If {\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha , then $$4x^2-4xy\cos \alpha +y^2$$ is equal to :

If the letter of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

If the roots of the equation $$x^2-bx+c=0$$ be two consecutive integers, then $$b^2-4c$$ equals

The value of $$a$$ for which the sum of the squares of the roots of the equation $$x^2-\left(a-2\right)x-a-1=0$$ assume the least value is :

If the roots of the equation $$x^2-bx+c=0$$ be two consecutive integers, then $$b^2-4c$$ equals

The value of $$a$$ for which the sum of the squares of the roots of the equation $$x^2-\left(a-2\right)x-a-1=0$$ assume the least value is

In a triangle PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right) and \tan \left( {{Q \over 2}} \right) are the roots of $$a{x}^2+bx+c=0,a\ne 0$$ then

If both the roots of the quadratic equation $$x^2-2kx+k^2+k-5=0$$ are less than 5, then $$k$$ lies in the interval

The locus of a point $$P\left(\alpha ,\beta \right)$$ moving under the condition that the line $$y=\alpha x+\beta$$ is tangent to the hyperbola {{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1 is :

If in a $$\Delta ABC$$, the altitudes from the vertices $$A,B,C$$ on opposite sides are in H.P, then $$\sin A,\sin B,\sin C$$ are in :

In a triangle $$ABC$$, let \angle C = {\pi \over 2}. If $$r$$ is the inradius and $$R$$ is the circumradius of the triangle $$ABC$$, then $$2(r+R)$$ equals :

If the angel $$\theta$$ between the line {{x + 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 2} and the plane $$2x-y+\sqrt{\lambda }z+4=0$$ is such that \sin \,\,\theta = {1 \over 3} then value of $$\lambda$$ is :

If the plane $$2ax-3ay+4az+6=0$$ passes through the midpoint of the line joining the centres of the spheres $$x^2+y^2+z^2+6x-8y-2z=13$$ and $$x^2+y^2+z^2-10x+4y-2z=8$$ then a equals :

The distance between the line $$\overrightarrow{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda \left(i-j+4k\right),$$ and the plane $$\overrightarrow{r}.\left(\hat{i}+5\hat{j}+\hat{k}\right)=5$$ is

The angle between the lines $$2x=3y=-z$$ and $$6x=-y=-4z$$ is :

The plane $$x+2y-z=4$$ cuts the sphere $$x^2+y^2+z^2-x+z-2=0$$ in a circle of radius

Let x1, x2,...........,xn be n observations such that $$\sum x_i^2=400$$ and $$\sum x_i=80$$. Then a possible value of n among the following is

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately :

If $$x=\sum _{n=0}^{\infty }{a}^n,y=\sum _{n=0}^{\infty }{b}^n,z=\sum _{n=0}^{\infty }{c}^n,$$ where a, b, c are in A.P and $$\left|a\right|<1,\left|b\right|<1,\left|c\right|<1$$ then x, y, z are in

The sum of the series 1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + ....... ad inf. is