List of practice Questions

Evaluate : \( \int_{-\pi/6}^{\pi/2} (\sin |x| + \cos |x|) dx \)

Solve the following differential equation : \( x \frac{dy}{dx} = y - x \sin^2 \left( \frac{y}{x} \right) \), given that y(1) = \( \frac{\pi}{6} \).

Find the general solution of the differential equation : \( y \log y \frac{dx}{dy} + x = \frac{2}{y} \).

Solve the following linear programming problem graphically :
Maximize \( Z = 4500x + 5000y \)
Subject to constraints
\( x + y \leq 250 \)
\( 25x + 40y \leq 7000 \)
\( x \geq 0, y \geq 0 \)

Find the absolute maximum value of \( f(x) = \cos x + \sin^2 x, x \in [0, \pi] \).

If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.

If \( \vec{AB} = \hat{j} + \hat{k} \) and \( \vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k} \) represent the two vectors along the sides \( AB \) and \( AC \) of \( \Delta ABC \), prove that the median \( \vec{AD} = \frac{\vec{AB} + \vec{AC}}{2} \), where \( D \) is midpoint of \( BC \). Hence, find the length of median \( AD \).

Mother, Father and Son line up at random for a family picture. Let events E : Son on one end and F : Father in the middle. Find P(E/F).

Assertion (A) : A line can have direction cosines \( \) Reason (R) : \( \cos \theta = 1 \) is possible for \( \theta = 0 \).

Find the co-ordinates of the point on line \( x = \frac{y-1}{2} = \frac{z-2}{3} \) whose \( y \) coordinate is 3 times the \( x \) coordinate.

Check whether \( f : Z \times Z \rightarrow Z \times Z \) (where \( Z \) is the set of integers) defined as \( f(x, y) = (2y, 3x) \) is injective or not.

If \( x = e^{\sin^{-1} t}, y = e^{\cos^{-1} t} \), find \( \frac{dy}{dx} \) at \( t = \frac{1}{\sqrt{2}} \).

Which of the following expressions will give the area of region bounded by the curve \( y = x^2 \) and line \( y = 16 \)?

The general solution of the differential equation: \( x^2 dy + y^2 dx = 0 \) is

The integrating factor of the differential equation \( 2x \frac{dy}{dx} - y = 3 \) is

If \( |\vec{a}| = 5 \) and \( -2 \leq \lambda \leq 1 \), then the sum of greatest and the smallest value of \( |\lambda \vec{a}| \) is

Vector of magnitude 3 making equal angles with \( x \) and \( y \) axes and perpendicular to \( z \) axis is

If \( A = \begin{bmatrix} \tan x & \cot x \\ -\cot x & \tan x \end{bmatrix} \) and \( A + A' = 2I \), then value of \( x \in [0, \pi/2] \) is

If \( \begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86 \), then the sum of all possible values of \( a \) is

If \( x + y = xy \), then \( \frac{dy}{dx} \) is

\( \int \frac{dx}{\sec x + \tan x} \) is equal to

A relation \( R \) on set \( A = \{1, 2, 3\} \) is defined as \( R = \{(1, 3), (3, 3), (1, 1), (2, 2), (3, 1)\ \) is}

If \( A \) and \( B \) are square matrices of same order, then which of the following statements is/are always true?
(i) $(A+B)(A-B)=A^2-B^2$
(ii) $AB=BA$
(iii) $(A+B)^2=A^2+AB+BA+B^2$
(iv) $AB=0\Rightarrow A=0\text{ or }B=0$}

If \( A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix} \) is a symmetric matrix, then the value of \( 3a + b + c \) is

Direction cosines of line \( \frac{1 - x}{0} = y = z \) are