List of top Mathematics Questions asked in UP Board XII

Solve: \(\int \frac{3x - 5}{x^3 - x^2 - x + 1} dx\)
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 the system of equations
x + y + z = 2,
2x + y - 3z = 0,
x - y + z - 4 = 0 by matrix method.
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?
Show that (3\(\hat{i}\) - 4\(\hat{j}\) - 4\(\hat{k}\)), (2\(\hat{i}\) - \(\hat{j}\) + \(\hat{k}\)) and (\(\hat{i}\) - 3\(\hat{j}\) - 5\(\hat{k}\)) are the position vectors of vertices of a right angle triangle.
Find the interval in which the given function \( f(x) = x^2 - 4x + 6 \) is (i) Increasing (ii) Decreasing.
Find the differential coefficient of \(y = x^{\cos x} + (\sin x)^x\) with respect to x.
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 shortest distance between the lines \(\vec{r} = (2\hat{i} + \hat{j} – \hat{k}) + \mu(3\hat{i} – 5\hat{j} + 2\hat{k})\) and \(\vec{r} = (\hat{i} + \hat{j}) + \lambda(2\hat{i} - \hat{j} + \hat{k})\).
Find the minimum value of Z = 3x + 7y by the graphical method under the following constraints:
\(x + y \le 8, 3x + 5y \ge 0, x \ge 0, y \ge 0\)
If \( \vec{a}, \vec{b} \) and \( \vec{c} \) are vectors and \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), then find the value of \( (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) \).
Find the value of \( \int_{-\pi/2}^{\pi/2} \sin^2 x \, dx \).
Solve the differential equation \( \frac{dy}{dx} = e^x \cos x \).
Find the Cartesian equation of a line which passes through point (3, -2, -5) and is parallel to the vector \( (3\hat{i} + 2\hat{j} - 2\hat{k}) \).
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.
If \( y = A \cos \theta + B \sin \theta \), then prove that \( \frac{d^2y}{d\theta^2} = -y \).
Show that \( f(x) = |x| \) is continuous for all values of x.
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 \( y = A + Be^x \), then prove that \( \frac{d^2y}{dx^2} - \frac{dy}{dx} = 0 \), where A and B are constants.
Solve the differential equation \( \frac{dy}{dx} = \frac{x^2 - 1}{y^2 + 1} \).
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 \( x = \frac{\pi}{2} \).