IPMAT Indore 2022

For $0\lt\theta\lt\frac{\pi}{4}$, let $a=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \cos \theta\right), b=\left((\cos \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right), c=\left((\sin \theta)^{\cos \theta}\right)\left(\log _{2} \cos \theta\right)$ and $d=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right)$. Then, the median value in the sequence $a, b, c, d$ is

The set of real values of $x$ for which the inequality $\log _{27} 8 \leq \log _{3} x \lt 9^{\frac{1}{\log _{2} 3}}$ holds is

The set of all possible values of $f(x)$ for which $(81)^{x}+(81)^{f(x)}=3$ is

Ayesha is standing atop a vertical tower $200 m$ high and observes a car moving away from the tower on a straight, horizontal road from the foot of the tower. At 11:00 AM, she observes the angle of depression of the car to be $45^{\circ}$. At 11:02 AM, she observes the angle of depression of the car to be $30^{\circ}$. The speed at which the car is moving is approximately

The value of $k$ for which the following lines are concurrent is

$x-y-1=0 \newline 2x+3y-12=0 \newline 2x-3y+k=0$

The sum of the squares of all the roots of the equation $x^{2}+|x+4|+|x-4|-35=0$ is

The lengths of the sides of a triangle are $x, 21$ and $40$, where $x$ is the shortest side. A possible value of $x$ is:

If the five-digit number abcde is divisible by 6, then which of the following numbers is not necessarily divisible by 6?

In how many ways can the letters of the word MANAGEMENT be arranged such that no two vowels appear together?

In a room, there are $n$ persons whose average height is $160 \ \text{cm}$. If $m$ more persons, whose average height is $172 \ \text{cm}$, enter the room, then the average height of all persons in the room becomes $164 \ \text{cm}$. Then $m: n$ is

The sum of the first 15 terms in an arithmetic progression is 200, while the sum of the next 15 terms is 350. Then the common difference is

Suppose $a, b$ and $c$ are integers such that $a>b>c>0$, and $A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]$. Then the value of the determinant of $A$ is

In a right-angled triangle ABC, the hypotenuse AC is of length 13 cm. A line drawn connecting the midpoints D and E of sides AB and AC is found to be 6 cm in length. The length of BC is

If $A=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 0\end{array}\right]$ then $A^{2022}$ is

Let $A=\{1,2,3\}$ and $B=\{a, b\}$. Assuming all relations from set $A$ to set $B$ are equally likely, what is the probability that a relation from $A$ to $B$ is also a function?

In a 400-metre race, Ashok beats Bipin and Chandan respectively by 15 seconds and 25 seconds. If Ashok beats Bipin by 150 metres, by how many metres does Bipin beat Chandan in the race?

In a bowl containing 60 ml orange juice, 40 ml of water is poured. Thereafter, 100 ml of apple juice is poured to make a fruit punch. Madhu drinks 50 ml of this fruit punch and comments that the proportion of orange juice needs to be higher for better taste. How much orange juice should be poured into the fruit punch that remained, in order to bring up the level of orange juice to 50 percentage?

The curve represented by the equation $\dfrac{x^{2}}{\sin \sqrt{2}-\sin \sqrt{3}}+\dfrac{y^{2}}{\cos \sqrt{2}-\cos \sqrt{3}}=1$ is

A set of all possible values the function $f(x)=\dfrac{x}{|x|}$, where $x \neq 0$, takes is

When the square of the difference of two natural numbers is subtracted from the square of the sum of the same two numbers and the result is divided by four, we get

The cost of a piece of jewellery is proportional to the square of its weight. A piece of jewellery weighing 10 grams is INR 3600. The cost of a piece of jewellery of the same kind weighing 4 grams is

Let $A$ and $B$ be two sets such that the Cartesian product $A \times B$ consists of four elements. If two elements of $A \times B$ are $(1,4)$ and $(4,1)$, then

If $f\left(x^{2}+f(y)\right)=x f(x)+y$ for all non-negative integers $x$ and $y$, then the value of $[f(0)]^{2}+f(0)$ equals _________.

If one of the factors of the number $3^{7} 2^{8} 17^{3}$ is randomly chosen, then the probability that the chosen factor will be a perfect square is

The number of four-digit integers which are greater than 1000 and divisible by both 2 and 3, but not by 5, is

Number of days Bhabha and Cathy work together in a week is

Which among the following employees do not work together on any of the days?

One of the days Alex works on is

The consecutive days on which Ethan works are

Employees who work on Sunday are