Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If
T = {x $$ \in $$ R | f(x) = f(0)}, then the sum of squares of all the elements of T is :

Let a, b c $$ \in $$ R be such that a² + b² + c² = 1. If
$$a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}} \right) = c\cos \left( {\theta + {{4\pi } \over 3}} \right)$$,
where $${\theta = {\pi \over 9}}$$, then the angle between the vectors $$a\widehat i + b\widehat j + c\widehat k$$ and $$b\widehat i + c\widehat j + a\widehat k$$ is :

Let the latus ractum of the parabola y² = 4x be the common chord to the circles C₁ and C₂ each of them having radius 2$$\sqrt 5 $$. Then, the distance between the centres of the circles C₁ and C₂ is :

Let R₁ and R₂ be two relation defined as follows :
R₁ = {(a, b) $$ \in $$ R² : a² + b² $$ \in $$ Q} and
R₂ = {(a, b) $$ \in $$ R² : a² + b² $$ \notin $$ Q},
where Q is the set of all rational numbers. Then :

If the value of the integral
$$\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}}}}} dx$$

is $${k \over 6}$$, then k is equal to :

Let e₁ and e₂ be the eccentricities of the ellipse,
$${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$$(b < 5) and the hyperbola,
$${{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1$$ respectively satisfying e₁e₂ = 1. If $$\alpha $$
and $$\beta $$ are the distances between the foci of the
ellipse and the foci of the hyperbola
respectively, then the ordered pair ($$\alpha $$, $$\beta $$) is equal to :

Let S be the set of all integer solutions, (x, y, z), of the system of equations
x – 2y + 5z = 0
–2x + 4y + z = 0
–7x + 14y + 9z = 0
such that 15 $$ \le $$ x² + y² + z² $$ \le $$ 150. Then, the number of elements in the set S is equal to ______ .

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to _________ .

The total number of 3-digit numbers, whose sum of digits is 10, is __________.

The probability that a randomly chosen 5-digit number is made from exactly two digits is :

If x³dy + xy dx = x²dy + 2y dx; y(2) = e and
x > 1, then y(4) is equal to :

$$\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \right)}^{{1 \over 3}}} - {{\left( {4x} \right)}^{{1 \over 3}}}}}$$ ($$a$$ $$ \ne $$ 0) is equal to :

If the surface area of a cube is increasing at a rate of 3.6 cm²/sec, retaining its shape; then the rate of change of its volume (in cm³/sec), when the length of a side of the cube is 10 cm, is :

If a $$\Delta $$ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates :

Let x_i (1 $$ \le $$ i $$ \le $$ 10) be ten observations of a random variable X. If
$$\sum\limits_{i = 1}^{10} {\left( {{x_i} - p} \right)} = 3$$ and $$\sum\limits_{i = 1}^{10} {{{\left( {{x_i} - p} \right)}^2}} = 9$$
where 0 $$ \ne $$ p $$ \in $$ R, then the standard deviation of these observations is :

If z₁ , z₂ are complex numbers such that
Re(z₁) = |z₁ – 1|, Re(z₂) = |z₂ – 1| , and
arg(z₁ - z₂) = $${\pi \over 6}$$, then Im(z₁ + z₂ ) is equal to :

If $$\int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx$$ = A(x)$${\tan ^{ - 1}}\left( {\sqrt x } \right)$$ + B(x) + C,
where C is a constant of integration, then the ordered pair (A(x), B(x)) can be :

If the term independent of x in the expansion of
$${\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9}$$ is k, then 18 k is equal to :

The set of all real values of $$\lambda $$ for which the quadratic equations,
($$\lambda $$² + 1)x² – 4$$\lambda $$x + 2 = 0 always have exactly one root in the interval (0, 1) is :