The value of the integral $\int _ { \pi / 2 } ^ { 5 \pi / 2 } \frac { e ^ { \tan ^ { - 1 } ( \sin x ) } } { e ^ { \tan ^ { - 1 } ( \sin x ) } + e ^ { \tan ^ { - 1 } ( \cos x ) } } d x$ equals
(a) 1 .
(B) $\pi$.
(C) $e$.
(D) none of these.

3. Verify that as $n$ varies, all these functions respect conditions a), b) and c). Let $A(n)$ and $B(n)$ be the areas of the coloured parts of the tiles obtained from such functions $a_n$ and $b_n$, calculate $\lim_{n \rightarrow +\infty} A(n)$ and $\lim_{n \rightarrow +\infty} B(n)$ and interpret the results in geometric terms.
The customer decides to order 5,000 tiles with the design derived from $a_2(x)$ and 5,000 with the one derived from $b_2(x)$. The painting is carried out by a mechanical arm that, after depositing the colour, returns to the initial position by flying over the tile along the diagonal. Due to a malfunction, during the production of the 10,000 tiles there is a 20\% probability that the mechanical arm drops a drop of colour at a random point along the diagonal, thus staining the newly produced tile.

7. Determine $a$ such that
$$\int_a^{a+1} (3x^2 + 3) \, dx$$
is equal to 10.

Ministry of Education, University and Research

7. A resistor with resistance $R$ is traversed by a current varying in time with intensity $I ( t ) = I _ { 0 } \frac { a } { t }$, with $t > 0$ and the positive constants $I _ { 0 }$ and $a$ expressed, respectively, in amperes and in seconds. Knowing that the power dissipated in the resistor due to the Joule effect is $P ( t ) = R I ^ { 2 } ( t )$, determine its average value on the interval $[ 2 a ; 3 a ]$.

6. Write a polynomial function $f$ such that the line with equation $y = 2 x + 3$ is tangent to the graph of $f$ at its point with abscissa 0 and $\int _ { 0 } { 3 } f ( x ) d x = 9$.

Ministry of Education and Merit

A002－FINAL STATE EXAMINATION OF THE SECOND CYCLE OF EDUCATION

6. Let $f ( x ) = x - [ x ]$, for every real number $x$, where $[ x ]$ is the integral part of $x$. Then $\int 1 - 1 f ( x ) d x$ is :
(A) 1
(B) 2
(C) 0
(D) $1 / 2$

8. If for a real number $y , [ y ]$ is the greatest integer less than or equal to $y$, then the value of the integral $\int \sqcap / 23 \sqcap / 2 [ 2 \sin x . d x ]$ is:
(A) $- \pi$
(B) 0
(C) $- \pi / 2$
(D) $\pi / 2$

13. Consider an infinite geometric series with first term and common ratio $r$. If its sum is 4 and the second term is $3 / 4$, then :
(A) $a = 4 / 7 , r = 3 / 7$
(B) $\mathrm { a } = 2 , \mathrm { r } = 3 / 8$
(C) $\mathrm { a } = 3 / 2 , \mathrm { r } = 1 / 2$
(D) $\mathrm { a } = 3 , \mathrm { r } = 1 / 4$

19. A pole stands vertically inside a triangular park $\triangle A B C$. If the angle of elevation of the top of the pole from each corner of the park is same, then in $\triangle \mathrm { ABC }$ the foot of the pole is at the :
(A) centroid
(B) circumcentre
(C) incentre
(D) orthocenter.

34. For all $\in ( 0,1 )$ :
(A) $e x < 1 + x$
(B) loge $( 1 + x ) < x$
(C) $\sin x > x$
(D) loge $x > x$.

35. The value of the integral $\int \mathrm { e } - 1 \mathrm { e } 2 | \log \mathrm { x } / \mathrm { x } | \mathrm { dx }$ is :
(A) $3 / 2$
(B) $5 / 2$
(C) 3
(D) 5

11. The value of $\int - ח ( \cos 2 x / 1 + a x ) d x , a > 0$, is

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(A) $\sqcap$
(B) ап
(C) $\pi / 2$
(D) $2 \sqcap$

27. Let $\mathrm { T } > 0$ be a fixed real number. Suppose f is a continuous function such that for all $x \varepsilon R . f ( x + T )$ If $I = \int _ { T 0 } f ( x ) . d x$ then the value of $\int _ { 3 } { } ^ { 3 + 3 T }$ is
(A) $( 3 / 2 ) \mathrm { I }$
(B) I
(C) 3 I
(D) 6 I

28. The integral $\int _ { 1 / 2 - 1 / 2 } ( [ x ] + \ln ( 1 + x / 1 + x ) ) d x$ equals
(A) $- 1 / 2$
(B) 0
(C) 1
(D) $2 \ln ( 1 / 2 )$

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1. If vector $a$ and bare two vectors such that $a \rightarrow + 2 b \rightarrow$ and $5 a \rightarrow - 4 b \rightarrow$ are perpendicular to each other then the angle between vector $a$ and $b$ is
   (A) $\quad 45 ^ { \circ }$
   (B) $\quad 60 ^ { 0 }$
   (C) $\quad \cos ^ { - 1 } 1 / 3$
   (D) $\quad \cos ^ { - 1 } 2 / 7$
2. Let vector $\mathrm { V } = 2 \mathrm { i } ^ { \rightarrow } + \mathrm { j } ^ { \rightarrow } - \mathrm { k } ^ { \rightarrow }$ and $\mathrm { W } ^ { \rightarrow } = \mathrm { i } ^ { \rightarrow } + 3 \mathrm { k } ^ { \rightarrow }$. If vector U is a unit vector, then the maximum value of the scalar triple product $\left[ \mathrm { U } ^ { \rightarrow } \mathrm { V } ^ { \rightarrow } \mathrm { W } ^ { \rightarrow } \right]$ is
   (A) - 1
   (B) $\quad \sqrt { } 10 + \sqrt { } 6$
   (C) $\sqrt { } 59$
   (D) $\sqrt { } 60$

If the function $\mathrm { f } : [ 0,4 ] - - > \mathrm { R }$ is differentiable then show that (i) For $\mathrm { a } , \mathrm { b } \hat { \mathrm { I } } ( 0,4 ) , ( \mathrm { f } ( 4 ) ) ^ { 2 } = \mathrm { f } ( \mathrm { a } ) \mathrm { f } ( \mathrm { b } )$ (ii) $\left. \int _ { 0 } ^ { 4 } \mathrm { f } ( \mathrm { t } ) \mathrm { dt } = 2 \left[ \alpha \mathrm { f } \left( \alpha ^ { 2 } \right) + \beta \mathrm { f } ( \beta ) ^ { 2 } \right] \right] \forall 0 < \alpha , \beta < 2$.

14. Evaluate $\int _ { - \pi / 3 } ^ { \pi / 3 } \frac { \pi + 4 x ^ { 3 } } { 2 - \cos \left( | x | + \frac { \pi } { 3 } \right) } d x$.
Sol. $I = \int _ { - \pi / 3 } ^ { \pi / 3 } \frac { \left( \pi + 4 x ^ { 3 } \right) d x } { 2 - \cos \left( | x | + \frac { \pi } { 3 } \right) }$ $2 \mathrm { I } = \int _ { - \pi / 3 } ^ { \pi / 3 } \frac { 2 \pi \mathrm { dx } } { 2 - \cos \left( | \mathrm { x } | + \frac { \pi } { 3 } \right) } = \int _ { 0 } ^ { \pi / 3 } \frac { 2 \pi \mathrm { dx } } { 2 - \cos \left( \mathrm { x } + \frac { \pi } { 3 } \right) }$ $\mathrm { I } = \int _ { \pi / 3 } ^ { 2 \pi / 3 } \frac { 2 \pi \mathrm { dt } } { 2 - \cos \mathrm { t } } \Rightarrow \mathrm { I } = 2 \pi \int _ { \pi / 3 } ^ { 2 \pi / 3 } \frac { \sec ^ { 2 } \frac { \mathrm { t } } { 2 } \mathrm { dt } } { 1 + 3 \tan ^ { 2 } \frac { \mathrm { t } } { 2 } } = 2 \pi \int _ { 1 / \sqrt { 3 } } ^ { \sqrt { 3 } } \frac { 2 \mathrm { dt } } { 1 + 3 \mathrm { t } ^ { 2 } } = \frac { 4 \pi } { 3 } \int _ { 1 / \sqrt { 3 } } ^ { \sqrt { 3 } } \frac { \mathrm { dt } } { \left( \frac { 1 } { \sqrt { 3 } } \right) ^ { 2 } + \mathrm { t } ^ { 2 } }$ $\mathrm { I } = \frac { 4 \pi } { 3 } \sqrt { 3 } \left[ \tan ^ { - 1 } \sqrt { 3 } \mathrm { t } \right] _ { 1 / \sqrt { 3 } } ^ { \sqrt { 3 } } = \frac { 4 \pi } { \sqrt { 3 } } \left[ \tan ^ { - 1 } 3 - \frac { \pi } { 4 } \right] = \frac { 4 \pi } { \sqrt { 3 } } \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$.

14. Evaluate $\int _ { - \pi / 3 } ^ { \pi / 3 } \frac { \pi + 4 x ^ { 3 } } { 2 - \cos \left( | x | + \frac { \pi } { 3 } \right) } d x$.
Sol. $I = \int _ { - \pi / 3 } ^ { \pi / 3 } \frac { \left( \pi + 4 x ^ { 3 } \right) d x } { 2 - \cos \left( | x | + \frac { \pi } { 3 } \right) }$ $2 \mathrm { I } = \int _ { - \pi / 3 } ^ { \pi / 3 } \frac { 2 \pi \mathrm { dx } } { 2 - \cos \left( | \mathrm { x } | + \frac { \pi } { 3 } \right) } = \int _ { 0 } ^ { \pi / 3 } \frac { 2 \pi \mathrm { dx } } { 2 - \cos \left( \mathrm { x } + \frac { \pi } { 3 } \right) }$ $\mathrm { I } = \int _ { \pi / 3 } ^ { 2 \pi / 3 } \frac { 2 \pi \mathrm { dt } } { 2 - \cos \mathrm { t } } \Rightarrow \mathrm { I } = 2 \pi \int _ { \pi / 3 } ^ { 2 \pi / 3 } \frac { \sec ^ { 2 } \frac { \mathrm { t } } { 2 } \mathrm { dt } } { 1 + 3 \tan ^ { 2 } \frac { \mathrm { t } } { 2 } } = 2 \pi \int _ { 1 / \sqrt { 3 } } ^ { \sqrt { 3 } } \frac { 2 \mathrm { dt } } { 1 + 3 \mathrm { t } ^ { 2 } } = \frac { 4 \pi } { 3 } \int _ { 1 / \sqrt { 3 } } ^ { \sqrt { 3 } } \frac { \mathrm { dt } } { \left( \frac { 1 } { \sqrt { 3 } } \right) ^ { 2 } + \mathrm { t } ^ { 2 } }$ $\mathrm { I } = \frac { 4 \pi } { 3 } \sqrt { 3 } \left[ \tan ^ { - 1 } \sqrt { 3 } \mathrm { t } \right] _ { 1 / \sqrt { 3 } } ^ { \sqrt { 3 } } = \frac { 4 \pi } { \sqrt { 3 } } \left[ \tan ^ { - 1 } 3 - \frac { \pi } { 4 } \right] = \frac { 4 \pi } { \sqrt { 3 } } \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$.

