If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :

The coefficient of x⁻⁵ in the binomial expansion of

$${\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + 1}} - {{x - 1} \over {x - {x^{{1 \over 2}}}}}} \right)^{10}},$$ where x $$ \ne $$ 0, 1, is :

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B₁ and a particular girl G₁ never sit adjacent to each other, is :

For two 3 × 3 matrices A and B, let A + B = 2B^T and 3A + 2B = I₃, where B^T is the transpose of B and I₃ is 3 × 3 identity matrix. Then :

The equation
Im $$\left( {{{iz - 2} \over {z - i}}} \right)$$ + 1 = 0, z $$ \in $$ C, z $$ \ne $$ i
represents a part of a circle having radius equal to :

The sum of all the real values of x satisfying the equation
2^{(x$$-$$1)(x2 + 5x $$-$$ 50)} = 1 is :

The function f : N $$ \to $$ N defined by f (x) = x $$-$$ 5 $$\left[ {{x \over 5}} \right],$$ Where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60^o. If the area of the quadrilateral is $$4\sqrt 3 $$, then the perimeter of the quadrilateral is :

The sum of 100 observations and the sum of their squares are 400 and 2475, respectively. Later on, three observations, 3, 4 and 5, were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations 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 :

A value of x satisfying the equation sin[cot⁻¹ (1+ x)] = cos [tan⁻¹ x], is :

If the vector $$\overrightarrow b = 3\widehat j + 4\widehat k$$ is written as the sum of a vector $$\overrightarrow {{b_1}} ,$$ paralel to $$\overrightarrow a = \widehat i + \widehat j$$ and a vector $$\overrightarrow {{b_2}} ,$$ perpendicular to $$\overrightarrow a ,$$ then $$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} $$ is equal to :

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 :

The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, −1) and (−2, 2) is :

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 :

The value of k for which the function

$$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr {k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} \cr } } \right.$$

is continuous at x = $${\pi \over 2},$$ is :

The function f defined by

f(x) = x³ $$-$$ 3x² + 5x + 7 , is :

If $$\,\,\,$$ f$$\left( {{{3x - 4} \over {3x + 4}}} \right)$$ = x + 2, x $$ \ne $$ $$-$$ $${4 \over 3}$$, and

$$\int {} $$f(x) dx = A log$$\left| {} \right.$$1 $$-$$ x $$\left| {} \right.$$ + Bx + C,

then the ordered pair (A, B) is equal to :

(where C is a constant of integration)

Let f be a polynomial function such that

f (3x) = f ' (x) . f '' (x), for all x $$ \in $$ R. Then :

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 30^o with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is :

A line drawn through the point P(4, 7) cuts the circle x² + y² = 9 at the points A and B. Then PA⋅PB is equal to :

If    $$\int\limits_1^2 {{{dx} \over {{{\left( {{x^2} - 2x + 4} \right)}^{{3 \over 2}}}}}} = {k \over {k + 5}},$$ then k is equal to :