List of top Mathematics Questions asked in UP Board XII

Find the interval in which the function \[ f(x) = \frac{3}{10}x^4 - \frac{4}{5}x^3 - 3x^2 + \frac{36}{5}x + 11 \] is (A) increasing, (B) decreasing.

Prove that: \[ \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right). \]

(a) If \( x^y = e^{x-y} \), then prove that \[ \frac{dy}{dx} = \frac{\log_e x}{(1 + \log_e x)^2}. \]

(c) If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A \mid B) = \frac{2}{5} \), then find \( P(A \cap B) \).

(b) If \( \sin^{-1} \left( \frac{1}{2} \right) = \tan^{-1} x \), then find the value of \( x \).

(d) If \( P(A) = \frac{3}{13} \), \( P(B) = \frac{5}{13} \), and \( P(A \cap B) = \frac{2}{13} \), find the value of \( P(B | A) \):

A die is thrown two times. It is found that the sum of the appeared numbers is 6. Find the conditional that the number 4 appeared at least one time.

There are two children in a family. If it is known that at least one child is a boy, find the that both children are boys.

Prove that the \( f(x) = x^2 \) is continuous at \( x \neq 0 \).

Differentiate the \( \sin mx \) with respect to \( x \).

The principal value of the \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) will be:

Minimize Z = 5x + 3y \text{ subject to the constraints} \[ 4x + y \geq 80, \quad x + 5y \geq 115, \quad 3x + 2y \leq 150, \quad x \geq 0, \quad y \geq 0. \]

{(a)Function \( f : \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = 5x, \forall x \in \mathbb{R} \). Select the correct answer:}

(c) If the unit vectors \( \vec{a}, \vec{b}, \vec{c} \) are such that \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), find the value of \( \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \):

(b) If \( A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix} \), find \( A^{-1} \).

Find the value of \[ \int \frac{\sec^2 2x}{(\cot x - \tan x)^2} \, dx. \]

If the matrices \( X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} \) and \( X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix} \), then find the matrices \( X \) and \( Y \).

If \( y = (\cot x)^{\sin x} + x^x \), then find \( \frac{dy}{dx} \).

If \[ A = \begin{bmatrix} 3 & 1 & 1 \\ 15 & 6 & 5 \\ 5 & 2 & 2 \end{bmatrix} \] then find \( A^{-1} \).

Solve the following system of equations by using matrix method: \[ 3x - 2y + 3z = 8 \] \[ 2x + y - z = 1 \] \[ 4x - 3y + 2z = 4 \]

(d) If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \), show that \( A^2 - 5A + 7I = 0 \) and find \( A^{-1} \):

(d) If \[ \begin{bmatrix} 2x - y & x + 2y \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}, \] then the values of \( x \) and \( y \) will be:

(e) If \( y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that \[ \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a}. \]

(b) Show that \[ \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \] :