List of top Mathematics Questions

If the equation of a line passing through \[ \left(\frac13,-\frac12\right) \] and making an angle of \(60^\circ\) with the line \[ 2x-3y+4=0 \] is \[ (2+3\sqrt3)x+by=c, \] then \(c=\)

If \(O\) is the origin and \(H\) is the orthocenter of a triangle formed by the lines \[ x+y=1, \] \[ 6x^2-13xy+6y^2=0, \] then \[ OH= \]

If the angular bisectors of the lines \[ 3x-4y-5=0 \] and \[ 8x-6y+1=0 \] are \[ x+y+c=0 \] and \[ x-y+k=0, \] then \[ 7(c+k)= \]

The origin is shifted to the point \((1,1)\) by translation of axes and then the axes are rotated through an angle \[ \frac{\pi}{3} \] about \((1,1)\) in the positive direction. After these two transformations, if the point \[ P(3,2) \] becomes \(P(\alpha,\beta)\), then \[ 4\alpha+2\beta= \]

In a binomial distribution \[ P(X=2)\div P(X=23)=\left(\frac{2}{3}\right)^{21}, \] then the mean of the binomial distribution is

If \(P(3,4)\) is a fixed point and \(Q\) is a variable point on the circle \[ x^2+y^2=16, \] then the locus of the midpoint of \(PQ\) is

If \(A\), \(B\), \(C\) are independent events of a random experiment such that \[ P(A)=\frac34,\qquad P(B)=\frac56,\qquad P(C)=\frac23, \] then the probability that exactly one of the events occurs is

From the set \[ \{2,3,5,7\} \] two numbers are selected one after the other with replacement. If \(X\) is the random variable representing the absolute difference of the two numbers selected, then the mean of \(X\) is

The variance of the data \[ 4,\;7,\;8,\;10,\;13,\;16,\;19 \] is

If a number \(x\) is to be chosen randomly from the set \[ \{1,2,3,\ldots,30\}, \] then the probability of getting an \(x\) that is a multiple of \(3\) such that \[ \left(x-\frac{26}{x}\right)>25 \] is

Let \[ \overrightarrow{OA}=\hat{i}+2\hat{j}-4\hat{k} \] and \[ \overrightarrow{OB}=3\hat{i}-4\hat{j}-2\hat{k} \] be the position vectors of points \(A\) and \(B\). If a point \(C\) divides the line segment \(AB\) in the ratio \(1:3\) externally, then the position vector of a point which divides \(OC\) in the ratio \(4:1\) internally is

In \(\triangle ABC\), if \[ \overrightarrow{AB}=2\hat{i}-\hat{j}+2\hat{k} \] and \[ \overrightarrow{AC}=3\hat{i}-3\hat{j}+4\hat{k}, \] then the triangle \(ABC\) is

If the unit vector which is perpendicular to the normals drawn to the planes \[ \vec r\cdot(2\hat{i}+\hat{j}-\hat{k})=3 \] and \[ \vec r\cdot(6\hat{i}+3\hat{j}+2\hat{k})=4 \] is \[ x\hat{i}+y\hat{j}+z\hat{k}, \] then \(x+y+z=\)

If \[ A(0,1,-2),\quad B(-1,2,-3),\quad C(2,-3,4) \] and \[ D(3,4,5) \] are the vertices of a tetrahedron \(ABCD\), then the volume of the tetrahedron is

If \[ \vec{a}=x\hat{i}+2\hat{j}-\hat{k}\quad (x>0) \] and \[ \vec{b}=2\hat{i}-\hat{j}+2\hat{k} \] are two vectors such that \[ |\vec{a}-2\vec{b}|=|2\vec{a}+\vec{b}|, \] then \(x=\)

In \(\triangle ABC\), if \[ c=9,\qquad s=10,\qquad \Delta=10\sqrt2, \] then \[ \sin\frac{C}{2} = \]

Evaluate \[ \tan^{-1}\left(\frac12\right) +\tan^{-1}\left(\frac18\right) +\tan^{-1}\left(\frac1{18}\right) +\tan^{-1}\left(\frac1{32}\right). \]

In a triangle \(ABC\), if \[ r_1=\frac{12\sqrt5}{5}, \qquad r_2=3\sqrt5, \qquad r_3=4\sqrt5, \] then \[ 5s^2= \]

In a triangle \(ABC\), if \[ a=2,\qquad b=4,\qquad \cos C=-\frac{5}{16}, \] then the circumradius \(R\) is

If \(\sin\theta\neq 0\) and \[ \frac{1}{2}\sin\theta,\quad \cos\theta,\quad \cot\theta \] are in geometric progression, then the number of values of \(\theta\) lying in the interval \((-2\pi,2\pi)\) is

If \[ \tanh x=\frac13, \] then \[ 6\sinh^4x+2\cosh^4x+2\sinh^2x+\cosh^2x= \]

For \(\theta\in\left(0,\frac{\pi}{2}\right)\), if the complete range of \[ (\cot^2\theta-\cos^2\theta)(\tan^2\theta-\sin^2\theta) \] is \((\alpha,\beta]\), then \(\beta-\alpha=\)

Evaluate \[ \frac{\cos\theta}{1-\cos\theta} + \frac{\sec\theta}{1+\sec\theta}. \]