List of top Mathematics Questions asked in UP Board XII

If \( y = A \cos \theta + B \sin \theta \), then prove that \( \frac{d^2y}{d\theta^2} = -y \).
Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \).
If \( A = \{1, 2\} \) and \( B = \{3, 4, 5\} \), then find all number of relations from A to B.
If \( R_1 \) and \( R_2 \) be two equivalence relations on a set A, then prove that \( (R_1 \cap R_2) \) also be an equivalence relation on A.
Solve the differential equation \( \frac{dy}{dx} = e^x \cos x \).
If \( y = A + Be^x \), then prove that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \), where A and B are constants.
Show that \( f(x) = |x| \) is continuous for all values of x.
Solve the differential equation \( \frac{dy}{dx} = \frac{x^2 - 1}{y^2 + 1} \).
The value of \( \int \cos^2 x \, dx \) will be:
If A is a square matrix and \( A^2 = A \), then \( (A + I)^3 - 7A \) will be:
A function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = 3x \) for all \( x \in \mathbb{R} \). Then function \( f \) will be:
If the vector \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) is perpendicular to the vector \( 5\hat{i} - \lambda\hat{j} + 2\hat{k} \), the value of \( \lambda \) is:
Find the particular solution of the differential equation \( \frac{dy}{dx} + y \cot x = 2x + x^2 \cot x, (x \neq 0) \). It is given that \( y = 0 \) if \( x = \frac{\pi}{2} \).
At which point is the slope of the curve \( y^2 = 4x \) equal to the slope of the line \( y = x + 3 \)?
Prove that: \( \int_0^{\pi} \log(1 + \cos x) dx = -\pi \log_e 2 \)
Solve the following system of linear equations by matrix method:
\( x + y + z = 6 \)
\( y + 3z = 11 \)
\( x + z = 2y \)
Evaluate: \( \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \)
Solve the differential equation \( ydx - (x + 2y^2) dy = 0 \).
Find the angle between the pair of lines:
\( \vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k}) \) and
\( \vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k}) \).
Find the shortest distance between the lines:
\( \frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1} \) and
\( \frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1} \).
Find the maximum value of Z = 8x + 5y under the constraints \( 5x + 3y \le 15 \), \( 2x + 5y \le 10 \) and \( x \ge 0, y \ge 0 \) by graphical method.
Find the area of the region enclosed by the parabola \( y^2 = 4ax \) and its latus rectum.
There are three groups of children having 3 girls and one boy, 2 girls and 2 boys, one girl and 3 boys respectively. One child is selected at random from each group. Find the probability that the three selected children have one girl and 2 boys.
Find the equations of the tangent and normal to the curve \( x^{2/3} + y^{2/3} = 2 \) at the point (1, 1).
Find the area of a parallelogram whose adjacent sides are given by vectors \( \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} \).