List of top Mathematics Questions

If \( \vec{a} = 2\vec{i} - \vec{j} + 6\vec{k} \); \( \vec{b} = \vec{i} - \vec{j} + \vec{k} \) and \( \vec{c} = 3\vec{j} - \vec{k} \), then \( \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \)
If \( \tan A + \tan B + \tan C = \tan A \tan B \tan C \) and \( A + B + C = \pi \), then which of the following is true?
In triangle $ ABC $, $ 2A + C = 300^\circ $. If the circumradius is 8 times the inradius, then $ \sin\frac{C}{2} = ? $
Evaluate the integral: \[ I = \int_{\pi/6}^{\pi/3} \cos^{-4} x \, dx \]

$\left( 1 + \sqrt{5} + i \sqrt{10 - 2\sqrt{5}} \right)^5$

If the angle between the lines joining the origin to the points of intersection of \( x + 2y + \lambda = 0 \) and \( 2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0 \) is \( \dfrac{\pi}{2} \), then a value of \( \lambda \) is:
Evaluate $ \lim_{x \to 2} \frac{x^2 - 4}{x - 2} $.
If the equation of the circle having the common chord to the circles $$ x^2 + y^2 + x - 3y - 10 = 0 $$ and $$ x^2 + y^2 + 2x - y - 20 = 0 $$ as its diameter is $$ x^2 + y^2 + \alpha x + \beta y + \gamma = 0, $$ then find $ \alpha + 2\beta + \gamma $.
If the average number of accidents occurring at a particular junction on a highway in a week is 5, then the probability that at most one accident occurs in a particular week is:
$\int \frac{\sin 2x}{\sin^2 x + 3 \cos x - 3} \, dx =$
Identify the correct option from the following:
If g is the inverse of the function f(x) and \( g(x) = x + \tan x \) then, \( f'(x) = \)
The radius of the circle passing through the points of intersection of the circles \( x^2+y^2+2x+4y+1=0 \), \( x^2+y^2-2x-4y-4=0 \), and intersecting the circle \( x^2+y^2=6 \) orthogonally is:
If the roots of the quadratic equation \( 2x^2 - 4x + k = 0 \) are real and equal, find the value of \( k \).
If $$ \int \frac{a \cos x + 3 \sin x}{5 \cos x + 2 \sin x} dx = \frac{26}{29} x - \frac{k}{29} \log |5 \cos x + 2 \sin x| + c, $$ then find $ |a + k| $.
If \[ \int \frac{dx}{(x \tan x + 1)^2} = f(x) + c, \] then \(\lim_{x \to \frac{\pi}{2}} f(x)\) is?
If $x>\sqrt{3$ and $\frac{x^2+1}{(x^2+2)(x^2+3)}$ is expanded in terms of powers of $x$, then the coefficient of $x^{-8}$ is}
If \( \alpha, \beta, \gamma \) are the roots of the equation \[ x^3 + px^2 + qx + r = 0, \] then \[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) =\ ? \]
If \( \alpha, \beta \) are the acute angles such that \( \frac{\sin \alpha}{\sin \beta} = \frac{6}{5} \) and \( \frac{\cos \alpha}{\cos \beta} = \frac{9}{5\sqrt{5}} \) then \( \sin \alpha = \)
The slope of one of the direct common tangents drawn to the circles \(x^2 + y^2 - 2x + 4y + 1 = 0\) and \(x^2 + y^2 - 4x - 2y + 4 = 0\) is
The tangents drawn to the hyperbola \(5x^2 - 9y^2 = 90\) through a variable point \(P\) make angles \(\alpha\) and \(\beta\) with its transverse axis. If \(\alpha\) and \(\beta\) are complementary angles, then the locus of \(P\) is
Evaluate: \( \int_0^{400\pi} \sqrt{1 - \cos 2x} \, dx \)
Given \(f(x) = x^2 - 5x + 4\). Out of first 20 natural numbers, if a number \(x\) is chosen at random, then the probability that the chosen \(x\) satisfies the inequality \(f(x)>10\) is
Evaluate the limit: \[ \lim_{x \to \infty} \frac{3x+4\cos^2x}{\sqrt{x^2-5\sin^2x}} \]
Consider the following statements
Statement-I: A function \( f: A \rightarrow B \) is said to be one-one if and only if \[ f(x) = f(y) \Rightarrow x = y \]
Statement-II: A relation \( f: A \rightarrow B \) is said to be a function if \[ x = y \Rightarrow f(x) \neq f(y) \]
Then which one of the following is true?
If \( \frac{1}{2} \leq \frac{x^2+x+a}{x^2-x+a} \leq 2 \ \forall x \in \mathbb{R} \), then \( a = \)