Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4.

Then $$\mathop {\lim }\limits_{x \to a} {{xf(a) - af(x)} \over {x - a}}$$ equals :

Let A(1, 4) and B(1, $$-$$5) be two points. Let P be a point on the circle
(x $$-$$ 1)² + (y $$-$$ 1)² = 1 such that (PA)² + (PB)² have maximum value, then the points, P, A and B lie on :

Consider the following system of equations :

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

2x + 6y $$-$$ 11z = b

x $$-$$ 2y + 7z = c,

where a, b and c are real constants. Then the system of equations :

A natural number has prime factorization given by n = 2^x3^y5^z, where y and z are such
that y + z = 5 and y^$$-$$1 + z^$$-$$1 = $${5 \over 6}$$, y > z. Then the number of odd divisions of n, including 1, is :

If 0 < a, b < 1, and tan^$$-$$1a + tan^$$-$$1b = $${\pi \over 4}$$, then the value of

$$(a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....$$ is :

Let $$f(x) = \int\limits_0^x {{e^t}f(t)dt + {e^x}} $$ be a differentiable function for all x$$\in$$R. Then f(x) equals :

For x > 0, if $$f(x) = \int\limits_1^x {{{{{\log }_e}t} \over {(1 + t)}}dt} $$, then $$f(e) + f\left( {{1 \over e}} \right)$$ is equal to :

Let $$A = \{ 1,2,3,....,10\} $$ and $$f:A \to A$$ be defined as

$$f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right.$$

Then the number of possible functions $$g:A \to A$$ such that $$gof = f$$ is :

Let A₁ be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A₂ be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = $${\pi \over 2}$$ in the first quadrant. Then,

Let $$f(x) = {\sin ^{ - 1}}x$$ and $$g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}$$. If $$g(2) = \mathop {\lim }\limits_{x \to 2} g(x)$$, then the domain of the function fog is :

Let f : R $$ \to $$ R be defined as

$$f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$$ If f(x) is continuous on R, then a + b equals :

If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x² + y² = 1 is a circle of radius r, then r is equal to :

If vectors $$\overrightarrow {{a_1}} = x\widehat i - \widehat j + \widehat k$$ and $$\overrightarrow {{a_2}} = \widehat i + y\widehat j + z\widehat k$$ are collinear, then a possible unit vector parallel to the vector $$x\widehat i + y\widehat j + z\widehat k$$ is :

A seven digit number is formed using digits 3, 3, 4, 4, 4, 5, 5. The probability, that number so formed is divisible by 2, is :

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is _________.

If the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 2 & 0 \cr 3 & 0 & { - 1} \cr } } \right]$$ satisfies the equation

$${A^{20}} + \alpha {A^{19}} + \beta A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & 0 \cr 0 & 0 & 1 \cr } } \right]$$ for some real numbers $$\alpha$$ and $$\beta$$, then $$\beta$$ $$-$$ $$\alpha$$ is equal to ___________.

Let $$\alpha$$ and $$\beta$$ be two real numbers such that $$\alpha$$ + $$\beta$$ = 1 and $$\alpha$$$$\beta$$ = $$-$$1. Let p_n = ($$\alpha$$)ⁿ + ($$\beta$$)ⁿ, p_n$$-$$1 = 11 and p_n+1 = 29 for some integer n $$ \ge $$ 1. Then, the value of p$$_n^2$$ is ___________.

Let z be those complex numbers which satisfy

| z + 5 | $$ \le $$ 4 and z(1 + i) + $$\overline z $$(1 $$-$$ i) $$ \ge $$ $$-$$10, i = $$\sqrt { - 1} $$.

If the maximum value of | z + 1 |² is $$\alpha$$ + $$\beta$$$$\sqrt 2 $$, then the value of ($$\alpha$$ + $$\beta$$) is ____________.

If the arithmetic mean and geometric mean of the p^th and q^th terms of the
sequence $$-$$16, 8, $$-$$4, 2, ...... satisfy the equation
4x² $$-$$ 9x + 5 = 0, then p + q is equal to __________.

If $${I_{m,n}} = \int\limits_0^1 {{x^{m - 1}}{{(1 - x)}^{n - 1}}dx} $$, for m, $$n \ge 1$$, and
$$\int\limits_0^1 {{{{x^{m - 1}} + {x^{n - 1}}} \over {{{(1 + x)}^{m + 1}}}}} dx = \alpha {I_{m,n}}\alpha \in R$$, then $$\alpha$$ equals ___________.

Let X₁, X₂, ......., X₁₈ be eighteen observations such
that $$\sum\limits_{i = 1}^{18} {({X_i} - } \alpha ) = 36$$ and $$\sum\limits_{i = 1}^{18} {({X_i} - } \beta {)^2} = 90$$, where $$\alpha$$ and $$\beta$$ are distinct real numbers. If the standard deviation of these observations is 1, then the value of | $$\alpha$$ $$-$$ $$\beta$$ | is ____________.

Let a be an integer such that all the real roots of the polynomial
2x⁵ + 5x⁴ + 10x³ + 10x² + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.