List of top Mathematics Questions

If \( \int \frac{e^x (1 + \sin x)}{1 + \cos x} \, dx = e^x f(x) + C \), then \( f(x) \) is equal to:
Simplify \( i^{57} + \frac{1}{i^{25}} \) and find its value:
Evaluate the integral: \[ I = \int \frac{x + 3}{(x + 4)^2} e^x \,dx \]
What is the angle between the two straight lines \( y = (2 - \sqrt{3})x + 5 \) and \( y = (2 + \sqrt{3})x - 7 \)?
The shortest distance between the lines \[ \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{-1} \] and \[ \frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3} \] is:
The middle term in the expansion of \( \left(\frac{10}{x} + \frac{x}{10}\right)^{10} \) is:
Bag P contains 6 red and 4 blue balls, and bag Q contains 5 red and 6 blue balls. A ball is transferred from bag P to bag Q and then a ball is drawn from bag Q. What is the probability that the ball drawn is blue?
If \( P(B) = \frac{3}{5} \), \( P(A \mid B) = \frac{1}{2} \), and \( P(A \cup B) = \frac{4}{5} \), then the value of \( P(A \cup B)' + P(A' \cup B) \) is:
If \( \sec^2 \theta = \frac{4}{3} \), then the general value of \( \theta \) is:
If \[ A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix} \] and \[ kA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix}, \] then the values of \( k \), \( a \), and \( b \) respectively are:
If \( f(x) = \frac{x}{\sqrt{1 + x^2}} \), then \( (f \circ f)(x) \) is:
Find the mean deviation about the mean for the data: \( 4, 7, 8, 9, 10, 12, 13, 17 \)

Solution of \( 2^x + 2^{|x|} \geq 2\sqrt{2} \) is:

The area enclosed between the curve \( y = \log_e(x + e) \) and the coordinate axes is:
The derivative of \( e^{x^3} \) with respect to \( \log x \) is:
If the function \[ f(x) = \begin{cases} 1, & x \leq 2 \\ ax + b, & 2 < x < 4 \\ 7, & x \geq 4 \end{cases} \] is continuous at \( x = 2 \) and \( x = 4 \), then the values of \( a \) and \( b \) are:
Evaluate: \[ \cos^{-1} \frac{1}{2} + \sin^{-1} (1) + \tan^{-1} \frac{1}{\sqrt{3}} \]
The general solution of the differential equation given by: \[ \tan^{-1} x + \tan^{-1} y = c \]
The number of nonzero terms in the expansion of \[ (1 + 3\sqrt{2}x)^9 + (1 - 3\sqrt{2}x)^9 \] is:

Consider a curve \( y = y(x) \) in the first quadrant as shown in the figure. Let the area \( A_1 \) be twice the area \( A_2 \). The normal to the curve perpendicular to the line \[ 2x - 12y = 15 \] does NOT pass through which point? 
 

The determinant of the matrix:

is:

The domain of the function \[ f(x) = \frac{1}{\sqrt{9 - x^2}} \] is:
If \( n(A) = 4 \) and \( n(B) = 7 \), then the difference between the maximum and minimum value of \( n(A \cup B) \) is:
If \[ f(x) = \frac{x + |x|}{x} \] then the value of \[ \lim_{x \to 0} f(x) \] is:
Which of the following is an empty set ?