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2021 CAT IIM Exam Slot 2: 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 2. The entire exam in pdf format can be downloaded from this link: 2021 CAT Quant Slot 2: Quantitative Ability.


Question 1: For all possible integers $n$ satisfying $2.25 \le 2+2^{n+2} \le 202$, the number of integer values of $3+3^{n+1}$ is:






Question 2: Three positive integers $x$, $y$, and $z$ are in arithmetic progression. If $y-x>2$ and $xyz=5(x+y+z)$, then $z-x$ equals:

  1. $\quad 8$

  2. $\quad 12$

  3. $\quad 14$

  4. $\quad 10$






Question 3: For a $4$-digit number, the sum of its digits in the thousands, hundreds, and tens places is $14$, the sum of its digits in the hundreds, tens, and units places is $15$, and the tens-place digit is $4$ more than the units-place digit. Then the highest possible $4$-digit number satisfying the above conditions is:






Question 4: Raj invested ₹$10000$ in a fund. At the end of the first year he incurred a loss, but his balance was more than ₹$5000$. This balance, when invested for another year, grew and the percentage of growth in the second year was five times the percentage of loss in the first year. If Raj’s overall gain from the initial investment over the two-year period is $35\%$, then the percentage of loss in the first year is:

  1. $\quad 5$

  2. $\quad 15$

  3. $\quad 17$

  4. $\quad 10$






Question 5: The number of ways of distributing $15$ identical balloons, $6$ identical pencils, and $3$ identical erasers among $3$ children, such that each child gets at least four balloons and one pencil, is:






Question 6: Two trains $A$ and $B$ were moving in opposite directions, their speeds being in the ratio $5:3$. The front end of $A$ crossed the rear end of $B$ $46$ seconds after the front ends of the trains had crossed each other. It took another $69$ seconds for the rear ends of the trains to cross each other. The ratio of the lengths of train $A$ to that of train $B$ is:

  1. $\quad 3:2$

  2. $\quad 5:3$

  3. $\quad 2:3$

  4. $\quad 2:1$





Question 7: Suppose one of the roots of the equation $ax^2-bx+c=0$ is $2+\sqrt3$, where $a$, $b$, and $c$ are rational and $a\ne0$. If $b=c^3$, then $|a|$ equals:

  1. $\quad 1$

  2. $\quad 2$

  3. $\quad 3$

  4. $\quad 4$





Question 8: From a container filled with milk, $9$ litres are drawn and replaced with water. Next, another $9$ litres are drawn and replaced with water. If the final ratio of milk to water is $16:9$, then the capacity of the container, in litres, is:





Question 9: If a rhombus has area $12\text{ cm}^2$ and side length $5\text{ cm}$, then the length (in cm) of its longer diagonal is:

  1. $\quad \sqrt{37}+\sqrt{13}$

  2. $\quad \sqrt{13}+\sqrt{12}$

  3. $\quad \dfrac{\sqrt{37}+\sqrt{13}}{2}$

  4. $\quad \dfrac{\sqrt{13}+\sqrt{12}}{2}$





Question 10: Let $D$ and $E$ be points on sides $AB$ and $AC$ of $\triangle ABC$, such that $AD:DB=2:1$ and $AE:EC=2:3$. If the area of $\triangle ADE$ is $8\text{ cm}^2$, then the area of $\triangle ABC$ (in cm$^2$) is:





Question 11: If $\log_2\![3+\log_3\!\{4+\log_4(x-1)\}]-2=0$, then $4x$ equals:





Question 12: The sides $AB$ and $CD$ of a trapezium $ABCD$ are parallel, with $AB$ smaller. $P$ is the midpoint of $CD$, and $ABPD$ is a parallelogram. If the difference between the areas of $ABPD$ and $\triangle BPC$ is $10\text{ cm}^2$, then the area (in cm$^2$) of $ABCD$ is:

  1. $\quad 30$

  2. $\quad 40$

  3. $\quad 25$

  4. $\quad 20$





Question 13: For all real $x$, the range of $f(x)=\dfrac{x^2+2x+4}{2x^2+4x+9}$ is:

  1. $\quad \left[\dfrac{4}{9},\dfrac{8}{9}\right]$

  2. $\quad \left[\dfrac{3}{7},\dfrac{8}{9}\right)$

  3. $\quad \left(\dfrac{3}{7},\dfrac{1}{2}\right)$

  4. $\quad \left[\dfrac{3}{7},\dfrac{1}{2}\right)$





Question 14: For a sequence of real numbers $x_1,x_2,\ldots,x_n$, if $x_1-x_2+x_3-\ldots+(-1)^{n+1}x_n=n^2+2n$ for all natural numbers $n$, then $x_{49}+x_{50}$ equals:

  1. $\quad 200$

  2. $\quad 2$

  3. $\quad -200$

  4. $\quad -2$





Question 15: For a real number $x$, the condition $|3x-20|+|3x-40|=20$ necessarily holds if:

  1. $\quad 10<x<15$

  2. $\quad 9<x<14$

  3. $\quad 7<x<12$

  4. $\quad 6<x<11$





Question 16: Anil can paint a house in $60$ days while Bimal can paint it in $84$ days. Anil starts painting, and after $10$ days, Bimal and Charu join him. Together, they finish the job in $14$ more days. If they are paid ₹$21000$ in total, then Charu’s share, in ₹, is:

  1. $\quad 9000$

  2. $\quad 9200$

  3. $\quad 9100$

  4. $\quad 9150$





Question 17: A box has $450$ balls, each either white or black, there being as many metallic white balls as metallic black balls. If $40\%$ of the white balls and $50\%$ of the black balls are metallic, then the number of non-metallic balls in the box is:





Question 18: In a football tournament, a player has played a certain number of matches, and $10$ more are to be played. If he scores $1$ goal in those $10$ matches, his average will be $0.15$; if he scores $2$, it will be $0.2$. The number of matches he has already played is:





Question 19: A person buys tea of three different qualities at ₹$800$, ₹$500$, ₹$300$ per kg, respectively, and the amounts bought are in the proportion $2:3:5$. She mixes all the tea and sells one-sixth of the mixture at ₹$700$ per kg. The price (₹ per kg) at which she should sell the remaining tea to make an overall profit of $50\%$ is:

  1. $\quad 653$

  2. $\quad 688$

  3. $\quad 692$

  4. $\quad 675$





Question 20: Consider the pair of equations: $x^2-xy-x=22$ and $y^2-xy+y=34$. If $x>y$, then $x-y$ equals:

  1. $\quad 6$

  2. $\quad 4$

  3. $\quad 7$

  4. $\quad 8$





Question 21: Anil, Bobby, and Chintu jointly invest in a business and agree to share the overall profit in proportion to their investments. Anil’s share of investment is $70\%$. His share of profit decreases by ₹$420$ if the overall profit goes down from $18\%$ to $15\%$. Chintu’s share of profit increases by ₹$80$ if the overall profit goes up from $15\%$ to $17\%$. The amount, in ₹, invested by Bobby is:

  1. $\quad 2000$

  2. $\quad 2400$

  3. $\quad 2200$

  4. $\quad 1800$





Question 22: Two pipes $A$ and $B$ are attached to an empty water tank. Pipe $A$ fills the tank while pipe $B$ drains it. If pipe $A$ is opened at $2$ pm and pipe $B$ is opened at $3$ pm, then the tank becomes full at $10$ pm. Instead, if pipe $A$ is opened at $2$ pm and pipe $B$ is opened at $4$ pm, then the tank becomes full at $6$ pm. If pipe $B$ is not opened at all, then the time (in minutes) taken to fill the tank is:

  1. $\quad 144$

  2. $\quad 140$

  3. $\quad 264$

  4. $\quad 120$




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