List of top Mathematics Questions

Evaluate: \[ \cos x \cos 7x - \cos 5x \cos 13x \]
If \( \tan \theta = \dfrac{1}{3} \), then find \( \cos 2\theta \).
Given below is the probability distribution of a discrete random variable \( X \):
\[ \begin{array}{c|cccccc} X=x & 1 & 2 & 3 & 4 & 5 & 6
\hline P(X=x) & k & 0 & 2k & 5k & k & 3k \end{array} \] Then find \( P(X \ge 4) \).
The dual of the statement pattern \( \sim p \land (q \lor t) \) is (where \( t \) is a tautology and \( c \) is a contradiction)
If \( \tan^{-1}\!\left(\dfrac{1-x}{1+x}\right) - \dfrac{1{2}\tan^{-1}x = 0 \), for \( x>0 \), then the value of \( x \) is}
In \( \triangle ABC \) with usual notations, \( a = 4 \), \( b = 3 \), \( \angle A = 60^\circ \), then \( c \) is a root of the equation
For a sequence if \( S_n = \dfrac{5^n - 2^n}{2^n} \), then its fourth term is
If the vectors \( \vec{a} = \hat{i} - 2\hat{j} + \hat{k} \), \( \vec{b} = 2\hat{i} - 5\hat{j} + p\hat{k} \) and \( \vec{c} = 5\hat{i} - 9\hat{j} + 4\hat{k} \) are coplanar, then the value of \( p \) is
The equation of the line passing through the point \( (2, 3, -4) \) and perpendicular to the XOZ plane is
If the function \( f \) defined by \[ f(x) = \begin{cases} K(x - x^2), & 0<x<1
0, & \text{otherwise} \end{cases} \] is the p.d.f. of a random variable \( X \), then the value of \( P(X<\tfrac{1}{2}) \) is
If \( f : \mathbb{R} \rightarrow \mathbb{R} \), \( g : \mathbb{R} \rightarrow \mathbb{R} \) are two functions defined by \( f(x) = 2x - 3 \), \( g(x) = x^3 + 5 \), then find \( (f \circ g)^{-1(x) \).}
Evaluate the integral: \[ \int_{-a}^{a} x^2 \left( \frac{e^{x^3} - e^{-x^3}}{e^{x^3} + e^{-x^3}} \right) dx \]
The shortest distance between the lines \[ \vec{r} = (1 - t)\hat{i} + (t - 2)\hat{j} + (3 - 2t)\hat{k} \] and \[ \vec{r} = (p + 1)\hat{i} + (2p - 1)\hat{j} + (2p + 1)\hat{k} \] is
The probability that a person who undergoes a certain operation will survive is 0.2. If 5 patients undergo similar operations, find the probability that exactly four will survive.
If \[ A = \begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}, \] then \[ B^{-1} A^{-1} = \]
If \( A = \left[ \begin{array}{cc} 1 & 2 \\ 1 & 0 \end{array} \right] \), \( B = \left[ \begin{array}{cc} 1 & 2\\ 2 & 1 \end{array} \right] \), then \( (AB)^{-1} \) is

Among the given statements below 
 (a) \( \neg p \vee (\neg p \vee q) \) 
 (b) \( \neg q \wedge (\neg p \vee \neg q) \) 
 (c) \( (\neg p \vee \neg q) \wedge (p \vee \neg q) \) 
 (d) \( (\neg p \vee \neg q) \vee (p \vee \neg q) \)
 .............. is a tautology.
 

If the function given by \( f(x) = \left( \frac{4x + 1}{1 - 4x} \right)^{\frac{1}{3}} \) for \( x \neq 0 \) is continuous at \( x = 0 \), the value of \( f(0) \) is
If \( f(x) = \frac{2x + 3}{3x - 2}, \, x \neq \frac{2}{3}, \) then the function \( f \circ f \) is
The value of \( \sin^{-1} \left( -\frac{1}{2} \right) + \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) \) is

The integrating factor of the differential equation \(\sin y \frac{dy}{dx} = \cos y (1 - x \cos y)\) is 
 

The domain of the function \( f(x) = \sqrt{x} \) is
For every value of \( x \), the function \( f(x) = \frac{1}{a^x}, a>0 \) is
If \( f(x) = \sin^{-1} \left( \sqrt{\frac{1 - x}{2}} \right) \), then \[ f'(x) = \]
Evaluate \[ \int_0^1 \frac{x^2 - 2}{x^2 + 1} \, dx \]