CAT 2019 Quantitative Aptitude (QA) Master Hub

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 $10$ am and reaches $B$ at $11$ am. Starting from $10:01$ am, every minute a motor cycle leaves $A$ and moves towards $B$. Forty-five such motor cycles reach $B$ by $11$ am. 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