2020 CAT IIM Exam Slot 2: 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 2. The entire exam in pdf format can be downloaded from this link: 2020 CAT Quant Slot 2: Quantitative Ability.

Question 1: The number of integers that satisfy the equality $(x^2 – 5x + 7)^{x+1} = 1$ is:

Question 2: Let the $m$-th and $n$-th terms of a geometric progression be $\dfrac{3}{4}$ and $12$, respectively, where $m < n$. If the common ratio of the progression is an integer $r$, then the smallest possible value of $r + n - m$ is:

$-4$

$-2$

Question 3: Students in a college have to choose at least two subjects from chemistry, mathematics and physics. The number of students choosing all three subjects is $18$, choosing mathematics as one of their subjects is $23$, and choosing physics as one of their subjects is $25$. The smallest possible number of students who could choose chemistry as one of their subjects is:

$22$

$20$

$19$

$21$

Question 4: The distance from B to C is thrice that from A to B. Two trains travel from A to C via B. The speed of train 2 is double that of train 1 while traveling from A to B, and their speeds are interchanged while traveling from B to C. The ratio of the time taken by train 1 to that taken by train 2 in travelling from A to C is:

$1:4$

$7:5$

$4:1$

$5:7$

Question 5: Let $C_1$ and $C_2$ be concentric circles such that the diameter of $C_1$ is $2$ cm longer than that of $C_2$. If a chord of $C_1$ has length $6$ cm and is a tangent to $C_2$, then the diameter, in cm, of $C_1$ is:

Question 6: In a car race, car A beats car B by $45$ km, car B beats car C by $50$ km, and car A beats car C by $90$ km. The distance (in km) over which the race has been conducted is:

$450$

$475$

$550$

$500$

Question 7: The sum of the perimeters of an equilateral triangle and a rectangle is $90$ cm. The area, $T$, of the triangle and the area, $R$, of the rectangle (both in sq cm) satisfy $R = T^2$. If the sides of the rectangle are in the ratio $1:3$, then the length, in cm, of the longer side of the rectangle is:

$24$

$27$

$21$

$18$

Question 8: For the same principal amount, the compound interest for two years at $5\%$ per annum exceeds the simple interest for three years at $3\%$ per annum by ₹$1125$. Then the principal amount in rupees is:

Question 9: In a group of $10$ students, the mean of the lowest $9$ scores is $42$ while the mean of the highest $9$ scores is $47$. For the entire group, the maximum possible mean exceeds the minimum possible mean by:

Question 10: A and B are two points on a straight line. Ram runs from A to B while Rahim runs from B to A. After crossing each other, Ram and Rahim reach their destinations in one minute and four minutes respectively. If they start at the same time, the ratio of Ram’s speed to Rahim’s speed is:

$\sqrt{2}$

$\dfrac{1}{2}$

$2\sqrt{2}$

Question 11: In May, John bought the same amount of rice and wheat as he had bought in April, but spent ₹$150$ more due to price increase of rice and wheat by $20\%$ and $12\%$, respectively. If John had spent ₹$450$ on rice in April, then how much did he spend on wheat in May?

₹$590$

₹$580$

₹$570$

₹$560$

Question 12: A sum of money is split among Amal, Sunil, and Mita so that the ratio of the shares of Amal and Sunil is $3:2$, while the ratio of the shares of Sunil and Mita is $4:5$. If the difference between the largest and smallest shares is ₹$400$, then Sunil’s share, in rupees, is:

Question 13: How many 4-digit numbers, each greater than $1000$ and having all digits distinct, are there with $7$ coming before $3$?

Question 14: John takes twice as much time as Jack to finish a job. Jack and Jim together take one-third of the time John takes alone. John takes 3 days more than the three working together. In how many days will Jim finish the job working alone?

Question 15: The value of $\log_a\!\left(\dfrac{a}{b}\right) + \log_b\!\left(\dfrac{b}{a}\right)$, for $1 < a \le b$, cannot be equal to:

$-1$

$-0.5$

Question 16: Let $f(x)=x^2+ax+b$ and $g(x)=f(x+1)-f(x-1)$. If $f(x)\ge0$ for all real $x$, and $g(20)=72$, then the smallest possible value of $b$ is:

$16$

Question 17: Let $C$ be a circle of radius $5$ m centered at $O$. Let $PQ$ be a chord of $C$ passing through points $A$ and $B$, where $A$ is $4$ m north of $O$ and $B$ is $3$ m east of $O$. The length of $PQ$, in m, is nearest to:

$8.8$

$6.6$

$7.2$

$7.8$

Question 18: Anil buys 12 toys and labels each with the same selling price. He sells 8 toys at 20 % discount and 4 toys at an additional 25 % discount on the discounted price. He gets a total of ₹ 2112 and makes 10 % profit. With no discounts, his percentage of profit would have been:

$55$

$50$

$54$

$60$

Question 19: From an interior point of an equilateral triangle, perpendiculars are drawn to all three sides. The sum of the lengths of the three perpendiculars is $s$. The area of the triangle is:

$\dfrac{s^2}{2\sqrt{3}}$

$\dfrac{\sqrt{3}s^2}{2}$

$\dfrac{2s^2}{\sqrt{3}}$

$\dfrac{s^2}{\sqrt{3}}$

Question 20: Aron bought some pencils and sharpeners. Spending the same amount as Aron, Aditya bought twice as many pencils and 10 fewer sharpeners. If a sharpener costs ₹ 2 more than a pencil, then the minimum possible number of pencils bought by both together is:

$30$

$36$

$27$

$33$

Question 21: If $x$ and $y$ are positive real numbers satisfying $x+y=102$, then the minimum possible value of $2601\!\left(1+\dfrac{1}{x}\right)\!\left(1+\dfrac{1}{y}\right)$ is:

Question 22: The number of pairs of integers $(x, y)$ satisfying $x ≥ y ≥ -20$ and $2x + 5y = 99$ is:

Question 23: Two circular tracks $T_1$ and $T_2$ of radii 100 m and 20 m touch at point A. Starting from A simultaneously, Ram and Rahim walk on $T_1$ and $T_2$ at 15 km/h and 5 km/h respectively. The number of full rounds Ram makes before meeting Rahim again for the first time is:

Question 24: If $x$ and $y$ are non-negative integers such that $x+9=z$, $y+1=z$, and $x+y \lt z+5$, then the maximum possible value of $2x+y$ equals:

Question 25: In how many ways can a pair of integers $(x, a)$ be chosen such that $x^2 − 2|x| + |a − 2| = 0$?

Question 26: For real $x$, the maximum possible value of $\dfrac{x}{\sqrt{1+x^4}}$ is:

$\dfrac{1}{\sqrt{3}}$

$\dfrac{1}{2}$

$\dfrac{1}{\sqrt{2}}$

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