question stringlengths 14 4.47k | reasoning stringlengths 0 21.2k ⌀ | answer stringlengths 0 1.98k | source stringclasses 12
values |
|---|---|---|---|
If 2 tic equals 3 tacs and 2 tacs equal 7 tocs, what is the ratio of one tic to one toc?
A: 15/2
B: 6/5
C: 21/4
D: 3/10
E: 1/15 | 2Tic=3*Tac and 2*Tac=7*Toc;
4*Tic=6*Tac and 6*Tac=21*Toc --> 4*Tic=21*Toc --> Tic/Toc=21/4.
Answer: C. | C: 21/4 | AQUA-RAT |
A trader mixes 80 kg of tea at 15 per kg with 20 kg of tea at cost price of 20 per kg. In order to earn a profit of 25%, what should be the sale price of the mixed tea?
A: 23.75
B: 22
C: 20
D: 19.20
E: None of these | C.P. of mixture = 80×15+20×20/80+20=16
∴ S.P. = (100+25)/100×16=20
Answer C | C: 20 | AQUA-RAT |
Machine A working alone can complete a job in 10 hours. Machine A+B working alone can do the same job in 16 hours. How long will it take machine B working at their respective constant rates to complete the job?
A: 20/3
B: 80/3
C: 50/4
D: 90/11
E: 23/7 | Machines ( A ) ( B ) (A+B)
---Time - 10 - ( - ) -(16)
---Rate 16--- 6 -- 10
--Work 160 --160 -160
A+B = (160/6) = 80/3
B | B: 80/3 | AQUA-RAT |
Given three lines $l_1: 2x - y + a = 0$ $(a > 0)$, $l_2: -4x + 2y + 1 = 0$, and $l_3: x + y - 1 = 0$, and the distance between $l_1$ and $l_2$ is $\frac{7}{10}\sqrt{5}$.
1. Determine the value of $a$;
2. Calculate the angle $\theta$ between $l_3$ and $l_1$;
3. Determine whether there is a point $P$ that simultaneously ... | 1. The line $l_2$ can be rewritten as $l_2: 2x - y - \frac{1}{2} = 0$. The distance $d$ between $l_1$ and $l_2$ can be calculated as follows:
$$
d = \frac{|a - (-\frac{1}{2})|}{\sqrt{2^2 + (-1)^2}} = \frac{7\sqrt{5}}{10}.
$$
This simplifies to:
$$
\frac{|a + \frac{1}{2}|}{\sqrt{5}} = \frac{7\sqrt{5}}{... | a = 3 | cn_k12 |
Find the HCF of
2/3,4/6,8/27
A: 2/27
B: 8/3
C: 2/3
D: 2/27
E: 1/7 | Explanation:
Whenever we have to solve this sort of question, remember the formula.
HCF =
HCF of Numerators/LCM of Denominators
So answers will be option 1
Option A | A: 2/27 | AQUA-RAT |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 2$ and for all real numbers $x$ and $y,$ it satisfies:
\[ 2f(x^2 - y^2) = (x - y)(f(x) + f(y)) \]
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | 1. Setting $x = y$ in the functional equation gives $2f(0) = 0$ hence, $f(0) = 0$.
2. Setting $x = -1$ and $y = 0$ yields:
\[ 2f(1) = -f(-1) \implies 2 \times 2 = -f(-1) \implies f(-1) = -4 \]
3. Let $x = 1$ and $y = -1$:
\[ 2f(1 - (-1)^2) = (1 + 1)(f(1) + f(-1)) \]
\[ 2f(0) = 2(f(1) - 4) \]
\[ 0 = 2(2 - 4) = -4 \]
T... | 0 | synthetic_math |
There exists a real number $m$ such that the equation
\[\begin{pmatrix} 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} + s \begin{pmatrix} 3 \\ m \end{pmatrix}\] does not have any solutions in $t$ and $s$. Find $m$. | To find $m$, we set up the conditions for the lines described by the vector equations to be parallel. These lines are given by:
\[\begin{pmatrix} 1 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}\]
and
\[\begin{pmatrix} -1 \\ 4 \end{pmatrix} + s \begin{pmatrix} 3 \\ m \end{pmatrix}\].
The direction vector... | -4.5 | synthetic_math |
Chromosomes further condensed in which stage?
A: zygum stage
B: zooflagellate stage
C: diakinesis stage
D: tracheophyte stage | During the diakinesis stage, chromosomes further condense. The chiasmata remain intact during this stage. The remainder of this stage is similar to prometaphase of mitosis: the nucleoli disappear, the nuclear membrane disintegrates, and the meiotic spindle begins to form. | C: diakinesis stage | SciQ |
If a tap could fill entire tank in 18 hrs due to leakage, then in how much time
tank can be emptied by leakage if tap can fill entire tank in 12 hrs without leakage
A: 12hrs
B: 24hrs
C: 36hrs
D: 48hrs
E: 52hrs | time take to fill[withleakage] =18hrs so workdone in 1hr=1/18
time tkae to fill[without leakage]=12hrs so workdone in 1hr=1/12
if u subtract both u'll get time taken by leakage to empty...
1/18-1/12=1/36 so 36hrs
ANSWER:C | C: 36hrs | AQUA-RAT |
What are the three fundamental phases of matter?
A: solid, liquid, and gas
B: fast, slow, normal
C: big, small. and medium
D: air, water, and land | is the phase (or state) of matter. The three fundamental phases of matter are solid, liquid, and gas (see Figure 1.1 "The Phases of Matter"). Chemical properties are characteristics of matter that describe how matter changes form in the presence of other matter. Does a sample of matter burn? Burning is a chemical prope... | A: solid, liquid, and gas | SciQ |
If point A (1, 1) and point B (2, -1) are on opposite sides of the line $x+y-a=0$, then the range of values for $a$ is ______. | Given that point A (1, 1) and point B (2, -1) are on opposite sides of the line $x+y-a=0$, we can conclude that the product of their respective substitutions into the equation of the line is negative:
$$(1+1-a)(2-1-a) < 0,$$
Simplifying the left-hand side:
$$(2-a)(1-a) < 0.$$
Upon reordering, we have:
$$(a-1)(a-2)... | a \in (1, 2) | cn_k12 |
Fifteen points are selected on the positive $x$-axis, \( X^+ \), and six points are selected on the positive $y$-axis, \( Y^+ \). All possible segments connecting points from \( X^+ \) to \( Y^+ \) are drawn. What is the maximum possible number of points of intersection of these segments that could lie in the interior ... | 1. **Understanding the Setup**:
- 15 points on \( X^+ \).
- 6 points on \( Y^+ \), explicitly non-integer to increase scenario complexity.
- Total segments: \( 15 \times 6 = 90 \).
2. **Conditions for Intersection**:
- Intersection occurs when two segments \( (x_1, y_2) \) and \( (x_2, y_1) \) where \( x_1... | - (B) 1575 | synthetic_amc |
Failure rate of contraceptive method is determined by ?
A: Pearl index
B: Half life
C: Number of accidental pregnancies
D: Period of contraceptive practice continued | Ans. is 'a' i.e., Pearl index | A: Pearl index | MedMCQA |
Given points \( P(-3, 7) \) and \( Q(5, y) \) in a coordinate plane, for what value of \( y \) is the slope of the line through \( P \) and \( Q \) equal to \( \frac{1}{2} \)? | The slope of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated as:
\[ \frac{y_2 - y_1}{x_2 - x_1} = \text{slope} \]
Plugging in the given points and slope:
\[ \frac{y - 7}{5 - (-3)} = \frac{1}{2} \]
The denominator becomes \( 5 + 3 = 8 \), thus:
\[ \frac{y - 7}{8} = \frac{1}{2} \]
Now, solve for... | 11 | synthetic_math |
The game of blackjack is played with a deck consisting of 13 cards (one of each of the numbers 2-10, one jack, one queen, one king, and one of ace) in each of four suits: clubs, diamonds, hearts, and spades.
In a game of blackjack, if your first card is a spade, what are the odds that it is a jack?
A: 1/52
B: 1/13
C: 5... | The question should ask about the probability not the odds.
There are 13 spades, the probability that the spade we already have is a jack is therefore 1/13.
Answer: B. | B: 1/13 | AQUA-RAT |
Which of the following is an Interstitial type of a hernia
A: Richter's hernia
B: Spigelian hernia
C: Femoral hernia
D: Epigastric hernia | An interstitial hernia occurs when a hernia extends between the layers of muscle and not directly through themThis is typical of a Spigelian hernia. | B: Spigelian hernia | MedMCQA |
Crohn's Disease most commonly affects the -
A: Ileum
B: Rectum
C: Duodenum
D: Stomach | Ans. is 'a' i.e., Ileum Commonest sites of various intstinal diseasesUlcerative colitisRectumCrohn's diseaseIleumCA colonRectumDiverticulosisSigmoid colon | A: Ileum | MedMCQA |
The gene responsible for folic acid transpo is situated on:
A: Chromosome 10
B: Chromosome 5
C: X Chromosome
D: Chromosome 21 | . Chromosome 21 | D: Chromosome 21 | MedMCQA |
What explains why water molecules tend to stick together?
A: helium bonds
B: mineral bonds
C: covalent bonds
D: hydrogen bonds | Hydrogen bonds between water molecules explain some of water’s properties. For example, hydrogen bonds explain why water molecules tend to stick together. Have you ever watched water drip from a leaky faucet or from a melting icicle? If you have, then you know that water always falls in drops rather than as separate mo... | D: hydrogen bonds | SciQ |
In an examination a pupil's average marks were 63 per paper. If he had obtained 30 more marks for his Geography paper and 2 more marks for his History paper, his average per paper would have been 65. How many papers were there in the examination?
A: 8
B: 9
C: 10
D: 16
E: 12 | Let the number of papers be x. Then,
63x + 30 + 2 = 65x
= 2x = 32
= x = 16.
ANSWER:D | D: 16 | AQUA-RAT |
Let $a, b \in \mathbb{R}$, if $a > b$, then ( )
A: $\dfrac{1}{a} < \dfrac{1}{b}$
B: $\lg a > \lg b$
C: $2^{a} > 2^{b}$
D: $a^{2} > b^{2}$ | Given $a, b \in \mathbb{R}$, if $a > b$,
when $a=1$, $b=-1$, thus option A is not valid,
when $a=-1$, $b=-2$, option B is meaningless, thus B is not valid,
since $y=2^{x}$ is an increasing function, it follows that $2^{a} > 2^{b}$, thus option C is valid,
when $a=1$, $b=-1$, option D is not valid,
therefore, ... | C: $2^{a} > 2^{b}$ | cn_k12 |
Given $\sin \alpha =\frac{2}{3}$, then $\cos (\pi -2\alpha )=$ _________________ | **Analysis**
This question examines the double angle formula and its application.
We can directly calculate the conclusion using the double angle formula.
**Solution**
Since $\sin \alpha= \frac{2}{3}$,
we have $\cos (\pi-2\alpha)=-\cos 2\alpha$
$=-1+2\sin^2\alpha$
$=-1+2×\left( \frac{2}{3}\right)^2= -\frac{1}{9}... | -\frac{1}{9} | cn_k12 |
What is a mixture of metal with one or more other elements?
A: an alkali metal
B: an alloy
C: a metalloid
D: a halloid | An alloy is a mixture of a metal with one or more other elements. The other elements may be metals, nonmetals, or both. An alloy is formed by melting a metal and dissolving the other elements in it. The molten solution is then allowed to cool and harden. Alloys generally have more useful properties than pure metals. Se... | B: an alloy | SciQ |
Given vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(m,1)$. If vector $\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$, find the value of $m$. | **Step 1**: Understand the problem.
This problem requires finding the value of a real number $m$. It is important to carefully read the problem, focusing on the rules for operations with plane vector coordinates and the properties of perpendicular vectors.
**Step 2**: Find the sum of the vectors.
Given vectors $\overr... | 7 | cn_k12 |
If log x + log y = log (x+y), then,
A: x=y
B: xy=1
C: y= (x-1)/x
D: y=x/(x-1)
E: y=1/(x-1) | log x + log y = log (x+y)
=> log (x+y) = log (xy)
=> x+y = xy => y(x-1) = x
=> y= x/(x-1)
ANSWER:D | D: y=x/(x-1) | AQUA-RAT |
Square $ BDEC$ with center $ F$ is constructed to the out of triangle $ ABC$ such that $ \angle A \equal{} 90{}^\circ$ , $ \left|AB\right| \equal{} \sqrt {12}$ , $ \left|AC\right| \equal{} 2$ . If $ \left[AF\right]\bigcap \left[BC\right] \equal{} \left\{G\right\}$ , then $ \left|BG\right|$ will be $\textbf... | 1. **Identify the given information and set up the coordinate system:**
- Given: $\angle A = 90^\circ$, $|AB| = \sqrt{12}$, $|AC| = 2$.
- Place $A$ at the origin $(0,0)$, $B$ at $(0, \sqrt{12})$, and $C$ at $(2, 0)$.
2. **Construct the square $BDEC$ with center $F$:**
- Since $BDEC$ is a square, $D$ and $E$ a... | 2 | aops_forum |
What are four children born at one birth called?
A: triplets
B: twins
C: quadruplets
D: kittens | QUADruplets are four children born at one birth. | C: quadruplets | SciQ |
Let p be a prime number. If N = 8p, how many distinct even divisors does N have?
A: 1
B: 4
C: 5
D: 2
E: 3 | The correct answer is D. There are two: 2 and 2p.
8p = 2*2*2p = (2*2)*2p. Since we care only about distinct divisors, we ignore the fact that 2 appears three times. | D: 2 | AQUA-RAT |
A+B+C+D=D+E+F+G=G+H+I=17 given A=4. Find value of G and H?
A: 2,9
B: 3,9
C: 1,9
D: 4,9
E: 5,9 | A+B+C+D=17
4+2+6+5=17
D=5
D+E+F+G=17
5+3+8+1=17
G=1
G+H+I=17
1+9+7=17
G=1,H=9.
ANSWER:C | C: 1,9 | AQUA-RAT |
the percentage discount offered for"Buy 6 get 1 free is"?
A: 14.29%
B: 15.29%
C: 17.29%
D: 19.29%
E: 12.29% | Total Items = 7
Discounted = 1
Percentage = 1/7*100
=14.29%
ANSWER:A | A: 14.29% | AQUA-RAT |
Lola and Tara decide to race to the top of a 50 story building. Tara takes the elevator and Lola runs up the stairs. Lola can run up 1 story in 12 seconds. The elevator goes up a story in 10 seconds but stops for 4 seconds on every single floor. How long would it take for the slower one of Lola and Tara to reach the to... | Let's calculate the time it takes for both Lola and Tara to reach the top floor.
For Lola:
She runs up 1 story in 12 seconds. To reach the top of a 50-story building, she would take:
50 stories * 12 seconds/story = 600 seconds
For Tara:
The elevator goes up a story in 10 seconds and stops for 4 seconds on every floor... | 692 | orca_math |
Let \\(0 < a < b < 1\\), then the following inequality that holds is ( )
A: \\(a^{3} > b^{3}\\)
B: \\( \\dfrac{1}{a} < \\dfrac{1}{b}\\)
C: \\(a^{b} > 1\\)
D: \\(\\lg (b-a) < 0\\) | Since \\(0 < a < b < 1\\), based on the basic properties of inequalities, we know that \\(a^{3} < b^{3}\\), thus option A is incorrect; \\(\\dfrac{1}{a} > \\dfrac{1}{b} \\), so option B is incorrect; according to the graph and properties of the exponential function, we know \\(a^{b} < 1\\), so option C is incorrect; gi... | \text{D} | cn_k12 |
A train 240 m long passed a pole in 24 sec. How long will it take to pass a platform 650 m long?
A: 80
B: 89
C: 78
D: 67
E: 56 | Speed = 240/24 = 10 m/sec.
Required time = (240 + 650)/10 = 89 sec.
Answer: Option B | B: 89 | AQUA-RAT |
20.036 divided by 0.04 gives :
A: 50.09
B: 5.06
C: 50.06
D: 100.9
E: 500.9 | = 20.036/0.04
= 2003.6/4
= 500.9
Answer is E. | E: 500.9 | AQUA-RAT |
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