CAT 2019 Slot 1 • Quantitative Aptitude

A person invested a total amount of Rs $15$ lakh. A part of it was invested in a fixed deposit earning $6\%$ annual interest, and the remaining amount was invested in two other deposits in the ratio $2: 1$, earning annual interest at the rates of $4\%$ and $3\%$, respectively. If the total annual interest income is Rs $76000$ then the amount (in Rs lakh) invested in the fixed deposit was

In a race of three horses, the first beat the second by $11$ metres and the third by $90$ metres. If the second beat the third by $80$ metres, what was the length, in metres, of the racecourse?

If $(5.55)^x= (0.555)^y = 1000$, then the value of $\frac{1}{x} - \frac{1}{y}$ is

The income of Amala is $20 \%$ more than that of Bimala and $20 \%$ less than that of Kamala. If Kamala's income goes down by $4 \%$ and Bimala's goes up by $10 \%$, then the percentage by which Kamala's income would exceed Bimala's is nearest to

Consider a function $f$ satisfying $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x}) \mathrm{f}(\mathrm{y})$ where $x , y$ are positive integers, and $f(1)=2$. If $f(\mathrm{a}+1)+f(\mathrm{a}+2)+\ldots \ldots+f(\mathrm{a}+\mathrm{n})=$ $16\left(2^{n}-1\right)$ then a is equal to

For any positive integer $n$, let $f(n)=n(n+1)$ if $n$ is even, and $f(n)=n+3$ if $n$ is odd. If $m$ is a positive integer such that $8 f(m+1)-f(m)=2$, then $m$ equals

In a class, $60\%$ of the students are girls and the rest are boys. There are $30$ more girls than boys. If $68\%$ of the students, including $30$ boys, pass an examination, the percentage of the girls who do not pass is

If $a_1 + a_2 + a_3 + \dots + a_n = 3 \times (2^{n+1} - 2)$, for every $n \ge 1$, then $a_{11}$ equals

Two cars travel the same distance starting at $10:00$ am and $11:00$ am, respectively, on the same day. They reach their common destination at the same point of time. If the first car travelled for at least $6$ hours, then the highest possible value of the percentage by which the speed of the second car could exceed that of the first car is

The number of solution to the equation $|x|(6x^2 + 1) = 5x^2$ is

Let $T$ be the triangle formed by the straight line $3x + 5y - 45 = 0$ and the coordinate axes. Let the circumcircle of $T$ have radius of length $L,$ measured in the same unit as the coordinate axes. Then, the integer closest to $L$ is

If the rectangular faces of a brick have their diagonals in the ratio $3: 2 \sqrt{3}: \sqrt{15}$, then the ratio of the length of the shortest edge of the brick to that of its longest edge is

Let $x$ and $y$ be positive real numbers such that $\log _{5}(x+y)+\log _{5}(x-y)=3$, and $\log _{2} y-\log _{2} x=1-\log _{2} 3$. Then $x y$ equals

A club has $256$ members of whom $144$ can play football, $123$ can play tennis, and $132$ can play cricket. Moreover, $58$ members can play both football and tennis, $25$ can play both cricket and tennis, while $63$ can play both football and cricket. If every member can play at least one game, then the number of members who can play only tennis is

The product of two positive numbers is $616$. If the ratio of the difference of their cubes to the cube of their difference is $157: 3$, then the sum of the two numbers is

In a circle of radius $11$ cm, $CD$ is a diameter and $AB$ is a chord of length $20.5$ cm. If $AB$ and $CD$ intersect at a point $E$ inside the circle and $CE$ has length $7$ cm, then the difference of the lengths of $BE$ and $AE$, in cm, is

Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in $13$ days, then how many men can finish the job in $13$ days?

Amala, Bina, and Gouri invest money in the ratio $3: 4: 5$ in fixed deposits having respective annual interest rates in the ratio $6: 5:4$. What is their total interest income (in Rs) after a year, if Bina's interest income exceeds Amala's by Rs $250$?

The product of the distinct roots of $|x^2 − x − 6| = x + 2$ is

The number of the real roots of the equation $2\cos (x(x + 1)) = 2^x + 2^{-x}$ is

A chemist mixes two liquids $1$ and $2$. One litre of liquid $1$ weighs $1 \mathrm{~kg}$ and one litre of liquid $2$ weighs $800 \mathrm{gm}$. If half litre of the mixture weighs $480 \mathrm{gm}$, then the percentage of liquid $1$ in the mixture, in terms of volume, is

$$AB$ is a diameter of a circle of radius $5 \mathrm{~cm}$. Let $P$ and $Q$ be two points on the circle so that the length of $PB$ is $6 \mathrm{~cm}$, and the length of $AP$ is twice that of $AQ$ . Then the length, in cm , of $QB$ is nearest to

Corners are cut off from an equilateral triangle $T$ to produce a regular hexagon $H$. Then, the ratio of the area of $H$ to the area of $T$ is

Let $S$ be the set of all points $(x, y)$ in the x-y plane such that $|x|+|y| \le 2$ and $|x| \ge 1$. Then, the area, in square units, of the region represented by $S$ equals:

Ramesh and Gautam are among $22$ students who write an examination. Ramesh scores $82.5$. The average score of the $21$ students other than Gautam is $62$. The average score of all the $22$ students is one more than the average score of the $21$ students other than Ramesh. The score of Gautam is