The number of real values of $$\lambda$$ for which the system of linear equations 2x + 4y $$-$$ $$\lambda$$z = 0 4x + $$\lambda$$y + 2z = 0 $$\lambda$$x + 2y + 2z = 0 has infinitely many solutions, is :

Let A be any 3 $$\times$$ 3 invertible matrix. Then which one of the following is not always true ?

If A = \left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right], then adj(3A2 + 12A) is equal to

If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :

From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for these committees to have more women than men, is :

Let E and F be two independent events. The probability that both E and F happen is {1 \over {12}} and the probability that neither E nor F happens is {1 \over {2}}, then a value of {{P\left( E \right)} \over {P\left( F \right)}} is :

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :

Three persons P, Q and R independently try to hit a target. I the probabilities of their hitting the target are {3 \over 4},{1 \over 2} and {5 \over 8} respectively, then the probability that the target is hit by P or Q but not by R is :

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P (Exactly one of C or A occurs) = {1 \over 4} and P(All the three events occur simultaneously) = {1 \over {16}}. Then the probability that at least one of the events occurs, is :

If two different numbers are taken from the set {0, 1, 2, 3, ........, 10}; then the probability that their sum as well as absolute difference are both multiple of 4, is :

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is :

If for x \in \left( {0,{1 \over 4}} \right), the derivatives of {\tan ^{ - 1}}\left( {{{6x\sqrt x } \over {1 - 9{x^3}}}} \right) is $$\sqrt{x}.g\left(x\right)$$, then $$g\left(x\right)$$ equals

Let f be a polynomial function such that f (3x) = f ' (x) . f '' (x), for all x $$\in$$ R. Then :

If y = $${\left[x+\sqrt{x^2-1}\right]}^{15}+{\left[x-\sqrt{x^2-1}\right]}^{15},$$ then (x2 $$-$$ 1) {{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}} is equal to :

The curve satisfying the differential equation, ydx $$-$$(x + 3y2)dy = 0 and passing through the point (1, 1), also passes through the point :

If 2x = y{^{{1 \over 5}}} + y{^{ - {1 \over 5}}} and (x2 $$-$$ 1) {{{d^2}y} \over {d{x^2}}} + $$\lambda$$x {{dy} \over {dx}} + ky = 0, then $$\lambda$$ + k is equal to :

If \left( {2 + \sin x} \right){{dy} \over {dx}} + \left( {y + 1} \right)\cos x = 0 and y(0) = 1, then y\left( {{\pi \over 2}} \right) is equal to :

Let k be an integer such that the triangle with vertices (k, – 3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :

A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30o with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is :

Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If $$\angle$$BPC = $$\beta$$, then tan$$\beta$$ is equal to: