List of top Mathematics Questions

The number of values of \(z \in \mathbb{C}\), satisfying the equations \(|z - (4 + 8i)| = \sqrt{10}\) and \(|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}\), is:

Consider the relation R on the set \(\{-2, -1, 0, 1, 2\}\) defined by \((a, b) \in R\) if and only if \(1 + ab>0\). Then, among the statements:
I. The number of elements in R is 17
II. R is an equivalence relation

The sum of squares of all the real solutions of the equation \(\log_{(x+1)}(2x^2 + 5x + 3) = 4 - \log_{(2x+3)}(x^2 + 2x + 1)\) is equal to ________.

In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.

Prove that :
\(\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta\).

Let the midpoints of the sides of a triangle \(ABC\) be \( \left(\frac{5}{2},7\right), \left(\frac{5}{2},3\right)\) and \( (4,5) \). If its incentre is \((h,k)\), then \(3h+k\) is equal to:

Let the image of the point \(P(0,-5,0)\) in the line \[ \frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2} \] be the point \(R\) and the image of the point \(Q(0,-\frac12,0)\) in the line \[ \frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1} \] be the point \(S\). Then the square of the area of the parallelogram \(PQRS\) is ______.

Let the line \(x-y=4\) intersect the circle \(C:(x-4)^2+(y+3)^2=9\) at the points \(Q\) and \(R\). If \(P(\alpha,\beta)\) is a point on \(C\) such that \(PQ=PR\), then \((6\alpha+8\beta)^2\) is equal to ______.

Let \[ f(x)= \begin{cases} x^3+8, & x<0,\\ x^2-4, & x\ge0, \end{cases} \qquad g(x)= \begin{cases} (x-8)^{1/3}, & x<0,\\ (x+4)^{1/2}, & x\ge0. \end{cases} \] Then the number of points where the function \(g\circ f\) is discontinuous is ______.

The value of the integral \[ \int_{-1}^{1}\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right)dx \] is equal to :

If \[ (1-x^3)^{10}=\sum_{r=0}^{10}a_r x^r(1-x)^{30-2r}, \] then \( \dfrac{9a_9}{a_{10}} \) is equal to ________.

The area of the region \( \{(x,y): x^2-8x \le y \le -x\} \) is :

Let \(f:\mathbb{R}\to\mathbb{R}\) be such that \(f(x+y)=f(x)f(y)\) for all \(x,y\in\mathbb{R}\) and \(f(0)\neq0\). Let \(g:[1,\infty)\to\mathbb{R}\) be a differentiable function such that \[ x^2g(x)=\int_1^x\big(t^2f(t)-tg(t)\big)\,dt \] Then \(g(2)\) is equal to :

Let \[ A= \begin{bmatrix} 1 & 3 & -1\\ 2 & 1 & \alpha\\ 0 & 1 & -1 \end{bmatrix} \] be a singular matrix. Let \[ f(x)=\int_{0}^{x}(t^2+2t+3)\,dt,\quad x\in[1,\alpha]. \] If \(M\) and \(m\) are respectively the maximum and the minimum values of \(f\) in \([1,\alpha]\), then \(3(M-m)\) is equal to :

The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \( \frac{\sqrt3}{2} \) and its directrices are \( x=\pm \frac{4\sqrt6}{3} \). Let \( H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\), and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is :

If \( \sin\!\left(\tan^{-1}(x\sqrt2)\right)=\cot\!\left(\sin^{-1}\!\sqrt{1-x^2}\right),\; x\in(0,1) \), then the value of \(x\) is :

Let \[ \lim_{x\to2}\frac{\tan(x-2)\,[x^2+(p-2)x-2p]}{(x-2)^2}=5 \] for some \(p,r\in\mathbb{R}\). If the set of all possible values of \(q\), such that the roots of the equation \(rx^2-px+q=0\) lie in \( (0,2) \), be the interval \( (\alpha,\beta) \), then \(4(\alpha+\beta)\) equals :

The shortest distance between the lines \[ \frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} \] and \[ \frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} \] is :

Let \(x=-9\) be a directrix of an ellipse \(E\), whose centre is at the origin and eccentricity is \( \frac13 \). Let \(P(\alpha,0), \alpha>0\), be a focus of \(E\) and \(AB\) be a chord passing through \(P\). Then the locus of the mid point of \(AB\) is :

Let \( \vec a = 2\hat i + 3\hat j + 3\hat k \) and \( \vec b = 6\hat i + 3\hat j + 3\hat k \). Then the square of the area of the triangle with adjacent sides determined by the vectors \( (2\vec a + 3\vec b) \) and \( (\vec a - \vec b) \) is :

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

Let \(C\) be a circle having centre in the first quadrant and touching the \(x\)-axis at a distance of \(3\) units from the origin. If the circle \(C\) has an intercept of length \(6\sqrt{3}\) on \(y\)-axis, then the length of the chord of the circle \(C\) on the line \(x-y=3\) is:

Let the mean and the variance of seven observations \(2,4,\alpha,8,\beta,12,14\), \( \alpha < \beta \), be \(8\) and \(16\) respectively. Then the quadratic equation whose roots are \(3\alpha+2\) and \(2\beta+1\) is :

Let \( A = \begin{bmatrix} 1 & 0 & 0\\ 3 & 1 & 0\\ 9 & 3 & 1 \end{bmatrix} \) and \( B = [b_{ij}], 1 \le i,j \le 3 \). If \( B = A^{99} - I \), then the value of \( \dfrac{b_{31}-b_{21}}{b_{32}} \) is: