List of practice Questions

Evaluate \( \tan \left[ \cos^{-1} \left( \tan \frac{3\pi}{4} \right) \right] \).

Find the angle between the following pair of lines : \( \frac{x-2}{3} = \frac{y+5}{2} = \frac{1-z}{-6} \) and \( \frac{x-7}{1} = \frac{y}{2} = \frac{6-z}{-2} \).

Determine the interval(s) in which \( f(x) = \frac{x{3} + \frac{3}{x}, x \neq 0 \) is increasing.}

Differentiate \( x^x \) with respect to \( x \log x \).

The order and degree of differential equation \( y = \left( \frac{d^2y}{dx^2} \right)^2 - \lambda \frac{dy{dx} \) is :}

The corner points of the feasible region determined by the system of linear constraints are \( (0, 0) \), \( (0, 40) \), \( (20, 40) \), \( (60, 20) \) and \( (60, 0) \). If the objective function of an LPP is \( Z = 4x + 3y \), then the maximum value is :

An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :

The general solution for the differential equation \( \frac{dy}{dx} = e^{3x-y} \) is :

Derivative of \( \cos^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right) \), \( -\frac{\pi}{4} < x < \frac{\pi}{4} \) with respect to \( x \) is :

If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :

If \( 3P(A) = P(B) = \frac{3{5} \) and \( P(A|B) = \frac{2}{3} \), then \( P(A \cup B) \) is :}

If the area of \( \triangle ABC \) with vertices \( A(3, 1) \), \( B(-2, 1) \) and \( C(0, k) \) is 5 sq. units, then values of \( k \) are :

\( \int \sqrt{\frac{1 + \cos x}{1 - \cos x}} \, dx \) is equal to :

If \( \int_0^1 (6x^2 - 4x + k) \, dx = 0 \), then the value of \( k \) is :

If position vector \( \vec{p} \) of a point \( (24, n) \) is such that \( |\vec{p}| = 25 \), then the value of \( n \) is :

For any square matrix \( A \) with real entries, if \( A + A' \) is a symmetric matrix then :

A matrix \( B = [b_{ij}]_{m \times m} \) is said to be a diagonal matrix, if :

If \( A \) is a non-singular matrix, then which of the following is not true ?

For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?

Let \( A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix} \). If \( A + B + C = O \), then matrix \( C \) is :

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost Rs.60, 2 apples, 4 oranges and 6 bananas cost Rs.90, whereas 6 apples, 2 oranges and 3 bananas cost Rs.70. Using matrix method, find the cost of one fruit of each kind.

If \( y\sqrt{x^2+1} = \log(\sqrt{x^2+1} - x) \), show that \( (x^2+1)\frac{dy{dx} + xy + 1 = 0 \).}

Find the differential of \( x^{\cot x} + \frac{2x^2-3}{2x^2-x+2} \) with respect to \( x \).

Sketch the curve \( \{(x, y) : 100x^2 + 25y^2 = 2500\ \) and find the area of the region enclosed by it, using integration.}