This page organizes my video explanations to all of the 22 quantitative ability questions that are part of the 2022 CAT exam Slot 3. The entire exam in pdf format can be downloaded from this link: 2022 CAT Quant Slot 3: Quantitative Ability.
Question 1: A donation box can receive only cheques of ₹$100$, ₹$250$, and ₹$500$. On one good day, the donation box was found to contain exactly $100$ cheques amounting to a total sum of ₹$15250$. Then, the maximum possible number of cheques of ₹$500$ that the donation box may have contained, is
Question 2: If $c = \displaystyle \frac{16x}{y} + \frac{49y}{x}$ for some non-zero real numbers $x$ and $y$, then $c$ cannot take the value
- $\quad -60$
- $\quad -50$
- $\quad 60$
- $\quad 70$
Question 3: If $(3+2\sqrt{2})$ is a root of the equation $ax^2+bx+c=0$, and $(4+2\sqrt{3})$ is a root of the equation $ay^2+my+n=0$, where $a, b, c, m$ and $n$ are integers, then the value of $\left(\displaystyle \frac{b}{m} + \frac{c-2b}{n}\right)$ is
- $\quad 0$
- $\quad 3$
- $\quad 4$
- $\quad 1$
Question 4: Suppose the medians $BD$ and $CE$ of a triangle $ABC$ intersect at a point $O$. If area of triangle $ABC$ is $108$ sq. cm., the area of the triangle $EOD$, in sq. cm., is
Question 5: Bob can finish a job in $40$ days, if he works alone. Alex is twice as fast as Bob and thrice as fast as Cole in the same job. Suppose Alex and Bob work together on the first day, Bob and Cole work together on the second day, Cole and Alex work together on the third day, and then they continue the work by repeating this three-day roster, with Alex and Bob working together on the fourth day, and so on. Then, the total number of days Alex would have worked when the job gets finished, is
Question 6: A glass contains $500$ cc of milk and a cup contains $500$ cc of water. From the glass, $150$ cc of milk is transferred to the cup and mixed thoroughly. Next, $150$ cc of this mixture is transferred from the cup to the glass. Now, the amount of water in the glass and the amount of milk in the cup are in the ratio
- $\quad 1:1$
- $\quad 10:3$
- $\quad 10:13$
- $\quad 3:10$
Question 7: Consider six distinct natural numbers such that the average of the two smallest numbers is $14$, and the average of the two largest numbers is $28$. Then, the maximum possible value of the average of these six numbers is
- $\quad 22.5$
- $\quad 23$
- $\quad 23.5$
- $\quad 24$
Question 8: Let $r$ be a real number and
$$f(x) = 2x-r \quad \textrm{if $x \geq r$}$$
$$f(x)=r \quad \textrm{if $x \lt r$} $$
Then, the equation $f(x) = f(f(x))$ holds for all real values of $x$ where
- $\quad x \leq r$
- $\quad x>r$
- $\quad x \geq r$
- $\quad x \neq r$
Question 9: Two ships are approaching a port along straight routes at constant speeds. Initially, the two ships and the port formed an equilateral triangle with sides of length $24$ km. When the slower ship travelled $8$ km, the triangle formed by the new positions of the two ships and the port became right-angled. When the faster ship reaches the port, the distance, in km, between the other ship and the port will be
- $\quad 4$
- $\quad 6$
- $\quad 12$
- $\quad 8$
Question 10: Nitu has an initial capital of ₹$20,000$. Out of this, she invests ₹$8,000$ at $5.5\%$ in bank A, ₹$5,000$ at $5.6\%$ in bank B and the remaining amount at $x\%$ in bank $C$, each rate being simple interest per annum. Her combined annual interest income from these investments is equal to $5\%$ of the initial capital. If she had invested her entire initial capital in bank $C$ alone, then her annual interest income, in rupees, would have been
- $\quad 900$
- $\quad 800$
- $\quad 1000$
- $\quad 700$
Question 11: The minimum possible value of $\displaystyle \frac{x^2-6x+10}{3-x}$, for $x \lt 3$, is
- $\quad -2$
- $\quad 2$
- $\quad \displaystyle \frac{1}{2}$
- $\quad \displaystyle -\frac{1}{2}$
Question 12: In an examination, the average marks of students in sections $A$ and $B$ are $32$ and $60$, respectively. The number of students in section $A$ is $10$ less than that in section $B$. If the average marks of all the students across both the sections combined is an integer, then the difference between the maximum and minimum possible number of students in section $A$ is
Question 13: If $\displaystyle \left(\sqrt{\frac{7}{5}}\right)^{3x-y}=\frac{875}{2401}$ and $\displaystyle \left(\frac{4 a}{b}\right)^{6 x-y}=\left(\frac{2 a}{b}\right)^{y-6 x}$, for all non-zero real values of $a$ and $b$, then the value of $x+y$ is
Question 14: A group of $N$ people worked on a project. They finished $35\%$ of the project by working $7$ hours a day for $10$ days. Thereafter, $10$ people left the group and the remaining people finished the rest of the project in $14$ days by working $10$ hours a day. Then the value of $N$ is
- $\quad 150$
- $\quad 36$
- $\quad 140$
- $\quad 23$
Question 15: Moody takes $30$ seconds to finish riding an escalator if he walks on it at his normal speed in the same direction. He takes $20$ seconds to finish riding the escalator if he walks at twice his normal speed in the same direction. If Moody decides to stand still on the escalator, then the time, in seconds, needed to finish riding the escalator is
Question 16: In a triangle $ABC$, $AB=AC=8$ cm. A circle drawn with $BC$ as diameter passes through $A$. Another circle drawn with center at $A$ passes through $B$ and $C$. Then the area, in sq. cm, of the overlapping region between the two circles is
- $\quad 16(\pi-1)$
- $\quad 32\pi$
- $\quad 32(\pi-1)$
- $\quad 16\pi$
Question 17: Suppose $k$ is any integer such that the equation $2x^2+kx+5=0$ has no real roots and the equation $x^2+(k-5)x+1=0$ has two distinct real roots for $x$. Then, the number of possible values of $k$ is
- $\quad 7$
- $\quad 9$
- $\quad 8$
- $\quad 13$
Question 18: The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in $1421$, including itself, is
- $\quad 2442$
- $\quad 3333$
- $\quad 2592$
- $\quad 2222$
Question 19: The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length $1$ cm, $2$ cm and $4$ cm, then the total number of possible lengths of the fourth side is
- $\quad 5$
- $\quad 4$
- $\quad 3$
- $\quad 6$
Question 20: Two cars travel from different locations at constant speeds. To meet each other after starting at the same time, they take $1.5$ hours if they travel towards each other, but $10.5$ hours if they travel in the same direction. If the speed of the slower car is $60$ km/hr, then the distance traveled, in km, by the slower car when it meets the other car while traveling towards each other, is
- $\quad 100$
- $\quad 90$
- $\quad 150$
- $\quad 120$
Question 21: A school has less than $5000$ students and if the students are divided equally into teams of either $9$ or $10$ or $12$ or $25$ each, exactly $4$ are always left out. However, if they are divided into teams of $11$ each, no one is left out. The maximum number of teams of $12$ each that can be formed out of the students in the school is
Question 22: The average of all $3$-digit terms in the arithmetic progression $38, 55, 72, \ldots ,$ is
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