Let [t] denote the greatest integer $$\le$$ t. If for some $$\lambda$$ $$\in$$ R - {1, 0}, \mathop {\lim }\limits_{x \to 0} \left| {{{1 - x + \left| x \right|} \over {\lambda - x + \left[ x \right]}}} \right| = L, then L is equal to :

\mathop {\lim }\limits_{x \to 0} {\left( {{{3{x^2} + 2} \over {7{x^2} + 2}}} \right)^{{1 \over {{x^2}}}}} is equal to

Let S be the set of all functions ƒ : [0,1] $$\to$$ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $$\in$$ (0,1), depending on ƒ, such that

Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $$\in$$ (a, b), ƒ'(x) > 0 and ƒ''(x) < 0, then for any c $$\in$$ (a, b), {{f(c) - f(a)} \over {f(b) - f(c)}} is greater than :

If f(x) = \left\{ {\matrix{ {{{\sin (a + 2)x + \sin x} \over x};} & {x 0} \cr } } \right. is continuous at x = 0, then a + 2b is equal to :

Let [t] denote the greatest integer $$\le$$ t and \mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A. Then the function, f(x) = [x2]sin($$\pi$$x) is discontinuous, when x is equal to :

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 :

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 :

2$$\pi$$ - \left( {{{\sin }^{ - 1}}{4 \over 5} + {{\sin }^{ - 1}}{5 \over {13}} + {{\sin }^{ - 1}}{{16} \over {65}}} \right) is equal to :

If S is the sum of the first 10 terms of the series {\tan ^{ - 1}}\left( {{1 \over 3}} \right) + {\tan ^{ - 1}}\left( {{1 \over 7}} \right) + {\tan ^{ - 1}}\left( {{1 \over {13}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {21}}} \right) + .... then tan(S) is equal to :

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is :

If the number of five digit numbers with distinct digits and 2 at the 10th place is 336 k, then k is equal to :

Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number of red lines is 99 times the number of blue lines, then the value of n is :

The value of (2.1P0 – 3.2P1 + 4.3P2 .... up to 51th term) + (1! – 2! + 3! – ..... up to 51th term) is equal to :

Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is :

The number of ordered pairs (r, k) for which 6.35Cr = (k2 - 3). 36Cr + 1, where k is an integer, is :

If a, b and c are the greatest value of 19Cp, 20Cq and 21Cr respectively, then :

If $$\alpha$$ and $$\beta$$ are the roots of the equation x2 + px + 2 = 0 and {1 \over \alpha } and {1 \over \beta } are the roots of the equation 2x2 + 2qx + 1 = 0, then \left( {\alpha - {1 \over \alpha }} \right)\left( {\beta - {1 \over \beta }} \right)\left( {\alpha + {1 \over \beta }} \right)\left( {\beta + {1 \over \alpha }} \right) is equal to :

Let $$\alpha$$ and $$\beta$$ be the roots of x2 - 3x + p=0 and $$\gamma$$ and $$\delta$$ be the roots of x2 - 6x + q = 0. If $$\alpha ,\beta ,\gamma ,\delta$$ form a geometric progression.Then ratio (2q + p) : (2q - p) is: