2019 CAT IIM Exam Slot 2: Quantitative Ability

This page organizes my video explanations to all of the 34 quantitative ability questions that are part of the 2019 CAT exam Slot 2. The entire exam in pdf format can be downloaded from this link: 2019 CAT Quant Slot 2: Quantitative Ability.

Question 1: The real root of the equation $2^{6x} + 2^{3x+2} – 21 = 0$ is:

$\dfrac{\log_{2} 3}{3}$

$\log_{2} 9$

$\dfrac{\log_{2} 7}{3}$

$\log_{2} 27$

Question 2: The average of $30$ integers is $5$. Among these $30$ integers, there are exactly $20$ which do not exceed $5$. What is the highest possible value of the average of these $20$ integers?

$4.5$

$3.5$

Question 3: Let $a,b,x,y$ be real numbers such that $a^{2}+b^{2}=25$, $x^{2}+y^{2}=169$, and $ax+by=65$. If $k=ay-bx$, then

$k=0$

$k>\dfrac{5}{13}$

$k=\dfrac{5}{13}$

$0 \lt k\le\dfrac{5}{13}$

Question 4: In a triangle $ABC$, medians $AD$ and $BE$ are perpendicular to each other, and have lengths $12$ cm and $9$ cm, respectively. Then, the area of triangle $ABC$, in sq cm, is

$80$

$68$

$72$

$78$

Question 5: Let $a_1,a_2,\ldots$ satisfy $a_1-a_2+a_3-a_4+\cdots+(-1)^{n-1}a_n=n$ for $n\ge1$. Then $a_{51}+a_{52}+\cdots+a_{1023}$ equals

$-1$

$10$

Question 6: How many factors of $2^{4}\!\times\!3^{5}\!\times\!10^{4}$ are perfect squares greater than 1?

Question 7: Two circles, each of radius $4$ cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

$\dfrac{\pi}{3}$

$\dfrac{1}{\sqrt{2}}$

$\sqrt{2}$

Question 8: What is the largest positive integer such that $\dfrac{n^2 + 7n + 12}{n^2 – n – 12}$ is also a positive integer?

$16$

$12$

Question 9: In $2010$, a library contained a total of $11500$ books in two categories — fiction and non-fiction. In $2015$, the library contained a total of $12760$ books in these two categories. During this period, there was $10\%$ increase in the fiction category while there was $12\%$ increase in the non-fiction category. How many fiction books were in the library in $2015$??

$6600$

$6160$

$6000$

$5500$

Question 10: Let $f$ satisfy $f(mn)=f(m)f(n)$ for all positive integers $m,n$. If $f(1),f(2),f(3)$ are positive integers, $f(1) \lt f(2)$ and $f(24)=54$, then $f(18)$ equals:

Question 11: Let $A$ and $B$ be two regular polygons having $a$ and $b$ sides, respectively. If $b = 2a$ and each interior angle of $B$ is $\dfrac{3}{2}$ times each interior angle of $A$, then each interior angle, in degrees, of a regular polygon with $a + b$ sides is

Question 12: A cyclist leaves $A$ at $10$ a.m. and reaches $B$ at $11$ a.m. Starting from $10{:}01$ a.m., every minute a motorcycle leaves $A$ and moves towards $B$. Forty-five such motorcycles reach $B$ by $11$ a.m. All motorcycles have the same speed. If the cyclist had doubled his speed, how many motorcycles would have reached $B$ by the time the cyclist reached $B$?

$22$

$20$

$15$

$23$

Question 13: Let $A$ be a real number. Then the roots of the equation $x^2 – 4x – \log_2 A = 0$ are real and distinct if and only if

$A<\dfrac{1}{16}$

$A>\dfrac{1}{8}$

$A>\dfrac{1}{16}$

$A<\dfrac{1}{8}$

Question 14: John jogs on track $A$ at $6$ kmph and Mary jogs on track $B$ at $7.5$ kmph. The total length of tracks $A$ and $B$ is $325$ metres. While John makes $9$ rounds of track $A$, Mary makes $5$ rounds of track $B$. In how many seconds will Mary make one round of track $A$?

Question 15: Anil alone can do a job in $20$ days while Sunil alone can do it in $40$ days. Anil starts the job, and after $3$ days, Sunil joins him. Again, after a few more days, Bimal joins them and they together finish the job. If Bimal has done $10\%$ of the job, then in how many days was the job done?

$13$

$12$

$15$

$14$

Question 16: In an examination, Rama’s score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by $6$. The revised scores of Anjali, Mohan, and Rama were in the ratio $11:10:3$. Then Anjali’s score exceeded Rama’s score by

$26$

$32$

$24$

$35$

Question 17: In an examination, the score of $A$ was $10\%$ less than that of $B$, the score of $B$ was $25\%$ more than that of $C$, and the score of $C$ was $20\%$ less than that of $D$. If $A$ scored $72$, then the score of $D$ was

Question 18: The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being $20$ cm. The vertical height of the pyramid, in cm, is

$10\sqrt{2}$

$8\sqrt{3}$

$12$

$5\sqrt{5}$

Question 19: If $x$ is a real number, then $\sqrt{\log_{e}\!\dfrac{4x-x^{2}}{3}}$ is real number if and only if

$-3\le x\le3$

$1\le x\le2$

$1\le x\le3$

$-1\le x\le3$

Question 20: Let $\triangle ABC$ be a right-angled triangle with hypotenuse $BC$ of length $20$ cm. If $AP$ is perpendicular on $BC$, then the maximum possible length of $AP$, in cm, is

$10$

$8\sqrt{2}$

$6\sqrt{2}$

Question 21: Two ants $A$ and $B$ start from a point $P$ on a circle at the same time, with $A$ moving clockwise and $B$ moving anticlockwise. They meet for the first time at $10{:}00$ am when $A$ has covered $60\%$ of the track. If $A$ returns to $P$ at $10{:}12$ am, then $B$ returns to $P$ at

$10{:}27$ am

$10{:}25$ am

$10{:}45$ am

$10{:}18$ am

Question 22: How many pairs $(m,n)$ of positive integers satisfy the equation ${m^2 + 105 = n^2}$?

Question 23: The salaries of Ramesh, Ganesh and Rajesh were in the ratio $6:5:7$ in $2010$, and in the ratio $3:4:3$ in $2015$. If Ramesh’s salary increased by $25\%$ during $2010$–$2015$, then the percentage increase in Rajesh’s salary during this period is closest to

$10$

Question 24: A man makes complete use of $405$ cc of iron, $783$ cc of aluminium, and $351$ cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius $3$ cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

$1044(4+\pi)$

$8464\pi$

$928\pi$

$1026(1+\pi)$

Question 25: The quadratic equation $x^2 + bx + c = 0$ has two roots $4a$ and $3a$, where $a$ is an integer. Which of the following is a possible value of $b^2 + c$?

$3721$

$549$

$361$

$427$

Question 26: In a six-digit number, the sixth (rightmost) digit is the sum of the first three digits, the fifth digit is the sum of the first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of the fifth and sixth digits. Then, the largest possible value of the fourth digit is

Question 27: Mukesh purchased $10$ bicycles in $2017$, all at the same price. He sold six of these at a profit of $25\%$ and the remaining four at a loss of $25\%$. If he made a total profit of $\text{Rs. }2000$, then his purchase price of a bicycle, in rupees, was

$2000$

$6000$

$8000$

$4000$

Question 28: The number of common terms in the two sequences $15,19,23,\ldots,415$ and $14,19,24,\ldots,464$ is

$20$

$18$

$21$

$19$

Question 29: If $(2n + 1) + (2n + 3) + (2n + 5) + \ldots + (2n + 47) = 5280$, then what is the value of $1 + 2 + 3 + \ldots + n$?

Question 30: The strength of a salt solution is $p\%$ if $100$ ml of the solution contains $p$ grams of salt. Each of three vessels $A, B, C$ contains $500$ ml of salt solution of strengths $10\%$, $22\%$, and $32\%$, respectively. Now, $100$ ml of the solution in vessel A is transferred to vessel B. Then, $100$ ml of the solution in vessel B is transferred to vessel C. Finally, $100$ ml of the solution in vessel C is transferred to vessel A. The strength, in percentage, of the resulting solution in vessel A is

$15$

$12$

$13$

$14$

Question 31: If $5^x – 3^y = 13438$ and $5^{x – 1} + 3^{y + 1} = 9686$, then $x + y$ equals

Question 32: Amal invests $\text{Rs. }12000$ at $8\%$ interest, compounded annually, and $\text{Rs. }10000$ at $6\%$ interest, compounded semi-annually$, $ both investments being for one year. Bimal invests his money at $7.5\%$ simple interest for one year. If Amal and Bimal get the same amount of interest, then the amount, in rupees, invested by Bimal is

Question 33: A shopkeeper sells two tables, each procured at cost price $p$, to Amal and Asim at a profit of $20\%$ and at a loss of $20\%$, respectively. Amal sells his table to Bimal at a profit of $30\%$, while Asim sells his table to Barun at a loss of $30\%$. If the amounts paid by Bimal and Barun are $x$ and $y$, respectively, then $\dfrac{x – y}{p}$ equals

$1.2$

$0.7$

$0.5$

Question 34: John gets $\text{Rs. }57$ per hour of regular work and $\text{Rs. }114$ per hour of overtime work. He works altogether $172$ hours and his income from overtime hours is $15\%$ of his income from regular hours. Then, for how many hours did he work overtime?

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