List of top Questions asked in UP Board XII

Find the maximum value of \( Z = 4x + y \) under the given constraints by graphical method: \[ x + y \leq 50, \quad 3x + y \leq 90, \quad x \geq 0, \quad y \geq 0. \]
Find the value of \( p \) such that the lines \[ \frac{1 - x}{3} = \frac{7y - 14}{2p} = \frac{z - 3}{2} \] and \[ \frac{7 - 7x}{3p} = \frac{y - 5}{1} = \frac{6 - z}{5} \] are mutually perpendicular.
Show that the points \( (2, -1, 1) \), \( (1, -3, -5) \) and \( (3, -4, -4) \) are the vertices of a right-angled triangle.
Find such two positive numbers whose sum is 15 and sum of their squares is minimum.
Find the area of the circle \( x^2 + y^2 = a^2 \) surrounded by it.
Find the value of \( \int \frac{x+2}{2x^2+6x+5} \,dx \).
If the position vectors of the points \( A, B, C, D \) are successively \( \hat{i} + \hat{j} + \hat{k} \), \( 2\hat{i} + 5\hat{j} \), \( 3\hat{i} + 2\hat{j} - 3\hat{k} \) and \( \hat{i} - 6\hat{j} + \hat{k} \), then find the angle between the lines \( AB \) and \( CD \).
If \( y = x^3 + \tan x \), then find \( \frac{d^2 y}{dx^2} \).
Find the area of a parallelogram whose adjacent sides are \( \vec{a} = 3\hat{i} + \hat{j} + 4\hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + \hat{k} \).
Find the value of \( \int x^2 e^{x^3} dx \).
Solve the differential equation \( y \log y \, dx - x \, dy = 0 \).
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 \( \theta \).
Find the order of the differential equation \[ \frac{d^3y}{dx^3} + x^2 \left(\frac{d^2y}{dx^2}\right)^3 + \frac{dy}{dx} + y = 0. \]
Find the value of \( \frac{dy}{dx} \) of the curve \( x = t^2 + 3t - 8 \), \( y = 2t^2 - 2t - 5 \) at the point \( (2, -1) \).
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.
Prove that \[ \int_0^{\pi/2} \log \sin x \, dx = -\frac{\pi}{2} \log 2. \]
Solve the system of linear equations by matrix method: \[ 3x - 2y + 3z = 8 \] \[ 2x + y - z = 1 \] \[ 4x - 3y + 2z = 4 \]
Find the area bounded by the circle \[ x^2 + y^2 - 2x = 8. \]
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 value of \[ \int_0^{\pi/2} \frac{x}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx. \]
Find the value of \[ \int \frac{x}{(x-a)(x-b)(x-c)} \, dx. \]
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 \[ \frac{1}{2} \left| \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \right|. \]
(b) Show that the points \( A(2,3,4) \), \( B(-1,-2,1) \), and \( C(5,8,7) \) are collinear.
Minimize \( Z = 3x + 9y \) under the following constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x \leq y, \quad x \geq 0, \quad y \geq 0. \]
(c) Find the value of \( \int \sqrt{5 + 2x + x^2} \, dx \).