Recent questions tagged cat2023-set3

Let $\triangle A B C$ be an isosceles triangle such that $A B$ and $A C$ are of equal length. $A D$ is the altitude from $A$ on $B C$ and $B E$ is the altitude from $B$ o...

A rectangle with the largest possible area is drawn inside a semicircle of radius $2 \mathrm{~cm}$. Then, the ratio of the lengths of the largest to the smallest side of ...

In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is

For a real number $x$, if $\frac{1}{2}, \frac{\log _{3}\left(2^{x}-9\right)}{\log _{3} 4}$, and $\frac{\log _{5}\left(2^{x}+\frac{17}{2}\right)}{\log _{5} 4}$ are in an a...

A quadratic equation $x^{2}+b x+c=0$ has two real roots. If the difference between the reciprocals of the roots is $\frac{1}{3}$, and the sum of the reciprocals of the sq...

The sum of the first two natural numbers, each having $15$ factors (including $1$ and the number itself), is

Let $n$ be any natural number such that $5^{n-1}<3^{n+1}$. Then, the least integer value of $m$ that satisfies $3^{n+1}<2^{n+m}$ for each such $n$, is

A boat takes $2$ hours to travel downstream a river from port $A$ to port $B$, and $3$ hours to return to port $A$. Another boat takes a total of $6$ hours to travel from...

Anil mixes cocoa with sugar in the ratio $3: 2$ to prepare mixture $A$, and coffee with sugar in the ratio $7: 3$ to prepare mixture $B$. He combines mixtures $A$ and $B$...

Rahul, Rakshita and Gurmeet, working together, would have taken more than $7$ days to finish a job. On the other hand, Rahul and Gurmeet, working together would have take...

The population of a town in $2020$ was $100000$. The population decreased by $y \%$ from the year $2020$ to $2021$, and increased by $x \%$ from the year $2021$ to $2022$...

There are three persons $A, B$ and $C$ in a room. If a person $D$ joins the room, the average weight of the persons in the room reduces by $x \mathrm{~kg}$. Instead of $\...

A merchant purchases a cloth at a rate of Rs. $100$ per meter and receives $5 \mathrm{~cm}$ length of cloth free for every $100 \mathrm{~cm}$ length of cloth purchased by...

Gautam and Suhani, working together, can finish a job in $20$ days. If Gautam does only $60 \%$ of his usual work on a day, Suhani must do $150 \%$ of her usual work on t...

The number of coins collected per week by two coin-collectors $A$ and $B$ are in the ratio $3: 4$. If the total number of coins collected by $A$ in $5$ weeks is a multipl...

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...

Let $\triangle A B C$ be an isosceles triangle such that $A B$ and $A C$ are of equal length. $A D$ is the altitude from $A$ on $B C$ and $B E$ is the altitude from $B$ o...

A rectangle with the largest possible area is drawn inside a semicircle of radius $2 \mathrm{~cm}$. Then, the ratio of the lengths of the largest to the smallest side of ...

In a regular polygon, any interior angle exceeds the exterior angle by $120$ degrees. Then, the number of diagonals of this polygon is