2019 CAT IIM Exam Slot 1: Quantitative Ability

This page organizes my video explanations to all of the 34 quantitative ability questions that are part of the 2019 CAT exam Slot 1. The entire exam in pdf format can be downloaded from this link: 2019 CAT Quant Slot 1: Quantitative Ability.

Question 1: In a class, $60\%$ of the students are girls and the rest are boys. There are $30$ more girls than boys. If $68\%$ of the students, including $30$ boys, pass an examination, the percentage of the girls who do not pass is:

Question 2: If $(5.55)^{x} = (0.555)^{y} = 1000$, then the value of $\dfrac{1}{x} – \dfrac{1}{y}$ is:

$\dfrac{1}{3}$

$\dfrac{2}{3}$

Question 3: With rectangular axes of coordinates, the number of paths from $(1,1)$ to $(8,10)$ via $(4,6)$, where each step from $(x,y)$ is either to $(x,y+1)$ or to $(x+1,y)$, is:

Question 4: A club has $256$ members of whom $144$ can play football, $123$ can play tennis, and $132$ can play cricket. Moreover, $58$ can play both football and tennis, $25$ can play both cricket and tennis, and $63$ can play both football and cricket. Every member plays at least one game. The number of members who can play only tennis is:

$32$

$43$

$38$

$45$

Question 5: In a circle of radius $11$ cm, $CD$ is a diameter and $AB$ is a chord of length $20.5$ cm. If $AB$ and $CD$ intersect at a point $E$ inside the circle and $CE=7$ cm, then the difference of the lengths of $BE$ and $AE$ (in cm) is:

$1.5$

$3.5$

$0.5$

$2.5$

Question 6: Meena scores $40\%$ in an examination and after review, even though her score is increased by $50\%$, she fails by $35$ marks. If her post-review score is increased by $20\%$, she will have $7$ marks more than the passing score. The percentage score needed for passing the examination is

$75$

$80$

$60$

$70$

Question 7: Corners are cut off from an equilateral triangle $T$ to produce a regular hexagon $H$. Then, the ratio of the area of $H$ to that of $T$ is:

$5:6$

$3:4$

$2:3$

$4:5$

Question 8: Let $T$ be the triangle formed by the straight line $3x + 5y – 45 = 0$ and the coordinate axes. Let the circumcircle of $T$ have radius of length $L$, measured in the same unit as the coordinate axes. Then, the integer closest to $L$ is

Question 9: For any positive integer $n$, let
$f(n)=\begin{cases} n(n+1), &\text{if $n$ even;}\\ n+3,&\text{if $n$ odd.}\end{cases}$
If $m$ is a positive integer such that $8f(m + 1) – f(m) = 2$, then $m$ equals

Question 10: If the population of a town is $p$ in the beginning of any year, then it becomes $3 + 2p$ in the beginning of the next year. If the population in the beginning of $2019$ is $1000$, then the population in the beginning of $2034$ will be

$(1003)^{15} + 6$

$(977)^{15} – 3$

$(1003)2^{15} – 3$

$(977)2^{14} + 3$

Question 11: A person invested a total amount of Rs $15$ lakh. A part of it was invested in a fixed deposit earning $6\%$ annual interest, and the remaining amount was invested in two other deposits in the ratio $2:1$, earning annual interest at the rates of $4\%$ and $3\%$, respectively. If the total annual interest income is Rs $76{,}000$, then the amount (in Rs lakh) invested in the fixed deposit was:

Question 12: The product of two positive numbers is $616$. If the ratio of the difference of their cubes to the cube of their difference is $157:3$, then the sum of the two numbers is:

$50$

$85$

$95$

$58$

Question 13: On selling a pen at $5\%$ loss and a book at $15\%$ gain, Karim gains Rs $7$. If he sells the pen at $5\%$ gain and the book at $10\%$ gain, he gains Rs $13$. What is the cost price of the book in rupees?

$80$

$85$

$100$

$95$

Question 14: Two cars travel the same distance starting at $10{:}00$ a.m. and $11{:}00$ a.m., respectively. They reach their common destination at the same time. If the first car travelled for at least $6$ hours, the highest possible value of the percentage by which the speed of the second car could exceed that of the first is:

$20$

$10$

$35$

$25$

Question 15: At their usual efficiency levels, $A$ and $B$ together finish a task in $12$ days. If $A$ had worked half as efficiently as she usually does, and $B$ had worked thrice as efficiently as he usually does, the task would have been completed in $9$ days. How many days would $A$ take to finish the task if she works alone at her usual efficiency?

