List of top Questions asked in UP Board XII

Where is the headquarters of the Reserve Bank of India?
Who presented the Central Budget on 23rd July, 2024?
If the value of marginal propensity to consume is 0.8, what will be the value of marginal propensity to save?
Who is called the father of Economics?
"What ought to be?" is a subject matter of
If \( A = \begin{bmatrix} 3 & 3 & 1 \\ 3 & 4 & 1 \\ 4 & 3 & 1 \end{bmatrix} \), then verify that \( A \, (\text{adj } A) = |A| I \) and find \( A^{-1} \).
Prove that: \(\int_0^{\pi/4} \log_e(1 + \tan \theta) \, d\theta = \frac{\pi}{8} \log_e 2\).
Solve the integral \(\int \frac{\sec^2 2\theta}{(\cot \theta - \tan \theta)^2} \, d\theta\).
Show that \( y = c_1 e^{ax} \cos(bx) + c_2 e^{ax} \sin(bx) \), where \( c_1, c_2 \) are constants, is a solution of the differential equation \( \frac{d^2y}{dx^2} - 2a\frac{dy}{dx} + (a^2 + b^2)y = 0 \).
If \( A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} \), then verify that \( A \cdot \text{adj}(A) = |A| I \) and find \( A^{-1} \).
If \( x\sqrt{1+y} + y\sqrt{1+x} = 0 \) for \( -1<x<1 \), then prove that \( \frac{dy}{dx} = -\frac{1}{(1+x)^2} \).
Express the matrix \( A = \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} \) as the sum of a symmetric and a skew-symmetric matrix.
Does the relation defined by \( R = \{(x, y) : y \text{ is divisible by } x\} \) on the set \( A = \{1, 2, 3, 4, 5, 6\} \), an equivalence relation?
Does the function \( f(x) = \begin{cases} x+5 & \text{if } x \le 1 \\ x-5 & \text{if } x > 1 \end{cases} \) continuous at \( x = 1 \)?
If \( \begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix} \), then find the values of \(x, y, z\).
A relation \(R = \{(a, b) : a = b - 1,\; b \geq 3\}\) is defined on set \(\mathbb{N}\), then
If \(y = 600 e^{-7x} + 500 e^{7x}\), then show that \(\frac{d^2y}{dx^2} = 49y\).
Find the area bounded by the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).
There are 10 white and 5 black balls in a bag. Two balls are drawn one by one. First ball is not placed back before the second is taken out. Assume that the taking out of each ball from the bag is equally likely. What is the probability that both balls taken out are white?
Prove that \(|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|\) is always true for any two vectors \(\vec{a}\) and \(\vec{b}\).
Solve: \(\int \frac{3x - 5}{x^3 - x^2 - x + 1} dx\)
Solve the system of equations
x + y + z = 2,
2x + y - 3z = 0,
x - y + z - 4 = 0 by matrix method.
Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.
There are two children in a family. It is known that there is at least one child is a boy. Then find the probability that both children are boys.
Find the interval in which the given function \( f(x) = x^2 - 4x + 6 \) is (i) Increasing (ii) Decreasing.