We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$ and $\sigma(x) = \sum_{k=1}^{+\infty} \frac{x^k}{k^2}$.
Conclude that $$\int_0^{\pi} \Psi(x)^2 \mathrm{d}x = 4\pi\left(\ln\left(\frac{a+b}{2}\right)\right)^2 + 2\pi\sigma(\rho^2)$$

123- In the figure below, the two shaded areas are equal, $C$ is which of the following?
[Figure: Graph showing $y = \sqrt{x}$ with shaded region between $x = C$ and $x = 4$]
(1) $\dfrac{4}{3}$ (2) $\dfrac{16}{9}$ (3) $2$ (4) $\dfrac{9}{4}$
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123. The area bounded by the curve $y = \sqrt{1 - \cos 2x}$ and the $x$-axis over one period is:
(1) $2$ (2) $2\sqrt{2}$ (3) $3$ (4) $3\sqrt{2}$

125. The area of a regular octagon inscribed in a circle of radius $2$ is:
(1) $8\sqrt{2}$ (2) $8(\sqrt{2}-1)$ (3) $4(1+\sqrt{2})$ (4) $4(2+\sqrt{2})$
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126. In a rectangle with side lengths 3 and 4 units, from each vertex, a perpendicular is drawn to the other diagonal of this rectangle. What is the area of the resulting parallelogram?
[Figure: Rectangle with diagonals and perpendiculars drawn]

• [(1)] $5/25$
• [(2)] $5/75$
• [(3)] $6$
• [(4)] $7/5$

123- What is the mean value (average) of the function $f(x) = \dfrac{2x-1}{\sqrt{x}}$ on the interval $[1, 4]$?
(1) $\dfrac{17}{9}$ (2) $\dfrac{7}{3}$ (3) $\dfrac{22}{9}$ (4) $\dfrac{8}{3}$

20 -- What is the minimum distance from points on the curve $y = \sqrt{x - [x^2]}$ to the line $x - y + 2 = 0$?
(1) $\dfrac{\sqrt{5}}{5}$ (2) $\dfrac{3\sqrt{5}}{8}$ (3) $\dfrac{\sqrt{5}}{10}$ (4) $\dfrac{3\sqrt{5}}{10}$

Let $f(x) = \int_0^1 |t - x|\, t\, dt$ for $x \in [0,1]$. Find $f(x)$ and sketch its graph.

The equation $x^{2} + (b/a)x + (c/a) = 0$ has two real roots $\alpha$ and $\beta$. If $a > 0$, then the area under the curve $f(x) = x^{2} + (b/a)x + (c/a)$ between $\alpha$ and $\beta$ is
(a) $(b^{2} - 4ac)/2a$
(b) $(b^{2} - 4ac)^{3/2}/6a^{3}$
(c) $-(b^{2} - 4ac)^{3/2}/6a^{3}$
(d) $-(b^{2} - 4ac)/2a$

Let $n$ be a positive integer. Define $$f ( x ) = \min \{ | x - 1 | , | x - 2 | , \ldots , | x - n | \}.$$ Then $\int _ { 0 } ^ { n + 1 } f ( x ) d x$ equals
(A) $\frac { ( n + 4 ) } { 4 }$
(B) $\frac { ( n + 3 ) } { 4 }$
(C) $\frac { ( n + 2 ) } { 2 }$
(D) $\frac { ( n + 2 ) } { 4 }$

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $(2, 3)$
(D) is greater than 3.

The area of the region bounded by the straight lines $x = \frac { 1 } { 2 }$ and $x = 2$, and the curves given by the equations $y = \log _ { e } x$ and $y = 2 ^ { x }$ is
(a) $\frac { 1 } { \log _ { e } 2 } ( 4 + \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$
(b) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2$
(c) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$
(d) none of the above.

The area of the region bounded by the straight lines $x = \frac { 1 } { 2 }$ and $x = 2$, and the curves given by the equations $y = \log _ { e } x$ and $y = 2 ^ { x }$ is
(a) $\frac { 1 } { \log _ { e } 2 } ( 4 + \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$
(b) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2$
(c) $\frac { 1 } { \log _ { e } 2 } ( 4 - \sqrt { 2 } ) - \frac { 5 } { 2 } \log _ { e } 2 + \frac { 3 } { 2 }$
(d) none of the above.

The area of the region bounded by the straight lines $x = \frac{1}{2}$ and $x = 2$, and the curves given by the equations $y = \log_e x$ and $y = 2^x$ is
(A) $\frac{1}{\log_e 2}(4 + \sqrt{2}) - \frac{5}{2} \log_e 2 + \frac{3}{2}$
(B) $\frac{1}{\log_e 2}(4 - \sqrt{2}) - \frac{5}{2} \log_e 2$
(C) $\frac{1}{\log_e 2}(4 - \sqrt{2}) - \frac{5}{2} \log_e 2 + \frac{3}{2}$
(D) none of the above

The area of the region bounded by the curve $y = \tan x$, the $x$-axis and the tangent to the curve $y = \tan x$ at $x = \frac{\pi}{4}$ is
(A) $\log_e 2 - \frac{1}{2}$
(B) $\log_e 2 + \frac{1}{2}$
(C) $\frac{1}{2}\left(\log_e 2 - \frac{1}{2}\right)$
(D) $\frac{1}{2}\left(\log_e 2 + \frac{1}{2}\right)$.

The area of the region in the plane $\mathbb { R } ^ { 2 }$ given by points $( x , y )$ satisfying $| y | \leq 1$ and $x ^ { 2 } + y ^ { 2 } \leq 2$ is
(A) $\pi + 1$
(B) $2 \pi - 2$
(C) $\pi + 2$
(D) $2 \pi - 1$.

34. For which of the following values of $m$ is the area of the region bounded by the curve $y = x - x 2$ and he line $y = m x$ equals 9/2?
(A) - 4
(B) - 2
(C) 2
(D) 4

7. Let $\mathrm { b } ^ { 1 } 0$ and for $\mathrm { j } = 0,1,2$, $\_\_\_\_$ n , let Sj be the area of the region bounded by the y -axis and the curve xmy $= \sin$ by, $\quad \pi / \mathrm { b } \leq \mathrm { y } \leq ( ( \mathrm { j } + 1 ) \pi ) / \mathrm { b }$.
Show that S0, S1, S2, $\_\_\_\_$ Sn are in geometric progression. Also, find their sum for $\mathrm { a } = - 1$ and $\mathrm { b } = \pi$.

7. Find the area of the region bounded by the curves
$$y = x ^ { 2 } , y = 1 / 22 - x ^ { 2 } 1 / 2 \text { and } y = 2$$
which lies to the right of the line $\mathrm { x } = 1$ ?

18. The area bounded by the curves $y = | x | - 1$ and $y = - | x | + 1$ is
(A) 1
(B) 2
(C) $\quad 2 \sqrt { } 2$
(D) 4

22. The area of bounded by the curves $y = \sqrt { } x , 2 y + 3 = x$ and $x$-axis in the $1 ^ { \text {st } }$ quadrant is :
(a) 9
(b) $27 / 4$
(c) 36
(d) 18

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1. Coefficient of $t ^ { 24 }$ in $\left( 1 + t ^ { 2 } \right) ^ { 12 } \left( 1 + t ^ { 12 } \right) \left( 1 + t ^ { 24 } \right)$ is:
   (a) $\quad { } ^ { 12 } \mathrm { C } _ { 6 } + 3$
   (b) $\quad { } ^ { 12 } \mathrm { C } _ { 6 } + 1$
   (c) $\quad { } ^ { 12 } \mathrm { C } _ { 6 }$
   (d) $\quad { } ^ { 12 } \mathrm { C } _ { 6 } + 2$
2. The value of ' $a$ ' so that the volume of parallelepiped formed by $\hat { i } + a \hat { j } + k , \hat { j } + a k$ and aî $+ k$ because minimum is:
   (a) $\quad - 3$
   (b) 3
   (c) $1 / \sqrt { } 3$
   (d) $\sqrt { } 3$
3. If the system of equations $x + a y = 0 , a z + y = 0$ and $a x + z = 0$ has infinite solutions, then the value of $a$ is
   (a) $\quad - 1$
   (b) 1
   (c) 0
   (d) no real values
4. If $y ( t )$ is a solution of $( 1 + t ) d y / d t - t y = 1$ and $y ( 0 ) = - 1$, then $y ( 1 )$ is equal to:
   (a) $\quad - 1 / 2$
   (b) $\mathrm { e } + \frac { 1 } { 2 }$
   (c) $\mathrm { e } - \frac { 1 } { 2 }$
   (d) $\quad 1 / 2$
5. Tangent is drawn to ellipse $x ^ { 2 } / 27 + y ^ { 2 } = 1$ at $( 3 \sqrt { } 3 \cos \theta , \sin \theta )$ (where $\theta \hat { \mathrm { I } } ( 0$, $\Pi / 2$ ). Then the value of $\theta$ such that sum of intercepts on axes made by this tangent is minimum, is
   (a) $\mathrm { p } / 3$
   (b) $\quad p / 6$
   (c) $\mathrm { p } / 8$
   (d) $\mathrm { p } / 4$
6. Orthocentre of triangle with vertices $( 0,0 ) , ( 3,4 )$ and $( 4,0 )$ is:
   (a) $\quad ( 3,4 / 5 )$
   (b) $( 3,12 )$
   (c) $( 3,3 / 4 )$
   (d) $( 3,9 )$

The area of the region between the curves $y = \sqrt{\frac{1+\sin x}{\cos x}}$ and $y = \sqrt{\frac{1-\sin x}{\cos x}}$ bounded by the lines $x = 0$ and $x = \frac{\pi}{4}$ is
(A) $\int_0^{\sqrt{2}-1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(B) $\int_0^{\sqrt{2}-1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(C) $\int_0^{\sqrt{2}+1} \frac{4t}{(1+t^2)\sqrt{1-t^2}}\,dt$
(D) $\int_0^{\sqrt{2}+1} \frac{t}{(1+t^2)\sqrt{1-t^2}}\,dt$

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
The area of the region bounded by the curve $y = f ( x )$, the $x$-axis, and the lines $x = a$ and $x = b$, where $- \infty < a < b < - 2$, is
(A) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$
(B) $\quad - \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x + b f ( b ) - a f ( a )$
(C) $\int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$
(D) $- \int _ { a } ^ { b } \frac { x } { 3 \left( ( f ( x ) ) ^ { 2 } - 1 \right) } d x - b f ( b ) + a f ( a )$

Area of the region bounded by the curve $y = e ^ { x }$ and lines $x = 0$ and $y = e$ is
(A) $e - 1$
(B) $\int _ { 1 } ^ { e } \ln ( e + 1 - y ) d y$
(C) $e - \int _ { 0 } ^ { 1 } e ^ { x } d x$
(D) $\int _ { 1 } ^ { e } \ln y \, d y$

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The area bounded by the curve $y = f ( x )$ and the lines $x = 0 , y = 0$ and $x = t$, lies in the interval
A) $\left( \frac { 3 } { 4 } , 3 \right)$
B) $\left( \frac { 21 } { 64 } , \frac { 11 } { 16 } \right)$
C) $( 9,10 )$
D) $\left( 0 , \frac { 21 } { 64 } \right)$