List of top Mathematics Questions asked in Gujarat Common Entrance Test

If the value of \( \cos \alpha \) is \( \frac{\sqrt{3}}{2} \), then \( A + A = I \), where \[ A = \begin{bmatrix} \sin\alpha & -\cos\alpha \\ \cos\alpha & \sin\alpha \end{bmatrix}. \]

If \( A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix} \), find \( I + A^2 \), where \( I \) is the identity matrix.
If \[ \left| \begin{pmatrix} 2017 & 2018 \\ 2019 & 2020 \end{pmatrix} \right| + \left| \begin{pmatrix} 2021 & 2022 \\ 2023 & 2024 \end{pmatrix} \right| = 2k \] find \( k^3 \).
Evaluate the integral \( \int \frac{1}{\sqrt{4x - x^2}} \, dx \).
Evaluate the integral \( \int e^x \left( \frac{1 + \sin x}{1 - \sin x} \right) \, dx \).
If \( x = a(1 - \cos\theta) \), \( y = a(\theta + \sin\theta) \), then \( \frac{dy}{dx} \) is: ...
Evaluate \( \int \frac{e^{2x} - 1}{e^{2x} + 1} \, dx \).
Global maximum value of the function \( f(x) = \sin x + \cos x, \quad x \in [0, \pi] \) is: ...
If \( A \) is a square matrix such that \( A^2 = A \), then \( (I - A)^3 - (I + A)^2 = \) __________.
Find \( \sin^{-1} \left( \sin \frac{23\pi}{6} \right) = \) __________.
The value of \(\tan^{-1}(-1) + \sec^{-1}(-2) + \sin^{-1}\left( \frac{1}{\sqrt{2}} \right)\)is __________.
If \( f: \mathbb{Z} \to \mathbb{Z} \), defined by \( f(x) = x^3 + 2 \), then the function \( f \) is __________.
The relation \( R = \{(a, a), (b, b), (c, c), (a, c)\ \) defined on the set \( \{a, b, c\} \) is _____________.}
If a pair of dice is thrown, the probability of getting an even prime number on each die is __________.
If \( x + y \leq 55 \) and \( x + y \geq 10 \), with \( x \geq 0 \), \( y \geq 0 \), then the minimum value of the objective function \( z = 7x + 3y \) is: \hfill \break
If the vertices of the finite feasible solution region are \( (0, 6), (3, 3), (9, 9), (0, 12) \), then the maximum value of the objective function \( z = 6x + 12y \) is __________.
If \( P(B) \neq 0 \) and \( P(A | B) = 1 \) for two events \( A \) and \( B \), then ____________.
Given events \( A \) and \( B \) are absolute and \( P(A) = p \), \( P(B) = \frac{1{2} \), and \( P(A \cup B) = \frac{3}{5} \), the value of \( p \) is __________.}
If \( y = \tan^{-1} x \), then __________.
If the area of \( \triangle PQR \) with vertices \( P(k, 1), Q(2, 4), R(1, 1) \) is 3 square units, find \( k \).