CAT (QA) PYQs

Among $100$ students, $x_{1}$ have birthdays in January, $x_{2}$ have birthday in February, and so on. $$\newline$$ If $x_{0}=\max \left(x_{1}, x_{2}, \ldots , x_{12}\right)$, then the smallest possible value of $x_{0}$ is

A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then the half of the remaining stock along with one extra to the second and continues giving away in this fashion. His total stock exhausts after he takes care of $5$ children. How many toffees were there in his stock initially?

Veeru invested Rs 10000 at 5% simple annual interest, and exactly after two years, Joy invested Rs 8000 at 10% simple annual interest. How many years after Veeru's investment, will their balances, i.e., principal plus accumulated interest, be equal?

If y is a negative number such that $2^{y^2log_3 5} = 5^{\log_2 3}$, then y equals

Leaving home at the same time, Amal reaches the office at $10:15 am$ if he travels at $8 km/hr$, and at $9:40 am$ if he travels at $15 km/hr$. Leaving home at 9:10, at what speed, in $km/hr,$ must be travel so as to reach office exactly at $10 am?$

A train travelled at one-thirds of its usual speed, and hence reached the destination $30$ minutes after the scheduled time. On its return journey, the train initially travelled at its usual speed for $5$ minutes but then stopped for $4$ minutes for an emergency. The percentage by which the train must now increase its usual speed so as to reach the destination at the scheduled time, is nearest to

The number of real-valued solutions of the equation $2^x + 2^{-x} = 2 - (x - 2)^2$ is

A solution, of volume $40$ litres, has dye and water in the proportion $2: 3$. Water is added to the solution to change this proportion to $2: 5$. If one-fourths of this diluted solution is taken out, how many litres of dye must be added to the remaining solution to bring the proportion back to $2: 3$?

Let $A, B$ and $C$ be three positive integers such that the sum of $A$ and the mean of $B$ and $C$ is $5.$ In addition, the sum of $B$ and the mean of $A$ and $C$ is $7.$ Then the sum of $A$ and $B$ is

How many $3-$digit numbers are there, for which the product of their digits is more than $2$ but less than $7?$

A straight road connects points $A$ and $B.$ Car $1$ travels from $A$ to $B$ and Car $2$ travels from $B$ to $A$, both leaving at the same time. After meeting each other, they take $45$ minutes and $20$ minutes, respectively, to complete their journeys. If Car $1$ travels at the speed of $60 km/hr$, then the speed of Car $2,$ in km/hr, is

If $\log _{4} 5=\left(\log _{4} y\right)\left(\log _{6} \sqrt{ } 5\right)$, then $y$ equals

A solid right circular cone of height $27$ cm is cut into pieces along a plane parallel to its base at a height of $18$ cm from the base. If the difference in volume of the two pieces is $225$ cc, the volume, in cc, of the original cone is

Two persons are walking beside a railway track at respective speeds of $2$ and $4 \mathrm{~km}$ per hour in the same direction. A train came from behind them and crossed them in $90$ and $100$ seconds, respectively. The time, in seconds, taken by the train to cross a stationary object?

A circle is inscribed in a rhombus with diagonals $12$ cm and $16$ cm. The ratio of the area of circle to the area of rhombus is

A circle is inscribed in a rhombus with diagonals $12 \mathrm{~cm}$ and $16 \mathrm{~cm}$. The ratio of the area of circle to the area of rhombus is

If $x=(4096)^{7+4 \sqrt{3}}$, then which of the following equals $64$ ?

On a rectangular metal sheet of area $135 \mathrm{sq}$ in, a circle is painted such that the circle touches two opposite sides. If the are the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

The number of distinct real roots of the equation $(x + \frac{1}{x})^2 - 3(x + \frac{1}{x}) + 2 = 0$ equals

If $a, b$ and $c$ are positive integers such that $ab = 432, bc = 96$ and $c < 9,$ then the smallest possible value of $a + b + c$ is

The area of the region satisfying the inequalities $|\mathrm{x}|-\mathrm{y} \leq 1, \mathrm{y} \geq 0$ and $\mathrm{y} \leq 1$ is

In a group of people, $28\%$ of the members are young while the rest are old. If $65\%$ of the members are literates, and $25\%$ of the literates are young, then the percentage of old people among the illiterates is nearest to

The mean of all $4-$digit even natural numbers of the form $'aabb',$ where $a > 0,$ is

An alloy is prepared by mixing three metals $A, B$ and $C$ in the proportion $3 : 4 :7$ by volume. Weights of the same volume of the metals $A, B$ and $C$ are in the ratio $5:2: 6.$ In $130$ kg of the alloy, the weight, in kg, of the metal $C$ is

A person spent Rs $50000$ to purchase a desktop computer and a laptop computer. He sold the desktop at $20%$ profit and the laptop at $10%$ loss. If overall he made a $2%$ profit then the purchase price, in rupees, of the desktop is

How many distinct positive integer-valued solutions exist to the equation $\left(x^{2}-7 x+11\right)^{\left(x^{2}-13 x+42\right)}=1 ?$