$18$

$12$

$24$

$36$

Question 16: If $a_1+a_2+\dots+a_n = 3(2^{\,n+1}-2)$, then $a_{11}$ equals:

Question 17: The number of the real roots of the equation $2\cos\!\big(x(x+1)\big)=2^{x}+2^{-x}$ is

Infinite

Question 18: The income of Amala is $20\%$ more than that of Bimala and $20\%$ less than that of Kamala. If Kamala’s income goes down by $4\%$ and Bimala’s goes up by $10\%$, then the percentage by which Kamala’s income would exceed Bimala’s is nearest to:

$28$

$29$

$31$

$32$

Question 19: In a race of three horses, the first beat the second by $11$ metres and the third by $90$ metres. If the second beat the third by $80$ metres, what was the length, in metres, of the racecourse?

Question 20: If $a_{1},a_{2},\ldots$ are in A.P., then
$$
\dfrac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\dfrac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\cdots+\dfrac{1}{\sqrt{a_{n}}+\sqrt{a_{n+1}}}
$$
is equal to:

$\dfrac{n}{\sqrt{a_{1}}+\sqrt{a_{n+1}}}$

$\dfrac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n}}}$

$\dfrac{n}{\sqrt{a_{1}}-\sqrt{a_{n+1}}}$

$\dfrac{n-1}{\sqrt{a_{1}}+\sqrt{a_{n-1}}}$

Question 21: $AB$ is a diameter of a circle of radius $5$ cm. Let $P$ and $Q$ be two points on the circle so that the length of $PB$ is $6$ cm, and the length of $AP$ is twice that of $AQ$. Then the length, in cm, of $QB$ is nearest to:

$8.5$

$9.3$

$9.1$

$7.8$

Question 22: One can use three different transports which move at $10$, $20$, and $30$ kmph, respectively. To reach from $A$ to $B$, Amal took each mode of transport for $\dfrac{1}{3}$ of his total journey time, while Bimal took each mode of transport for $\dfrac{1}{3}$ of the total distance. The percentage by which Bimal’s travel time exceeds Amal’s travel time is nearest to

$22$

$19$

$21$

$20$

Question 23: Amala, Bina, and Gouri invest money in the ratio $3:4:5$ in fixed deposits having respective annual interest rates in the ratio $6:5:4$. What is their total interest income (in rupees) after a year, if Bina’s interest income exceeds Amala’s by Rs $250$?

$7000$

$6000$

$6350$

$7250$

Question 24: If $m$ and $n$ are integers such that $(\sqrt{2})^{19}\,3^{4}\,4^{2}\,9^{m}\,8^{n} \;=\; 3^{n}\,16^{m}\,\sqrt[4]{64}$, then $m$ is

$-16$

$-24$

$-12$

$-20$

Question 25: A chemist mixes two liquids $1$ and $2$. One litre of liquid $1$ weighs $1$ kg and one litre of liquid $2$ weighs $800$ gm. If half litre of the mixture weighs $480$ gm, then the percentage of liquid $1$ in the mixture, in terms of volume, is

$70$

$85$

$80$

$75$

Question 26: Let $x$ and $y$ be positive real numbers such that $\log_{5}(x+y)+\log_{5}(x-y)=3$, and $\log_{2}y-\log_{2}x=1-\log_{2}3$. Then $xy$ equals

$25$

$150$

$250$

$100$

Question 27: If the rectangular faces of a brick have their diagonals in the ratio $3:2\sqrt{3}:\sqrt{15}$, then the ratio of the length of the shortest edge of the brick to that of its longest edge is

$1:\sqrt{3}$

$2:\sqrt{5}$

$\sqrt{2}:\sqrt{3}$

$\sqrt{3}:2$

Question 28: Let $S$ be the set of all points $(x,y)$ in the $xy$-plane such that $|x|+|y|\le 2$ and $|x|\ge 1$. Then, the area, in square units, of the region represented by $S$ equals

Question 29: The number of solutions of the equation $|x|\,(6x^{2}+1)=5x^{2}$ is

Question 30: Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in $13$ days, then how many men can finish the job in $13$ days?

Question 31: The product of the distinct roots of $|x^{2} – x – 6| = x + 2$ is

$-4$

$-16$

$-8$

$-24$

Question 32: The wheels of bicycles $A$ and $B$ have radii $30$ cm and $40$ cm, respectively. While traveling a certain distance, each wheel of $A$ required $5000$ more revolutions than each wheel of $B$. If bicycle $B$ traveled this distance in $45$ minutes, then its speed, in km per hour, was

$18\pi$

$16\pi$

$12\pi$

$14\pi$

Question 33: Consider a function $f(x+y) = f(x)\,f(y)$ where $x, y$ are positive integers, and $f(1) = 2$. If $f(a+1) + f(a+2) + \dots + f(a+n) = 16(2^{n} – 1)$, then $a$ is equal to

Question 34: Ramesh and Gautam are among $22$ students who write an examination. Ramesh scores $82.5$. The average score of the $21$ students other than Gautam is $62$. The average score of all the $22$ students is one more than the average score of the $21$ students other than Ramesh. The score of Gautam is

$51$

$53$

$49$

$48$

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