List of top Mathematics Questions asked in UP Board XII

At which point is the slope of the curve \( y^2 = 4x \) equal to the slope of the line \( y = x + 3 \)?
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 \( x = \frac{\pi}{2} \).
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:
Solve the following system of linear equations by matrix method:
\( x + y + z = 6 \)
\( y + 3z = 11 \)
\( x + z = 2y \)
Find the area of the region enclosed by the parabola \( y^2 = 4ax \) and its latus rectum.
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 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
\( \vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k}) \).
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).
Solve the differential equation \( ydx - (x + 2y^2) dy = 0 \).
The volume of a cube is increasing at the rate of 9 cm\(^3\)/s. If the length of its edge is 10 cm, then its surface area is increasing with which rate?
Find the angle between the pair of lines \( \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \) and \( \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \).
Show that the given function \( f(x) = \cos x \) is increasing in \( (\pi, 2\pi) \).
Find the area of a parallelogram whose adjacent sides are given by vectors \( \vec{a} = 3\hat{i} + \hat{j} + 2\hat{k} \) and \( \vec{b} = \hat{i} + 2\hat{j} - 2\hat{k} \).
If \( [a_{ij}] = 2i - j \), then determine a matrix A of order \( 2 \times 3 \).
If \( y = x^{x^{x^{.... \text{ad inf}}}} \), then prove that \( x\frac{dy}{dx} = \frac{y^2}{1 - y \log x} \).
Find the principal value of \( \sec^{-1}(-\sqrt{2}) \).
Find the order and degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + y \cdot \sin x = 0 \).
If P(A) = 0.3, P(B) = 0.4, then find P(A/B) if A and B are independent events.
The value of \( \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \) is :
If \( A = \begin{bmatrix} \cos\alpha & -\sin\alpha
\sin\alpha & \cos\alpha \end{bmatrix} \) and \( A + A' = I \), then find the value of \( \alpha \).
The order of the differential equation \( \frac{d^3y}{dx^3} + \frac{d^2y}{dx^2} + y \cos x = 0 \) is :
Solve the system of equations by matrix method:
\( 3x - 2y + 3z = 8 \)
\( 2x + y - z = 1 \)
\( 4x - 3y + 2z = 4 \)
The value of \( \int_1^{\sqrt{3}} \frac{dx}{1 + x^2} \) will be :