24. $\quad \int _ { 0 } ^ { \pi / 2 } \sin x d x$ is equal to
(A) $\frac { \pi } { 8 } ( 1 + \sqrt { 2 } )$
(B) $\frac { \pi } { 4 } ( 1 + \sqrt { 2 } )$
(C) $\frac { \pi } { 8 \sqrt { 2 } }$
(D) $\frac { \pi } { 4 \sqrt { 2 } }$
Sol. (A)
$$\begin{aligned} & \int _ { 0 } ^ { \pi / 2 } \sin x d x = \frac { \frac { \pi } { 2 } + 0 } { 4 } \left( \sin ( 0 ) + \sin \left( \frac { \pi } { 2 } \right) + 2 \sin \left( \frac { 0 + \frac { \pi } { 2 } } { 2 } \right) \right) \\ & = \frac { \pi } { 8 } ( 1 + \sqrt { 2 } ) \end{aligned}$$

1. Data could not be retrieved.
2. If $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) < 0 \forall \mathrm { x } \in ( \mathrm { a } , \mathrm { b } )$ and c is a point such that $\mathrm { a } < \mathrm { c } < \mathrm { b }$, and (c, $\left. \mathrm { f } ( \mathrm { c } ) \right)$ is the point lying on the curve for which $\mathrm { F } ( \mathrm { c } )$ is maximum, then $\mathrm { f } ^ { \prime } ( \mathrm { c } )$ is equal to
   (A) $\frac { f ( b ) - f ( a ) } { b - a }$
   (B) $\frac { 2 ( f ( b ) - f ( a ) ) } { b - a }$
   (C) $\frac { 2 f ( b ) - f ( a ) } { 2 b - a }$
   (D) 0

Sol. (A)
$$\begin{aligned} & \left( F ^ { \prime } ( c ) = ( b - a ) f ^ { \prime } ( c ) + f ( a ) - f ( b ) \right. \\ & F ^ { \prime \prime } ( c ) = f ^ { \prime \prime } ( c ) ( b - a ) < 0 \\ & \Rightarrow F ^ { \prime } ( c ) = 0 \Rightarrow f ^ { \prime } ( c ) = \frac { f ( b ) - f ( a ) } { b - a } \end{aligned}$$

Comprehension III

Let ABCD be a square of side length 2 units. $\mathrm { C } _ { 2 }$ is the circle through vertices $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and $\mathrm { C } _ { 1 }$ is the circle touching all the sides of the square ABCD . L is a line through A .

38. Match the following
(i) $\int _ { 0 } ^ { \pi / 2 } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \left( \cos \mathrm { x } \cot \mathrm { x } - \log ( \sin \mathrm { x } ) ^ { \sin \mathrm { x } } \right) \mathrm { dx }$
(A) 1
(ii) Area bounded by $- 4 y ^ { 2 } = x$ and $x - 1 = - 5 y ^ { 2 }$
(B) 0
(iii) Cosine of the angle of intersection of curves $y = 3 ^ { x - 1 } \log x$ and
$$y = x ^ { x } - 1 \text { is }$$
(C) $6 \ln 2$
(iv) Data could not be retrieved.
(D) $4 / 3$
Sol. (i) $I = \int _ { 0 } ^ { \pi / 2 } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \left( \cos \mathrm { x } \cdot \cot \mathrm { x } - \log ( \sin \mathrm { x } ) ^ { \sin \mathrm { x } } \right) \mathrm { dx }$
$$\Rightarrow \quad \mathrm { I } = \int _ { 0 } ^ { \pi / 2 } \frac { \mathrm {~d} } { \mathrm { dx } } ( \sin \mathrm { x } ) ^ { \cos \mathrm { x } } \mathrm { dx } = 1 .$$
(ii) The points of intersection of $- 4 y ^ { 2 } = x$ and $x - 1 = - 5 y ^ { 2 }$ is $( - 4 , - 1 )$ and $( - 4,1 )$
Hence required area $= 2 \left[ \mid \int _ { 0 } ^ { 1 } \left( 1 - 5 y ^ { 2 } \right) d y - \int _ { 0 } ^ { 1 } - 4 y ^ { 2 } d y \right] \left\lvert \, = \frac { 4 } { 3 } \right.$.
(iii) The point of intersection of $y = 3 ^ { x - 1 } \log x$ and $y = x ^ { x } - 1$ is $( 1,0 )$
Hence $\frac { d y } { d x } = \frac { 3 ^ { x - 1 } } { x } + 3 ^ { x - 1 } \log 3 \cdot \log x . \left. \quad \frac { d y } { d x } \right| _ { ( 1,0 ) } = 1$ for $\mathrm { y } = \mathrm { x } ^ { \mathrm { x } } - \left. 1 \cdot \frac { \mathrm { dy } } { \mathrm { dx } } \right| _ { ( 1,0 ) } = 1$ If $\theta$ is the angle between the curve then $\tan \theta = 0 \Rightarrow \cos \theta = 1$.
(iv) $\frac { \mathrm { dy } } { \mathrm { dx } } = \left( \frac { 2 } { \mathrm { x } + \mathrm { y } } \right)$ $\Rightarrow \frac { \mathrm { dx } } { \mathrm { dy } } - \frac { \mathrm { x } } { 2 } = \frac { \mathrm { y } } { 2 }$ $\Rightarrow \quad \mathrm { xe } ^ { - \mathrm { y } / 2 } = \frac { 1 } { 2 } \int \mathrm { y } \cdot \mathrm { e } ^ { - \mathrm { y } / 2 } \mathrm { dy }$ $\Rightarrow \quad x + y + 2 = k ^ { \mathrm { y } / 2 } = 3 \mathrm { e } ^ { \mathrm { y } / 2 }$.

