CAT 2019 QA Questions: Slot 1 & Slot 2

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

The quadratic equation $x² + bx + c = 0$ has two roots $4a$ and $3a,$ where a is an integer. Which of the following is a possible value of $b² + c$?

Two ants $A$ and $B$ start from a point $P$ on a circle at the same time, with $A$ moving clock-wise and $B$ moving anti-clockwise. 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

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

What is the largest positive integer such that $\frac{n^2 + 7n + 12}{n^2 - n - 12}$ is also a positive integer?

Let $A$ be a real number. Then the roots of the equation $x^2 - 4x - \log_2A = 0$ are real and distinct if and only if

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 Rs. $2000,$ then his purchase price of a bicycle, in Rupees, was

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?

Let $f$ be a function such that $f(mn) = f(m)f(n)$ for every positive integers m and n. If $f(1) , f(2)$ and $f(3)$ are positive integers, $f(1) < f(2)$ , and $f(24) = 54$ , then $f(18)$ equals

Let $a_1, a_2, ...$ be integers such that $a_1 - a_2 + a_3 - a_4 + ... + (-1)^{n - 1} a_n = n$ , for all $n \ge 1$. Then $a_{51} + a_{52} + ... + a_{1023}$ equals

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

A cyclist leaves $A$ at 10am and reaches $B$ at 11am. Starting from $10:01$ am, every minute a motor cycle leaves $A$ and moves towards $B$. Forty-five such motor cycles reach $B$ by 11am. All motor cycles have the same speed. If the cyclist had doubled his speed, how many motor cycles would have reached $B$ by the time the cyclist reached $B$?

Let $ABC$ 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

In a triangle $ABC$ , medians $AD$ and $BE$ are perpendicular to each other, and have lengths $12 \mathrm{~cm}$ and $9 \mathrm{~cm}$ , respectively. Then, the area of triangle $ABC$ , in sq cm, is

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

The real root of the equation $2^{6x} + 2^{3x+2} - 21 = 0$ is

If $x$ is a real number, then $\sqrt{\log_e \frac{4x-x^2}{3}}$ is a real number if and only if

John gets Rs 57 per hour of regular work and 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?

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

If $5^{x}- 3^{y} = 13438$ and $5^{x-1}+ 3^{y+1} = 9686$ , then $x+y$ equals

A man makes complete use of $405 \mathrm{cc}$ of iron, $783 \mathrm{cc}$ of aluminium, and $351 \mathrm{cc}$ 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 $3 \mathrm{~cm}$. 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

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

How many factors of $2^4 \times 3^5 \times 10^4$ are perfect squares which are greater than $1?$

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 $A$?

Let $a, b, x, y$ be real number such $a² + b² = 25, x² + y² = 169$ , and $ax$ + by = $65.$ If $k = ay - bx,$ then