2024 CAT Exam Slot 2: 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 2. The entire exam in pdf format can be downloaded from this link: 2024 CAT Quant Slot 2: Quantitative Ability.

Question 1: If $(x+6\sqrt{2})^{\frac{1}{2}} – (x-6\sqrt{2})^{\frac{1}{2}} = 2\sqrt{2}$, then $x$ equals

Video Explanation

Question 2: A bus starts at $9$ am and follows a fixed route every day. One day, it traveled at a constant speed of $60$ km per hour and reached its destination $3.5$ hours later than its scheduled arrival time. Next day, it traveled two-thirds of its route in one-third of its total scheduled travel time, and the remaining part of the route at $40$ km per hour to reach just on time. The scheduled arrival time of the bus is

$\quad 7:30$ pm

$\quad 7:00$ pm

$\quad 10:30$ pm

$\quad 9:00$ pm

Video Explanation

Question 3: All the values of $x$ satisfying the inequality $\displaystyle \frac{1}{x+5} \leq \displaystyle \frac{1}{2x-3}$ are

$\quad -5 \lt x \lt \frac{3}{2}$ or $\frac{3}{2} \lt x \leq 8$

$\quad -5 \lt x \lt \frac{3}{2}$ or $x > \frac{3}{2}$

$\quad x \lt -5$ or $x > \frac{3}{2}$

$\quad x \lt -5$ or $\frac{3}{2} \lt x \leq 8$

Video Explanation

Question 4: When $3^{333}$ is divided by 11, the remainder is

$\quad 1$

$\quad 6$

$\quad 5$

$\quad 10$

Video Explanation

Question 5: If $m$ and $n$ are natural numbers such that $n > 1$, and $m^n = 2^{25} \times 3^{40}$, then $m-n$ equals

$\quad 209942$

$\quad 209947$

$\quad 209932$

$\quad 209937$

Video Explanation
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Question 6: The roots $\alpha$, $\beta$ of the equation $3x^2+\lambda x – 1 = 0$, satisfy $\displaystyle \frac{1}{\alpha^2} + \frac{1}{\beta^2} = 15$. The value of $(\alpha^3+\beta^3)^2$, is

$\quad 16$

$\quad 9$

$\quad 1$

$\quad 4$

Video Explanation

Question 7: If $x$ and $y$ satisfy the equations $|x|+x+y=15$ and $x+|y|-y=20$, then $(x-y)$ equals

$\quad 5$

$\quad 10$

$\quad 20$

$\quad 15$

Video Explanation

Question 8: Anil invests Rs $22000$ for $6$ years in a scheme with $4\%$ interest per annum, compounded half-yearly. Separately, Sunil invests a certain amount in the same scheme for $5$ years, and then reinvests the entire amount he receives at the end of $5$ years, for one year at $10\%$ simple interest. If the amounts received by both at the end of $6$ years are equal, then the initial investment, in rupees, made by Sunil is

$\quad 20640$

$\quad 20808$

$\quad 20860$

$\quad 20480$

Video Explanation

Question 9: A vessel contained a certain amount of a solution of acid and water. When $2$ litres of water was added to it, the new solution had $50\%$ acid concentration. When $15$ litres of acid was further added to this new solution, the final solution had $80\%$ acid concentration. The ratio of water and acid in the original solution was

$\quad 3:5$

$\quad 5:3$

$\quad 3.4:5$

$\quad 4.5:4$

Video Explanation

Question 10: The coordinates of the three vertices of a triangle are: $(1, 2)$, $(7, 2)$, and $(1, 10)$. Then the radius of the incircle of the triangle is

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Question 11: Bina incurs $19\%$ loss when she sells a product at Rs. $4860$ to Shyam, who in turn sells this product to Hari. If Bina would have sold this product to Shyam at the purchase price of Hari, she would have obtained $17\%$ profit. Then, the profit, in rupees, made by Shyam is

Video Explanation

Question 12: Amal and Vimal together can complete a task in $150$ days, while Vimal and Sunil together can complete the same task in $100$ days. Amal starts working on the task and works for $75$ days, then Vimal takes over and works for $135$ days. Finally, Sunil takes over and completes the remaining task in $45$ days. If Amal had started the task alone and worked on all days, Vimal had worked on every second day, and Sunil had worked on every third day, then the number of days required to complete the task would have been

Video Explanation

Question 13: A function $f$ maps the set of natural numbers to whole numbers, such that $f(xy)=f(x)f(y)+f(x)+f(y)$ for all $x$, $y$ and $f(p)=1$ for every prime number $p$. Then, the value of $f(160000)$ is

$\quad 4095$

$\quad 8191$

$\quad 2047$

$\quad 1023$

Video Explanation

Question 14: When Rajesh’s age was same as the present age of Garima, the ratio of their ages was $3 : 2$. When Garima’s age becomes the same as the present age of Rajesh, the ratio of the ages of Rajesh and Garima will become

$\quad 5:4$

$\quad 2.2:1$

$\quad 4:3$

$\quad 3:2$

Video Explanation

Question 15: The sum of the infinite series $\displaystyle \frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right) + \left(\frac{1}{5}\right)^2 \left(\left(\frac{1}{5}\right)^2-\left(\frac{1}{7}\right)^2\right) + \left(\frac{1}{5}\right)^3 \left(\left(\frac{1}{5}\right)^3-\left(\frac{1}{7}\right)^3\right)+ \cdots $ is equal to

$\quad \displaystyle \frac{7}{408}$

$\quad \displaystyle \frac{5}{408}$

$\quad \displaystyle \frac{7}{816}$

$\quad \displaystyle \frac{7}{816}$

Video Explanation

Question 16: A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up $40\%$ of his stock. That day, he sells half of the mangoes, $96$ bananas and $40\%$ of the apples. At the end of the day, he ends up selling $50\%$ of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is

Video Explanation

Question 17: Three circles of equal radii touch(but not cross) each other externally. Two other circles, $X$ and $Y$, are drawn such that both touch(but not cross) each of the three previous circles. If the radius of $X$ is more than that of $Y$, the ratio of the radii of $X$ and $Y$ is

$\quad 4+\sqrt{3}:1$

$\quad 2+\sqrt{3}:1$

$\quad 4+2\sqrt{3}:1$

$\quad 7+4\sqrt{3}:1$

Video Explanation

Question 18: A company has $40$ employees whose names are listed in a certain order. In the year $2022$, the average bonus of the first $30$ employees was Rs. $40000$, of the last $30$ employees was Rs. $60000$, and of the first $10$ and last $10$ employees together was Rs. $50000$. Next year, the average bonus of the first $10$ employees increased by $100\%$, of the last $10$ employees increased by $200\%$ and of the remaining employees was unchanged. Then, the average bonus, in rupees, of all the $40$ employees together in the year $2023$ was

$\quad 90000$

$\quad 95000$

$\quad 85000$

$\quad 80000$

Video Explanation

Question 19: $ABCD$ is a trapezium in which $AB$ is parallel to $CD$. The sides $AD$ and $BC$ when extended, intersect at point $E$. If $AB = 2$ cm, $CD = 1$ cm, and perimeter of $ABCD$ is $6$ cm, then the perimeter, in cm, of $\triangle AEB$ is

$\quad 10$

$\quad 9$

$\quad 8$

$\quad 7$

Video Explanation

Question 20: If $x$ and $y$ are real numbers such that $4x^2+4y^2-4xy-6y+3=0$, then the value of $(4x+5y)$ is

Video Explanation

Question 21: $P$, $Q$, $R$ and $S$ are four towns. One can travel between $P$ and $Q$ along $3$ direct paths, between $Q$ and $S$ along $4$ direct paths, and between $P$ and $R$ along $4$ direct paths. There is no direct path between $P$ and $S$, while there are few direct paths between $Q$ and $R$, and between $R$ and $S$. One can travel from $P$ to $S$ either via $Q$, or via $R$, or via $Q$ followed by $R$, respectively, in exactly $62$ possible ways. One can also travel from $Q$ to $R$ either directly, or via $P$, or via $S$, in exactly $27$ possible ways. Then, the number of direct paths between $Q$ and $R$ is

Video Explanation

Question 22: If $a$, $b$ and $c$ are positive real numbers such that $a > 10 \geq b \geq c$ and $\dfrac{\log_{8} (a+b)}{\log_{2} c} + \dfrac{\log_{27} (a-b)}{\log_{3} c} = \dfrac{2}{3}$, then the greatest possible integer value of $a$ is

Video Explanation