List of top Mathematics Questions asked in UP Board XII

Prove that: \[ \left| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} \right| = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \]
If \( y = e^{a \cos^{-1} x}, -1 \leq x \leq 1 \), then prove that \( (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} - a^2 y = 0 \).
If \[ A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}, B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}, \] and \[ 2A + 3X = 5B, \text{then find the matrix} \, X. \]
Let a relation \( R = \{(a, b) : (a - b) \text{ is a multiple of 5} \} \) be defined on the set \( \mathbb{Z} \) (set of integers). Prove that \( R \) is an equivalence relation.
Show that the points \( A(2, 3, 4) \), \( B(-1, -2, 1) \) and \( C(5, 8, 7) \) are collinear.
Evaluate: \[ \int \sqrt{x^2 + 2x + 5} \, dx \]
Find the equation of tangent at the point \( (am^2, am^3) \) on the curve \[ ay^2 = x^3. \]
If \( x = a \cos^2 t, y = b \sin^2 t \), then find \( \frac{dy}{dx} \).
Find the general solution of \[ \frac{dy}{dx} = \frac{x - 1}{2 + y}. \]
Prove that the function \( f : \mathbb{R} \to \mathbb{R}^+ \) defined by \( f(x) = e^x \) is one-one.
If \[ \begin{bmatrix} x + z \\ y + z \\ x + y + z \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \\ 9 \end{bmatrix}, \] then find the value of \( x \), \( y \), and \( z \).
Find the maximum value of \( Z = 3x + 4y \) under the constraints \[ x + y \leq 4, x \geq 0, y \geq 0. \]
If vectors \( 2\hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular to each other, then find the value of \( \lambda \).
The order of the differential equation $2x^2 \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + y = 0$ will be:
If \( A = \{a, b, c\} \) and \( B = \{1, 2\} \), then find the number of relations from \( A \) to \( B \).
The value of $\int x e^x \, dx$ will be:
Differential coefficient of $\cos^{-1}(e^x)$ will be:
The value of $\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})$ will be:
A relation $R = \{(a, b) : a = b - 1, b > 4\}$ is defined on set $\mathbb{N}$, then correct answer will be:

If \[ A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}, \] \(\text{then prove that}\) \[ A \cdot \text{adj} \, A = |A| \cdot I. \]

If \[\begin{bmatrix} 2x - y & x + 2y \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}\]
 then the value of $x$ and $y$ will be: 
 

Find the maximization of \( z = x + y \), under the following constraints: \[ x - y \leq -1, -x + y \leq 0, x \geq 0, y \geq 0. \]
Let \[ A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}, B = \begin{pmatrix} 5 & 2 \\ 7 & 4 \end{pmatrix}, C = \begin{pmatrix} 2 & 5 \\ 3 & 8 \end{pmatrix}. \] Then find a matrix D such that \( CD - AB = 0 \).
Solve the system of linear equations by matrix method: \[ 3x + 2y + 3z = 5, \] \[ -2x + y + z = -4, \] \[ -x + 3y - 2z = 3. \]
Differentiate: \( y = x^{x} \).