Q3. A given object takes n times the time to slide down $45 ^ { \circ }$ rough inclined plane as it takes the time to slide down an identical perfectly smooth $45 ^ { \circ }$ inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is : (1) $\sqrt { 1 - \frac { 1 } { n ^ { 2 } } }$ (2) $1 - n ^ { 2 }$ (3) $1 - \frac { 1 } { n ^ { 2 } }$ (4) $\sqrt { 1 - n ^ { 2 } }$
Q61. The sum of all possible values of $\theta \in [ - \pi , 2 \pi ]$, for which $\frac { 1 + i \cos \theta } { 1 - 2 i \cos \theta }$ is purely imaginary, is equal (1) $3 \pi$ (2) $2 \pi$ (3) $5 \pi$ (4) $4 \pi$
Q62. The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to : (1) 179 (2) 177 (3) 181 (4) 175
Q63. In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac { 70 } { 3 }$ and the product of the third and fifth terms is 49 . Then the sum of the $4 ^ { \text {th } } , 6 ^ { \text {th } }$ and $8 ^ { \text {th } }$ terms is equal to : (1) 96 (2) 91 (3) 84 (4) 78
Q64. If the term independent of $x$ in the expansion of $\left( \sqrt { \mathrm { a } } x ^ { 2 } + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 10 }$ is 105 , then $\mathrm { a } ^ { 2 }$ is equal to : (1) 2 (2) 4 (3) 6 (4) 9
Q66. If the image of the point $( - 4,5 )$ in the line $x + 2 y = 2$ lies on the circle $( x + 4 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 }$, then r is equal to: (1) 2 (2) 3 (3) 1 (4) 4
Q67. If the line segment joining the points $( 5,2 )$ and $( 2 , a )$ subtends an angle $\frac { \pi } { 4 }$ at the origin, then the absolute value of the product of all possible values of $a$ is : (1) 6 (2) 8 (3) 2 (4) - 4
Q68. Let $A = \{ 2,3,6,8,9,11 \}$ and $B = \{ 1,4,5,10,15 \}$. Let $R$ be a relation on $A \times B$ defined by ( $a , b$ ) $R ( c , d )$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is (1) an equivalence relation. (2) reflexive and symmetric but not transitive. (3) transitive but not symmetric. (4) reflexive but not symmetric.
Q70. If the system of equations $x + 4 y - z = \lambda , 7 x + 9 y + \mu z = - 3,5 x + y + 2 z = - 1$ has infinitely many solutions, then $( 2 \mu + 3 \lambda )$ is equal to : (1) 3 (2) - 3 (3) - 2 (4) 2
Q71. Let $f ( x ) = \left\{ \begin{array} { c c c } - \mathrm { a } & \text { if } & - \mathrm { a } \leq x \leq 0 \\ x + \mathrm { a } & \text { if } & 0 < x \leq \mathrm { a } \end{array} \right.$ where $\mathrm { a } > 0$ and $\mathrm { g } ( x ) = ( f ( x \mid ) - | f ( x ) | ) / 2$. Then the function $g : [ - a , a ] \rightarrow [ - a , a ]$ is (1) neither one-one nor onto. (2) onto. (3) both one-one and onto. (4) one-one.
Q72. $\quad$ For $\mathrm { a } , \mathrm { b } > 0$, let $f ( x ) = \left\{ \begin{array} { c l } \frac { \tan ( ( \mathrm { a } + 1 ) x ) + \mathrm { b } \tan x } { x } , & x < 0 \\ 3 , & x = 0 \\ \frac { \sqrt { \mathrm { a } x + \mathrm { b } ^ { 2 } x ^ { 2 } } - \sqrt { \mathrm { a } x } } { \mathrm {~b} \sqrt { \mathrm { a } } \sqrt { x } } , & x > 0 \end{array} \right.$ be a continous function at $x = 0$. Then $\frac { \mathrm { b } } { \mathrm { a } }$ is equal to : (1) 6 (2) 4 (3) 5 (4) 8
Q73. If the function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 \mathrm { a } ^ { 2 } x + 1 , \mathrm { a } > 0$ has a local maximum at $x = \alpha$ and a local minimum at $x = \alpha ^ { 2 }$, then $\alpha$ and $\alpha ^ { 2 }$ are the roots of the equation : (1) $x ^ { 2 } - 6 x + 8 = 0$ (2) $x ^ { 2 } + 6 x + 8 = 0$ (3) $8 x ^ { 2 } + 6 x - 1 = 0$ (4) $8 x ^ { 2 } - 6 x + 1 = 0$
Q77. Let $\vec { a } = 4 \hat { i } - \hat { j } + \hat { k } , \vec { b } = 11 \hat { i } - \hat { j } + \hat { k }$ and $\vec { c }$ be a vector such that $( \vec { a } + \vec { b } ) \times \vec { c } = \vec { c } \times ( - 2 \vec { a } + 3 \vec { b } )$. If $( 2 \vec { a } + 3 \vec { b } ) \cdot \vec { c } = 1670$, then $| \vec { c } | ^ { 2 }$ is equal to : (1) 1609 (2) 1618 (3) 1600 (4) 1627
Q80. There are three bags $X , Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is : (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) $\frac { 5 } { 12 }$ (4) $\frac { 1 } { 3 }$
Q83. Let a ray of light passing through the point $( 3,10 )$ reflects on the line $2 x + y = 6$ and the reflected ray passes through the point $( 7,2 )$. If the equation of the incident ray is $a x + b y + 1 = 0$, then $a ^ { 2 } + b ^ { 2 } + 3 a b$ is equal to $\_\_\_\_$
Q84. Let S be the focus of the hyperbola $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 5 } = 1$, on the positive $x$-axis. Let C be the circle with its centre at $A ( \sqrt { 6 } , \sqrt { 5 } )$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to $\_\_\_\_$
Q86. Let $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathrm { N }$ and $\mathrm { a } < \mathrm { b } < \mathrm { c }$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9,25 , \mathrm { a } , \mathrm { b } , \mathrm { c }$ be 18,4 and $\frac { 136 } { 5 }$, respectively. Then $2 \mathrm { a } + \mathrm { b } - \mathrm { c }$ is equal to $\_\_\_\_$
Q87. Let A be the region enclosed by the parabola $y ^ { 2 } = 2 x$ and the line $x = 24$. Then the maximum area of the rectangle inscribed in the region A is $\_\_\_\_$
Q88. If $\int \frac { 1 } { \sqrt [ 5 ] { ( x - 1 ) ^ { 4 } ( x + 3 ) ^ { 6 } } } \mathrm {~d} x = \mathrm { A } \left( \frac { \alpha x - 1 } { \beta x + 3 } \right) ^ { B } + \mathrm { C }$, where C is the constant of integration, then the value of $\alpha + \beta + 20 \mathrm { AB }$ is $\_\_\_\_$
Q89. Let $\alpha | x | = | y | \mathrm { e } ^ { x y - \beta } , \alpha , \beta \in \mathbf { N }$ be the solution of the differential equation $x \mathrm {~d} y - y \mathrm {~d} x + x y ( x \mathrm {~d} y + y \mathrm {~d} x ) = 0$, $y ( 1 ) = 2$. Then $\alpha + \beta$ is equal to $\_\_\_\_$
Q90. Let $\mathrm { P } ( \alpha , \beta , \gamma )$ be the image of the point $\mathrm { Q } ( 1,6,4 )$ in the line $\frac { x } { 1 } = \frac { y - 1 } { 2 } = \frac { z - 2 } { 3 }$. Then $2 \alpha + \beta + \gamma$ is equal to $\_\_\_\_$