Recent questions tagged cat2024-set2

The roots $\alpha, \beta$ of the equation $3 x^{2}+\lambda x-1=0$, satisfy $\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}=15$. The value of $\left(\alpha^{3}+\beta^{3}\right)^...

A function $f$ maps the set of natural numbers to whole numbers, such that $f(x y)=f(x) f(y)+f(x)+f(y)$ for all $x, y$ and $f(p)=1$ for every prime number $p$. Then, the ...

If $a, b$ and $c$ are positive real numbers such that $a>10 \geq b \geq c$ and $\frac{\log _{8}(a+b)}{\log _{2} c}+\frac{\log _{27}(a-b)}{\log _{3} c}=\frac{2}{3}$, then ...

If $x$ and $y$ are real numbers such that $4 x^{2}+4 y^{2}-4 x y-6 y+3=0$, then the value of $(4 x+5 y)$ is

Three circles of equal radii touch (but not cross) each other externally. Two other circles, $X$ and $Y$, are drawn such that both touch (but not cross) each of the three...

The sum of the infinite series $\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^{2}\left(\left(\frac{1}{5}\right)^{2}-\left(\frac{1}{7}\right)^{2...

A vessel contained a certain amount of a solution of acid and water. When $2$ litres of water was added to it, the new solution had $50 \%$ acid concentration. When $15$ ...

If $m$ and $n$ are natural numbers such that $n>1$, and $m^{n}=2^{25} \times 3^{40}$, then $m-n$ equals$209942$$209947$$209932$$209937$

$\mathrm{ABCD}$ is a trapezium in which $\mathrm{A B}$ is parallel to $\mathrm{C D}$. The sides $\mathrm{A D}$ and $\mathrm{B C}$ when extended, intersect at point $\math...

The coordinates of the three vertices of a triangle are: $(1,2),(7,2)$, and $(1,10)$. Then the radius of the incircle of the triangle is

When Rajesh's age was same as the present age of Garima, the ratio of their ages was $3 : 2$. When Garima's age becomes the same as the present age of Rajesh, the ratio o...

A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up $40 \%$ of his st...

When $3^{333}$ is divided by $11$, the remainder is$5$$6$$1$$10$

Bina incurs $19 \%$ loss when she sells a product at Rs. $4860$ to Shyam, who in turn sells this product to Hari. If Bina would have sold this product to Shyam at the pur...

All the values of $x$ satisfying the inequality $\frac{1}{x+5} \leq \frac{1}{2 x-3}$ are$-5 < x<\frac{3}{2}$ or $\frac{3}{2}<x\leq 8$$x<-5$ or $x \frac{3}{2}$$x < -5$ or...

A company has $40$ employees whose names are listed in a certain order. In the year $2022$, the average bonus of the first $30$ employees was Rs. $40000$, of the last $30...

If $(x+6 \sqrt{2})^{\frac{1}{2}}-(x-6 \sqrt{2})^{\frac{1}{2}}=2 \sqrt{2}$, then $x$ equals

A bus starts at $9 \: \mathrm{am}$ and follows a fixed route every day. One day, it traveled at a constant speed of $60 \: \mathrm{km}$ per hour and reached its destinati...

$\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ are four towns. One can travel between $\mathrm{P}$ and $\mathrm{Q}$ along $3$ direct paths, between $\mathrm{Q}$ an...

Anil invests Rs $22000$ for $6$ years in a scheme with $4\%$ interest per annum, compounded half-yearly. Separately, Sunil invests a certain amount in the same scheme for...

Amal and Vimal together can complete a task in $150$ days, while Vimal and Sunil together can complete the same task in $100$ days. Amal starts working on the task and wo...

If $x$ and $y$ satisfy the equations $\mid x \mid+x+y=15$ and $x+\mid y \mid-y=20$, then $(x-y)$ equals$15$$10$$20$$5$

The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes ($3$ of them, each in the shape of rectangles - shaded in the diagram) of a g...

 The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes ($3$ of them, each in the shape of rectangles - shaded in the diagram) of a ...

The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes ($3$ of them, each in the shape of rectangles - shaded in the diagram) of a g...

The above is a schematic diagram of walkways (indicated by all the straight-lines) and lakes ($3$ of them, each in the shape of rectangles - shaded in the diagram) of a g...

The numbers $1,2,3,4,5,6,7,8,9$, and $10$ are placed in ten slots of the following grid based on the conditions below.Numbers in any row appear in an increasing order fro...

The numbers $1,2,3,4,5,6,7,8,9$, and $10$ are placed in ten slots of the following grid based on the conditions below.Numbers in any row appear in an increasing order fro...

The numbers $1,2,3,4,5,6,7,8,9$, and $10$ are placed in ten slots of the following grid based on the conditions below.Numbers in any row appear in an increasing order fro...

The numbers $1,2,3,4,5,6,7,8,9$, and $10$ are placed in ten slots of the following grid based on the conditions below.Numbers in any row appear in an increasing order fro...

The numbers $1,2,3,4,5,6,7,8,9$, and $10$ are placed in ten slots of the following grid based on the conditions below.Numbers in any row appear in an increasing order fro...

An online e-commerce firm receives daily integer product ratings from $1$ through $5$ given by buyers. The daily average is the average of the ratings given on that day. ...

An online e-commerce firm receives daily integer product ratings from $1$ through $5$ given by buyers. The daily average is the average of the ratings given on that day. ...

An online e-commerce firm receives daily integer product ratings from $1$ through $5$ given by buyers. The daily average is the average of the ratings given on that day. ...

An online e-commerce firm receives daily integer product ratings from $1$ through $5$ given by buyers. The daily average is the average of the ratings given on that day. ...

The two plots below give the following information about six firms $\text{A, B, C, D, E}$, and $\text{F}$ for $2019$ and $2023$.PAT: The firm's profits after taxes in Rs....

The two plots below give the following information about six firms $\text{A, B, C, D, E}$, and $\text{F}$ for $2019$ and $2023$.PAT: The firm's profits after taxes in Rs....

The two plots below give the following information about six firms $\text{A, B, C, D, E}$, and $\text{F}$ for $2019$ and $2023$.PAT: The firm's profits after taxes in Rs....

The two plots below give the following information about six firms $\text{A, B, C, D, E}$, and $\text{F}$ for $2019$ and $2023$.PAT: The firm's profits after taxes in Rs....

Eight gymnastics players numbered $1$ through $8$ underwent a training camp where they were coached by three coaches - Xena, Yuki, and Zara. Each coach trained at least t...