IPMAT Indore 2024

The terms of a geometric progression are real and positive. If the $p$-th term of the progression is $q$ and the $q$-th term is $p$, then the logarithm of the first term is

If the shortest distance of a given point to a given circle is $4 \, \text{cm}$ and the longest distance is $9 \, \text{cm}$, then the radius of the circle is

If $|x+1| + (y+2)^2 = 0$ and $ax - 3ay = 1$, then the value of $a$ is

The number of real solutions of the equation $x^2 - 10|x| - 56 = 0$ is

The greatest number among $2^{300}$, $3^{200}$, $4^{100}$, $2^{100} + 3^{100}$ is

The sum of a given infinite geometric progression is 80 and the sum of its first two terms is 35. Then the value of $n$ for which the sum of its first $n$ terms is closest to 100, is

Let $n$ be the number of ways in which 20 identical balloons can be distributed among 5 girls and 3 boys such that everyone gets at least one balloon and no girl gets fewer balloons than a boy does. Then

Let $a = \dfrac{(\log_7 4)(\log_7 5 - \log_7 2)}{\log_{7} 25 (\log_7 8 - \log_7 4)}$. Then the value of $5^a$ is

The smallest possible number of students in a class if the girls in the class are less than 50% but more than 48% is

The side AB of a triangle ABC is c. The median BD is of length k. If $\angle BDA = \theta$ and $\theta < 90^\circ$, then the area of triangle ABC is

Let $\triangle ABC$ be a triangle with $AB = AC$ and $D$ be a point on $BC$ such that $\angle BAD = 30^\circ$. If $E$ is a point on $AC$ such that $AD = AE$, then $\angle CDE$ equals

If $\log_4 x = a$ and $\log_{25} x = b$, then $\log_x 10$ is

If 5 boys and 3 girls sit randomly around a circular table, the probability that there will be at least one boy sitting between any two girls is

A fruit seller had a certain number of apples, bananas, and oranges at the start of the day. The number of bananas was 10 more than the number of apples, and the total number of bananas and apples was a multiple of 11. She was able to sell 70% of the apples, 60% of bananas, and 50% of oranges during the day. If she was able to sell 55% of the fruits she had at the start of the day, then the minimum number of oranges she had at the start of the day was

A boat goes 96 km upstream in 8 hours and covers the same distance moving downstream in 6 hours. On the next day it starts from point A, goes downstream for 1 hour, then upstream for 1 hour, and repeats this for four more times, that is, 5 upstream and 5 downstream journeys. Then the boat would be

The number of solutions of the equation $x_1 + x_2 + x_3 + x_4 = 50$, where $x_1, x_2, x_3, x_4$ are integers with $x_1 \geq 1, x_2 \geq 2, x_3 \geq 0, x_4 \geq 0$ is

The numbers $2^{2024}$ and $5^{2024}$ are expanded and their digits are written out consecutively on one page. The total number of digits written on the page is

If $\theta$ is the angle between the pair of tangents drawn from the point $(0,\frac{7}{2})$ to the circle $x^2 + y^2 - 14x + 16y + 88 = 0$, then $\tan \theta$ equals

The difference between the maximum real root and the minimum real root of the equation $(x^2 - 5)^4 + (x^2 - 7)^4 = 16$ is

The angle of elevation of the top of a pole from a point A on the ground is 30°. The angle of elevation changes to 45°, after moving 20 meters towards the base of the pole. Then the height of the pole, in meters, is

The number of values of $x$ for which $C \binom {17-x}{3x+1}$ is defined as an integer is

Let ABC be an equilateral triangle, with each side of length $k$. If a circle is drawn with diameter AB, then the area of the portion of the triangle lying inside the circle is

Sagarika divides her savings of $10000$ rupees to invest across two schemes A and B. Scheme A offers an interest rate of $10\%$ per annum, compounded half-yearly, while scheme B offers a simple interest rate of $12\%$ per annum. If at the end of first year, the value of her investment in scheme B exceeds the value of her investment in scheme A by $2310$ rupees, then the total interest, in rupees, earned by Sagarika during the first year of investment is

In a survey of 500 people, it was found that 250 owned a 4-wheeler but not a 2-wheeler, 100 owned a 2-wheeler but not a 4-wheeler, and 100 owned neither a 4-wheeler nor a 2-wheeler. Then the number of people who owned both is

For some non-zero real values of $a, b$ and $c$, it is given that $\left|\frac{c}{a}\right|=4,\left|\frac{a}{b}\right|=\frac{1}{3}$ and $\frac{b}{c}=-\frac{3}{4}$. If $a c>0$, then $\left(\frac{b+c}{a}\right)$ equals

The constituency in which B got lower number of votes compared to A and C is

The number of votes obtained by B in S2 is

The number of votes obtained by A in S5 is

Comparing the number votes obtained by A across different constituencies, the lowest number of votes were in constituency

Assume that A and C had formed an alliance and any voter who voted for either A or C would have voted for this alliance. Then the number of seats this alliance would have won is