List of top Mathematics Questions

If \( L_1 \) and \( L_2 \) are two parallel lines and \( \triangle ABC \) is an equilateral triangle as shown in the figure, then the area of triangle \( ABC \) is:

The sum of roots of the equation \[ |x - 1|^2 - 5|x - 1| + 6 = 0 \] is:

Evaluate the limit: \[ \lim_{x \to 0} \frac{\sin(2x) - 2\sin x}{x^3} \]

Find the number of matrices $A$ of order $3\times 2$ whose elements are from the set $\{\pm2,\pm1,0\}$, if $\mathrm{Tr}(A^TA)=5$.

Let \( A = \{-2,-1,0,1,2,3,4\} \) and \( R \) be a relation such that \[ R=\{(x,y): 2x+y \le -2,\ x\in A,\ y\in A\}. \] Let
\( l \) = number of elements in \( R \),
\( m \) = minimum number of elements to be added to \( R \) to make it reflexive,
\( n \) = minimum number of elements to be added to \( R \) to make it symmetric. Then \( (l+m+n) \) is:

The line \(y = x + 1\) intersects the ellipse \[ \frac{x^2}{2} + \frac{y^2}{1} = 1 \] at points \(A\) and \(B\). Find the angle subtended by the segment \(AB\) at the centre of the ellipse.

Let \[ k = \tan\!\left(\frac{\pi}{4} + \frac{1}{2}\cos^{-1}\!\frac{2}{3}\right) + \tan^{-1}\!\left(\frac{1}{2}\sin^{-1}\!\frac{2}{3}\right). \] Then the number of solutions of the equation \[ \sin^{-1}(kx - 1) = \sin^{-1}x - \cos^{-1}x \] is:

If \(\alpha, \beta\) are roots of the quadratic equation \[ \lambda x^2 - (\lambda+3)x + 3 = 0 \] and \(\alpha<\beta\) such that \[ \frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}, \] then find the sum of all possible values of \(\lambda\).

If the system of linear equations \[ \begin{cases} 3x + y + \beta z = 3 \\ 2x + \alpha y - z = 1 \\ x + 2y + z = 4 \end{cases} \] has infinitely many solutions, then the value of \(22\beta - 9\alpha\) is:

Let \(A\) be a matrix of order \(3 \times 3\) and \(|A| = 5\). If \[ \left|\,2\,\text{adj}\big(3A\,\text{adj}(2A)\big)\right| = 2^{\alpha}\cdot 3^{\beta}\cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N}, \] then the value of \(\alpha + \beta + \gamma\) is:

Let $z = (1+i)(1+2i)(1+3i)\cdots(1+ni)$ and $|z|^2 = 44200$. Find the value of $n$.

\[ \int_0^1 \cot^{-1} \left( x^2 + x + 1 \right) \, dx \] is equal to:

Let \( f(x) = \lim_{n \to \infty} \left( \frac{1}{n^3} \sum_{k=1}^{n} \left\lfloor \frac{k^2}{3^x} \right\rfloor \right) \), where \( \left\lfloor . \right\rfloor \) denotes the greatest integer function, then \( 12 \sum_{j=1}^{\infty} f(j) \) is equal to:

The number of solutions of the equation \( \tan 3x = \cot x \) in \( x \in [0, 2\pi] \) is:

The area bounded by \[ 2 - 4x \leq y \leq x^2 + 4 \quad \text{and} \quad x = \frac{1}{2}, \, x \geq 0, \, y \geq 0 \] (in square units) is:

Let \( z \) be the complex number satisfying \( |z - 5| \leq 3 \) and having maximum possible positive argument. Then the value of \[ \left| \frac{5z - 12}{5iz + 16} \right|^2 \] is equal to:

Let \( f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f^{(3)}(3) \) for all \( x \in \mathbb{R} \). Then the value of \( f'(5) \) is:

The largest value of \( n \in \mathbb{N} \) such that \( 7^n \) divides \( 101! \) is ______.

The minimum value of \( (\sin^{-1} x)^2 + (\cos^{-1} x)^2 \) in \( x \in \left[ \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}} \right] \) is \( \frac{a \pi^2}{b} \), then \( a + b \) is:

If \( a_1 = 1 \) and for \( n \geq 1 \), \[ a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \] then \[ \left| \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \right| \] is equal to

The value of \[ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\pi + 4x^{11}}{1 - \sin\left( \left| x \right| + \frac{\pi}{6} \right)} dx \] is