This page organizes my video explanations to all of the 22 quantitative ability questions that are part of the 2023 CAT exam Slot 3. The entire exam in pdf format can be downloaded from this link: 2023 CAT Quant Slot 3: Quantitative Ability.
Question 1: For some real numbers $a$ and $b$, the system of equations $x+y=4$ and $(a+5)x+(b^2-15)y=8b$ has infinitely many solutions for $x$ and $y$. Then, the maximum possible value of $ab$ is
- $\quad 25$
- $\quad 15$
- $\quad 55$
- $\quad 33$
Question 2: Let $n$ and $m$ be two positive integers such that there are exactly $41$ integers greater than $8^m$ and less than $8^n$, which can be expressed as powers of $2$. Then, the smallest possible value of $n+m$ is
- $\quad 42$
- $\quad 16$
- $\quad 14$
- $\quad 44$
Question 3: If $x$ is a positive real number such that $x^8 + \left(\displaystyle \frac{1}{x}\right)^8 = 47$, then the value of $x^9 + \left(\displaystyle \frac{1}{x}\right)^9$ is
- $\quad 40\sqrt{5}$
- $\quad 30\sqrt{5}$
- $\quad 36\sqrt{5}$
- $\quad 34\sqrt{5}$
Question 4: For a real number $x$, if $\displaystyle \frac{1}{2}$, $\displaystyle \frac{\log_3 {(2^x-9)}}{\log_3 4}$, and $\displaystyle \frac{\log_5 {(2^x+\frac{17}{2})}}{\log_5 4}$ are in an arithmetic progression, then the common difference is
- $\quad \log_4 7$
- $\quad \log_4 {\left(\displaystyle \frac{3}{2}\right)}$
- $\quad \log_4 {\left(\displaystyle \frac{23}{2}\right)}$
- $\quad \log_4 {\left(\displaystyle \frac{7}{2}\right)}$
Question 5: A quadratic equation $x^2+bx+c=0$ has two real roots. If the difference between the reciprocals of the roots is $\displaystyle \frac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $\displaystyle \frac{5}{9}$, then the largest possible value of $b+c$ is
Question 6: The sum of the first two natural numbers, each having $15$ factors (including $1$ and the number itself), is
Question 7: Let $n$ be any natural number such that $5^{n-1} \lt 3^{n+1}$. Then, the least integer value of $m$ that satisfies $3^{n+1} \lt 2^{n+m}$ for each such $n$, is
Question 8: A boat takes $2$ hours to travel downstream a river from port $A$ to port $B$, and $3$ hours to return to port $A$. Another boat takes a total of $6$ hours to travel from port $B$ to port $A$ and return to port $B$. If the speeds of the boats and the river are constant, then the time, in hours taken by the slower boat to travel from port $A$ to port $B$ is
- $\quad 3(\sqrt{5}-1)$
- $\quad 12(\sqrt{5}-2)$
- $\quad 3(3-\sqrt{5})$
- $\quad 3(3+\sqrt{5})$
Question 9: The population of a town in $2020$ was $100000$ . The population decreased by $y\%$ from the year $2020$ to $2021$, and increased by $x\%$ from the year $2021$ to $2022$, where $x$ and $y$ are two natural numbers. If population in $2022$ was greater than the population in $2020$ and the difference between $x$ and $y$ is $10$ , then the lowest possible population of the town in $2021$ was
- $\quad 72000$
- $\quad 73000$
- $\quad 75000$
- $\quad 74000$
Question 10: Anil mixes cocoa with sugar in the ratio $3:2$ to prepare mixture $A$, and coffee with sugar in the ratio $7:3$ to prepare mixture $B$. He combines mixtures $A$ and $B$ in the ratio $2:3$ to make a new mixture $C$. If he mixes $C$ with an equal amount of milk to drink, then the percentage of sugar in this drink will be
- $\quad 17$
- $\quad 16$
- $\quad 24$
- $\quad 21$
Question 11: A merchant purchases a cloth at a rate of Rs.$100$ per meter and receives $5$ cm length of cloth free for every $100$ cm length of cloth purchased by him. He sells the same cloth at a rate of Rs.$110$ per meter but cheats his customers by giving $95$ cm length of cloth for every $100$ cm length of cloth purchased by the customers. If the merchant provides a $5\%$ discount, the resulting profit earned by him is
- $\quad 19.7\%$
- $\quad 16\%$
- $\quad 4.2\%$
- $\quad 15.5\%$
Question 12: There are three persons $A$, $B$ and $C$ in a room. If a person $D$ joins the room, the average weight of the persons in the room reduces by $x$ kg. Instead of $D$, if person $E$ joins the room, the average weight of the persons in the room increases by $2x$ kg. If the weight of $E$ is $12$ kg more than that of $D$, then the value of $x$ is
- $\quad 1.5$
- $\quad 1$
- $\quad 0.5$
- $\quad 2$
Question 13: Rahul, Rakshita and Gurmeet, working together, would have taken more than $7$ days to finish a job. On the other hand, Rahul and Gurmeet, working together would have taken less than $15$ days to finish the job. However, they all worked together for $6$ days, followed by Rakshita, who worked alone for $3$ more days to finish the job. If Rakshita had worked alone on the job then the number of days she would have taken to finish the job, cannot be
- $\quad 20$
- $\quad 21$
- $\quad 17$
- $\quad 16$
Question 14: 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
Question 15: The number of coins collected per week by two coin-collectors $A$ and $B$ are in the ratio $3 : 4$. If the total number of coins collected by $A$ in $5$ weeks is a multiple of $7$, and the total number of coins collected by $B$ in $3$ weeks is a multiple of $24$, then the
minimum possible number of coins collected by $A$ in one week is
Question 16: Gautam and Suhani, working together, can finish a job in $20$ days. If Gautam does only $60\%$ of his usual work on a day, Suhani must do $150\%$ of her usual work on that day to exactly make up for it. Then, the number of days required by the faster worker to complete the job working alone is
Question 17: Let $\triangle ABC$ be an isosceles triangle such that $AB$ and $AC$ are of equal length. $AD$ is the altitude from $A$ on $BC$ and $BE$ is the altitude from $B$ on $AC$. If $AD$ and $BE$ intersect at $O$ such that $\angle AOB=105^{\circ}$, then $\displaystyle \frac{AD}{BE}$ equals
- $\quad 2 \cos{15^{\circ}}$
- $\quad 2 \sin{15^{\circ}}$
- $\quad \sin{15^{\circ}}$
- $\quad \cos{15^{\circ}}$
Question 18: A rectangle with the largest possible area is drawn inside a semicircle of radius $2$ cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is
- $\quad 2:1$
- $\quad \sqrt{5}:1$
- $\quad 1:1$
- $\quad \sqrt{2}:1$
Question 19: In a regular polygon, any interior angle exceeds the exterior angle by $120$ degrees. Then, the number of diagonals of this polygon is
Question 20: Let $a_n = 46+8n$ and $b_n = 98+4n$ be two sequences for natural numbers $n \leq 100$. Then, the sum of all terms common to both the sequences is
- $\quad 15000$
- $\quad 14602$
- $\quad 14798$
- $\quad 14900$
Question 21: The value of $1 + \left(1+\displaystyle \frac{1}{3}\right) \displaystyle \frac{1}{4} + \left(1+\displaystyle \frac{1}{3}+ \frac{1}{9}\right)\displaystyle \frac{1}{16} + \left(1+\displaystyle \frac{1}{3}+ \frac{1}{9} + \frac{1}{27}\right)\displaystyle \frac{1}{64}+ \cdots $, is
- $\quad \displaystyle \frac{15}{8}$
- $\quad \displaystyle \frac{16}{11}$
- $\quad \displaystyle \frac{27}{12}$
- $\quad \displaystyle \frac{15}{13}$
Question 22: Suppose $f(x,y)$ is a real-valued function such that $f(3x+2y, 2x-5y)=19x$, for all real numbers $x$ and $y$. The value of $x$ for which $f(x,2x)=27$, is