Let S be the set of all $$\lambda $$ $$ \in $$ R for which the system of linear equations

2x – y + 2z = 2
x – 2y + $$\lambda $$z = –4
x + $$\lambda $$y + z = 4

has no solution. Then the set S :

Area (in sq. units) of the region outside

$${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$$ and inside the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$$ is :

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :

If a function f(x) defined by

$$f\left( x \right) = \left\{ {\matrix{ {a{e^x} + b{e^{ - x}},} & { - 1 \le x < 1} \cr {c{x^2},} & {1 \le x \le 3} \cr {a{x^2} + 2cx,} & {3 < x \le 4} \cr } } \right.$$

be continuous for some $$a$$, b, c $$ \in $$ R and f'(0) + f'(2) = e, then the value of of $$a$$ is :

Let $$\alpha $$ > 0, $$\beta $$ > 0 be such that
$$\alpha $$³ + $$\beta $$² = 4. If the maximum value of the term independent of x in
the binomial expansion of $${\left( {\alpha {x^{{1 \over 9}}} + \beta {x^{ - {1 \over 6}}}} \right)^{10}}$$ is 10k,
then k is equal to :

Let A be a 2 $$ \times $$ 2 real matrix with entries from {0, 1} and |A| $$ \ne $$ 0. Consider the following two statements :

(P) If A $$ \ne $$ I₂ , then |A| = –1
(Q) If |A| = 1, then tr(A) = 2,

where I₂ denotes 2 $$ \times $$ 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A. Then :

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ______.

If $$\mathop {\lim }\limits_{x \to 1} {{x + {x^2} + {x^3} + ... + {x^n} - n} \over {x - 1}}$$ = 820,
(n $$ \in $$ N) then the value of n is equal to _______.

Let $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be three unit vectors such that
$${\left| {\overrightarrow a - \overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$$ = 8.

Then $${\left| {\overrightarrow a + 2\overrightarrow b } \right|^2}$$ + $${\left| {\overrightarrow a + 2\overrightarrow c } \right|^2}$$ is equal to ______.

The integral $$\int\limits_0^2 {\left| {\left| {x - 1} \right| - x} \right|dx} $$
is equal to______.

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :

The value of

$${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$$ is :

The domain of the function
f(x) = $${\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)$$ is (– $$\infty $$, -a]$$ \cup $$[a, $$\infty $$). Then a is equal to :

Let $$\alpha $$ and $$\beta $$ be the roots of the equation
5x² + 6x – 2 = 0. If S_n = $$\alpha $$ⁿ + $$\beta $$ⁿ, n = 1, 2, 3...., then :

If R = {(x, y) : x, y $$ \in $$ Z, x² + 3y² $$ \le $$ 8} is a relation on the set of integers Z, then the domain of R^–1 is :

Let X = {x $$ \in $$ N : 1 $$ \le $$ x $$ \le $$ 17} and
Y = {ax + b: x $$ \in $$ X and a, b $$ \in $$ R, a > 0}. If mean
and variance of elements of Y are 17 and 216
respectively then a + b is equal to :

Let y = y(x) be the solution of the differential equation,
$${{2 + \sin x} \over {y + 1}}.{{dy} \over {dx}} = - \cos x$$, y > 0,y(0) = 1.
If y($$\pi $$) = a and $${{dy} \over {dx}}$$ at x = $$\pi $$ is b, then the ordered pair (a, b) is equal to :