Let $f : R \rightarrow R$ be a continuous function such that $f ( x ) + f ( x + 1 ) = 2$ for all $x \in R$. If $I _ { 1 } = \int _ { 0 } ^ { 8 } f ( x ) d x$ and $I _ { 2 } = \int _ { - 1 } ^ { 3 } f ( x ) d x$, then the value of $I _ { 1 } + 2 I _ { 2 }$ is equal to $\_\_\_\_$.

Let the curve $y = y ( x )$ be the solution of the differential equation, $\frac { d y } { d x } = 2 ( x + 1 )$. If the numerical value of area bounded by the curve $y = y ( x )$ and $x$-axis is $\frac { 4 \sqrt { 8 } } { 3 }$, then the value of $y ( 1 )$ is equal to $\_\_\_\_$.

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined as $f ( x ) = a \sin \left( \frac { \pi [ x ] } { 2 } \right) + [ 2 - x ] , a \in \mathbb { R }$, where $[ t ]$ is the greatest integer less than or equal to $t$. If $\lim _ { x \rightarrow - 1 } f ( x )$ exists, then the value of $\int _ { 0 } ^ { 4 } f ( x ) d x$ is equal to
(1) - 1
(2) - 2
(3) 1
(4) 2

For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10,10]$ by $f(x) = \begin{cases} x - \lfloor x \rfloor, & \text{if } \lfloor x \rfloor \text{ is odd} \\ 1 + \lfloor x \rfloor - x, & \text{if } \lfloor x \rfloor \text{ is even} \end{cases}$ Then, the value of $\frac { \pi ^ { 2 } } { 10 } \int _ { - 10 } ^ { 10 } f(x) \cos(\pi x)\, dx$ is
(1) 4
(2) 2
(3) 1
(4) 0

Let $f ( x ) = 2 + | x | - | x - 1 | + | x + 1 | , x \in R$. Consider $( S 1 ) : f ^ { \prime } \left( - \frac { 3 } { 2 } \right) + f ^ { \prime } \left( - \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 3 } { 2 } \right) = 2$ $( S 2 ) : \int _ { - 2 } ^ { 2 } f ( x ) d x = 12$ Then,
(1) both ( $S 1$ ) and ( $S 2$ ) are correct
(2) both $( S 1 )$ and $( S 2 )$ are wrong
(3) only ( $S 1$ ) is correct
(4) only ( $S 2$ ) is correct

The value of the integral $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { d x } { \left( 1 + e ^ { x } \right) \left( \sin ^ { 6 } x + \cos ^ { 6 } x \right) }$ is equal to
(1) $2 \pi$
(2) 0
(3) $\pi$
(4) $\frac { \pi } { 2 }$

Let $f : R \rightarrow R$ be a function defined by : $f ( x ) = \left\{ \begin{array} { c c } \max _ { t \leq x } \left\{ t ^ { 3 } - 3 t \right\} ; & x \leq 2 \\ x ^ { 2 } + 2 x - 6 ; & 2 < x < 3 \\ { [ x - 3 ] + 9 ; } & 3 \leq x \leq 5 \\ 2 x + 1 ; & x > 5 \end{array} \right.$ Where $[ t ]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I = \int _ { - 2 } ^ { 2 } f ( x ) d x$. Then the ordered pair ( $m , I$ ) is equal to
(1) $\left( 3 , \frac { 27 } { 4 } \right)$
(2) $\left( 3 , \frac { 23 } { 4 } \right)$
(3) $\left( 4 , \frac { 27 } { 4 } \right)$
(4) $\left( 4 , \frac { 23 } { 4 } \right)$

$\lim_{n \rightarrow \infty} \frac{1}{2^n} \left( \frac{1}{\sqrt{1 - \frac{1}{2^n}}} + \frac{1}{\sqrt{1 - \frac{2}{2^n}}} + \frac{1}{\sqrt{1 - \frac{3}{2^n}}} + \ldots + \frac{1}{\sqrt{1 - \frac{2^n - 1}{2^n}}} \right)$ is equal to
(1) $\frac{1}{2}$
(2) 1
(3) 2
(4) $-2$

If $f ( \alpha ) = \int _ { 1 } ^ { \alpha } \frac { \log _ { 10 } t } { 1 + t } d t , \alpha > 0$, then $f \left( e ^ { 3 } \right) + f \left( e ^ { - 3 } \right)$ is equal to
(1) 9
(2) $\frac { 9 } { 2 }$

If $[t]$ denotes the greatest integer $\leq t$, then the value of $\int_0^1 \left[2x - \left|3x^2 - 5x + 2\right| + 1\right] dx$ is
(1) $\frac{\sqrt{37} + \sqrt{13} - 4}{6}$
(2) $\frac{\sqrt{37} - \sqrt{13} - 4}{6}$
(3) $\frac{-\sqrt{37} - \sqrt{13} + 4}{6}$
(4) $\frac{-\sqrt{37} + \sqrt{13} + 4}{6}$

$I = \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( \frac { 8 \sin x - \sin 2 x } { x } \right) d x$. Then
(1) $\frac { \pi } { 2 } < I < \frac { 3 \pi } { 4 }$
(2) $\frac { \pi } { 5 } < I < \frac { 5 \pi } { 12 }$
(3) $\frac { 5 \pi } { 12 } < I < \frac { \sqrt { 2 } } { 3 } \pi$
(4) $\frac { 3 \pi } { 4 } < I < \pi$

$\int _ { 0 } ^ { 2 } \left( \left| 2 x ^ { 2 } - 3 x \right| + \left[ x - \frac { 1 } { 2 } \right] \right) d x$, where $[ t ]$ is the greatest integer function, is equal to
(1) $\frac { 7 } { 6 }$
(2) $\frac { 19 } { 12 }$
(3) $\frac { 31 } { 12 }$
(4) $\frac { 3 } { 2 }$

$\lim _ { n \rightarrow \infty } \left( \frac { n ^ { 2 } } { \left( n ^ { 2 } + 1 \right) ( n + 1 ) } + \frac { n ^ { 2 } } { \left( n ^ { 2 } + 4 \right) ( n + 2 ) } + \frac { n ^ { 2 } } { \left( n ^ { 2 } + 9 \right) ( n + 3 ) } + \ldots + \frac { n ^ { 2 } } { \left( n ^ { 2 } + n ^ { 2 } \right) ( n + n ) } \right)$ is equal to
(1) $\frac { \pi } { 8 } + \frac { 1 } { 4 } \ln 2$
(2) $\frac { \pi } { 4 } + \frac { 1 } { 8 } \ln 2$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 8 } \ln 2$
(4) $\frac { \pi } { 8 } + \ln \sqrt { 2 }$

The value of $\int _ { 0 } ^ { \pi } \frac { e ^ { \cos x } \sin x } { ( 1 + \cos ^ { 2 } x )( e ^ { \cos x } + e ^ { - \cos x } ) } \mathrm { d } x$ is equal to

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\int_{-3}^{101} \left([\sin(\pi x)] + e^{[\cos(2\pi x)]}\right) dx$ is equal to
(1) $\frac{52(1-e)}{e}$
(2) $\frac{52}{e}$
(3) $\frac{52(2+e)}{e}$
(4) $\frac{104}{e}$

The integral $\int _ { 0 } ^ { 1 } \frac { 1 } { 7 ^ { \left[ \frac { 1 } { x } \right] } } d x$, where $[ \cdot ]$ denotes the greatest integer function, is equal to
(1) $1 - 6 \ln \left( \frac { 6 } { 7 } \right)$
(2) $1 + 6 \ln \left( \frac { 6 } { 7 } \right)$
(3) $1 - 7 \ln \left( \frac { 6 } { 7 } \right)$
(4) $1 + 7 \ln \left( \frac { 6 } { 7 } \right)$

If $b_n = \int_0^{\frac{\pi}{2}} \frac{\cos^2 nx}{\sin x} dx$, $n \in \mathbb{N}$, then
(1) $b_3 - b_2, b_4 - b_3, b_5 - b_4$ are in an A.P. with common difference $-2$
(2) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference $2$
(3) $b_3 - b_2, b_4 - b_3, b_5 - b_4$ are in a G.P.
(4) $\frac{1}{b_3 - b_2}, \frac{1}{b_4 - b_3}, \frac{1}{b_5 - b_4}$ are in an A.P. with common difference $-2$

$\int _ { 0 } ^ { 20 \pi } ( | \sin x | + | \cos x | ) ^ { 2 } \, d x$ is equal to:
(1) $10 \pi + 4$
(2) $10 \pi + 2$
(3) $20 \pi - 2$
(4) $20 \pi + 2$

The value of the integral $\frac { 48 } { \pi ^ { 4 } } \int _ { 0 } ^ { \pi } \left( \frac { 3 \pi x ^ { 2 } } { 2 } - x ^ { 3 } \right) \frac { \sin x } { 1 + \cos ^ { 2 } x } \, dx$ is equal to $\_\_\_\_$.

If $n ( 2 n + 1 ) \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n } d x = 1177 \int _ { 0 } ^ { 1 } \left( 1 - x ^ { n } \right) ^ { 2 n + 1 } d x$, then $n \in N$ is equal to $\_\_\_\_$.

Let $f ( x ) = \min \{ [ x - 1 ] , [ x - 2 ] , \ldots , [ x - 10 ] \}$ where $[ t ]$ denotes the greatest integer $\leq t$. Then $\int _ { 0 } ^ { 10 } f ( x ) d x + \int _ { 0 } ^ { 10 } ( f ( x ) ) ^ { 2 } d x + \int _ { 0 } ^ { 10 } | f ( x ) | d x$ is equal to $\_\_\_\_$ .

The value of the integral $\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \frac { \sin ( 6 x ) } { \sin x } d x$ is equal to $\_\_\_\_$.

The value of $\int_0^1 \frac{d}{dx}\left[\tan^{-1}\left(\frac{1}{1-x+x^2}\right)\right]dx$ is
(1) $\frac{\pi}{4}$
(2) $\tan^{-1}(2)$
(3) $\frac{\pi}{2} - \tan^{-1}(2)$
(4) $\frac{\pi}{4} - \tan^{-1}(2)$

Let f be a continuous function satisfying $\int _ { 0 } ^ { t ^ { 2 } } \left( f(x) + x^2 \right) dx = \frac { 4 } { 3 } t ^ { 3 } , \forall t > 0$. Then $f\left( \frac { \pi ^ { 2 } } { 4 } \right)$ is equal to
(1) $\pi ^ { 2 } \left( 1 - \frac { \pi ^ { 2 } } { 16 } \right)$
(2) $- \pi \left( 1 + \frac { \pi ^ { 3 } } { 16 } \right)$
(3) $\pi \left( 1 - \frac { \pi ^ { 3 } } { 16 } \right)$
(4) $- \pi ^ { 2 } \left( 1 + \frac { \pi ^ { 2 } } { 16 } \right)$

Let $f ( x ) = x + \frac { a } { \pi ^ { 2 } - 4 } \sin x + \frac { b } { \pi ^ { 2 } - 4 } \cos x , x \in \mathbb { R }$ be a function which satisfies $f ( x ) = x + \int _ { 0 } ^ { \pi / 2 } \sin ( x + y ) f ( y ) d y$. Then $( a + b )$ is equal to
(1) $- \pi ( \pi + 2 )$
(2) $- 2 \pi ( \pi + 2 )$
(3) $- 2 \pi ( \pi - 2 )$
(4) $- \pi ( \pi - 2 )$