CAT 2020 Slot 3 • Quantitative Aptitude

Two alcohol solutions, $A$ and $B$, are mixed in the proportion $1:3$ by volume. The volume of the mixture is then doubled by adding solution $A$ such that the resulting mixture has $72 \%$ alcohol. If solution $A$ has $60 \%$ alcohol, then the percentage of alcohol in solution $B$ is

Anil, Sunil, and Ravi run along a circular path of length $3 \mathrm{~km}$, starting from the same point at the same time, and going in the clockwise direction. If they run at speeds of $15 \mathrm{~km} / \mathrm{hr}, 10 \mathrm{~km} / \mathrm{hr}$, and $8 \mathrm{~km} / \mathrm{hr}$, respectively, how much distance in km will Ravi have run when Anil and Sunil meet again for the first time at the starting point?

The area, in sq. units, enclosed by the lines $x = 2$, $y = |x - 2| + 4$, the $X-$axis and the $Y-$axis is equal to

$$A$ and $B$ are two railway stations $90 \mathrm{~km}$ apart. A train leaves $A$ at $9:00$ am, heading towards $B$ at a speed of $40 \mathrm{~km} / \mathrm{hr}$. Another train leaves $B$ at $10:30$ am, heading towards $A$ at a speed of $20 \mathrm{~km} / \mathrm{hr}$. The trains meet each other at

In the final examination, Bishnu scored $52 \%$ and Asha scored $64 \%$. The marks obtained by Bishnu is $23$ less, and that by Asha is $34$ more than the marks obtained by Ramesh. The marks obtained by Geeta, who scored $84 \%$, is

If $\log_a 30 = A$, $\log_a(5/3) = - B$ and $\log_2 a = 1 /3$, then $\log_3 a$ equals

In a trapezium $\mathrm{ABCD}, \mathrm{AB}$ is parallel to $\mathrm{DC}, \mathrm{BC}$ is perpendicular to DC and $\angle \mathrm{BAD}=45^{\circ}$. If $\mathrm{DC}=5 \mathrm{~cm}, \mathrm{BC}=4 \mathrm{~cm}$, the area of the trapezium in sq cm is

If $f(x + y) = f (x) f (y)$ and $f(5) = 4$, then $f (10) - f (-10)$ is equal to

$$\small \dfrac{2\times4\times8\times16}{(\log_2 4)^2 (\log_4 8)^3 (\log_8 16)^4}$ is equal to

Let $N, x$ and $y$ be positive integers such that $N = x + y,$ $2 < x < 10$ and $14 < y < 23$. If $N > 25$, then how many distinct values are possible for $N?$

A man buys $35$ kg of sugar and sets a marked price in order to make a $20\%$ profit. He sells $5$ kg at this price, and $15$ kg at a $10\%$ discount. Accidentally, $3$ kg of sugar is wasted. He sells the remaining sugar by raising the marked price by $p$ percent so as to make an overall profit of $15\%$. Then $p$ is nearest to

Vimla starts for office every day at $9$ am and reaches exactly on time if she drives at her usual speed of $40 \mathrm{~km} / \mathrm{hr}$. She is late by $6$ minutes if she drives at $35 \mathrm{~km} / \mathrm{hr}$. One day, she covers two-thirds of her distance to office in onethirds of her usual time to reach office, and then stops for $8$ minutes. The speed, in km/hr, at which she should drive the remaining distance to reach office exactly on time is

How many integers in the set $(100,101,102, \ldots, 999)$ have at least one digit repeated?

A batsman played $n + 2$ innings and got out on all occasions. His average score in these $n + 2$ innings was 29 runs and he scored 38 and 15 runs in the last two innings. The batsman scored less than 38 runs in each of the first $n$ innings. In these $n$ innings, his average score was 30 runs and lowest score was $x$ runs. The smallest possible value of $x$ is

Let $k$ be a constant. The equations $kx + y = 3$ and $4x + ky = 4$ have a unique solution if and only if

Dick is thrice as old as Tom and Harry is twice as old as Dick. If Dick's age is $1$ year less than the average age of all three, then Harry's age, in years, is

The vertices of a triangle are $(0,0)$, and $(4,0)$ and $(3,9)$. The area of the circle passing through these three points is

A person invested a certain amount of money at $10\%$ annual interest, compounded half-yearly. After one and a half years, the interest and principal together became Rs $18522$. The amount, in rupees, that the person had invested is

If $a, b, c$ are non-zero and $14^a = 36^b = 84^c$, then $6b \left( \frac{1}{c} - \frac{1}{a} \right)$ is equal to

The points $(2, 1)$ and $(-3, -4)$ are opposite vertices of a parallelogram. If the other two vertices lie on the line $x + 9y + c = 0$, then $c$ is

Let $m$ and $n$ be natural numbers such that $n$ is even and $0.2<\frac{\mathrm{m}}{20}, \frac{\mathrm{n}}{\mathrm{m}}, \frac{\mathrm{n}}{11}<0.5$. Then $m-2n$ equals

If $x_1 = -1$ and $x_m = x_{m+1} + (m + 1)$ for every positive integer $m,$ then $x_{100}$ equals

How many of the integers $1, 2, ..., 120$, are divisible by none of $2, 5$ and $7$?

A contractor agreed to construct a $6$ km road in $200$ days. He employed $140$ persons for the work. After $60$ days, he realized that only $1.5$ km road has been completed. How many additional people would he need to employ in order to finish the work exactly on time?

How many pairs $(a, b)$ of positive integers are there such that $a \le b$ and $ab =$ $4^{2017}$?