When a missile is fired from a ship, the probability that it is intercepted is $${1 \over 3}$$ and the probability that the missile hits the target, given that it is not intercepted, is $${3 \over 4}$$. If three missiles are fired independently from the ship, then the probability that all three hit the target, is :

The equation of the line through the point (0, 1, 2) and perpendicular to the line

$${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over { - 2}}$$ is :

The value of $$\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx$$, where [ t ] denotes the greatest integer $$ \le $$ t, is :

The integer 'k', for which the inequality x² $$-$$ 2(3k $$-$$ 1)x + 8k² $$-$$ 7 > 0 is valid for every x in R, is :

The total number of positive integral solutions (x, y, z) such that xyz = 24 is :

If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is $${{{x^2} - 4x + y + 8} \over {x - 2}}$$, then this curve also passes through the point :

Let f, g : N $$ \to $$ N such that f(n + 1) = f(n) + f(1) $$\forall $$ n$$\in$$N and g be any arbitrary function. Which of the following statements is NOT true?

Let the lines (2 $$-$$ i)z = (2 + i)$$\overline z $$ and (2 $$+$$ i)z + (i $$-$$ 2)$$\overline z $$ $$-$$ 4i = 0, (here i² = $$-$$1) be normal to a circle C. If the line iz + $$\overline z $$ + 1 + i = 0 is tangent to this circle C, then its radius is :

$$\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}$$ is equal to :

Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations l + m $$-$$ n = 0 and l² + m² $$-$$ n² = 0. Then the value of sin⁴$$\alpha$$ + cos⁴$$\alpha$$ is :

The value of the integral
$$\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta $$ is :

The coefficients a, b and c of the quadratic equation, ax² + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :

The image of the point (3, 5) in the line x $$-$$ y + 1 = 0, lies on :

The locus of the point of intersection of the lines $$\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0$$ and $$\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 0$$ is a conic, whose eccentricity is _________.

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A⁴ is equal to __________.

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is _____________.

If $$A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over 2}} \right)} & 0 \cr } } \right]$$ and
$$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]$$, then $$13({a^2} + {b^2})$$ is equal to

Let A₁, A₂, A₃, ....... be squares such that for each n $$ \ge $$ 1, the length of the side of A_n equals the length of diagonal of A_n+1. If the length of A₁ is 12 cm, then the smallest value of n for which area of A_n is less than one, is __________.

If the system of equations

kx + y + 2z = 1

3x $$-$$ y $$-$$ 2z = 2

$$-$$2x $$-$$2y $$-$$4z = 3

has infinitely many solutions, then k is equal to __________.

The number of points, at which the function
f(x) = | 2x + 1 | $$-$$ 3| x + 2 | + | x² + x $$-$$ 2 |, x$$\in$$R is not differentiable, is __________.

Let $$\overrightarrow a = \widehat i + 2\widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j$$ and $$\overrightarrow c = \widehat i - \widehat j - \widehat k$$ be three given vectors. If $$\overrightarrow r $$ is a vector such that $$\overrightarrow r \times \overrightarrow a = \overrightarrow c \times \overrightarrow a $$ and $$\overrightarrow r .\,\overrightarrow b = 0$$, then $$\overrightarrow r .\,\overrightarrow a $$ is equal to __________.

Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x⁶ is unity and it has extrema at x = $$-$$1 and x = 1. If $$\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^3}}} = 1$$, then $$5.f(2)$$ is equal to _________.