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2024 CAT Exam Slot 3: Quantitative Ability

 
This page organizes my video explanations to all of the 22 quantitative ability questions that are part of the 2024 CAT exam Slot 3. The entire exam in pdf format can be downloaded from this link: 2024 CAT Quant Slot 3: Quantitative Ability.

Question 1: A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of $120^{\circ}$ at the center. Then, the area, in square meters, of the smaller region is

  1. $\quad 20\left(\displaystyle \frac{4\pi}{3} + \sqrt{3}\right)$

  2. $\quad 20\left(\displaystyle \frac{4\pi}{3} – \sqrt{3}\right)$

  3. $\quad 25\left(\displaystyle \frac{4\pi}{3} + \sqrt{3}\right)$

  4. $\quad 25\left(\displaystyle \frac{4\pi}{3} – \sqrt{3}\right)$

Video Explanation


 

Question 2: If $(a+b\sqrt{3})^2 = 52+30\sqrt{3}$, where $a$ and $b$ are natural numbers, then $a+b$ equals

  1. $\quad 8$

  2. $\quad 10$

  3. $\quad 9$

  4. $\quad 7$

Video Explanation


 

Question 3: The number of distinct real values of $x$, satisfying the equation $\textrm{max}\{x,2\} – \textrm{min}\{x,2\} = |x+2|-|x-2|$, is

Video Explanation


 

Question 4: The average of three distinct real numbers is $28$. If the smallest number is increased by $7$ and the largest number is reduced by $10$, the order of the numbers remains unchanged, and the new arithmetic mean becomes $2$ more than the middle number, while the difference between the largest and the smallest numbers becomes $64$. Then, the largest number in the original set of three numbers is

Video Explanation


 

Question 5: Aman invests Rs $4000$ in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after $3$ years to the value of the investment after $5$ years is $25 : 36$, then the minimum number of years required for the value of the investment to exceed Rs $20000$ is

Video Explanation


 

Question 6: Rajesh and Vimal own $20$ hectares and $30$ hectares of agricultural land, respectively, which are entirely covered by wheat and mustard crops. The cultivation area of wheat and mustard in the land owned by Vimal are in the ratio of $5 : 3$. If the total cultivation area of wheat and mustard are in the ratio $11 : 9$, then the ratio of cultivation area of wheat and mustard in the land owned by Rajesh is

  1. $\quad 7:9$

  2. $\quad 2.3:7$

  3. $\quad 3.1:1$

  4. $\quad 4.4:3$

Video Explanation


 

Question 7: If $10^{68}$ is divided by $13$, the remainder is

  1. $\quad 9$

  2. $\quad 4$

  3. $\quad 5$

  4. $\quad 8$

Video Explanation


 

Question 8: The number of distinct integer solutions $(x, y)$ of the equation $|x+y|+|x-y|=2$, is

Video Explanation


 

Question 9: A train travelled a certain distance at a uniform speed. Had the speed been $6$ km per hour more, it would have needed $4$ hours less. Had the speed been $6$ km per hour less, it would have needed $6$ hours more. The distance, in km, travelled by the train is

  1. $\quad 800$

  2. $\quad 640$

  3. $\quad 720$

  4. $\quad 780$

Video Explanation


 

Question 10: Consider the sequence $t_1=1$, $t_2=-1$ and $t_n = \left(\displaystyle \frac{n-3}{n-1}\right) t_{n-2}$ for $n \geq 3$. Then, the value of the sum $\displaystyle \frac{1}{t_2} + \frac{1}{t_4} + \frac{1}{t_6} + \cdots + \frac{1}{t_{2022}}+\frac{1}{t_{2024}}$, is

  1. $\quad -1024144$

  2. $\quad -1023132$

  3. $\quad -1026169$

  4. $\quad -1022121$

Video Explanation


 

Question 11: If $3^a = 4$, $4^b=5$, $5^c=6$, $6^d=7$, $7^e=8$ and $8^f=9$, then the value of the product $abcdef$ is

Video Explanation


 

Question 12: After two successive increments, Gopal’s salary became $187.5\%$ of his initial salary. If the percentage of salary increase in the second increment was twice of that in the first increment, then the percentage of salary increase in the first increment was

  1. $\quad 27.5$

  2. $\quad 30$

  3. $\quad 25$

  4. $\quad 20$

Video Explanation


 

Question 13: For any non-zero real number $x$, let $f(x)+2f\left(\frac{1}{x}\right) = 3x$. Then, the sum of all possible values of $x$ for which $f(x)=3$, is

  1. $\quad 3$

  2. $\quad -3$

  3. $\quad -2$

  4. $\quad 2$

Video Explanation


 

Question 14: A certain amount of water was poured into a $300$ litre container and the remaining portion of the container was filled with milk. Then an amount of this solution was taken out from the container which was twice the volume of water that was earlier poured into it, and water was poured to refill the container again. If the resulting solution contains $72\%$ milk, then the amount of water, in litres, that was initially poured into the container was

Video Explanation


 

Question 15: In a group of $250$ students, the percentage of girls was at least $44\%$ and at most $60\%$. The rest of the students were boys. Each student opted for either swimming or running or both. If $50\%$ of the boys and $80\%$ of the girls opted for swimming while $70\%$ of the boys and $60\%$ of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

  1. $\quad 75$ and $90$, respectively

  2. $\quad 72$ and $80$, respectively

  3. $\quad 72$ and $88$, respectively

  4. $\quad 75$ and $96$, respectively

Video Explanation


 

Question 16: The sum of all distinct real values of $x$ that satisfy the equation $10^x + \displaystyle \frac{4}{10^x} = \frac{81}{2}$, is

  1. $\quad 3\log_{10} 2$

  2. $\quad \log_{10} 2$

  3. $\quad 4\log_{10} 2$

  4. $\quad 2\log_{10} 2$

Video Explanation


 

Question 17: A regular octagon $ABCDEFGH$ has sides of length $6$ cm each. Then the area, in sq. cm, of the square $ACEG$ is

  1. $\quad 36(1+\sqrt{2})$

  2. $\quad 72(2+\sqrt{2})$

  3. $\quad 72(1+\sqrt{2})$

  4. $\quad 36(2+\sqrt{2})$

Video Explanation


 

Question 18: For some constant real numbers $p$, $k$, and $a$, consider the following system of linear equations in $x$ and $y$:
$$ px – 4y=2 $$
$$ 3x+ky=a $$
A necessary condition for the system to have no solution for $(x,y)$, is

  1. $\quad ap-6=0$

  2. $\quad kp+12 \neq 0$

  3. $\quad ap+6=0$

  4. $\quad 2a+k \neq 0$

Video Explanation


 

Question 19: Gopi marks a price on a product in order to make $20\%$ profit. Ravi gets $10\%$ discount on this marked price, and thus saves Rs $15$. Then, the profit, in rupees, made by Gopi by selling the product to Ravi, is

  1. $\quad 20$

  2. $\quad 25$

  3. $\quad 15$

  4. $\quad 10$

Video Explanation


 

Question 20: The midpoints of sides $AB$, $BC$, and $AC$ in $\triangle ABC$ are $M$, $N$, and $P$, respectively. The medians drawn from $A$, $B$, and $C$ intersect the line segments $MP$, $MN$ and $NP$ at $X$, $Y$, and $Z$, respectively. If the area of $\triangle ABC$ is $1440$ sq cm, then the area, in sq cm, of $\triangle XYZ$ is

Video Explanation


 

Question 21: The number of all positive integers up to $500$ with non-repeating digits is

Video Explanation


 

Question 22: Sam can complete a job in $20$ days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is

  1. $\quad \displaystyle \frac{3}{20}$

  2. $\quad \displaystyle \frac{3}{10}$

  3. $\quad \displaystyle \frac{1}{5}$

  4. $\quad \displaystyle \frac{1}{20}$

Video Explanation