2021 CAT IIM Exam Slot 3: Quantitative Ability

This page organizes my video explanations to all of the 22 quantitative ability questions that are part of the 2021 CAT exam Slot 3. The entire exam in pdf format can be downloaded from this link: 2021 CAT Quant Slot 3: Quantitative Ability.

Question 1: Bank $A$ offers $6\%$ interest rate per annum compounded half-yearly. Bank $B$ and Bank $C$ offer simple interest but the annual interest rate offered by Bank $C$ is twice that of Bank $B$. Raju invests a certain amount in Bank $B$ for a certain period and Rupa invests ₹$10{,}000$ in Bank $C$ for twice that period. The interest that would accrue to Raju during that period is equal to the interest that would have accrued had he invested the same amount in Bank $A$ for one year. The interest accrued, in INR, to Rupa is:

$1436$

$2436$

$3436$

$2346$

Question 2: If $f(x)=x^2-7x$ and $g(x)=x+3$, then the minimum value of $f(g(x))-3x$ is:

$-15$

$-20$

$-16$

$-12$

Question 3: In a tournament, a team has played $40$ matches so far and won $30\%$ of them. If they win $60\%$ of the remaining matches, their overall win percentage will be $50\%$. Suppose they win $90\%$ of the remaining matches, then the total number of matches won by the team in the tournament will be:

$86$

$78$

$80$

$84$

Question 4: If $3x+2|y|+y=7$ and $x+|x|+3y=1$, then $x+2y$ is:

$-\dfrac{4}{3}$

$\dfrac{8}{3}$

Question 5: A shop owner bought a total of $64$ shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR $50$ less than that of a large shirt. She paid a total of INR $5000$ for the large shirts, and total of INR $1800$ for the small shirts. Then, the price of a large shirt and a small shirt together, in INR, is:

$200$

$175$

$225$

$150$

Question 6: Mira and Amal walk along a circular track, starting from the same point at the same time. If they walk in the same direction, then in $45$ minutes, Amal completes exactly $3$ more rounds than Mira. If they walk in opposite directions, then they meet for the first time exactly after $3$ minutes. The number of rounds Mira walks in one hour is:

Question 7: If a certain weight of an alloy of silver and copper is mixed with $3$ kg of pure silver, the resulting alloy will have $90\%$ silver by weight. If the same weight of the initial alloy is mixed with $2$ kg of another alloy which has $90\%$ silver by weight, the resulting alloy will have $84\%$ silver by weight. Then, the weight of the initial alloy, in kg, is:

$3.5$

$2.5$

Question 8: If $n$ is a positive integer such that $(\sqrt[7]{10})(\sqrt[7]{10})^{2}\cdots(\sqrt[7]{10})^{n} > 999$, then the smallest value of $n$ is:

Question 9: The number of distinct pairs of integers $(m,n)$ satisfying $|1+mn| < |m+n| < 5$ is:

Question 10: One day, Rahul started a work at $9$ AM and Gautam joined him two hours later. They then worked together and completed the work at $5$ PM the same day. If both had started at $9$ AM and worked together, the work would have been completed $30$ minutes earlier. Working alone, the time Rahul would have taken, in hours, to complete the work is:

$12$

$12.5$

$11.5$

$10$

Question 11: Anil can paint a house in $12$ days while Barun can paint it in $16$ days. Anil, Barun, and Chandu undertake to paint the house for ₹$24{,}000$ and the three of them together complete the painting in $6$ days. If Chandu is paid in proportion to the work done by him, then the amount in INR received by him is:

Question 12: The cost of fencing a rectangular plot is ₹$200$ per ft along one side, and ₹$100$ per ft along the three other sides. If the area of the rectangular plot is $60000$ sq. ft, then the lowest possible cost of fencing all four sides, in INR, is:

$120000$

$100000$

$160000$

$90000$

Question 13: A four-digit number is formed by using only the digits $1$, $2$ and $3$ such that both $2$ and $3$ appear at least once. The number of all such four-digit numbers is:

Question 14: In a triangle $ABC$, $\angle BCA = 50^{\circ}$. $D$ and $E$ are points on $AB$ and $AC$, respectively, such that $AD=DE$. If $F$ is a point on $BC$ such that $BD=DF$, then $\angle FDE$, in degrees, is equal to:

$72$

$100$

$80$

$96$

Question 15: Consider a sequence of real numbers $x_1, x_2, x_3, \ldots$ such that $x_{n+1} = x_n + n-1$ for all $n \geq 1$. If $x_1=-1$ then $x_{100}$ is equal to:

$4850$

$4950$

$4849$

$4949$

Question 16: A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to $800$. The prices of the smallest size and the medium size are in the ratio of $2:5$. If the shop owner decides to increase the prices of the smallest and the medium ones by INR $6$ keeping the price of the largest size unchanged, the product then changes to $3200$. The sum of the original prices of three different sizes, in INR, is:

Question 17: For a real number $a$, if $\displaystyle \frac{\log_{15} {a} + \log_{32} {a}}{(\log_{15} a)(\log_{32} a)} = 4$ then $a$ must lie in the range:

$4<a<5$

$3<a<4$

$2<a<3$

$a>3$

Question 18: A park is shaped like a rhombus and has area $96$ sq m. If $40$ m of fencing is needed to enclose the park, the cost, in INR, of laying electric wires along its two diagonals, at the rate of ₹$125$ per m, is:

Question 19: One part of a hostel’s monthly expenses is fixed, and the other part is proportional to the number of its boarders. The hostel collects ₹$1600$ per month from each boarder. When the number of boarders is $50$, the profit of the hostel is ₹$200$ per boarder, and when the number of boarders is $75$, the profit of the hostel is ₹$250$ per boarder. When the number of boarders is $80$, the total profit of the hostel, in INR, will be:

$20800$

$20500$

$20200$

$20000$

Question 20: Let $ABCD$ be a parallelogram. The lengths of the side $AD$ and the diagonal $AC$ are $10$ cm and $20$ cm, respectively. If the angle $\angle ADC$ is equal to $30^{\circ}$ then the area of the parallelogram, in sq. cm, is:

$\displaystyle \frac{25(\sqrt{5}+\sqrt{15})}{2}$

$\displaystyle 25(\sqrt{5}+\sqrt{15})$

$\displaystyle 25(\sqrt{3}+\sqrt{15})$

$\displaystyle \frac{25(\sqrt{3}+\sqrt{15})}{2}$

Question 21: The total of male and female populations in a city increased by $25\%$ from $1970$ to $1980$. During the same period, the male population increased by $40\%$ while the female population increased by $20\%$. From $1980$ to $1990$, the female population increased by $25\%$. In $1990$, if the female population is twice the male population, then the percentage increase in the total of male and female populations in the city from $1970$ to $1990$ is:

$68.75$

$68.25$

$69.25$

$68.5$

Question 22: The arithmetic mean of scores of $25$ students in an examination is $50$. Five of these students top the examination with the same score. If the scores of the other students are distinct integers with the lowest being $30$, then the maximum possible score of the toppers is:

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