The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :

Consider the parabola with vertex \left( {{1 \over 2},{3 \over 4}} \right) and the directrix y = {1 \over 2}. Let P be the point where the parabola meets the line x = - {1 \over 2}. If the normal to the parabola at P intersects the parabola again at the point Q, then (PQ)2 is equal to :

\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {\left( {\sin \sqrt t } \right)dt} } \over {{x^3}}} is equal to :

The value of the integral, $$\underset{1}{\overset{3}{\int }}[x^2-2x-2]dx$$, where [x] denotes the greatest integer less than or equal to x, is :

Let f(x) be a differentiable function defined on [0, 2] such that f'(x) = f'(2 $$-$$ x) for all x$$\in$$ (0, 2), f(0) = 1 and f(2) = e2. Then the value of $$\underset{0}{\overset{2}{\int }}f(x)dx$$ is :

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $$\ne$$ 0 for all x $$\in$$ R. If \left| {\matrix{ {f(x)} & {f'(x)} \cr {f'(x)} & {f''(x)} \cr } } \right| = 0, for all x$$\in$$R, then the value of f(1) lies in the interval :

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

If {I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx}, then :

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

The value of \int\limits_{ - \pi /2}^{\pi /2} {{{{{\cos }^2}x} \over {1 + {3^x}}}} dx is :

The value of $$\sum _{n=1}^{100}\underset{n-1}{\overset{n}{\int }}{e}^{x-[x]}dx$$, where [ x ] is the greatest integer $$\le$$ x, is :

Let $$f(x)=\underset{0}{\overset{x}{\int }}{e}^{t}f(t)dt+e^x$$ 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 :

Consider the integral I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}, where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that $${\int }_0^{1}P(x)dx$$ = 1 and P(x) leaves remainder 5 when it is divided by (x $$-$$ 2). Then the value of 9(b + c) is equal to :

Which of the following statements is correct for the function g($$\alpha$$) for $$\alpha$$ $$\in$$ R such that g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx}

Let f : R $$\to$$ R be defined as f(x) = e$$-$$xsinx. If F : [0, 1] $$\to$$ R is a differentiable function with that F(x) = $${\int }_0^{x}f(t)dt$$, then the value of $${\int }_0^1(F^{\prime }(x)+f(x))e^{x}dx$$ lies in the interval

If the integral \int_0^{10} {{{[\sin 2\pi x]} \over {{e^{x - [x]}}}}} dx = \alpha {e^{ - 1}} + \beta {e^{ - {1 \over 2}}} + \gamma, where $$\alpha$$, $$\beta$$, $$\gamma$$ are integers and [x] denotes the greatest integer less than or equal to x, then the value of $$\alpha$$ + $$\beta$$ + $$\gamma$$ is equal to :

Let g(x) = $${\int }_0^{x}f(t)dt$$, where f is continuous function in [ 0, 3 ] such that {1 \over 3} $$\le$$ f(t) $$\le$$ 1 for all t$$\in$$ [0, 1] and 0 $$\le$$ f(t) $$\le$$ {1 \over 2} for all t$$\in$$ (1, 3]. The largest possible interval in which g(3) lies is :

Let a be a positive real number such that $${\int }_0^a{e}^{x-[x]}dx=10e-9$$ where [ x ] is the greatest integer less than or equal to x. Then a is equal to: