9. A quantity of an element exists. If each hour $\dfrac{1}{9}$ of the mass is lost, after how many minutes will $\dfrac{1}{6}$ of the element's mass remain? $\left(\log_2^5 = 1/4 \text{ and } \log_3^5 = 2/4\right)$
(1) $380$ (2) $360$ (3) $440$ (4) $420$

The set of all $x$ for which the function $f ( x ) = \log _ { 1 / 2 } \left( x ^ { 2 } - 2 x - 3 \right)$ is defined and monotone increasing is
(a) $( - \infty , 1 )$
(b) $( - \infty , - 1 )$
(c) $( 1 , \infty )$
(d) $( 3 , \infty )$

Find $\lim_{n\to\infty}\left(1 + \dfrac{1}{n}\right)^n$.

Let $f : \mathbb { R } \rightarrow [ 0 , \infty )$ be a continuous function such that $$f ( x + y ) = f ( x ) f ( y )$$ for all $x , y \in \mathbb { R }$. Suppose that $f$ is differentiable at $x = 1$ and $$\left. \frac { d f ( x ) } { d x } \right| _ { x = 1 } = 2$$ Then, the value of $f ( 1 ) \log _ { e } f ( 1 )$ is
(A) $e$.
(B) 2.
(C) $\log _ { e } 2$.
(D) 1 .

Let $$P(x) = 1 + 2x + 7x^2 + 13x^3, \quad x \in \mathbb{R}$$ Calculate for all $x \in \mathbb{R}$, $$\lim_{n \to \infty} \left(P\left(\frac{x}{n}\right)\right)^n$$

1) Based on the information obtainable from the graph in Figure 2, show, with appropriate reasoning, that the function:
$$f ( x ) = \sqrt { 2 } - \frac { e ^ { x } + e ^ { - x } } { 2 } \quad x \in \mathbb { R }$$
adequately represents the profile of the platform for $x \in [ - a ; a ]$; also determine the value of the endpoints $a$ and $-a$ of the interval.

Ministry of Education, University and Research

To visualize the complete profile of the platform on which the bicycle will be able to move, several copies of the graph of the function $f(x)$ relating to the interval $[-a; a]$ are placed side by side, as shown in Figure 3.
[Figure]
Figure 3

8. A country has food deficit of $10 \%$. Its population grows continuously at a rate of $3 \%$ per year. Its annual food production every year is $4 \%$ more than that of the last year. Assuming that the average food requirement per person remains constant, prove that the country will become self-sufficient in food after n years, where n is the smallest integer bigger than or equal to (In $\mathbf { 1 0 } - \mathbf { I n } \mathbf { 9 } ) / ( \mathbf { I n } ( \mathbf { 1 . 0 4 } ) - \mathbf { 0 . 0 3 } )$.

11. In a triangle $A B C , 2 a c \sin 1 / 2 ( A - B + C ) =$
(A) $a 2 + b 2 - c 2$
(B) $c 2 + a 2 - b 2$
(C) $b 2 - c 2 - a 2$
(D) $c 2 - a 2 - b 2$

33. If the line $x - 1 = 0$ is the directrix of the parabola $y 2 - k x + 8 = 0$ then one of the values of ... Powered By IITians k is :
(A) $1 / 8$
(B) 8
(C) 4
(D) $1 / 4$

9. Let $\theta \in \left( 0 , \frac { \pi } { 4 } \right)$ and $\mathrm { t } _ { 1 } = ( \tan \theta ) ^ { \tan \theta } , \mathrm { t } _ { 2 } = ( \tan \theta ) ^ { \cot \theta } , \mathrm { t } _ { 3 } = ( \cot \theta ) ^ { \tan \theta }$ and $\mathrm { t } _ { 4 } = ( \cot \theta ) ^ { \cot \theta }$, then
(A) $\mathrm { t } _ { 1 } > \mathrm { t } _ { 2 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 4 }$
(B) $\mathrm { t } _ { 4 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 }$
(C) $\mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 } > \mathrm { t } _ { 4 }$
(D) $\mathrm { t } _ { 2 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 4 }$

Sol. (B)

Given $\theta \in \left( 0 , \frac { \pi } { 4 } \right)$, then $\tan \theta < 1$ and $\cot \theta > 1$. Let $\tan \theta = 1 - \lambda _ { 1 }$ and $\cot \theta = 1 + \lambda _ { 2 }$ where $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are very small and positive. then $\mathrm { t } _ { 1 } = \left( 1 - \lambda _ { 1 } \right) ^ { 1 - \lambda _ { 1 } } , \mathrm { t } _ { 2 } = \left( 1 - \lambda _ { 1 } \right) ^ { 1 + \lambda _ { 2 } }$
$$t _ { 3 } = \left( 1 + \lambda _ { 2 } \right) ^ { 1 - \lambda _ { 1 } } \text { and } t _ { 4 } = \left( 1 + \lambda _ { 2 } \right) ^ { 1 + \lambda _ { 2 } }$$
Hence $\mathrm { t } _ { 4 } > \mathrm { t } _ { 3 } > \mathrm { t } _ { 1 } > \mathrm { t } _ { 2 }$.

