2020 CAT IIM Exam Slot 1: Quantitative Ability

This page organizes my video explanations to all of the 26 quantitative ability questions that are part of the 2020 CAT exam Slot 1. The entire exam in pdf format can be downloaded from this link: 2020 CAT Quant Slot 1: Quantitative Ability.

Question 1: If $\log_{4} 5 = (\log_{4} y)(\log_{6} \sqrt{5})$, then $y$ equals

Question 2: An alloy is prepared by mixing three metals $A$, $B$ and $C$ in the proportion $3 : 4 : 7$ by volume. Weights of the same volume of the metals $A$, $B$ and $C$ are in the ratio $5 : 2 : 6$. In $130$ kg of the alloy, the weight (in kg) of the metal $C$ is:

$96$

$84$

$70$

$48$

Question 3: Leaving home at the same time, Amal reaches office at $10{:}15$ a.m. if he travels at $8$ km/h, and at $9{:}40$ a.m. if he travels at $15$ km/h. Leaving home at $9{:}10$ a.m., at what speed (in km/h) must he travel so as to reach office exactly at $10$ a.m.?

$13$

$14$

$11$

$12$

Question 4: Let $A$, $B$ and $C$ be three positive integers such that the sum of $A$ and the mean of $B$ and $C$ is $5$. In addition, the sum of $B$ and the mean of $A$ and $C$ is $7$. Then the sum of $A$ and $B$ is:

$6$

Question 5: A solid right circular cone of height $27$ cm is cut into two pieces along a plane parallel to its base at a height of $18$ cm from the base. If the difference in volume of the two pieces is $225$ cc, the volume (in cc) of the original cone is:

$256$

$232$

$264$

$243$

Question 6: Two persons are walking beside a railway track at respective speeds of $2$ and $4$ km/h in the same direction. A train came from behind and crossed them in $90$ and $100$ seconds, respectively. The time, in seconds, taken by the train to cross an electric post is nearest to:

$78$

$82$

$87$

$75$

Question 7: A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then half of the remaining stock along with one extra to the second, and continues giving away in this fashion. His total stock exhausts after he takes care of $5$ children. How many toffees were there in his stock initially?

Question 8: If $a$, $b$ and $c$ are positive integers such that $ab = 432$, $bc = 96$ and $c < 9$, then the smallest possible value of $a + b + c$ is:

$46$

$59$

$49$

$56$

Question 9: Among $100$ students, $x_1$ have birthdays in January, $x_2$ in February, and so on. If $x_0 = \max(x_1,\ldots,x_{12})$, then the smallest possible value of $x_0$ is:

$10$

$9$

$12$

Question 10: The mean of all $4$-digit even natural numbers of the form ‘aabb’, where $a > 0$, is:

$4466$

$4864$

$5050$

$5544$

Question 11: The number of real-valued solutions of the equation $2^{x}+2^{-x}=2-(x-2)^{2}$ is:

$0$

infinite

Question 12: How many distinct positive integer-valued solutions exist to the equation $(x^{2}-7x+11)^{(x^{2}-13x+42)}=1$?

$6$

Question 13: In a group of people, $28\%$ of the members are young while the rest are old. If $65\%$ are literates, and $25\%$ of the literates are young, then the percentage of old people among the illiterates is nearest to:

$66$

$59$

$62$

$55$

Question 14: A straight road connects points $A$ and $B$. Car 1 travels from $A$ to $B$ and Car 2 travels from $B$ to $A$, both leaving at the same time. After meeting, they take $45$ minutes and $20$ minutes, respectively, to complete their journeys. If Car 1 travels at $60$ km/h, the speed of Car 2, in km/h, is:

$100$

$80$

$90$

$70$

Question 15: A person spent Rs $50,000$ to purchase a desktop and a laptop. He sold the desktop at $20\%$ profit and the laptop at $10\%$ loss. If overall he made a $2\%$ profit, the purchase price, in rupees, of the desktop is:

Question 16: If $f(5+x)=f(5-x)$ for every real $x$, and $f(x)=0$ has four distinct real roots, then the sum of these roots is:

$20$

$40$

$10$

Question 17: Veeru invested Rs $10,000$ at $5\%$ simple annual interest, and exactly after two years Joy invested Rs $8,000$ at $10\%$ simple annual interest. How many years after Veeru’s investment will their balances (principal + interest) be equal?

Question 18: On a rectangular metal sheet of area $135$ sq in, a circle is painted such that it touches two opposite sides. If the area of the sheet left unpainted is two-thirds of the painted area, then the perimeter of the rectangle (in inches) is:

$5\sqrt{\pi}\!\left(3+\dfrac{9}{\pi}\right)$

$3\sqrt{\pi}\!\left(\dfrac{5}{2}+\dfrac{6}{\pi}\right)$

$4\sqrt{\pi}\!\left(3+\dfrac{9}{\pi}\right)$

$3\sqrt{\pi}\!\left(5+\dfrac{12}{\pi}\right)$

Question 19: How many $3$-digit numbers are there for which the product of their digits is more than $2$ but less than $7$?

Question 20: The number of distinct real roots of the equation $\left(x+\dfrac{1}{x}\right)^{2}-3\left(x+\dfrac{1}{x}\right)+2=0$ equals:

Question 21: A solution of volume $40$ litres has dye and water in the ratio $2:3$. Water is added to change this to $2:5$. If one-fourth of this diluted solution is taken out, how many litres of dye must be added to bring the ratio back to $2:3$?

Question 22: The area of the region satisfying $|x|-y\le1$, $y\ge0$, and $y\le1$ is:

Question 23: A circle is inscribed in a rhombus with diagonals $12$ cm and $16$ cm. The ratio of the area of the circle to the area of the rhombus is:

$\dfrac{6\pi}{25}$

$\dfrac{2\pi}{15}$

$\dfrac{3\pi}{25}$

$\dfrac{5\pi}{18}$

Question 24: If $y$ is a negative number such that $2^{y^{2}\log_{3}5}=5^{\log_{2}3}$, then $y$ equals:

$-\log_{2}\!\left(\dfrac{1}{5}\right)$

$\log_{2}\!\left(\dfrac{1}{3}\right)$

$\log_{2}\!\left(\dfrac{1}{5}\right)$

$-\log_{2}\!\left(\dfrac{1}{3}\right)$

Question 25: A train travelled at one-third of its usual speed and hence reached $30$ minutes late. On return, it travelled at usual speed for $5$ minutes but stopped for $4$ minutes. The percentage by which it must now increase its speed to reach on time is nearest to:

$67$

$61$

$50$

$58$

Question 26: If $x=(4096)^{7+4\sqrt{3}}$, then which of the following equals $64$?

$\dfrac{x^{7}}{x^{2\sqrt{3}}}$

$\dfrac{x^{7}}{x^{4\sqrt{3}}}$

$\dfrac{x^{\frac{7}{2}}}{x^{2\sqrt{3}}}$

$\dfrac{x^{\frac{7}{2}}}{x^{\frac{4}{\sqrt{3}}}}$

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