List of practice Questions

Solve the following differential equation: \[ (\tan^{-1}y - x) \, dy = (1 + y^2) \, dx. \]
Find the equation of a line \( l_2 \) which is the mirror image of the line \( l_1 \) with respect to line \( l : \frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} \), given that line \( l_1 \) passes through the point \( P(1, 6, 3) \) and is parallel to line \( l \).
A relation \( R \) on set \( A = \{1, 2, 3, 4, 5\} \) is defined as \[ R = \{(x, y) : |x^2 - y^2| <8\}. \] Check whether the relation \( R \) is reflexive, symmetric, and transitive.
If \( \sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y) \), prove that \( \frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} \).
Find: \[ \int \frac{x^2}{(x^2 + 4)(x^2 + 9)} \, dx. \]
E and F are two independent events such that \( P(\overline{E}) = 0.6 \) and \( P(E \cup F) = 0.6 \). Find \( P(F) \) and \( P(\overline{E} \cup \overline{F}) \).
Find: \[ \int x \sqrt{1 + 2x} \, dx \]
Evaluate: \[ \int_0^{\frac{\pi^2}{4}} \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx \]
A linear programming problem deals with the optimization of a/an:
If \( P(A|B) = P(A'|B) \), then which of the following statements is true?
If the direction cosines of a line are \( \sqrt{3}k, \sqrt{3}k, \sqrt{3}k \), then the value of \( k \) is:
Let \( f(x) \) be a continuous function on \([a, b]\) and differentiable on \((a, b)\). Then, this function \( f(x) \) is strictly increasing in \((a, b)\) if:
If \( f(x) \) is an odd function, then: \[ \int_{-\pi/2}^{\pi/2} f(x) \cos^3 x \, dx \, \text{equals:} \]
Let \( \theta \) be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \) such that \( \sin \theta = \frac{3}{5} \). Then, \( \hat{a} \cdot \hat{b} \) is equal to:
If a matrix has 36 elements, the number of possible orders it can have, is:
The distance of point \( P(a, b, c) \) from the y-axis is:
The number of corner points of the feasible region determined by constraints \( x \geq 0, y \geq 0, x + y \geq 4 \) is:
If matrices \( A \) and \( B \) are of order \( 1 \times 3 \) and \( 3 \times 1 \) respectively, then the order of \( A'B' \) is:
The solution of the differential equation \( \frac{dy}{dx} = \frac{1}{\log y} \) is:
Who was Subbu’s principal?
A ball falls from a height of 20 m on the floor and rebounds to a height of 5 m. Time of contact is 0.02 s. Find the acceleration during impact
Replace * in each of the following by the correct number : 
  1. \(\frac{2}{7} = \frac{8}{*}\)
  2. \(\frac{5}{8} =\frac{10}{*}\)
  3. \(\frac{3}{5} =\frac{*}{20}\)
  4. \(\frac{45}{ 60}= \frac{15}{*}\)
  5. \(\frac{18}{24} =\frac{*}{4}\)
Find area bounded by $y^2 ≤ 2x$ and $y ≥ 4x – 1$
For the data:
Xi13579
fi42428𝞪8
If the mean of data is 5 and mean deviation about mean is M and variance is 𝝈2 , then 3𝞪/M+𝝈2 is?
Give three examples of shapes with no line of symmetry.