3{C_3} + $$ ....... upto 10 terms $$ = {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\left( {{C_0} + {{{C_1}} \over 2} + {{{C_2}} \over 3} + \,... | 286
Explanation:
<p>Given,</p>
<p>$${C_1} + 3\,.\,2{C_2} + 5\,.\,3{C_3} + $$ ...... upto 10 terms</p>
<p>$$ = {{\alpha \,.\,{2^{11}}} \over {{2^\beta } - 1}}$$ ($${C_0} + {{{C_1}} \over 2} + {{{C_2}} \over 3}$$ + ..... upto 10 terms)</p>
<p>Now,</p>
<p>L.H.S. :-</p>
<p>$${C_1} + 3\,.\,2{C_2} + 5\,.\,3{C_3} + $$ ......... |
<p>If the coefficients of $$x$$ and $$x^{2}$$ in the expansion of $$(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}, \mathrm{p}, \mathrm{q} \leq 15$$, are $$-3$$ and $$-5$$ respectively, then the coefficient of $$x^{3}$$ is equal to _____________.</p>
Options:
[] | 23
Explanation:
<p>Coefficient of x in $${(1 + x)^p}{(1 - x)^q}$$</p>
<p>$$ - {}^p{C_0}\,{}^q{C_1} + {}^p{C_1}\,{}^q{C_0} = - 3 \Rightarrow p - q = - 3$$</p>
<p>Coefficient of x<sup>2</sup> in $${(1 + x)^p}{(1 - x)^q}$$</p>
<p>$${}^p{C_0}\,{}^q{C_2} - {}^p{C_1}\,{}^q{C_1} + {}^p{C_2}\,{}^q{C_0} = - 5$$</p>
<p>$${{q... |
<p>$$\sum\limits_{\matrix{
{i,j = 0} \cr
{i \ne j} \cr
} }^n {{}^n{C_i}\,{}^n{C_j}} $$ is equal to</p>
Options:
[{"identifier": "A", "content": "$$2^{2 n}-{ }^{2 n} C_{n}$$"}, {"identifier": "B", "content": "$${2^{2n - 1}} - {}^{2n - 1}{C_{n - 1}}$$"}, {"identifier": "C", "content": "$$2^{2 n}-\\frac{1}{2}... | ["A"]
Explanation:
<p>$$\sum\limits_{i,\,j = 0\,\,i \ne j}^n {{}^n{C_i}\,{}^n{C_j} = \sum\limits_{i,\,j = 0}^n {{}^n{C_i}\,{}^n{C_j} - \sum\limits_{i = j}^n {{}^n{C_i}\,{}^n{C_j}} } } $$</p>
<p>$$ = \sum\limits_{j = 0}^n {{}^n{C_i}\,\sum\limits_{j = 0}^n {{}^n{C_j} - \sum\limits_{i = 0}^n {{}^n{C_i}\,{C_i}} } } $$</p>... |
<p>If $$1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})$$ is equal to $$2^{\mathrm{n}} \cdot \mathrm{m}$$, where $$\mathrm{m}$$ is odd, then $$\mathrm{n}+\mathrm{m}$$ is equal to __________.</p>
Options:
[] | 99
Explanation:
<p>$$l = 1 + (1 + {}^{49}{C_0} + {}^{49}{C_1}\, + \,....\, + \,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4}\, + \,....\, + \,{}^{50}{C_{50}})$$</p>
<p>As $${}^{49}{C_0} + {}^{49}{C_1}\, + \,.....\, + \,{}^{49}{C_{49}} = {2^{49}}$$</p>
<p>and $${}^{50}{C_0} + {}^{50}{C_2}\, + \,....\, + \,{}^{50}{C_{50}... |
<p>$$\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)$$ is equal to</p>
Options:
[{"identifier": "A", "content": "$$22 !-21 !$$"}, {"identifier": "B", "content": "$$22 !-2(21 !)$$"}, {"identifier": "C", "content": "$$21 !-2(20 !)$$"}, {"identifier": "D", "content": "$$21 !-20$$ !"}] | ["B"]
Explanation:
<p>Given,</p>
<p>$$\sum\limits_{r = 1}^{20} {({r^2} + 1)(r!)} $$</p>
<p>Let, $$f(r) = ({r^2} + 1)(r!)$$</p>
<p>$$ = ({r^2})(r!) + r!$$</p>
<p>$$ = r(r\,r!) + r!$$</p>
<p>$$ = r[(r + 1 - 1)r!] + r!$$</p>
<p>$$ = r[(r + 1)r! - r!] + r!$$</p>
<p>$$ = r[(r + 1)! - (r!)] + r!$$</p>
<p>$$ = r(r + 1)! - r(... |
<p>$$
\text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }$$ ________.</p>
Options:
[] | 221
Explanation:
<p>Given,</p>
<p>$$\sum\limits_{k = 1}^{10} {{k^2}{{\left( {{}^{10}{C_k}} \right)}^2} = 2200\,L} $$</p>
<p>$$ \Rightarrow \sum\limits_{k = 1}^{10} {{{\left( {k\,.\,{}^{10}{C_k}} \right)}^2} = 22000\,L} $$</p>
<p>$$ \Rightarrow \sum\limits_{k = 1}^{10} {{{\left( {k\,.\,{{10} \over k}\,.\,{}^9{C_{k - 1}... |
<p>The coefficient of $${x^{301}}$$ in $${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$$ is :</p>
Options:
[{"identifier": "A", "content": "$${}^{500}{C_{300}}$$"}, {"identifier": "B", "content": "$${}^{501}{C_{200}}$$"}, {"identifier": "C", "content": "$${}^{500}{C_{301}}$$"}, {"... | ["B"]
Explanation:
<p>The coefficient of $${x^{301}}$$ in
<br/><br/>$${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$$</p>
<p>$${}^{500}{C_{301}} + {}^{499}{C_{300}} + {}^{498}{C_{299}}\, + \,...\, + \,{}^{199}{C_0}$$</p>
<p>$$ = {}^{500}{C_{199}} + {}^{499}{C_{199}} + {}^{498}{C_... |
<p>If $$a_r$$ is the coefficient of $$x^{10-r}$$ in the Binomial expansion of $$(1 + x)^{10}$$, then $$\sum\limits_{r = 1}^{10} {{r^3}{{\left( {{{{a_r}} \over {{a_{r - 1}}}}} \right)}^2}} $$ is equal to </p>
Options:
[{"identifier": "A", "content": "3025"}, {"identifier": "B", "content": "4895"}, {"identifier": "C", "... | ["D"]
Explanation:
$$
\begin{aligned}
& \mathrm{a}_{\mathrm{r}}={ }^{10} \mathrm{C}_{10-\mathrm{r}}={ }^{10} \mathrm{C}_{\mathrm{r}} \\\\
& \Rightarrow \sum_{\mathrm{r}=1}^{10} \mathrm{r}^3\left(\frac{{ }^{10} \mathrm{C}_{\mathrm{r}}}{{ }^{10} \mathrm{C}_{\mathrm{r}-1}}\right)^2=\sum_{\mathrm{r}=1}^{10} \mathrm{r}^3\l... |
<p>If $${({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}}$$ then $$\alpha$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "30"}, {"identifier": "B", "content": "10"}, {"identifier": "C", "content": "15"}, {"identifie... | ["C"]
Explanation:
$$
\begin{aligned}
& \mathrm{S}=0 \cdot\left({ }^{30} \mathrm{C}_0\right)^2+1 \cdot\left({ }^{30} \mathrm{C}_1\right)^2+2 \cdot\left({ }^{30} \mathrm{C}_2\right)^2+\ldots \ldots+30 \cdot\left({ }^{30} \mathrm{C}_{30}\right)^2 \\\\
& \mathrm{S}=30 \cdot\left({ }^{30} \mathrm{C}_0\right)^2+29 \cdot\le... |
<p>The value of $$\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}} $$ is</p>
Options:
[{"identifier": "A", "content": "$${}^{44}{C_{23}}$$"}, {"identifier": "B", "content": "$${}^{45}{C_{23}}$$"}, {"identifier": "C", "content": "$${}^{44}{C_{22}}$$"}, {"identifier": "D", "content": "$${}^{45}{C_{24}}$$"}] | ["B"]
Explanation:
<p>$$\sum\limits_{r = 0}^{22} {{}^{22}{C_r}\,.\,{}^{23}{C_r}} $$</p>
<p>$$ = \sum\limits_{r = 0}^{22} {{}^{22}{C_r}\,{}^{23}{C_{23 - r}}} $$ [using $${}^n{C_r} = {}^n{C_{n - r}}$$]</p>
<p>$$ = {}^{22}{C_0}{}^{23}{C_{23}} + {}^{22}{C_1}{}^{23}{C_{22}}\, + \,...\, + \,{}^{22}{C_{21}}{}^{23}{C_2} + {}^... |
<p>Suppose $$\sum\limits_{r = 0}^{2023} {{r^2}{}~^{2023}{C_r} = 2023 \times \alpha \times {2^{2022}}} $$. Then the value of $$\alpha$$ is ___________</p>
Options:
[] | 1012
Explanation:
<p>Concept :</p>
<p>(1) $${}^n{C_r} = {n \over r}\,.\,{}^{n - 1}{C_{r - 1}}$$</p>
<p>Given,</p>
<p>$$\sum\limits_{r = 0}^{2023} {{r^2}\,.\,{}^{2023}{C_r}} $$</p>
<p>$$ = \sum\limits_{r = 0}^{2023} {{r^2}\,.\,{{2023} \over r}\,.{}^{2022}{C_{r - 1}}} $$</p>
<p>$$ = 2023\sum\limits_{r = 0}^{2023} {{r}\,... |
<p>If $$\frac{1}{n+1}{ }^{n} \mathrm{C}_{n}+\frac{1}{n}{ }^{n} \mathrm{C}_{n-1}+\ldots+\frac{1}{2}{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{0}=\frac{1023}{10}$$ then $$n$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "7"}, {"ident... | ["A"]
Explanation:
$$\frac{1}{n+1}{ }^{n} \mathrm{C}_{n}+\frac{1}{n}{ }^{n} \mathrm{C}_{n-1}+\ldots+\frac{1}{2}{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{0}=\frac{1023}{10}$$
<br/><br/>$$
\begin{aligned}
& \Rightarrow \sum_{r=0}^n \frac{1}{r+1}{ }^n C_r=\frac{1023}{10} \\\\
& \quad\left( \because{ }^{n+1} C_{r+1}=\fra... |
<p>The sum, of the coefficients of the first 50 terms in the binomial expansion of $$(1-x)^{100}$$, is equal to</p>
Options:
[{"identifier": "A", "content": "$${ }^{99} \\mathrm{C}_{49}$$"}, {"identifier": "B", "content": "$${ }^{101} \\mathrm{C}_{50}$$"}, {"identifier": "C", "content": "$$-{ }^{99} \\mathrm{C}_{49}$$... | ["C"]
Explanation:
$$
\begin{aligned}
& \left({ }^{100} C_0-{ }^{100} C_1+{ }^{100} C_2-\ldots . .{ }^{100} C_{49}\right)+{ }^{100} C_{50} \\\\
& +\left(-{ }^{100} C_{51}+{ }^{100} C_{52}-\ldots .+{ }^{100} C_{100}\right)=0 \\\\
& \lambda_1+{ }^{100} C_{50}+\lambda_2=0 \\\\
& \lambda_1=-\frac{1}{2}{ }^{100} C_{50} \qu... |
<p>If the coefficients of three consecutive terms in the expansion of $$(1+x)^{n}$$ are in the ratio $$1: 5: 20$$, then the coefficient of the fourth term is</p>
Options:
[{"identifier": "A", "content": "3654"}, {"identifier": "B", "content": "1827"}, {"identifier": "C", "content": "5481"}, {"identifier": "D", "conten... | ["A"]
Explanation:
$$
\begin{aligned}
& \text { Given: }{ }^n \mathrm{C}_{r-1}:{ }^n \mathrm{C}_r:{ }^n \mathrm{C}_{r+1} \\\\
& =1: 5: 20 \\\\
& \Rightarrow \frac{n !}{(r-1) !(n-r+1) !} \times \frac{r !(n-r) !}{n !}=\frac{1}{5} \\\\
& \Rightarrow \frac{r}{(n-r+1)}=\frac{1}{5} \\\\
& \Rightarrow 5 r=n-r+1 \\\\
& \Right... |
<p>If $${ }^{2 n} C_{3}:{ }^{n} C_{3}=10: 1$$, then the ratio $$\left(n^{2}+3 n\right):\left(n^{2}-3 n+4\right)$$ is :</p>
Options:
[{"identifier": "A", "content": "$$27: 11$$"}, {"identifier": "B", "content": "$$2: 1$$"}, {"identifier": "C", "content": "$$35: 16$$"}, {"identifier": "D", "content": "$$65: 37$$"}] | ["B"]
Explanation:
$$
\begin{aligned}
& \text {We have, }{ }^{2 n} C_3:{ }^n C_3=10: 1 \\\\
& \Rightarrow \frac{{ }^{2 n} C_3}{{ }^n C_3}=\frac{10}{1} \\\\
& \Rightarrow \frac{(2 n) !}{3 !(2 n-3) !} \times \frac{3 !(n-3) !}{n !}=\frac{10}{1} \\\\
& \Rightarrow \frac{(2 n)(2 n-1)(2 n-2)}{(n)(n-1)(n-2)}=\frac{10}{1} \\\... |
<p>The coefficient of $$x^{18}$$ in the expansion of $$\left(x^{4}-\frac{1}{x^{3}}\right)^{15}$$ is __________.</p>
Options:
[] | 5005
Explanation:
$\begin{aligned} T_{r+1}= & { }^{15} C_r\left(x^4\right)^{15-r}\left(-\frac{1}{x^3}\right)^r={ }^{15} C_r(-1)^r x^{60-4 r-3 r} \\\\ = & { }^{15} C_r(-1)^r x^{60-7 r}\end{aligned}$
<br/><br/>$\begin{aligned} \therefore 60-7 r =18 \\\\ \Rightarrow 7 r =42 \\\\ \Rightarrow r =6\end{aligned}$
<br... |
${ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}$ if and only if :
Options:
[{"identifier": "A", "content": "$2 \\sqrt{2}<\\mathrm{k}<2 \\sqrt{3}$"}, {"identifier": "B", "content": "$2 \\sqrt{2}<\\mathrm{k} \\leq 3$"}, {"identifier": "C", "content": "$2 \\sqrt{3}<\\mathrm{k}<3 \\sqrt{3}$"}, {"identifier": "D", "content... | ["B"]
Explanation:
<p>$${ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}}=(\mathrm{k}^2-8){ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}+1}$$</p>
<p>$$\underbrace{\mathrm{r}+1 \geq 0, \quad \mathrm{r} \geq 0}_{\mathrm{r} \geq 0}$$</p>
<p>$$\begin{aligned}
& \frac{{ }^{n-1} C_r}{{ }^n C_{r+1}}=k^2-8 \\
& \frac{r+1}{n}=k^2-8 \\
& \Ri... |
If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :
Options:
[{"identifier": "A", "content": "$\\mathrm{B}=\\mathrm{A}^3$"}, {"identifier": "B", "content": "$3 \\mathrm... | ["D"]
Explanation:
<p>Sum of coefficients in the expansion of $$\left(1-3 \mathrm{x}+10 \mathrm{x}^2\right)^{\mathrm{n}}=\mathrm{A}$$</p>
<p>then $$A=(1-3+10)^n=8^n$$ (put $$x=1$$)<?p>
<p>and sum of coefficients in the expansion of</p>
<p>$$\begin{aligned}
& \left(1+x^2\right)^n=B \\
& \text { then } B=(1+1)^n=2^n \\
... |
<p>Let the coefficient of $$x^r$$ in the expansion of $$(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots \ldots \ldots .+(x+2)^{n-1}$$ be $$\alpha_r$$. If $$\sum_\limits{r=0}^n \alpha_r=\beta^n-\gamma^n, \beta, \gamma \in \mathbb{N}$$, then the value of $$\beta^2+\gamma^2$$ equals _________.</p>
Options:
[] | 25
Explanation:
<p>$$\begin{aligned}
& (x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3} \\
& (x+2)^2+\ldots \ldots .+(x+2)^{n-1} \\
& \sum \alpha_r=4^{n-1}+4^{n-2} \times 3+4^{n-3} \times 3^2 \ldots \ldots+3^{n-1} \\
& =4^{n-1}\left[1+\frac{3}{4}+\left(\frac{3}{4}\right)^2 \ldots .+\left(\frac{3}{4}\right)^{n-1}\right] \\
& =... |
<p>In the expansion of $$(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$$, the sum of the coefficients of $x^3$ and $$x^{-13}$$ is equal to __________.</p>
Options:
[] | 118
Explanation:
<p>$$\begin{aligned}
& (1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5 \\
& =(1+x)\left(1-x^2\right)\left(\left(1+\frac{1}{x}\right)^3\right)^5 \\
& =\frac{(1+x)^2(1-x)(1+x)^{15}}{x^{15}} \\
& =\frac{(1+x)^{17}-x(1+x)^{17}}{x^{15}}
\end{aligned}$$</p>
<p>$$=\operatornam... |
<p>$$\text { If } \frac{{ }^{11} C_1}{2}+\frac{{ }^{11} C_2}{3}+\ldots+\frac{{ }^{11} C_9}{10}=\frac{n}{m} \text { with } \operatorname{gcd}(n, m)=1 \text {, then } n+m \text { is equal to }$$ _______.</p>
Options:
[] | 2041
Explanation:
<p>$$\begin{aligned}
& \sum_{\mathrm{r}=1}^9 \frac{{ }^{11} \mathrm{C}_{\mathrm{r}}}{\mathrm{r}+1} \\
& =\frac{1}{12} \sum_{\mathrm{r}=1}^9{ }^{12} \mathrm{C}_{\mathrm{r}+1} \\
& =\frac{1}{12}\left[2^{12}-26\right]=\frac{2035}{6} \\
& \therefore \mathrm{m}+\mathrm{n}=2041
\end{aligned}$$</p> |
<p>Suppose $$2-p, p, 2-\alpha, \alpha$$ are the coefficients of four consecutive terms in the expansion of $$(1+x)^n$$. Then the value of $$p^2-\alpha^2+6 \alpha+2 p$$ equals</p>
Options:
[{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "6"}, {"identifier": "D",... | []
Explanation:
<p>$$2-p, p, 2-\alpha, \alpha$$</p>
<p>Binomial coefficients are</p>
<p>$$\begin{aligned}
& { }^n C_r,{ }^n C_{r+1},{ }^n C_{r+2},{ }^n C_{r+3} \text { respectively } \\
\Rightarrow \quad & { }^n C_r+{ }^n C_{r+1}=2 \\
\Rightarrow \quad & { }^{n+1} C_{r+1}=2 \quad \ldots . .(1)
\end{aligned}$$</p>
<p>A... |
<p>Let $$\alpha=\sum_\limits{k=0}^n\left(\frac{\left({ }^n C_k\right)^2}{k+1}\right)$$ and $$\beta=\sum_\limits{k=0}^{n-1}\left(\frac{{ }^n C_k{ }^n C_{k+1}}{k+2}\right)$$ If $$5 \alpha=6 \beta$$, then $$n$$ equals _______.</p>
Options:
[] | 10
Explanation:
<p>$$\begin{aligned}
\alpha= & \sum_{k=0}^n \frac{{ }^n C_k \cdot{ }^n C_k}{k+1} \cdot \frac{n+1}{n+1} \\
& =\frac{1}{n+1} \sum_{k=0}^n{ }^{n+1} C_{k+1} \cdot{ }^n C_{n-k} \\
\alpha & =\frac{1}{n+1} \cdot{ }^{2 n+1} C_{n+1} \\
\beta & =\sum_{k=0}^{n-1} C_k \cdot \frac{{ }^n C_{k+1}}{k+2} \frac{n+1}{n+1... |
<p>The sum of the coefficient of $$x^{2 / 3}$$ and $$x^{-2 / 5}$$ in the binomial expansion of $$\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9$$ is</p>
Options:
[{"identifier": "A", "content": "19/4"}, {"identifier": "B", "content": "69/16"}, {"identifier": "C", "content": "63/16"}, {"identifier": "D", "content": "2... | ["D"]
Explanation:
<p>$$
\begin{aligned}
& T_{r+1}={ }^9 C_r\left(\frac{x^{-2 / 5}}{2}\right)^r\left(x^{2 / 3}\right)^{9-r} \\
& ={ }^9 C_r \frac{1}{2^r} x^{\frac{2}{3}(9-r)+\left(\frac{-2 r}{5}\right)} \\
& ={ }^9 C_r \cdot \frac{1}{2^r} \cdot x^{6-\frac{16 r}{15}}
\end{aligned}
$$</p>
<p>For coefficient of $$x^{2 / ... |
<p>The coefficient of $$x^{70}$$ in $$x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$$ is $${ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}$$. Then a possible value of $$\mathrm{p}+\mathrm{q}$$ is :</p>
Options:
[{"identifier": "A", "content": "61"}, {"identifier": "B", "content": ... | ["B"]
Explanation:
<p>$$x^2(1+x)^{98}+x^3(1+x)^{97}+\ldots+x^{54}(1+x)^{46}$$</p>
<p>It is a G.P. with first term $$=x^2(1+x)^{98}$$</p>
<p>and common ratio $$=\frac{x}{1+x}$$</p>
<p>sum of these term $$=x^2(1+x)^{98}\left(\frac{\left(\frac{x}{1+x}\right)^{53}-1}{\frac{x}{1+x}-1}\right)$$</p>
<p>$$=x^2(1+x)^{98}\left(... |
<p>Let $$a=1+\frac{{ }^2 \mathrm{C}_2}{3 !}+\frac{{ }^3 \mathrm{C}_2}{4 !}+\frac{{ }^4 \mathrm{C}_2}{5 !}+...., \mathrm{b}=1+\frac{{ }^1 \mathrm{C}_0+{ }^1 \mathrm{C}_1}{1 !}+\frac{{ }^2 \mathrm{C}_0+{ }^2 \mathrm{C}_1+{ }^2 \mathrm{C}_2}{2 !}+\frac{{ }^3 \mathrm{C}_0+{ }^3 \mathrm{C}_1+{ }^3 \mathrm{C}_2+{ }^3 \mathrm... | 8
Explanation:
<p>$$\begin{aligned}
& a=1+\frac{{ }^2 C_2}{3!}+\frac{{ }^3 C_2}{4!}+\frac{{ }^4 C_2}{5!}+\ldots \\
& b=1+\frac{{ }^1 C_0+{ }^1 C_1}{1!}+\frac{{ }^2 C_0+{ }^2 C_1+{ }^2 C_2}{2!}+\ldots \\
& b=1+\frac{2}{1!}+\frac{2^2}{2!}+\frac{2}{3!}+\ldots=e^2
\end{aligned}$$</p>
<p>Using $$e^x=1+\frac{x}{1!}+\frac{x^... |
<p>If the coefficients of $$x^4, x^5$$ and $$x^6$$ in the expansion of $$(1+x)^n$$ are in the arithmetic progression, then the maximum value of $$n$$ is:</p>
Options:
[{"identifier": "A", "content": "28"}, {"identifier": "B", "content": "21"}, {"identifier": "C", "content": "7"}, {"identifier": "D", "content": "14"}] | ["D"]
Explanation:
<p>$$\begin{aligned}
& (1+x)^n={ }^n C_0+{ }^n C_1 x^1+{ }^n C_2 x^2+\ldots{ }^n C_n x^n \\
& { }^n C_4,{ }^n C_5 \&{ }^n C_6 \text { are in A.P. } \\
& { }^n C_5-{ }^n C_4={ }^n C_6-{ }^n C_5 \\
& \Rightarrow \frac{n!}{5!(n-5)!}-\frac{n!}{4!(n-4)!}=\frac{n!}{6!(n-6)!}-\frac{n!}{5!(n-5)!} \\
& \Righ... |
The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length $$3$$$$a$$ is :
Options:
[{"identifier": "A", "content": "$${x^2}\\, + \\,{y^2} = 9{a^2}$$ "}, {"identifier": "B", "content": "$${x^2}\\, + \\,{y^2} = 16{a^2}$$"}, {"identifier": "C", "content": "$${x... | ["C"]
Explanation:
Let $$ABC$$ be an equilateral triangle, whose median is $$AD.$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267493/exam_images/qh2bpsmsygtrhhpn1feb.webp" loading="lazy" alt="AIEEE 2002 Mathematics - Circle Question 157 English Explanation">
<br><br... |
The point diametrically opposite to the point $$P(1, 0)$$ on the circle $${x^2} + {y^2} + 2x + 4y - 3 = 0$$ is :
Options:
[{"identifier": "A", "content": "$$(3, -4)$$"}, {"identifier": "B", "content": "$$(-3, 4)$$ "}, {"identifier": "C", "content": "$$(-3, -4)$$"}, {"identifier": "D", "content": "$$(3, 4)$$"}] | ["C"]
Explanation:
The given circle is $${x^2} + {y^2} + 2x + 4y - 3 = 0$$
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266849/exam_images/earxtruibjeyrajhopbc.webp" loading="lazy" alt="AIEEE 2008 Mathematics - Circle Question 144 English Explanation">
<br><br>Center... |
Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the point $$(1, 0)$$ to the distance from the point $$(-1, 0)$$ is equal to $${1 \over 3}$$. Then the circumcentre of the triangle ABC is at the point :
Options:
[{"identifier"... | ["A"]
Explanation:
Given that
<br><br>$$P\left( {1,0} \right),Q\left( { - 1,0} \right)$$
<br><br>and $${{AP} \over {AQ}} = {{BP} \over {BQ}} = {{CP} \over {CQ}} = {1 \over 3}$$
<br><br>$$ \Rightarrow 3AP = AQ$$
<br><br>$$\,\,\,\,\,\,$$ Let $$A = (x,y)$$ then $$3AP = AQ \Rightarrow 9A{P^2} = A{Q^2}$$
<br><br>$$ \Righ... |
Locus of the image of the point $$(2, 3)$$ in the line $$\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,\,k \in R,$$ is a :
Options:
[{"identifier": "A", "content": "circle of radius $$\\sqrt 2 $$."}, {"identifier": "B", "content": "circle of radius $$\\sqrt 3 $$."}, {"identifier": "C", "content": "s... | ["A"]
Explanation:
Intersection point of $$2x - 3y + 4 = 0$$
<br><br>and $$x-2y+3=0$$ is $$(1, 2)$$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l91npsqa/a856b112-8560-4eba-878d-d29d808c6c0a/1868e920-47fd-11ed-8757-0f869593f41f/file-1l91npsqb.png?format=png" data-orsrc="https://app-conten... |
The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60<sup>o</sup>. If the area of the quadrilateral is $$4\sqrt 3 $$, then the perimeter
of the quadrilateral is :
Options:
[{"identifier": "A", "content": "12.5 "}, {"identifier": "B", "content": "13.2"}, {"identifier": "C", "c... | ["C"]
Explanation:
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265487/exam_images/fnyfpg1ibpegkks9nf7r.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2017 (Online) 9th April Morning Slot Mathematics - Circle Question 124 English Explanation... |
A rectangle is inscribed in a circle with a diameter
lying along the line 3y = x + 7. If the two adjacent
vertices of the rectangle are (–8, 5) and (6, 5), then
the area of the rectangle (in sq. units) is :
Options:
[{"identifier": "A", "content": "72"}, {"identifier": "B", "content": "84"}, {"identifier": "C", "conte... | ["B"]
Explanation:
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263678/exam_images/ch1ypnxfax2wczso5gcp.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266115/exam_images/vwts7pqrwjuyf7wdy8m4.webp"><im... |
The diameter of the circle, whose centre lies on
the line x + y = 2 in the first quadrant and which
touches both the lines x = 3 and y = 2, is
_______ .
Options:
[] | 3
Explanation:
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263616/exam_images/ssi5vdlnvmrlmfb1iuud.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 3rd September Morning Slot Mathematics - Circle Question 96 English Explanation">
... |
Let PQ be a diameter of the circle x<sup>2</sup> + y<sup>2</sup> = 9. If $$\alpha $$ and $$\beta $$ are the lengths of the perpendiculars from P and Q on the straight line,<br/> x + y = 2 respectively, then the maximum value of $$\alpha\beta $$ is _____.
Options:
[] | 7
Explanation:
Let $$P(3\cos \theta ,\,3\sin \theta )$$<br><br>$$Q( - 3\cos \theta ,\, - 3\sin \theta )$$<br><br>$$\alpha = \left| {{{3\cos \theta + 3\sin \theta - 2} \over {\sqrt 2 }}} \right|$$<br><br>$$\beta = \left| {{{ - 3\cos \theta - 3\sin \theta - 2} \over {\sqrt 2 }}} \right|$$<br><br>$$\alpha \beta = ... |
Let a point P be such that its distance from the point (5, 0) is thrice the distance of P from the point ($$-$$5, 0). If the locus of the point P is a circle of radius r, then 4r<sup>2</sup> is equal to ________
Options:
[] | 56
Explanation:
Let P(h, k)<br><br>Given<br><br>PA = 3PB<br><br>PA<sup>2</sup> = 9PB<sup>2</sup><br><br>$$ \Rightarrow $$ (h $$-$$ 5)<sup>2</sup> + k<sup>2</sup> = 9[(h + 5)<sup>2</sup> + k<sup>2</sup>]<br><br>$$ \Rightarrow $$ 8h<sup>2</sup> + 8k<sup>2</sup> + 100h + 200 = 0<br><br>$$ \therefore $$ Locus<br><br>$${x^... |
In the circle given below, let OA = 1 unit, OB = 13 unit and PQ $$ \bot $$ OB. Then, the area of the triangle PQB (in square units) is :<br/><br/><img src="data:image/png;base64,UklGRnoKAABXRUJQVlA4IG4KAACwPQCdASr1ALAAPm0ylkgkIqIhJJKaoIANiWlu/Hx7KxnZ1z/qN/KO2P+xflJ16nnD2g9MKuOvk/zH8w/cL+r/tJ9+fz3/ff0Xxb+BX8P6hHrP+7/y/9... | ["B"]
Explanation:
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266436/exam_images/ms5tyuzkyy8vnwsy0d9h.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 26th February Morning Shift Mathematics - Circle Question 89 English Explanati... |
Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to :
Options:
[{"identifier": "A", "content": "{(4, 0), (0, 6)}"}, {"identifier": "B", "content": "$$\\{ (2 + 2\\sqrt 2 ... | ["D"]
Explanation:
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265883/exam_images/j4eb4u3olf4aobpzkvfj.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263397/exam_images/a0quemgl8enbtkiviozm.webp"><im... |
Let $$A = \{ (x,y) \in R \times R|2{x^2} + 2{y^2} - 2x - 2y = 1\} $$, $$B = \{ (x,y) \in R \times R|4{x^2} + 4{y^2} - 16y + 7 = 0\} $$ and $$C = \{ (x,y) \in R \times R|{x^2} + {y^2} - 4x - 2y + 5 \le {r^2}\} $$.<br/><br/>Then the minimum value of |r| such that $$A \cup B \subseteq C$$ is equal to
Options:
[{"identif... | ["C"]
Explanation:
$${S_1}:{x^2} + {y^2} - x - y - {1 \over 2} = 0;{C_1}\left( {{1 \over 2},{1 \over 2}} \right)$$<br><br>$${r_1} = \sqrt {{1 \over 4} + {1 \over 4} + {1 \over 2}} = 1$$<br><br>$${S_2}:{x^2} + {y^2} - 4y + {7 \over 4} = 0;{C_2}:(0,2)$$<br><br>$${r_2} = \sqrt {4 - {7 \over 4}} = {3 \over 2}$$<br><br>$... |
The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter d. Then d<sup>2</sup> is equal to _____________.
Options:
[] | 16
Explanation:
Let point P(x, y)
<br><br>A(0, 0), B(1, 0), C(0, 1), D(1, 1)
<br><br>(PA)<sup>2</sup> + (PB)<sup>2</sup> + (PC)<sup>2</sup> + (PD)<sup>2</sup> = 18
<br><br>$${x^2} + {y^2} + {x^2} + {(y - 1)^2} + {(x - 1)^2} + {y^2} + {(x - 1)^2} + {(y - 1)^2}$$ = 18<br><br>$$ \Rightarrow 4({x^2} + {y^2}) - 4y - 4x = 1... |
Let Z be the set of all integers,<br/><br/>$$A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\} $$<br/><br/>$$B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\} $$<br/><br/>$$C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\} $$<br/><br/>If the total number of relation from A $$\cap$$ B to A $$\cap$$ C i... | ["B"]
Explanation:
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265832/exam_images/dtgcpzs2ufin73ryqlsn.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264735/exam_images/vsklxgqplszmejwhrtui.webp"><im... |
<p>Let a triangle ABC be inscribed in the circle $${x^2} - \sqrt 2 (x + y) + {y^2} = 0$$ such that $$\angle BAC = {\pi \over 2}$$. If the length of side AB is $$\sqrt 2 $$, then the area of the $$\Delta$$ABC is equal to :</p>
Options:
[{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "$$\\left( {\\... | ["A"]
Explanation:
<p>Note:</p>
<p>For equation of circle $${x^2} + {y^2} + 2gx + 2fy + c = 0$$, center is $$( - g,\, - f)$$ and radius $$r = \sqrt {{g^2} + {f^2} - c} $$</p>
<p>Given,</p>
<p>equation of circle is</p>
<p>$${x^2} - \sqrt 2 (x + y) + {y^2} = 0$$</p>
<p>$$ \Rightarrow {x^2} + {y^2} - \sqrt 2 x - \sqrt 2 ... |
<p>A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x $$-$$ y + 4 = 0, then the area of R is ____________.</p>
Options:
[] | 16
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5qa5m7e/7e601531-e009-4eeb-99e6-ea8138a03412/cb3a40a0-0656-11ed-903e-c9687588b3f3/file-1l5qa5m7f.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5qa5m7e/7e601531-e009-4eeb-99e6-ea8138a03412/cb3a40a0-0656-11ed-... |
<p>$$\text { Let } S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: 9(x-3)^{2}+16(y-4)^{2} \leq 144\right\}$$ and
$$T=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:(x-7)^{2}+(y-4)^{2} \leq 36\right\}$$. Then $$n(S \cap T)$$ is equal to __________.</p>
Options:
[] | 27
Explanation:
$S=\left\{(x, y) \in \mathbb{N} \times \mathbb{N}: \frac{(x-3)^{2}}{16}+\frac{(y-4)^{2}}{9} \leq 1\right\}$ <br><br>represents all the integral points inside <br><br>and on the ellipse $\frac{(x-3)^{2}}{16}+\frac{(y-4)^{2}}{9}=1$, in first quadrant.
<br><br>
and $T=\left\{(x, y) \in \mathbb{R} \times \... |
Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $\mathrm{O}$ be the origin and $\mathrm{OC}$ be perpendicular to both $\mathrm{CP}$ and $\mathrm{CQ}$. If the area of the triangle $\mathrm{OCP}$ is $\frac{\sqrt{35}}{2}$, then $a_1^2+a_... | 24
Explanation:
<p>$$OC \,\bot \,CP$$ and $$OC \,\bot \, CQ$$</p>
<p>$$\Rightarrow PCQ$$ is a straight line</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1leol1wpz/d1fe7720-2e72-4e7b-baef-e1168e337218/c9a57f70-b795-11ed-b103-ed967fad3dff/file-1leol1wq0.png?format=png" data-orsrc="https://app-... |
<p>The points of intersection of the line $$ax + by = 0,(a \ne b)$$ and the circle $${x^2} + {y^2} - 2x = 0$$ are $$A(\alpha ,0)$$ and $$B(1,\beta )$$. The image of the circle with AB as a diameter in the line $$x + y + 2 = 0$$ is :</p>
Options:
[{"identifier": "A", "content": "$${x^2} + {y^2} + 5x + 5y + 12 = 0$$"}, ... | ["A"]
Explanation:
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lebxhw5x/9e3df15a-0f74-4149-87bf-67c2379ca244/4344ed40-b0a0-11ed-a0da-1fff956b892c/file-1lebxhw5y.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lebxhw5x/9e3df15a-0f74-4149-87bf-67c2379ca244/4344ed40-b0a0-11ed-... |
<p>Consider a circle $$C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5$$. Let its mirror image in the line $$y=2 x+1$$ be another circle $$C_{2}: 5 x^{2}+5 y^{2}-10 f x-10 g y+36=0$$. Let $$r$$ be the radius of $$C_{2}$$. Then $$\alpha+r$$ is equal to _________.</p>
Options:
[] | 2
Explanation:
We have,
<br/><br/>$$
\begin{aligned}
& C_1: x^2+y^2-4 x-2 y=\alpha-5 \\\\
& C_1:(x-2)^2+(y-1)^5-5=\alpha-5 \\\\
& C_1:(x-2)^2+(y-1)^2=(\sqrt{\alpha})^2
\end{aligned}
$$
<br/><br/>So, centre and radius of $C_1$ are $(2,1)$ and $\sqrt{\alpha}$ respectively
<br/><br/>Now, image of $(2,1)$ along the line $... |
<p>Let the point $$(p, p+1)$$ lie inside the region $$E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}$$. If the set of all values of $$\mathrm{p}$$ is the interval $$(a, b)$$, then $$b^{2}+b-a^{2}$$ is equal to ___________.</p>
Options:
[] | 3
Explanation:
Given region,
<br/><br/>$$
E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^2}, 0 \leq x \leq 3\right\}
$$
<br/><br/>Since, point $(p, p+1)$ lie on line $y=x+1$
<br/><br/>$\therefore$ Point of intersection of $y=x+1$ and $y=3-x$
<br/><br/>i.e.,
$x+1=3-x$
<br/><br/>$\Rightarrow$ $2 x=2 \Rightarrow x=1$
<br/><... |
<p>Let a circle passing through $$(2,0)$$ have its centre at the point $$(\mathrm{h}, \mathrm{k})$$. Let $$(x_{\mathrm{c}}, y_{\mathrm{c}})$$ be the point of intersection of the lines $$3 x+5 y=1$$ and $$(2+\mathrm{c}) x+5 \mathrm{c}^2 y=1$$. If $$\mathrm{h}=\lim _\limits{\mathrm{c} \rightarrow 1} x_{\mathrm{c}}$$ and ... | ["B"]
Explanation:
<p>$$\begin{aligned}
& 3 x+5 y=1 \\
& (2+c) x+5 c^2 y=1 \\
& 3 c^2 x+5 c^2 y=c^2
\end{aligned}$$</p>
<p>Subtracting</p>
<p>$$\begin{aligned}
& \left(2+c-3 c^2\right) x=1-c^2 \\
& x_c=\frac{1-c^2}{2+c-3 c^2}=\frac{(1-c)(1+c)}{(1-c)(3 c+2)}=\frac{c+1}{3 c+2} \\
& y=\frac{1-3 x}{5}=\frac{1-3\left(\frac... |
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