For a suitably chosen real constant a, let a

function, $$f:R - \left\{ { - a} \right\} \to R$$ be defined by

$$f(x) = {{a - x} \over {a + x}}$$. Further suppose that for any real number $$x \ne - a$$ and $$f(x) \ne - a$$,

(fof)(x) = x. Then $$f\left( { - {1 \over 2}} \right)$$ is equal to :

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, –4) in this line is :

The area (in sq. units) of the region enclosed
by the curves y = x² – 1 and y = 1 – x² is equal to :

Let z = x + iy be a non-zero complex number such that $${z^2} = i{\left| z \right|^2}$$, where i = $$\sqrt { - 1} $$ , then z lies on the :

The integral $$\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx$$ equals :

For all twice differentiable functions f : R $$ \to $$ R,
with f(0) = f(1) = f'(0) = 0

Let f : R $$ \to $$ R be a function defined by
f(x) = max {x, x²}. Let S denote the set of all points in R, where f is not differentiable. Then :

The set of all real values of $$\lambda $$ for which the function

$$f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lambda + \sin x} \right),x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$

has exactly one maxima and exactly one minima, is :

The sum of distinct values of $$\lambda $$ for which the system of equations

$$\left( {\lambda - 1} \right)x + \left( {3\lambda + 1} \right)y + 2\lambda z = 0$$
$$\left( {\lambda - 1} \right)x + \left( {4\lambda - 2} \right)y + \left( {\lambda + 3} \right)z = 0$$
$$2x + \left( {3\lambda + 1} \right)y + 3\left( {\lambda - 1} \right)z = 0$$

has non-zero solutions, is ________ .

Consider the data on x taking the values
0, 2, 4, 8,....., 2ⁿ with frequencies
ⁿC₀ , ⁿC₁ , ⁿC₂ ,...., ⁿC_n respectively. If the
mean of this data is $${{728} \over {{2^n}}}$$, then n is equal to _________ .

If $$\overrightarrow x $$ and $$\overrightarrow y $$ be two non-zero vectors such that $$\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|$$ and $${2\overrightarrow x + \lambda \overrightarrow y }$$ is perpendicular to $${\overrightarrow y }$$, then the value of $$\lambda $$ is _________ .

Suppose that a function f : R $$ \to $$ R satisfies
f(x + y) = f(x)f(y) for all x, y $$ \in $$ R and f(1) = 3.
If $$\sum\limits_{i = 1}^n {f(i)} = 363$$ then n is equal to ________ .

The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is ________ .

The common difference of the A.P.
b₁, b₂, … , b_m is 2 more than the common
difference of A.P. a₁, a₂, …, a_n. If
a₄₀ = –159, a₁₀₀ = –399 and b₁₀₀ = a₇₀, then b₁ is equal to :

The probabilities of three events A, B and C are given by
P(A) = 0.6, P(B) = 0.4 and P(C) = 0.5.
If P(A$$ \cup $$B) = 0.8, P(A$$ \cap $$C) = 0.3, P(A$$ \cap $$B$$ \cap $$C) = 0.2, P(B$$ \cap $$C) = $$\beta $$
and P(A$$ \cup $$B$$ \cup $$C) = $$\alpha $$, where 0.85 $$ \le \alpha \le $$ 0.95, then $$\beta $$ lies in the interval :

If $$\alpha $$ and $$\beta $$ are the roots of the equation
2x(2x + 1) = 1, then $$\beta $$ is equal to :

Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{ {\cos \theta } & {\sin \theta } \cr { - \sin \theta } & {\cos \theta } \cr } } \right]$$.

If B = A + A⁴ , then det (B) :

If the constant term in the binomial expansion of
$${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$$ is 405, then |k| equals :