List of top Mathematics Questions asked in UP Board XII

Solve the differential equation \( y \log y \, dx - x \, dy = 0 \).
If \( y = x^3 + \tan x \), then find \( \frac{d^2 y}{dx^2} \).
The angle between two vectors \( \vec{a} \) and \( \vec{b} \) is 0 and \( |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \) is given. Find the value of \( \theta \).
The value of the determinant \( \begin{vmatrix} -2 & 5
-4 & 1 \end{vmatrix} \) will be:
If matrix \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \) and \( A + A' = I \), then the value of \( \alpha \) will be:
The differential coefficient of the 6266899d2bbfcb1799af2d57 \( \sin(x^2 + 5) \) with respect to \( x \) will be:
Find the value of \( \frac{dy}{dx} \) of the curve \( x = t^2 + 3t - 8 \), \( y = 2t^2 - 2t - 5 \) at the point \( (2, -1) \).
Find the order of the differential equation \[ \frac{d^3y}{dx^3} + x^2 \left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} + y = 0. \]
Differentiate: \( y = (\tan x)^{\cot x + (\sin x)^{\cos x} \).}
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.}
If the position vectors of the vertices \( A, B, C \) of a triangle \( \Delta ABC \) are \( \vec{a}, \vec{b}, \vec{c} \), respectively, then prove that the area of \( \Delta ABC \) is \[ \frac{1}{2} \left| \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \right|. \]
If \[ A = \begin{bmatrix} 1 -1 0
2 3 -2
-2 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 -15 & 5
-1 6 & -2
1 -5 & 2 \end{bmatrix}, \] then find the value of \( (AB)^{-1} \).
Find the value of \[ \int \frac{x}{(x-a)(x-b)(x-c)} \, dx. \]
Solve the system of linear equations by matrix method: \[ 3x - 2y + 3z = 8 \] \[ 2x + y - z = 1 \] \[ 4x - 3y + 2z = 4 \]
Find the value of \[ \int_0^{\pi/2} \frac{x}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx. \]
There are 4 red and 4 black balls in a bag and another bag contains 2 red and 6 black balls. One bag is randomly selected and a red ball is drawn. Find the probability that the red ball is drawn from the first bag.
Find the particular solution of the differential equation \[ \frac{dy}{dx} + y \cot x = 2x \quad (x \neq 0), \quad \text{when } y = 0 \text{ if } x = \frac{\pi}{2}. \]
Find the area bounded by the circle \[ x^2 + y^2 - 2x = 8. \]
Prove that \[ \int_0^{\pi/2} \log \sin x \, dx = -\frac{\pi}{2} \log 2. \]
If \( x \sqrt{1 + y} + y \sqrt{1 + x} + x = 0 \) for \( -1<x<1 \), then prove that \( \frac{dy}{dx} = -\frac{1}{(1+x)^2 \).}
(c) Find the value of \( \int \sqrt{5 + 2x + x^2} \, dx \).
(d) If \( y = x^x \), then find \( \frac{dy}{dx} \).
Show that a relation \( R = \{(a, b) : (a - b) \text{ is a multiple of 5 \} \) on the set \( Z \) of integers is an equivalence relation.}
Minimize \( Z = 3x + 9y \) under the following constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]
Find the shortest distance between the lines: \[ \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}), \] \[ \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}). \]