This page organizes my video explanations to all of the 22 quantitative ability questions that are part of the 2022 CAT exam Slot 2. The entire exam in pdf format can be downloaded from this link: 2022 CAT Quant Slot 2: Quantitative Ability.
Question 1: In triangle $ABC$, altitudes $AD$ and $BE$ are drawn to the corresponding bases. If $\angle BAC = 45^{\circ}$ and $\angle ABC = \theta$, then $\displaystyle \frac{AD}{BE}$ equals
- $\quad \sqrt{2} \sin{\theta}$
- $\quad \sqrt{2} \cos{\theta}$
- $\quad \displaystyle \frac{(\sin{\theta}+\cos{\theta})}{\sqrt{2}}$
- $\quad 1$
Question 2: Working alone, the times taken by Anu, Tanu, and Manu to complete any job are in the ratio $5:8:10$. They accept a job which they can finish in $4$ days if they all work together for $8$ hours per day. However, Anu and Tanu work together for the first $6$ days, working $6$ hours $40$ minutes per day. Then, the number of hours that Manu will take to complete the remaining job working alone is
Question 3: Regular polygons $A$ and $B$ have number of sides in the ratio $1:2$ and interior angles in the ratio $3:4$. Then the number of sides of $B$ equals
Question 4: If $a$ and $b$ are non-negative real numbers such that $a+2b=6$, then the average of the maximum and minimum possible values of $(a+b)$ is
- $\quad 4$
- $\quad 4.5$
- $\quad 3.5$
- $\quad 3$
Question 5: Manu earns ₹$4000$ per month and wants to save an average of ₹$550$ per month in a year. In the first nine months, his monthly expenses was ₹$3500$, and he foresees that, tenth month onward, his monthly expense will increase to ₹$3700$. In order to meet his yearly savings target, his monthly earnings, in rupees, from the tenth month onward should be
- $\quad 4200$
- $\quad 4400$
- $\quad 4300$
- $\quad 4350$
Question 6: There are two containers of the same volume, first container half-filled with sugar syrup and the second container half-filled with milk. Half the content of the first container is transferred to the second container, and then the half of this mixture is transferred back to the first container. Next, half the content of the first container is transferred back to the second container. Then the ratio of sugar syrup and milk in the second container is
- $\quad 5:6$
- $\quad 5:4$
- $\quad 6:5$
- $\quad 4:5$
Question 7: On day one, there are $100$ particles in a laboratory experiment. On day $n$, where $n \geq 2$, one out of every $n$ particles produces another particle. If the total number of particles in the laboratory experiment increases to $1000$ on day $m$, then $m$ equals
- $\quad 19$
- $\quad 16$
- $\quad 17$
- $\quad 18$
Question 8: The average of a non-decreasing sequence of $N$ numbers $a_1,a_2,\cdots, a_N$ is $300$. If $a_1$ is replaced by $6a_1$, the new average becomes $400$. Then, the number of possible values of $a_1$ is
Question 9: Let $r$ and $c$ be real numbers. If $r$ and $-r$ are roots of $5x^3+cx^2-10x+9=0$, then $c$ equals
- $\quad \displaystyle -\frac{9}{2}$
- $\quad \displaystyle \frac{9}{2}$
- $\quad -4$
- $\quad 4$
Question 10: Suppose for all integers $x$, there are two functions $f$ and $g$ such that $f(x)+f(x-1)-1=0$ and $g(x)=x^2$. If $f(x^2-x)=5$, then the value of the sum $f(g(5))+g(f(5))$ is
Question 11: In an election, there were four candidates and $80\%$ of the registered voters casted their votes. One of the candidates received $30\%$ of the casted votes while the other three candidates received the remaining casted votes in the proportion $1 : 2 : 3$. If the winner of the election received $2512$ votes more than the candidate with the second highest votes, then the number of registered voters was
- $\quad 40192$
- $\quad 60288$
- $\quad 50240$
- $\quad 62800$
Question 12: The number of integers greater than $2000$ that can be formed with the digits $0, 1, 2, 3, 4, 5$, using each digit at most once, is
- $\quad 1440$
- $\quad 1200$
- $\quad 1420$
- $\quad 1480$
Question 13: For some natural number $n$, assume that $(15,000)!$ is divisible by $(n!)!$. The largest possible value of $n$ is
- $\quad 5$
- $\quad 7$
- $\quad 4$
- $\quad 6$
Question 14: The number of distinct integer values of $n$ satisfying $\displaystyle \frac{4 – \log_{2} n}{3 – \log_{4} n} \lt 0$, is
Question 15: In an examination, there were $75$ questions. $3$ marks were awarded for each correct answer, $1$ mark was deducted for each wrong answer and $1$ mark was awarded for each unattempted question. Rayan scored a total of $97$ marks in the examination. If the number of unattempted questions was higher than the number of attempted questions, then the maximum number of correct answers that Rayan could have given in the examination is
Question 16: Five students, including Amit, appear for an examination in which possible marks are integers between $0$ and $50$, both inclusive. The average marks for all the students is $38$ and exactly three students got more than $32$. If no two students got the same marks and Amit got the least marks among the five students, then the difference between the highest and lowest possible marks of Amit is
- $\quad 21$
- $\quad 24$
- $\quad 20$
- $\quad 22$
Question 17: The number of integer solutions of the equation $(x^2-10)^{(x^2-3x-10)} = 1$ is
Question 18: Mr. Pinto invests one-fifth of his capital at $6\%$, one-third at $10\%$ and the remaining at $1\%$, each rate being simple interest per annum. Then, the minimum number of years required for the cumulative interest income from these investments to equal or exceed his initial capital is
Question 19: Consider the arithmetic progression $3, 7, 11, \ldots$ and let $A_n$ denote the sum of the first $n$ terms of this progression. Then the value of $\displaystyle \frac{1}{25} \sum_{n=1}^{25} A_n$ is
- $\quad 404$
- $\quad 442$
- $\quad 455$
- $\quad 415$
Question 20: Let $f(x)$ be a quadratic polynomial in $x$ such that $f(x) \geq 0$ for all real numbers $x$. If $f(2)=0$ and $f(4)=6$, then $f(-2)$ is equal to
- $\quad 12$
- $\quad 36$
- $\quad 24$
- $\quad 6$
Question 21: The length of each side of an equilateral triangle $ABC$ is $3$ cm. Let $D$ be a point on $BC$ such that the area of triangle $ADC$ is half the area of triangle $ABD$. Then the length of $AD$, in cm, is
- $\quad \sqrt{6}$
- $\quad \sqrt{5}$
- $\quad \sqrt{8}$
- $\quad \sqrt{7}$
Question 22: Two ships meet mid-ocean, and then, one ship goes south and the other ship goes west, both travelling at constant speeds. Two hours later, they are $60$ km apart. If the speed of one of the ships is $6$ km per hour more than the other one, then the speed, in km per hour, of the slower ship is
- $\quad 12$
- $\quad 18$
- $\quad 20$
- $\quad 24$
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