List of top Mathematics Questions

\((1,1)\) is the focus of the parabola \[ y^{2}-4ax-2ay+a^{2}=0. \] If the circles \[ (x-\alpha)^2+(y-\beta)^2=r^2 \] touch the X-axis and the axis of the given parabola, then \[ \{(\alpha,\beta)\} \] is:

If the area of the triangle formed by the points \((0,0,0)\), \((1,1,1)\) and \((t,2t,3t)\) is \(\sqrt6\), then the sum of squares of all possible values of \(t\) is:

If line \(4x-3y+c=0\) makes a chord of length 10 on circle \(x^2+y^2-2x+4y-23=0\), then \(c=\):

If a circle inscribed in the parabola \(y^{2}=4ax\) passes through its focus, then the equation of the circle is:

If \(3x+4y-24=0\) and \(3x-4y-32=0\) are tangents to a circle and \(4x+3y-1=0\) is a normal, then \(r+h+k=\):

If two vertices of a quadrilateral are the centres of the circles \[ S\equiv x^{2}+y^{2}-2x-2y-2=0 \] and \[ S^{\prime}\equiv x^{2}+y^{2}-6x-6y+14=0 \] and the other two vertices of that quadrilateral are the points of intersection of these two circles, then the area of the quadrilateral is:

The tangent drawn at a point \(P\) on the circle \(x^{2}+y^{2}+6x+6y-2=0\) cuts the line \(5x-2y+6=0\) at a point \(Q\). If \(PQ=5\), then a point \(Q\) having integral coordinates is:

Let P be any point on circle \(x^2+y^2=16\) and \(A=(1,2)\). If the locus of point dividing AP in ratio 3:2 is a circle, its radius is:

If the coordinate axes are rotated about the origin through \(60^\circ\), the equation \(x^{2}+y^{2}-4x-8y+16=0\) becomes \(x^{2}+y^{2}+2Gx+2Fy+C=0\). Then \(G+F+C=\):

The line \((3a+1)x+(7a+2)y=17a+5\) represents concurrent lines. If \(d\) is distance from \((3,1)\) to line of slope 1 in this family, find \(2d^2\):

One of the pair of lines \(x^{2}-3y^{2}-4x-6\sqrt{3}y-5=0\) is \(x+by+c=0\) \((b<0)\). If the other line intersects the curve \(x^{2}-5y^{2}-4x=0\) at two points A and B, then \(\angle AOB=\):

In triangle ABC, B lies on positive x-axis, A=(-1,0), \(a=4\sqrt{3}\), \(\angle A=120^\circ\). If C has integer coordinate condition, distance of C from origin is:

A ray from (7,2) reflects on \(2x+y=1\) and passes through (3,10). Equation of incident ray is:

Bag A contains 5 white and 2 black balls. Bag B contains 2 white and 5 black balls. Two balls are randomly chosen from bag A and placed in bag B. Now a ball is drawn randomly from bag B and found that it is white. The probability that the two balls drawn from bag A are of different colour is:

Probability for a person A to have success in one trial is \(\frac{2}{5}\). In 7 Bernoulli trials, if the probability that A has \(k\) successes is maximum, then \(k = \):

The mean deviation from the median of the given frequency distribution is:

Let \(p\) be the probability of success in one trial, \(0 < p < 1\). If \(X\) is a random variable representing the number of trials until the first success occurs and \[ P(X=k)=\lambda(1-p)^{k-1},\quad k=1,2,3,\dots \] then \(\lambda = \):

If A, B, C are mutually exclusive and exhaustive events such that \(P(A):P(B):P(C)=1:l:m\), then \(P(A\cup B)+P(B\cup C)+P(C\cup A)+P(A\cup B\cup C)=\):

Let \(\overline{OA} = \overline{i} + 2\overline{j} + 2\overline{k}\), \(\overline{OB} = 3\overline{i} + 4\overline{k}\). If \(x\overline{i} + y\overline{j} + z\overline{k}\) is the vector along the bisector of \(\angle AOB\) and of length 2 units, then a possible value of \(x + y + z\) is:

If the line \(\overline{r}=\overline{a}+t\overline{b}\) lies on the plane \(\overline{r}\cdot\overline{n}=p\), then \(p=\):

If the shortest distance between the two skew lines \(\overline{r}=\overline{i}+\overline{j}+\overline{k}+t(3\overline{i}+2\overline{j}+\overline{k})\) and \(\overline{r}=\overline{i}-\overline{j}+x\overline{k}+s(\overline{i}+2\overline{j}+3\overline{k})\) is at most \(2\sqrt{6}\), then all values of \(x\) lie in the interval:

\(\overline{a}, \overline{b}, \overline{c}\) are non-coplanar vectors. If \(\overline{x}=2\overline{a}+3\overline{b}+4\overline{c}\), \(\overline{y}=3\overline{a}+4\overline{b}+5\overline{c}\), \(\overline{z}=4\overline{a}+5\overline{b}+6\overline{c}\), then \([\overline{x}\ \overline{y}\ \overline{z}]=\):

If \(\overline{AB}=\overline{i}+\overline{j}-2\overline{k}\), \(\overline{CB}=2\overline{i}-\overline{j}+\alpha\overline{k}\) \((\alpha\in Z)\) are two sides of a triangle ABC and the angle between these two sides is \(\frac{\pi}{3}\), then the length of its third side is:

In a triangle ABC, if \( a = 2 \), \( \sin A = \frac{2}{3} \), \( B = \frac{\pi}{3} \), then \( \sqrt{5}b - 3c = \):

If \( a \) is a real number and \( 2\sinh^2 x - 3\cosh x + a = 0 \) has a solution, then the range of \( a \) is: