CAT (QA) PYQs

Points $A, P, Q$ and $B$ lie on the same line such that $P, Q$ and $B$ are, respectively, $100 \mathrm{~km}, 200 \mathrm{~km}$ and $300 \mathrm{~km}$ away from $A$. Cars $1$ and $2$ leave $A$ at the same time and move towards $B.$ Simultaneously, car $3$ leaves $B$ and moves towards $A$. Car $3$ meets car $1$ at $Q$, and car $2$ at $P$. If each car is moving in uniform speed then the ratio of the speed of car $2$ to that of car $1$ is

A water tank has inlets of two types $A$ and $B$. All inlets of type $A$ when open, bring in water at the same rate. All inlets of type $B$, when open, bring in water at the same rate. The empty tank is completely filled in $30$ minutes if $10$ inlets of type $A$ and $45$ inlets of type $B$ are open, and in $1$ hour if $8$ inlets of type $A$ and $18$ inlets of type $B$ are open. In how many minutes will the empty tank get completely filled if $7$ inlets of type $A$ and $27$ inlets of type $B$ are open?

How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits?

A tank is emptied everyday at a fixed time point. Immediately thereafter, either pump $A$ or pump $B$ or both start working until the tank is full. On Monday, $A$ alone completed filling the tank at $8 \mathrm{pm}$. On Tuesday, B alone completed filling the tank at $6 \mathrm{pm}$. On Wednesday, $A$ alone worked till $5 \mathrm{pm}$, and then B worked alone from $5 \mathrm{pm}$ to $7 \mathrm{pm}$, to fill the tank. At what time was the tank filled on Thursday if both pumps were used simultaneously all along?

The area of a rectangle and the square of its perimeter are in the ratio $1:25$. Then the lengths of the shorter and longer sides of the rectangle are in the ratio

If the sum of squares of two numbers is $97$, then which one of the following cannot be their product?

If $a$ and $b$ are integers such that $2x^2 - ax + 2 > 0$ and $x^2 −bx + 8 \geq 0$ for all real numbers $x,$ then the largest possible value of $2a-6b$ is

The value of the sum $7 \times 11 + 11 \times 15 + 15 \times 19 + \dots + 95 \times 99$ is

If $A=\left\{6^{2 n}-35 n-1: n=1,2,3, \ldots\right\}$ and $B=\{35(n-1): n=1,2,3, \ldots\}$ then which of the following is true?

In a tournament, there are $43$ junior level and $51$ senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is $153$, while the number of boy versus boy matches in senior level is $276$. The number of matches a boy plays against a girl is

A chord of length $5 \mathrm{~cm}$ subtends an angle of $60^{\circ}$ at the centre of a circle. The length, in cm , of a chord that subtends an angle of $120^{\circ}$ at the centre of the same circle is

From a rectangle $ABCD$ of area $768 \text{ sq cm}$, a semicircular part with diameter $AB$ and area $72\pi \text{ sq cm}$ is removed. The perimeter of the leftover portion, in cm, is

Gopal borrows Rs. $X$ from Ankit at $8 \%$ annual interest. He then adds Rs. $Y$ of his own money and lends Rs. $X+Y$ to Ishan at $10 \%$ annual interest. At the end of the year, after returning Ankit's dues, the net interest retained by Gopal is the same as that accrued to Ankit. On the other hand, had Gopal lent Rs. $X+2Y$ to Ishan at $10 \%$, then the net interest retained by him would have increased by Rs. $150$. If all interests are compounded annually, then find the value of $X+Y$.

The smallest integer $n$ for which $4^n > 17^{19}$ holds, is closest to

A triangle $ABC$ has area $32$ sq units and its side $BC$ , of length $8$ units, lies on the line $\mathrm{x}=4$. Then the shortest possible distance between $A$ and the point $(0,0)$ is

For two sets $A$ and $B$, let $A \Delta B$ denote the set of elements which belong to $A$ or $B$ but not both. If $P=\{1,2,3,4\}, Q=\{2,3,5,6\},, R=\{1,3,7,8,9\}, S=$ $\{2,4,9,10\}$, then the number of elements in $(P \Delta Q) \Delta(R \Delta S)$ is

If $p^3 = q^4 = r^5 = s^6$, then the value of $\log_s(pqr)$ is equal to

There are two drums, each containing a mixture of paints $A$ and $B$ . In drum $1, \mathrm{~A}$ and B are in the ratio $18: 7$. The mixtures from drums $1$ and $2$ are mixed in the ratio $3: 4$ and in this final mixture, $A$ and $B$ are in the ratio $13: 7$. In drum $2$, then $A$ and $B$ were in the ratio

The scores of Amal and Bimal in an examination are in the ratio $11:14$. After an appeal, their scores increase by the same amount and their new scores are in the ratio $47 : 56$. The ratio of Bimal's new score to that of his original score is

The arithmetic mean of $x, y$ and $z$ is $80$, and that of $x, y, z, u$ and $v$ is $75$, where $u=(x+y) / 2$ and $v=(y+z) / 2$. If $x \geq z$, then the minimum possible value of $x$ is

The strength of a salt solution is $p\%$ if $100 \mathrm{ml}$ of the solution contains p grams of salt. If three salt solutions $A, B, C$ are mixed in the proportion $1: 2: 3$, then the resulting solution has strength $20 \%$. If instead the proportion is $3: 2: 1$, then the resulting solution has strength $30 \%$. A fourth solution, $D$, is produced by mixing $B$ and $C$ in the ratio $2: 7$. The ratio of the strength of $D$ to that of $A$ is

Points $A$ and $B$ are $150 \mathrm{~km}$ apart. Cars $1$ and $2$ travel from $A$ to $B$, but car $2$ starts from $A$ when car $1$ is already $20 \mathrm{~km}$ away from $A$. Each car travels at a speed of $100 \mathrm{kmph}$ for the first $50 \mathrm{~km}$, at $50 \mathrm{kmph}$ for the next $50 \mathrm{~km}$, and at $25 \mathrm{kmph}$ for the last $50 \mathrm{~km}$. The distance, in km, between car $2$ and $B$ when car $1$ reaches $B$ is

Let $f(x)=$ max{$5x$, $52-2x^2$}, where $x$ is any positive real number. Then the minimum possible value of $f(x)$ is

$\dfrac{1}{\log _{2} 100}-\dfrac{1}{\log _{4} 100}+\dfrac{1}{\log _{5} 100}-\dfrac{1}{\log _{10} 100}+\dfrac{1}{\log _{20} 100}-\dfrac{1}{\log _{25} 100}+\dfrac{1}{\log _{50} 100}= \ ?$

Let $t_1, t_2, \dots$ be real numbers such that $t_1 + t_2 +\dots+ t_n = 2n^2 + 9n + 13$, for every positive integer $n\ge2$. If $t_k =103$, then $k$ equals

A $20 \%$ ethanol solution is mixed with another ethanol solution, say, $S$ of unknown concentration in the proportion $1:3$ by volume. This mixture is then mixed with an equal volume of $20 \%$ ethanol solution. If the resultant mixture is a $31.25 \%$ ethanol solution, then the unknown concentration of $S$ is

On a triangle $A B C$, a circle with diameter $B C$ is drawn, intersecting $A B$ and $A C$ at points $P$ and $Q$, respectively. If the lengths of $A B, A C$, and $C P$ are $30 \mathrm{~cm}, 25 \mathrm{~cm}$, and $20 \mathrm{~cm}$ respectively, then the length of $BQ$ , in cm , is

The smallest integer $n$ such that $n^3 - 11n^2 + 32n - 28 > 0$ is

Let $a_{1}, a_{2}, \ldots, a_{52}$ be positive integers such that $a_{1}<a_{2}<\ldots<a_{52}$. Suppose, their arithmetic mean is one less than the arithmetic mean of $a_{2}, a_{3}$, $\ldots, a_{52}$. If $a_{52}=100$, then the largest possible value of $a_{1}$ is

A parallelogram $ABCD$ has area $48 \mathrm{sqcm}$. If the length of CD is $8 \mathrm{~cm}$ and that of $AD$ is s cm , then which one of the following is necessarily true?

If $N$ and $x$ are positive integers such that $N^{N}=2^{160}$ and $N^{2}+2 ^N$ is an integral multiple of $2^x$, then the largest possible $x$ is

Ramesh and Ganesh can together complete a work in $16$ days. After seven days of working together, Ramesh got sick and his efficiency fell by $30 \%$. As a result, they completed the work in $17$ days instead of $16$ days. If Ganesh had worked alone after Ramesh got sick, in how many days would he have completed the remaining work?

On a long stretch of east-west road, $A$ and $B$ are two points such that $B$ is $350$ km west of $A$. One car starts from $A$ and another from $B$ at the same time. If they move towards each other, then they meet after $1$ hour. If they both move towards east, then they meet in $7 \mathrm{hrs}$. The difference between their speeds, in km per hour, is

A jar contains a mixture of $175 \mathrm{ml}$ water and $700 \mathrm{ml}$ alcohol. Gopal takes out $10 \%$ of the mixture and substitutes it by water of the same amount. The process is repeated once again. The percentage of water in the mixture is now