List of top Higher Mathematics Questions

If \( y = \sqrt{e^x} \), \( x > 0 \), then \( \frac{dy}{dx} = \underline{\hspace{2cm}} \)

Match the pairs correctly: 

(i)  \( \int \tan x \, dx \)  \(\hspace{3.5cm}\)  \( \log |\sin x| + c \) 
(ii)  \( \int \cot x \, dx \)  \(\hspace{3.5cm}\)  \( \log |\csc x| - \cot x + c \) 
(iii)  \( \int \sec x \, dx \)  \(\hspace{3.5cm}\)  \( \log |\sec x + \tan x| + c \) 
(iv)  \( \int \csc x \, dx \)  \(\hspace{3.5cm}\)  \( -\log |\csc x + \cot x| + c \) 
(v)  \( \int \frac{\cos x}{\sin x} \, dx \)  \(\hspace{3.5cm}\)  \( \log |\sin x| + c \) 
(vi)  Derivative of \( \sin 2x \) with respect to \( x \)  \(\hspace{0.75cm}\)  \( 2 \cos 2x \) 

What is column matrix?

Order of differential equation \( xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} = 0 \) is 2.

If \( A = \{1, 2, 3\} \), \( B = \{4, 5, 6, 7\} \), and \( f = \{(1, 4), (2, 5), (3, 6)\} \) is a function from A to B, then show that \( f \) is one-one.

Examine that the relation \( R \) in the set \( \{1, 2, 3, 4\} \) given by \( R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\} \) is reflexive and transitive but not symmetric.

Prove that \( \sin^{-1}(-x) = -\sin^{-1}(x) \), where \( x \in [-1, 1] \)

Prove that \( \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \), where \( x \in [-1, 1] \)

If \( A = \begin{pmatrix} -2 & 3 \\ 1 & 2 \end{pmatrix} \) and \( B = \begin{pmatrix} -1 & 0 \\ 1 & 2 \end{pmatrix} \), then find \( (A + B) \).
If \( A = \begin{pmatrix} 1 & -4 \\ 3 & 3 \end{pmatrix} \) and \( B = \begin{pmatrix} -1 & 2 \\ 1 & 2 \end{pmatrix} \), then find \( AB \).

Show that the function given by \( f(x) = e^{3x} \) is increasing on \( R \).

Find the area of a parallelogram whose adjacent sides are given by the vectors \( \mathbf{a} = \hat{i} - \hat{j} + 3 \hat{k} \) and \( \mathbf{b} = 2 \hat{i} - 7 \hat{j} + \hat{k} \).

Find the area of the region bounded by the curve \( y = x^2 \,\,\, \text{and the line} \,\,\, y = 4. \)

Find the area of the region bounded by the curve \( y^2 = 4x \,\,\, \text{and the line} \,\,\, x = 3. \)

Find the general solution of the differential equation \[ x \frac{dy}{dx} + 2y = x^2 \,\,\, \text{where} \,\,\, (x \neq 0). \]

Prove that: \[ \left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right| = (a + b + c)^3 \]
Solve the equation: \[ \left| \begin{matrix} x+a & x & x \\ x & x+a & x \\ x & x & x+a \end{matrix} \right| = 0 \] where \( a \neq 0 \).
Evaluate the integral: \[ \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx \]

Find the shortest distance between parallel lines: \[ \vec{r_1} = \hat{i} + 2 \hat{j} - 4 \hat{k} + \lambda (2 \hat{i} + 3 \hat{j} + 6 \hat{k}) \,\,\, \text{and} \,\,\, \vec{r_2} = 3 \hat{i} + 3 \hat{j} - 5 \hat{k} + \mu (2 \hat{i} + 3 \hat{j} + 6 \hat{k}) \]

Find the shortest distance between the lines: \[ \vec{r_1} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - 2t) \hat{k} \,\,\, \text{and} \,\,\,\vec{r_2} = (s + 1) \hat{i} + (2s - 1) \hat{j} - (2s + 1) \hat{k} \]

Differentiate \( \sin(x^2) \) with respect to \( x^2 \).
Evaluate the integral: \[ \int_0^{\pi/2} \frac{\cos^5 x}{\sin^5 x + \cos^5 x} \, dx \]
Prove that if function \( f \) is differentiable at a point \( a \), then it is also continuous at that point.
Find the general solution of the differential equation \[ \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2}. \]
An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. Suppose that the probability of drawing each ball is the same. What is the probability that both drawn balls are black?