Let $m$ be the minimum possible value of $\log _ { 3 } \left( 3 ^ { y _ { 1 } } + 3 ^ { y _ { 2 } } + 3 ^ { y _ { 3 } } \right)$, where $y _ { 1 } , y _ { 2 } , y _ { 3 }$ are real numbers for which $y _ { 1 } + y _ { 2 } + y _ { 3 } = 9$. Let $M$ be the maximum possible value of ( $\log _ { 3 } x _ { 1 } + \log _ { 3 } x _ { 2 } + \log _ { 3 } x _ { 3 }$ ), where $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are positive real numbers for which $x _ { 1 } + x _ { 2 } + x _ { 3 } = 9$. Then the value of $\log _ { 2 } \left( m ^ { 3 } \right) + \log _ { 3 } \left( M ^ { 2 } \right)$ is

Let
$$\alpha = \sum _ { k = 1 } ^ { \infty } \sin ^ { 2 k } \left( \frac { \pi } { 6 } \right) .$$
Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function defined by
$$g ( x ) = 2 ^ { \alpha x } + 2 ^ { \alpha ( 1 - x ) }$$
Then, which of the following statements is/are TRUE ?
(A) The minimum value of $g ( x )$ is $2 ^ { \frac { 7 } { 6 } }$
(B) The maximum value of $g ( x )$ is $1 + 2 ^ { \frac { 1 } { 3 } }$
(C) The function $g ( x )$ attains its maximum at more than one point
(D) The function $g ( x )$ attains its minimum at more than one point

The sum of the series $\frac { 1 } { 2 ! } - \frac { 1 } { 3 ! } + \frac { 1 } { 4 ! } - \ldots$ upto infinity is
(1) $e ^ { - 2 }$
(2) $e ^ { - 1 }$
(3) $e ^ { - 1 / 2 }$
(4) $e ^ { 1 / 2 }$

If $f ( x ) = \left( \frac { 3 } { 5 } \right) ^ { x } + \left( \frac { 4 } { 5 } \right) ^ { x } - 1 , x \in R$, then the equation $f ( x ) = 0$ has:
(1) No solution
(2) More than two solutions
(3) One solution
(4) Two solutions

Let $f ( x ) = a ^ { x }$ $( a > 0 )$ be written as $f ( x ) = f _ { 1 } ( x ) + f _ { 2 } ( x )$, where $f _ { 1 } ( x )$ is an even function and $f _ { 2 } ( x )$ is an odd function. Then $f _ { 1 } ( x + y ) + f _ { 1 } ( x - y )$ equals:
(1) $2 f _ { 1 } ( x ) f _ { 1 } ( y )$
(2) $2 f _ { 1 } ( x + y ) f _ { 1 } ( x - y )$
(3) $2 f _ { 1 } ( x ) f _ { 2 } ( y )$
(4) $2 f _ { 1 } ( x + y ) f _ { 2 } ( x - y )$

Let $\sum _ { k = 1 } ^ { 10 } f ( a + k ) = 16 \left( 2 ^ { 10 } - 1 \right)$, where the function $f$ satisfies $f ( x + y ) = f ( x ) f ( y )$ for all natural numbers $x , y$ and $f ( 1 ) = 2$. Then the natural number ' $a$ ' is:
(1) 3
(2) 16
(3) 4
(4) 2

Let $S$ be the set of all real roots of the equation, $3^{x}\left(3^{x} - 1\right) + 2 = \left|3^{x} - 1\right| + \left|3^{x} - 2\right|$, then
(1) contains exactly two elements.
(2) is a singleton.
(3) is an empty set.
(4) contains at least four elements.

The number of real roots of the equation, $e ^ { 4 x } + e ^ { 3 x } - 4 e ^ { 2 x } + e ^ { x } + 1 = 0$ is:
(1) 1
(2) 3
(3) 2
(4) 4

Let $f : R \rightarrow R$ be such that for all $x \in R$, $\left( 2 ^ { 1 + x } + 2 ^ { 1 - x } \right)$, $f ( x )$ and $\left( 3 ^ { x } + 3 ^ { - x } \right)$ are in A.P., then the minimum value of $f ( x )$ is
(1) 2
(2) 3
(3) 0
(4) 4

The minimum value of $2 ^ { \sin x } + 2 ^ { \cos x }$ is:
(1) $2 ^ { - 1 + \frac { 1 } { \sqrt { 2 } } }$
(2) $2 ^ { - 1 + \sqrt { 2 } }$
(3) $2 ^ { 1 - \sqrt { 2 } }$
(4) $2 ^ { 1 - \frac { 1 } { \sqrt { 2 } } }$

The minimum value of $f ( x ) = a ^ { a ^ { x } } + a ^ { 1 - a ^ { x } }$, where $a , x \in R$ and $a > 0$, is equal to:
(1) $a + 1$
(2) $2 a$
(3) $a + \frac { 1 } { a }$
(4) $2 \sqrt { a }$

The sum of all real roots of equation $\left( e ^ { 2 x } - 4 \right) \left( 6 e ^ { 2 x } - 5 e ^ { x } + 1 \right) = 0$ is
(1) $\ln 4$
(2) $- \ln 3$
(3) $\ln 3$
(4) $\ln 5$

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x) = \log _ { \mathrm { e } } ( x ^ { 2 } + 1 ) - e ^ { - x } + 1$ and $g(x) = \frac { 1 - 2 e ^ { 2 x } } { e ^ { x } }$. Then, for which of the following range of $\alpha$, the inequality $f\left( g\left( \frac { ( \alpha - 1 ) ^ { 2 } } { 3 } \right) \right) > f\left( g\left( \alpha - \frac { 5 } { 3 } \right) \right)$ holds?
(1) $( - 2 , - 1 )$
(2) $(2, 3)$
(3) $(1, 2)$
(4) $( - 1, 1 )$

The equation $e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0, x \in R$ has:
(1) four solutions two of which are negative
(2) two solutions and both are negative
(3) no solution
(4) two solutions and only one of them is negative

Let $S = \{x : x \in \mathbb{R}$ and $\left(\sqrt{3} + \sqrt{2}\right)^{x^2 - 4} + \left(\sqrt{3} - \sqrt{2}\right)^{x^2 - 4} = 10\}$. Then $n(S)$ is equal to
(1) 2
(2) 4
(3) 6
(4) 0