2018 CAT IIM Exam Slot 1: Quantitative Ability

This page organizes my video explanations to all of the 34 quantitative ability questions that are part of the 2018 CAT exam Slot 1. The entire exam in pdf format can be downloaded from this link: 2018 CAT Quant Slot 1: Quantitative Ability.

Question 1: A trader sells $10$ litres of a mixture of paints $A$ and $B$, where the amount of $B$ in the mixture does not exceed that of $A$. The cost of paint $A$ per litre is Rs. $8$ more than that of paint $B$. If the trader sells the entire mixture for Rs. $264$ and makes a profit of $10\%$, then the highest possible cost of paint $B$, in Rs. per litre, is

$20$

$16$

$22$

$26$

Question 2: In a circle with centre $O$ and radius $1$ cm, an arc $AB$ makes an angle $60^\circ$ at $O$. Let $R$ be the region bounded by the radii $OA$, $OB$ and the arc $AB$. If $C$ and $D$ are two points on $OA$ and $OB$, respectively, such that $OC=OD$ and the area of triangle $OCD$ is half that of $R$, then the length of $OC$, in cm, is

$\left(\dfrac{\pi}{4}\right)^{\tfrac{1}{2}}$

$\left(\dfrac{\pi}{6}\right)^{\tfrac{1}{2}}$

$\left(\dfrac{\pi}{4\sqrt{3}}\right)^{\tfrac{1}{2}}$

$\left(\dfrac{\pi}{3\sqrt{3}}\right)^{\tfrac{1}{2}}$

Question 3: If $f(x+2)=f(x)+f(x+1)$ for all positive integers $x$, and $f(11)=91$, $f(15)=617$, then $f(10)$ equals

Question 4: The distance from $A$ to $B$ is $60$ km. Partha and Narayan start from $A$ at the same time and move towards $B$. Partha takes four hours more than Narayan to reach $B$. Moreover, Partha reaches the mid-point of $A$ and $B$ two hours before Narayan reaches $B$. The speed of Partha, in km per hour, is

Question 5: A CAT aspirant appears for a certain number of tests. His average score increases by $1$ if the first $10$ tests are not considered, and decreases by $1$ if the last $10$ tests are not considered. If his average scores for the first $10$ and the last $10$ tests are $20$ and $30$, respectively, then the total number of tests taken by him is

Question 6: Two types of tea, $A$ and $B$, are mixed and then sold at Rs. $40$ per kg. The profit is $10\%$ if $A$ and $B$ are mixed in the ratio $3:2$, and $5\%$ if this ratio is $2:3$. The cost prices, per kg, of $A$ and $B$ are in the ratio

$21:25$

$19:24$

$18:25$

$17:25$

Question 7: A wholesaler bought walnuts and peanuts, the price of walnut per kg being thrice that of peanut per kg. He then sold $8$ kg of peanuts at a profit of $10\%$ and $16$ kg of walnuts at a profit of $20\%$ to a shopkeeper. However, the shopkeeper lost $5$ kg of walnuts and $3$ kg of peanuts in transit. He then mixed the remaining nuts and sold the mixture at Rs. $166$ per kg, thus making an overall profit of $25\%$. At what price, in Rs. per kg, did the wholesaler buy the walnuts?

$84$

$86$

$96$

$98$

Question 8: When they work alone, $B$ needs $25\%$ more time to finish a job than $A$ does. They two finish the job in $13$ days in the following manner: $A$ works alone till half the job is done, then $A$ and $B$ work together for four days, and finally $B$ works alone to complete the remaining $5\%$ of the job. In how many days can $B$ alone finish the entire job?

$16$

$22$

$20$

$18$

Question 9: Given an equilateral triangle $T_1$ with side $24$ cm, a second triangle $T_2$ is formed by joining the midpoints of the sides of $T_1$. Then a third triangle $T_3$ is formed by joining the midpoints of the sides of $T_2$. If this process continues, the sum of the areas, in sq cm, of infinitely many such triangles $T_1,T_2,T_3,\ldots$ will be

$192\sqrt{3}$

$164\sqrt{3}$

$248\sqrt{3}$

$188\sqrt{3}$

Question 10: While multiplying three real numbers, Ashok took one of the numbers as $73$ instead of $37$. As a result, the product went up by $720$. Then the minimum possible value of the sum of squares of the other two numbers is

Question 11: If $x$ is a positive quantity such that $2^{x}=3^{\log_{5}2}$, then $x$ is equal to

$\log_{5}9$

$1+\log_{5}\dfrac{3}{5}$

$1+\log_{3}\dfrac{5}{3}$

$\log_{5}8$

Question 12: If $\log_{12}81=p$, then $3\!\left(\dfrac{4-p}{4+p}\right)$ is equal to

$\log_{2}8$

$\log_{6}8$

$\log_{4}16$

$\log_{6}16$

Question 13: A right circular cone, of height $12$ ft, stands on its base which has diameter $8$ ft. The tip of the cone is cut off with a plane which is parallel to the base and $9$ ft from the base. With $\pi=\dfrac{22}{7}$, the volume, in cubic ft, of the remaining part of the cone is

Question 14: How many numbers with two or more digits can be formed with the digits $1,2,3,4,5,6,7,8,9$ so that in every such number, each digit is used at most once and the digits appear in ascending order?

Question 15: John borrowed Rs. $2{,}10{,}000$ at $10\%$ per annum, compounded annually, repaid in two equal instalments after one and two years, respectively. Each instalment (in Rs.) is

Question 16: If $u^{2}+(u-2v-1)^{2}=-4v(u+v)$, then the value of $u+3v$ is

$\dfrac{1}{4}$

$\dfrac{1}{2}$

$-\dfrac{1}{4}$

Question 17: Point $P$ lies between points $A$ and $B$ such that the length of $BP$ is thrice that of $AP$. Car $1$ starts from $A$ towards $B$ and car $2$ starts from $B$ towards $A$ simultaneously. Car $2$ reaches $P$ one hour after car $1$ reaches $P$. If the speed of car $2$ is half that of car $1$, then the time, in minutes, taken by car $1$ in reaching $P$ from $A$ is

Question 18: Let $ABCD$ be a rectangle inscribed in a circle of radius $13$ cm. Which one of the following pairs can represent, in cm, the possible length and breadth of $ABCD$?

$25, 10$

$24, 12$

$25, 9$

$24, 10$

Question 19: In an examination, the maximum possible score is $N$ while the pass mark is $45\%$ of $N$. A candidate obtains $36$ marks, but falls short of the pass mark by $68\%$. Which one of the following is then correct?

$N \le 200$

$243 \le N \le 252$

$N \ge 253$

$201 \le N \le 242$

Question 20: Let $x, y, z$ be three positive real numbers in a geometric progression such that $x < y < z$. If $5x, 16y,$ and $12z$ are in an arithmetic progression, then the common ratio of the geometric progression is

$\dfrac{1}{6}$

$\dfrac{3}{6}$

$\dfrac{3}{2}$

$\dfrac{5}{2}$

Question 21: The number of integers $x$ such that $0.25 \lt 2^{x} \lt 200$, and $2^{x} + 2$ is perfectly divisible by either $3$ or $4$, is

Question 22: Each of $74$ students in a class studies at least one of the three subjects $H, E,$ and $P$. Ten students study all three subjects, while twenty study $H$ and $E$, but not $P$. Every student who studies $P$ also studies $H$ or $E$ or both. If the number of students studying $H$ equals that studying $E$, then the number of students studying $H$ is

Question 23: Train $T$ leaves station $X$ for station $Y$ at $3$ pm. Train $S$, traveling at three-quarters of the speed of $T$, leaves $Y$ for $X$ at $4$ pm. The two trains pass each other at a station $Z$, where the distance between $X$ and $Z$ is three-fifths of that between $X$ and $Y$. How many hours does train $T$ take for its journey from $X$ to $Y$?

Question 24: Points $E, F, G, H$ lie on the sides $AB, BC, CD,$ and $DA,$ respectively, of a square $ABCD$. If $EFGH$ is also a square whose area is $62.5\%$ of that of $ABCD$ and $CG$ is longer than $EB$, then the ratio of length of $EB$ to that of $CG$ is

$1:3$

$4:9$

$2:5$

$3:8$

Question 25: Given that $x^{2018}y^{2017}=\dfrac{1}{2}$ and $x^{2016}y^{2019}=8$, the value of $x^{2}+y^{3}$ is

$\dfrac{37}{4}$

$\dfrac{31}{4}$

$\dfrac{35}{4}$

$\dfrac{33}{4}$

Question 26: Raju and Lalitha originally had marbles in the ratio $4:9$. Then Lalitha gave some of her marbles to Raju. As a result, the ratio of the number of marbles with Raju to that with Lalitha became $5:6$. What fraction of her original number of marbles was given by Lalitha to Raju?

$\dfrac{1}{4}$

$\dfrac{1}{5}$

$\dfrac{6}{19}$

$\dfrac{7}{33}$

Question 27: If $\log_{2}(5+\log_{3}a)=3$ and $\log_{5}(4a+12+\log_{2}b)=3$, then $a+b$ is equal to

$32$

$59$

$67$

$40$

Question 28: Humans and robots can both perform a job but at different efficiencies. Fifteen humans and five robots working together take $30$ days to finish the job, whereas five humans and fifteen robots working together take $60$ days to finish it. How many days will fifteen humans working together (without any robot) take to finish it?

$40$

$32$

$36$

$45$

Question 29: In a parallelogram $ABCD$ of area $72$ sq. cm, the sides $CD$ and $AD$ have lengths $9$ cm and $16$ cm, respectively. Let $P$ be a point on $CD$ such that $AP$ is perpendicular to $CD$. Then the area, in sq. cm, of triangle $APD$ is

$18\sqrt{3}$

$24\sqrt{3}$

$32\sqrt{3}$

$12\sqrt{3}$

Question 30: In a circle, two parallel chords on the same side of a diameter have lengths $4$ cm and $6$ cm. If the distance between these chords is $1$ cm, then the radius of the circle, in cm, is

$\sqrt{13}$

$\sqrt{14}$

$\sqrt{11}$

$\sqrt{12}$

Question 31: A tank is fitted with pipes, some filling it and the rest draining it. All filling pipes fill at the same rate, and all draining pipes drain at the same rate. The empty tank gets completely filled in $6$ hours when $6$ filling and $5$ draining pipes are on, but this time becomes $60$ hours when $5$ filling and $6$ draining pipes are on. In how many hours will the empty tank get completely filled when one draining and two filling pipes are on?

Question 32: If among $200$ students, $105$ like pizza and $134$ like burger, then the number of students who like only burger can possibly be

$26$

$23$

$96$

$93$

Question 33: Let $f(x)=\min\{2x^{2}, 52-5x\}$, where $x$ is any positive real number. Then the maximum possible value of $f(x)$ is

Question 34: In an apartment complex, the number of people aged $51$ years and above is $30$ and there are at most $39$ people whose ages are below $51$ years. The average age of all the people in the apartment complex is $38$ years. What is the largest possible average age, in years, of the people whose ages are below $51$ years?

$25$

$26$

$27$

$28$

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