List of top Mathematics Questions

Let \( y = y(x) \) be the solution of the differential equation \( x^2 dy + (4x^2 y + 2\sin x)dx = 0 \), \( x>0 \), \( y\left(\frac{\pi}{2}\right) = 0 \). Then \( \pi^4 y\left(\frac{\pi}{3}\right) \) is equal to :
Let $z$ be the complex number satisfying $|z - 5| \le 3$ and having maximum positive principal argument. Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to :

If the area of the region \[ \{(x, y) : 1 - 2x \le y \le 4 - x^2,\ x \ge 0,\ y \ge 0\} \] is \[ \frac{\alpha}{\beta}, \] \(\alpha, \beta \in \mathbb{N}\), \(\gcd(\alpha, \beta) = 1\), then the value of \[ (\alpha + \beta) \] is : 

Let \(S\) and \(S'\) be the foci of the ellipse \[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \] and \(P(\alpha,\beta)\) be a point on the ellipse in the first quadrant. If \[ (SP)^2 + (S'P)^2 - SP \cdot S'P = 37, \] then \(\alpha^2 + \beta^2\) is equal to

Let $\alpha,\beta\in\mathbb{R}$ be such that the function \[ f(x)= \begin{cases} 2\alpha(x^2-2)+2\beta x, & x<1 \\ (\alpha+3)x+(\alpha-\beta), & x\ge1 \end{cases} \] is differentiable at all $x\in\mathbb{R}$. Then $34(\alpha+\beta)$ is equal to}
 

Let \( \vec{a} = -\hat{i} + \hat{j} + 2\hat{k} \), \( \vec{b} = \hat{i} - \hat{j} - 3\hat{k} \), \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} = \vec{c} \times \vec{a} \). Then \( (|\vec{a}|^2 - |\vec{b}|^2) \cdot \vec{d \) is equal to:}

Let \(S=\left\{ z\in\mathbb{C}:\left|\frac{z-6i}{z-2i}\right|=1 \text{ and } \left|\frac{z-8+2i}{z+2i}\right|=\frac{3}{5} \right\}.\)
Then $\sum_{z\in S}|z|^2$ is equal to
 

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + 2ax + (3a + 10) = 0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is :
Let the length of the latus rectum of an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $(a>b)$ be $30$. If its eccentricity is the maximum value of the function $f(t)=-\dfrac{3}{4}+2t-t^2$, then $(a^2+b^2)$ is equal to
The area of the region, inside the ellipse \(x^2+4y^2=4\) and outside the region bounded by the curves \(y=x-1\) and \(y=1-x\), is:
Let $[t]$ denote the greatest integer less than or equal to $t$.
If the function
\[ f(x)= \begin{cases} b^2 \sin\!\left[\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x+\sin x)\cos x\right]\right], & x < 0 \\ \dfrac{\sin x-\dfrac{1}{2}\sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases} \] is continuous at $x=0$, then $a^2+b^2$ is equal to
Let \( \vec{c} \) and \( \vec{d} \) be vectors such that \[ |\vec{c} + \vec{d}| = \sqrt{29} \] and \[ \vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d}. \] If \( \lambda_1, \lambda_2 \) (\( \lambda_1 > \lambda_2 \)) are the possible values of \[ (\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}), \] then the equation \[ K^2 x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \frac{\lambda_2}{2}\right)y^2 - 8x + 12y + \lambda_2 = 0 \] represents a circle, for \( K \) equal to
Let \( \vec{a} = \hat{i} - 2\hat{j} + 3\hat{k} \), \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \), \( \vec{c} = \lambda \hat{i} + \hat{j} + \hat{k} \) and \( \vec{v} = \vec{a} \times \vec{b} \). If \( \vec{v} \cdot \vec{c} = 11 \) and the length of the projection of \( \vec{b} \) on \( \vec{c} \) is \( p \), then find the value of \( 9p^2 \).
Let \( PQ \) be a chord of the hyperbola \[ \frac{x^2}{4} - \frac{y^2}{b^2} = 1, \] perpendicular to the \(x\)-axis such that \( OPQ \) is an equilateral triangle, \( O \) being the centre of the hyperbola. If the eccentricity of the hyperbola is \( \sqrt{3} \), then find the area of the triangle \( OPQ \).
If \[ f(x) = \begin{cases} \dfrac{a|x| + x^2 - 2(\sin|x|)(\cos|x|)}{x}, & x \ne 0, \\ b, & x = 0, \end{cases} \] is continuous at \( x = 0 \), then \( a + b \) is equal to.
The value of \[ \int_{-\pi/6}^{\pi/6} \left( \frac{\pi + 4x^{11}}{1 - \sin\left(|x| + \frac{\pi}{6}\right)} \right) dx \] is equal to
Let the circle \(x^2+y^2=4\) intersect the \(x\)-axis at points \(A(a,0)\) and \(B(b,0)\). Let \(P(2\cos\alpha,2\sin\alpha)\), \(0<\alpha<\frac{\pi}{2}\), and \(Q(2\cos\beta,2\sin\beta)\) be two points on the circle such that \((\alpha-\beta)=\frac{\pi}{2}\). Then the point of intersection of lines \(AQ\) and \(BP\) lies on:
Let the image of parabola $x^2=4y$ in the line $x-y=1$ be $(y+a)^2=b(x-c)$, where $a,b,c\in\mathbb{N}$. Then $a+b+c$ is equal to
The value of \[ \int_{-100}^{100} \frac{x+x^3+x^5}{1+x^2+x^4+x^6}\,dx \] is:
Let \( f(x)=x^3,\; x\in[-1,1] \). Then which of the following are correct?
Let \( f:[0,1]\to\mathbb{R} \) and \( g:[0,1]\to\mathbb{R} \) be defined as: \[ f(x)= \begin{cases} 1, & x \text{ rational}\\ 0, & x \text{ irrational} \end{cases} \quad g(x)= \begin{cases} 0, & x \text{ rational}\\ 1, & x \text{ irrational} \end{cases} \] Then:
Let \( f(x) \) be continuous on \( [0,5] \) and differentiable in \( (0,5) \). If \( f(0)=0 \) and \( |f'(x)| \le \frac{1{5} \) for all \( x \in (0,5) \), then \( \forall x \in [0,5] \):}
The value of \[ \sum_{r=1}^{20}\sqrt{\left|\pi\left(\int_0^r x|\sin \pi x|\,dx\right)\right|} \] is:
Let \( f \) be a function which is differentiable for all real \( x \). If \( f(2) = -4 \) and \( f'(x) \ge 6 \) for all \( x \in [2,4] \), then:
If \( P \) is a non-singular matrix of order \( 5\times5 \) and the sum of the elements of each row is 1, then the sum of the elements of each row in \( P^{-1} \) is: