List of top Mathematics Questions asked in UP Board XII

If the Cartesian equation of a line is \( \frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2} \), then find its equation in vector form.
Find the degree of the differential equation \( xy \frac{d^2y}{dx^2} + x \left(\frac{dy}{dx}\right)^2 - y\left(\frac{dy}{dx}\right) = 2 \)
Prove that the function f(x) = |x| is continuous at x = 0.
Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.
The value of \( \int_{0}^{\pi/2} \frac{dx}{1 + \sqrt{\tan x}} \) will be

Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where

\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]

Find the minimum value of  ( z = x + 3y ) under the following constraints: 

• x + y ≤ 8
• 3x + 5y ≥ 15
• x ≥ 0, y ≥ 0

(a) If orders of matrices A and B are \( p \times q \) and \( q \times r \) respectively, then order of AB is:
Find the principal value of \( \sin^{-1} \left( \sin \frac{7\pi}{4} \right) \).
(d) If vectors \( \mathbf{5i - \lambda j + 2k} \) and \( \mathbf{2i + 3j + 4k} \) are perpendicular, find \( \lambda \):
(b) At which point is the slope of the line \( y = x + 1 \) equal to the slope of the curve \( y^2 = 4x \)?
Evaluate: \[ I = \int_0^{\pi} \frac{x \, dx}{a^2 \cos^2 x + b^2 \sin^2 x} \]

There are 10 black and 5 white balls in a bag. Two balls are taken out, one after another, and the first ball is not placed back before the second is taken out. Assume that the drawing of each ball from the bag is equally likely. What is the probability that both the balls drawn are black? 

Solve the differential equation \[ \frac{dy}{dx} = e^x \sin x. \]
Find the probability of 53 Sundays in a leap year.
If \( y = \frac{x^2 + 3x + 4}{e^x \cos x} \), find \( \frac{dy}{dx} \).
Solve the differential equation \[ \frac{dy}{dx} = \frac{1 + x^2}{1 + y^2} \]

Find the value of \[ \int \frac{\sec^2 2x}{(\cot x - \tan x)^2} \, dx. \] 

(c) If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A \mid B) = \frac{2}{5} \), then find \( P(A \cap B) \).
Find the perpendicular unit vectors on the vectors \[ \mathbf{a} = 2\hat{i} - \hat{j} + \hat{k} \quad \text{and} \quad \mathbf{b} = 3\hat{i} + 4\hat{j} - \hat{k} \] and find the sine of the angle between them.
If \( y = Ae^x + B \) where \( A, B \) are constants, then show that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \).
(c) The value of the integral \( \int \sqrt{1 + \sin 2x} \,dx \) is:

(b) Order of the differential equation: $ 5x^3 \frac{d^3y}{dx^3} - 3\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0 $

Differentiate \( y = x^x + (\cos x)^{\tan x} \) with respect to \( x \).

The probabilities of solving a question by \( A \) and \( B \) independently are \( \frac{1}{2} \) and \( \frac{1}{3} \) respectively. If both of them try to solve it independently, find the probability that: