If \mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}, then the value of $$a$$ and $$b$$, are

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

If $$\left(1-p\right)$$ is a root of quadratic equation $$x^2+px+\left(1-p\right)=0$$ then its root are

If one root of the equation $$x^2+px+12=0$$ is 4, while the equation $$x^2+px+q=0$$ has equal roots, then the value of $${}^{\prime }{q}^{\prime }$$ is

The sides of a triangle are $$\sin \alpha ,\cos \alpha$$ and $$\sqrt{1+\sin \alpha \cos \alpha }$$ for some 0 < \alpha < {\pi \over 2}. Then the greatest angle of the triangle is :

A line makes the same angle $$\theta$$, with each of the $$x$$ and $$z$$ axis. If the angle $$\beta$$, which it makes with y-axis, is such that $${\sin }^2\beta =3{\sin }^2\theta ,$$ then $${\cos }^2\theta$$ equals :

If the straight lines $$x=1+s,y=-3$$$$-\lambda s,$$ $$z=1+\lambda s$$ and x = {t \over 2},y = 1 + t,z = 2 - t, with parameters $$s$$ and $$t$$ respectively, are co-planar, then $$\lambda$$ equals :

A line with direction cosines proportional to $$2,1,2$$ meets each of the lines $$x=y+a=z$$ and $$x+a=2y=2z$$ . The co-ordinates of each of the points of intersection are given by :

Distance between two parallel planes $$2x+y+2z=8$$ and $$4x+2y+4z+5=0$$ is :

The intersection of the spheres $$x^2+y^2+z^2+7x-2y-z=13$$ and $$x^2+y^2+z^2-3x+3y+4z=8$$ is the same as the intersection of one of the sphere and the plane

Consider the following statements: (a) Mode can be computed from histogram (b) Median is not independent of change of scale (c) Variance is independent of change of origin and scale. Which of these is/are correct?

In a series of 2n observations, half of them equal $$a$$ and remaining half equal $$–a$$. If the standard deviation of the observations is 2, then $$|a|$$ equals

Let $$T_r$$ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\, then a - d equals

The sum of the first n terms of the series {1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1)}^2}} \over 2} when n is even. When n is odd the sum is

The sum of series {1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........ is

A point on the parabola $$y^2=18x$$ at which the ordinate increases at twice the rate of the abscissa is

A function $$y=f(x)$$ has a second order derivative $$f^{\prime \prime }\left(x\right)=6\left(x-1\right).$$ If its graph passes through the point $$(2,1)$$ and at that point the tangent to the graph is $$y=3x-5$$, then the function is :

If $$2a+3b+6c=0$$, then at least one root of the equation $$a{x}^2+bx+c=0$$ lies in the interval