Let $I = \int_0^1 \frac{\sin x}{\sqrt{x}}\,dx$ and $J = \int_0^1 \frac{\cos x}{\sqrt{x}}\,dx$. Then which one of the following is true?
(A) $I > \frac{2}{3}$ and $J > 2$
(B) $I < \frac{2}{3}$ and $J < 2$
(C) $I < \frac{2}{3}$ and $J > 2$
(D) $I > \frac{2}{3}$ and $J < 2$

Match the statements in Column I with the values in Column II.
Column I
(A) $\int_{-\pi}^{\pi} \cos^2 x\,\frac{1}{1+a^x}\,dx$, $a > 0$
(B) $\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx$
(C) $\int_{-2}^{2} \frac{x^2}{1+5^x}\,dx$
(D) $\int_1^2 \frac{\sqrt{\ln(3-x)}}{\sqrt{\ln(3-x)}+\sqrt{\ln(x+1)}}\,dx$
Column II
(p) $\frac{1}{2}$
(q) $0$
(r) $\frac{\pi}{4}$
(s) $\frac{\pi}{2}$

Let $f$ be a non-negative function defined on the interval $[ 0,1 ]$. If
$$\int _ { 0 } ^ { x } \sqrt { 1 - \left( f ^ { \prime } ( t ) \right) ^ { 2 } } d t = \int _ { 0 } ^ { x } f ( t ) d t , \quad 0 \leq x \leq 1 ,$$
and $f ( 0 ) = 0$, then
(A) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(B) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) > \frac { 1 } { 3 }$
(C) $f \left( \frac { 1 } { 2 } \right) < \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$
(D) $f \left( \frac { 1 } { 2 } \right) > \frac { 1 } { 2 }$ and $f \left( \frac { 1 } { 3 } \right) < \frac { 1 } { 3 }$

If $$I_{n}=\int_{-\pi}^{\pi}\frac{\sin nx}{\left(1+\pi^{x}\right)\sin x}dx,\quad n=0,1,2,\ldots,$$ then
(A) $I_{n}=I_{n+2}$
(B) $\sum_{m=1}^{10}I_{2m+1}=10\pi$
(C) $\sum_{m=1}^{10}I_{2m}=0$
(D) $I_{n}=I_{n+1}$

The value of $\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 3 } } \int _ { 0 } ^ { x } \frac { t \ln ( 1 + t ) } { t ^ { 4 } + 4 } d t$ is
A) 0
B) $\frac { 1 } { 12 }$
C) $\frac { 1 } { 24 }$
D) $\frac { 1 } { 64 }$

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