IPMAT Indore 2019

The sum of the interior angles of a convex $n$-sided polygon is less than $2019^\circ$. The maximum possible value of $n$ is

Suppose that a, b, and c are real numbers greater than 1. Then the value of $\dfrac{1}{1+\log_{a^2 b} \frac{c}{a}} + \dfrac{1}{1+\log_{b^2 c} \frac{a}{b}} + \dfrac{1}{1+\log_{c^2 a} \frac{b}{c}}$ is

A real-valued function $f$ satisfies the relation $f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(y) + 6y$, for all real numbers $x$ and $y$, then the value of $f(8)$ is

Let $A, B, C$ be three $4 \times 4$ matrices such that $det \ A = 5, det \ B = -3$, and $det \ C = \frac{1}{2}$. Then the $det$ $2AB^{-1}C^3B^T$ is

If $A$ is a $3 \times 3$ non-zero matrix such that $A^2 = 0$ then the determinant of $(I + A)^{50} - 50A$ is equal to

Three friends divided some apples in the ratio $3 : 5 : 7$. After consuming 16 apples they found that the remaining number of apples with them was equal to the largest number of apples received by one of them at the beginning. The total number of apples these friends initially had was

A shopkeeper reduces the price of a pen by 25% as a result of which the sales quantity increased by 20%. If the revenue made by the shopkeeper decreases by x% then x is

For all real values of $x$, $\dfrac{3x^2 - 6x + 12}{x^2 + 2x + 4}$ lies between $1$ and $k$, and does not take any value above $k$. Then $k$ equals:

The maximum distance between the point $(-5, 0)$ and a point on the circle $x^2 + y^2 = 4$ is

If $x, y, z$ are positive real numbers such that $x^{12} = y^{16} = z^{24}$ and the three quantities $3 \log_y x, 4 \log_z y, n \log_x z$ are in arithmetic progression, then the value of $n$ is

The number of pairs $(x, y)$ satisfying the equation $\sin x + \sin y = \sin(x + y)$ and $|x| + |y| = 1$ is

The circle $x^2 + y^2 - 6x - 10y + k = 0$ does not touch or intersect the coordinate axes. If the point $(1, 4)$ does not lie outside the circle, and the range of $k$ is $(a, b]$, then $a + b$ is

If a $3 \times 3$ matrix is filled with +1's and -1's such that the sum of each row and column of the matrix is 1, then the absolute value of its determinant is

Let the set $P= \{2,3,4,..., 25\}$. For each $k ∈ P$, define $Q(k)= \{x ∈ P$ such that $x > k$ and $k$ divides $x\}$. Then the number of elements in the set $P - U_{k=2}^{25} Q(k)$ is

The number of whole metallic tiles that can be produced by melting and recasting a circular metallic plate, if each of the tiles has a shape of a right-angled isosceles triangle and the circular plate has a radius equal in length to the longest side of the tile (Assume that the tiles and plate are of uniform thickness, and there is no loss of material in the melting and recasting process) is

If $|x| <100$ and $|y| <100$, then the number of integer solutions of $(x, y)$ satisfying the equation $4x + 7y = 3$ is

The average of five distinct integers is 110 and the smallest number among them is 100. The maximum possible value of the largest integer is

Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The 6389th digit in this sequence is

The number of pairs of integers whose sums are equal to their products is

You have been asked to select a positive integer N which is less than 1000, such that it is either a multiple of 4, or a multiple of 6, or an odd multiple of 9. The number of such numbers is