Let $f : \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be a continuous function such that $$f ( 0 ) = 1 \text { and } \int _ { 0 } ^ { \frac { \pi } { 3 } } f ( t ) d t = 0$$ Then which of the following statements is (are) TRUE ? (A) The equation $f ( x ) - 3 \cos 3 x = 0$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$ (B) The equation $f ( x ) - 3 \sin 3 x = - \frac { 6 } { \pi }$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$ (C) $\lim _ { x \rightarrow 0 } \frac { x \int _ { 0 } ^ { x } f ( t ) d t } { 1 - e ^ { x ^ { 2 } } } = - 1$ (D) $\lim _ { x \rightarrow 0 } \frac { \sin x \int _ { 0 } ^ { x } f ( t ) d t } { x ^ { 2 } } = - 1$
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 $-2$
The value of $k \in \mathrm {~N}$ for which the integral $I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 - x ^ { k } \right) ^ { n } d x , n \in \mathbb { N }$, satisfies $147 I _ { 20 } = 148 I _ { 21 }$ is (1) 14 (2) 8 (3) 10 (4) 7
Computer Science applicants should turn to page 14. For a positive whole number $n$, the function $f _ { n } ( x )$ is defined by $$f _ { n } ( x ) = \left( x ^ { 2 n - 1 } - 1 \right) ^ { 2 } .$$ (i) On the axes provided opposite, sketch the graph of $y = f _ { 2 } ( x )$ labelling where the graph meets the axes. (ii) On the same axes sketch the graph of $y = f _ { n } ( x )$ where $n$ is a large positive integer. (iii) Determine $$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$ (iv) The positive constants $A$ and $B$ are such that $$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x \leqslant 1 - \frac { A } { n + B } \text { for all } n \geqslant 1 .$$ Show that $$( 3 n - 1 ) ( n + B ) \geqslant A ( 4 n - 1 ) n ,$$ and explain why $A \leqslant 3 / 4$. (v) When $A = 3 / 4$, what is the smallest possible value of $B$ ? [Figure]
Consider this statement about a function $f ( x )$ : $\left( ^ { * } \right)$ If $( f ( x ) ) ^ { 2 } \leq 1$ for all $- 1 \leq x \leq 1$ then $\int _ { - 1 } ^ { 1 } ( f ( x ) ) ^ { 2 } \mathrm {~d} x \leq \int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } x$ Which one of the following functions provides a counterexample to (*)?
$\mathrm { f } ( x )$ is a function for which $$\int _ { 0 } ^ { 3 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x + \int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$$ Which of the following claims about $\mathrm { f } ( x )$ is/are necessarily true? I $\mathrm { f } ( x ) \leq 0$ for some $x$ with $1 \leq x \leq 3$ II $\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x \leq \int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$ A neither of them B I only C II only D I and II
Which of the following statements about polynomials $f$ and $g$ is/are true? I If $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$, then $\int _ { 0 } ^ { x } \mathrm { f } ( t ) \mathrm { d } t \geq \int _ { 0 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t$ for all $x \geq 0$. II If $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$, then $\mathrm { f } ^ { \prime } ( x ) \geq \mathrm { g } ^ { \prime } ( x )$ for all $x \geq 0$. III If $\mathrm { f } ^ { \prime } ( x ) \geq \mathrm { g } ^ { \prime } ( x )$ for all $x \geq 0$, then $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$. A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
Place the following integrals in order of size, starting with the smallest. $$\begin{aligned}
& P = \int _ { 0 } ^ { 1 } 2 ^ { \sqrt { x } } \mathrm {~d} x \\
& Q = \int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x \\
& R = \int _ { 0 } ^ { 1 } ( \sqrt { 2 } ) ^ { x } \mathrm {~d} x
\end{aligned}$$ A $P < Q < R$ B $P < R < Q$ C $Q < P < R$ D $Q < R < P$ E $\quad R < P < Q$ F $R < Q < P$
Let $f$ be a polynomial with real coefficients. The integral $I _ { p , q }$ where $p < q$ is defined by $$I _ { p , q } = \int _ { p } ^ { q } ( f ( x ) ) ^ { 2 } - ( f ( | x | ) ) ^ { 2 } \mathrm {~d} x$$ Which of the following statements must be true? $1 I _ { p , q } = 0$ only if $0 < p$ $2 f ^ { \prime } ( x ) < 0$ for all $x$ only if $I _ { p , q } < 0$ for all $p < q < 0$ $3 \quad I _ { p , q } > 0$ only if $p < 0$ A none of them B 1 only C 2 only D 3 only E 1 and 2 only F 1 and 3 only G 2 and 3 only H 1, 2 and 3
Let $n$ be a natural number, $$\begin{aligned}
& f _ { n } : [ n , n + 1 ) \rightarrow \left[ 0 , \frac { 1 } { 2 ^ { n } } \right) \\
& f _ { n } ( x ) = \frac { ( x - n ) ^ { 2 } } { 2 ^ { n } }
\end{aligned}$$ The regions between the functions defined in this form and the x-axis are given shaded in the figure below. Accordingly, what is the sum of the areas of all shaded regions in square units? A) $\frac { 2 } { 3 }$ B) $\frac { 3 } { 4 }$ C) $\frac { 5 } { 6 }$ D) $\frac { 8 } { 9 }$ E) $\frac { 11 } { 12 }$
For an increasing and continuous function f defined on the set of real numbers, $$\begin{aligned}
& f ( 0 ) = 2 \\
& f ( 1 ) = 3 \\
& f ( 2 ) = 4
\end{aligned}$$ equalities are given. Accordingly, the value of the integral $\int _ { 0 } ^ { 2 } f ( x ) d x$ could be which of the following? A) 4 B) 4.5 C) 6 D) 7.5 E) 8
Let $m$ be a positive real number. In the rectangular coordinate plane, the region between the graph of a function $f$ defined on the closed interval $[-m, m]$ and the $x$-axis is divided into four regions and these regions are colored as shown in the figure. The areas of these regions, which are different from each other, are denoted by $A, B, C$ and $D$ as shown in the figure. $$\int_{-m}^{m} |f(x)|\, dx = \int_{-m}^{m} f(x)\, dx + \int_{0}^{m} 2 \cdot f(x)\, dx$$ Given that, which of the following is the integral $\int_{-m}^{m} f(x)\, dx$ equal to? A) $A + B$ B) $A + C$ C) $A + D$ D) $B + C$ E) $C + D$