Q88. Let $r _ { k } = \frac { \int _ { 0 } ^ { 1 } \left( 1 - x ^ { 7 } \right) ^ { k } d x } { \int _ { 0 } ^ { 1 } \left( 1 - x ^ { 7 } \right) ^ { k + 1 } d x } , k \in \mathbb { N }$. Then the value of $\sum _ { k = 1 } ^ { 10 } \frac { 1 } { 7 \left( r _ { k } - 1 \right) }$ is equal to $\_\_\_\_$

If $\int ( \cos x ) ^ { - 5 / 2 } ( \sin x ) ^ { - 11 / 2 } d x = \frac { p _ { 1 } } { q _ { 1 } } ( \cot x ) ^ { 9 / 2 } + \frac { p _ { 2 } } { q _ { 2 } } ( \cot x ) ^ { 5 / 2 } + \frac { p _ { 3 } } { q _ { 3 } } ( \cot x ) ^ { 1 / 2 } - \frac { p _ { 4 } } { q _ { 4 } } ( \cot x ) ^ { - 3 / 2 } + c$ (where c is constant of integration), then value of $\frac { 15 p _ { 1 } p _ { 2 } p _ { 3 } p _ { 4 } } { q _ { 1 } q _ { 2 } q _ { 3 } q _ { 4 } }$ is
(A) 16
(B) 14

Given that
$$\frac { d y } { d x } = 3 x ^ { 2 } - \frac { 2 - 3 x } { x ^ { 3 } } , \quad x \neq 0$$
and $y = 5$ when $x = 1$, find $y$ in terms of $x$.
A $y = \frac { 1 } { 3 } x ^ { 3 } + x ^ { - 2 } - 3 x ^ { - 1 } + 6 \frac { 2 } { 3 }$
B $y = x ^ { 3 } + \frac { 1 } { 2 } x ^ { - 2 } - 3 x ^ { - 1 } + 6 \frac { 1 } { 2 }$
C $y = x ^ { 3 } + x ^ { - 2 } - 3 x ^ { - 1 } + 6$
D $y = x ^ { 3 } + x ^ { - 2 } - x ^ { - 1 } + 4$
E $y = x ^ { 3 } + 2 x ^ { - 2 } - x ^ { - 1 } + 3$
F $y = 3 x ^ { 3 } + x ^ { - 2 } - x ^ { - 1 } + 2$

A sequence $u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots$ is defined as follows:
$$\begin{aligned} & u _ { 0 } = 1 \\ & u _ { n } = \int _ { 0 } ^ { 1 } 4 x u _ { n - 1 } d x \quad \text { for } n \geqslant 1 \end{aligned}$$
What is the value of $u _ { 1000 }$ ?
A $2 ^ { 1000 }$
B $4 ^ { 1000 }$
C $\frac { 4 } { 1000 ! }$
D $\frac { 4 } { 1001 ! }$
$\mathbf { E } \quad \frac { 2 ^ { 1000 } } { 1000 ! }$
F $\frac { 4 ^ { 1000 } } { 1000 ! }$
G $\frac { 2 ^ { 1000 } } { 1001 ! }$
$\mathbf { H } \frac { 4 ^ { 1000 } } { 1001 ! }$

Find the value of
$$\int _ { 1 } ^ { 4 } \frac { 3 - 2 x } { x \sqrt { x } } \mathrm {~d} x$$
A $- \frac { 13 } { 2 }$
B $- \frac { 85 } { 16 }$
C $- \frac { 13 } { 8 }$
D - 1
E $- \frac { 1 } { 4 }$
F $\frac { 7 } { 4 }$
G 7

Consider the two statements R: $\quad k$ is an integer multiple of $\pi$
$$\mathrm { S } : \quad \int _ { 0 } ^ { k } \sin 2 x \mathrm {~d} x = 0$$
Which of the following statements is true? A $R$ is necessary and sufficient for $S$. B R is necessary but not sufficient for $S$. C R is sufficient but not necessary for $S$. D $R$ is not necessary and not sufficient for $S$.

The functions $f$ and $g$ are defined and differentiable on the set of real numbers and satisfy
$$\begin{aligned} & \int_{1}^{2} f^{\prime}(3x)\, dx = 4 \\ & \int f(2x)\, dx = g(x) + C, \quad (C \text{ constant}) \end{aligned}$$
If $f(3) = 5$, what is the value of the derivative $g^{\prime}(3)$?
A) 1 B) 5 C) 9 D) 13 E) 17