2017 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 2017 CAT exam Slot 2. The entire exam in pdf format can be downloaded from this link: 2017 CAT Quant Slot 2: Quantitative Ability.

Question 1: The numbers $1, 2, \ldots, 9$ are arranged in a $3 \times 3$ square grid in such a way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value. If the top left and the top right entries of the grid are $6$ and $2$, respectively, then the bottom middle entry is:

Question 2: In a $10$ km race, $A$, $B$, and $C$, each running at uniform speed, get the gold, silver, and bronze medals, respectively. If $A$ beats $B$ by $1$ km and $B$ beats $C$ by $1$ km, then by how many metres does $A$ beat $C$?

Question 3: Bottle $1$ contains a mixture of milk and water in the ratio $7 : 2$ and Bottle $2$ contains a mixture of milk and water in the ratio $9 : 4$. In what ratio of volumes should the liquids in Bottle $1$ and Bottle $2$ be combined to obtain a mixture of milk and water in the ratio $3 : 1$?

$27 : 14$

$27 : 13$

$27 : 16$

$27 : 18$

Question 4: Arun drove from home to his hostel at $60$ miles per hour. While returning home he drove half way along the same route at a speed of $25$ miles per hour and then took a bypass road which increased his driving distance by $5$ miles, but allowed him to drive at $50$ miles per hour along this bypass road. If his return journey took $30$ minutes more than his onward journey, then the total distance travelled by him is:

$55$ miles

$60$ miles

$65$ miles

$70$ miles

Question 5: Out of the shirts produced in a factory, $15\%$ are defective, while $20\%$ of the rest are sold in the domestic market. If the remaining $8840$ shirts are left for export, then the number of shirts produced in the factory is:

$13600$

$13000$

$13400$

$14000$

Question 6: The average height of $22$ toddlers increases by $2$ inches when two of them leave this group. If the average height of these two toddlers is one-third the average height of the original $22$, then the average height (in inches) of the remaining $20$ toddlers is:

$30$

$28$

$32$

$26$

Question 7: The manufacturer of a table sells it to a wholesale dealer at a profit of $10\%$. The wholesale dealer sells the table to a retailer at a profit of $30\%$. Finally, the retailer sells it to a customer at a profit of $50\%$. If the customer pays $\text{Rs. }4290$ for the table, then its manufacturing cost (in $\text{Rs.}$) is:

$1500$

$2000$

$2500$

$3000$

Question 8: A tank has an inlet pipe and an outlet pipe. If the outlet pipe is closed, then the inlet pipe fills the empty tank in $8$ hours. If the outlet pipe is open, then the inlet pipe fills the empty tank in $10$ hours. If only the outlet pipe is open, then in how many hours will the full tank become half-full?

$20$

$30$

$40$

$45$

Question 9: Mayank buys some candies for $\text{Rs. }15$ a dozen and an equal number of different candies for $\text{Rs. }12$ a dozen. He sells all for $\text{Rs. }16.50$ a dozen and makes a profit of $\text{Rs. }150$. How many dozens of candies did he buy altogether?

$50$

$30$

$25$

$45$

Question 10: In a village, the production of food grains increased by $40\%$ and the per capita production of food grains increased by $27\%$ during a certain period. The percentage by which the population of the village increased during the same period is nearest to:

$16$

$13$

$10$

Question 11: If $a$, $b$, and $c$ are three positive integers such that $a$ and $b$ are in the ratio $3 : 4$ while $b$ and $c$ are in the ratio $2 : 1$, then which one of the following is a possible value of $(a + b + c)$?

$201$

$205$

$207$

$210$

Question 12: A motorbike leaves point $A$ at $1$ pm and moves towards point $B$ at a uniform speed. A car leaves point $B$ at $2$ pm and moves towards point $A$ at a uniform speed which is double that of the motorbike. They meet at $3{:}40$ pm at a point which is $168$ km away from $A$. What is the distance (in km) between $A$ and $B$?

$364$

$378$

$380$

$388$

Question 13: Amal can complete a job in $10$ days and Bimal can complete it in $8$ days. Amal, Bimal and Kamal together complete the job in $4$ days and are paid a total amount of $\text{Rs. }1000$ as remuneration. If this amount is shared by them in proportion to their work, then Kamal’s share (in $\text{Rs.}$) is:

$100$

$200$

$300$

$400$

Question 14: Consider three mixtures — the first having water and liquid $A$ in the ratio $1 : 2$, the second having water and liquid $B$ in the ratio $1 : 3$, and the third having water and liquid $C$ in the ratio $1 : 4$. These three mixtures of $A$, $B$, and $C$, respectively, are further mixed in the proportion $4 : 3 : 2$. Then the resulting mixture has:

$\ \textrm{The same amount of water and liquid }B$

$\ \textrm{The same amount of liquids }B\ \textrm{and }C$

$\ \textrm{More water than liquid }B$

$\ \textrm{More water than liquid }A$

Question 15: Let $ABCDEF$ be a regular hexagon with each side of length $1$ cm. The area (in sq cm) of a square with $AC$ as one side is:

$3\sqrt{2}$

$\sqrt{3}$

Question 16: The base of a vertical pillar with uniform cross-section is a trapezium whose parallel sides are of lengths $10$ cm and $20$ cm while the other two sides are of equal length. The perpendicular distance between the parallel sides of the trapezium is $12$ cm. If the height of the pillar is $20$ cm, then the total area (in sq cm) of all six surfaces of the pillar is:

$1300$

$1340$

$1480$

$1520$

Question 17: The points $(2, 5)$ and $(6, 3)$ are two endpoints of a diagonal of a rectangle. If the other diagonal has the equation $y = 3x + c$, then $c$ is:

$-5$

$-6$

$-7$

$-8$

Question 18: $ABCD$ is a quadrilateral inscribed in a circle with centre $O$. If $\angle COD = 120^\circ$ and $\angle BAC = 30^\circ$, then the value of $\angle BCD$ (in degrees) is:

Question 19: If three sides of a rectangular park have a total length of $400$ ft, then the area of the park is maximum when the length (in ft) of its longer side is:

Question 20: Let $P$ be an interior point of a right-angled isosceles triangle $ABC$ with hypotenuse $AB$. If the perpendicular distance of $P$ from each of $AB$, $BC$, and $CA$ is $4(\sqrt{2}-1)$ cm, then the area (in sq cm) of the triangle $ABC$ is:

Question 21: If the product of three consecutive positive integers is $15600$, then the sum of the squares of these integers is:

$1777$

$1785$

$1875$

$1877$

Question 22: If $x$ is a real number such that $\log_{3} 5 = \log_{5}(2 + x)$, then which of the following is true?

$0 < x < 3$

$23 < x < 30$

$x > 30$

$3 < x < 23$

Question 23: Let $f(x)=x^{2}$ and $g(x)=2^{x}$, for all real $x$. Then the value of $f\!\big(f(g(x)) + g(f(x))\big)$ at $x=1$ is:

$16$

$18$

$36$

$40$

Question 24: The minimum possible value of the sum of the squares of the roots of the equation $x^{2}+(a+3)x-(a+5)=0$ is:

Question 25: If $9^{x-\tfrac{1}{2}} – 2^{2x-2} = 4^{x} – 3^{2x-3}$, then $x$ is:

$\dfrac{3}{2}$

$\dfrac{2}{5}$

$\dfrac{3}{4}$

$\dfrac{4}{9}$

Question 26: If $\log(2^{a}3^{b}5^{c})$ is the arithmetic mean of $\log(2^{2}3^{3}5)$, $\log(2^{6}3^{1}5^{7})$, and $\log(2^{1}3^{2}5^{4})$, then $a$ equals:

Question 27: Let $a_{1},a_{2},a_{3},a_{4},a_{5}$ be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with $2a_{3}$. If the sum of the numbers in the new sequence is $450$, then $a_{5}$ is:

Question 28: How many different pairs $(a,b)$ of positive integers are there such that $a \le b$ and $\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{9}$?

Question 29: In how many ways can $8$ identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least $1$ pen, Bimal at least $2$ pens, and Kamal at least $3$ pens?

Question 30: How many four-digit numbers, which are divisible by $6$, can be formed using the digits $\{0,2,3,4,6\}$, such that no digit is used more than once and $0$ does not occur in the left-most position?

Question 31: If $f(ab)=f(a)f(b)$ for all positive integers $a$ and $b$, then the largest possible value of $f(1)$ is:

Question 32: Let $f(x)=2x-5$ and $g(x)=7-2x$. Then $|f(x)+g(x)|=|f(x)|+|g(x)|$ if and only if:

$\dfrac{5}{2} < x < \dfrac{7}{2}$

$x \le \dfrac{5}{2}$ or $x \ge \dfrac{7}{2}$

$x < \dfrac{5}{2}$ or $x \ge \dfrac{7}{2}$

$\dfrac{5}{2} \le x \le \dfrac{7}{2}$

Question 33: An infinite geometric progression $a_{1},a_{2},a_{3},\ldots$ has the property that $a_{n}=3(a_{n+1}+a_{n+2}+\ldots)$ for every $n\ge1$. If the sum $a_{1}+a_{2}+a_{3}+\ldots=32$, then $a_{5}$ is:

$\dfrac{1}{32}$

$\dfrac{2}{32}$

$\dfrac{3}{32}$

$\dfrac{4}{32}$

Question 34: If $a_{1}=\dfrac{1}{2\times5}$, $a_{2}=\dfrac{1}{5\times8}$, $a_{3}=\dfrac{1}{8\times11}$, $\ldots$, then $a_{1}+a_{2}+a_{3}+\ldots+a_{100}$ is:

$\dfrac{25}{151}$

$\dfrac{1}{2}$

$\dfrac{1}{4}$

$\dfrac{111}{55}$

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