Q4. The co-ordinates of a particle moving in $x - y$ plane are given by : $x = 2 + 4 \mathrm { t } , y = 3 \mathrm { t } + 8 \mathrm { t } ^ { 2 }$. The motion of the particle is : (1) uniformly accelerated having motion along a (2) uniform motion along a straight line. parabolic path. (3) uniformly accelerated having motion along a (4) non-uniformly accelerated. straight line.
Q61. If 2 and 6 are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, then the quadratic equation, whose roots are $\frac { 1 } { 2 a + b }$ and $\frac { 1 } { 6 a + b }$, is : (1) $2 x ^ { 2 } + 11 x + 12 = 0$ (2) $x ^ { 2 } + 8 x + 12 = 0$ (3) $4 x ^ { 2 } + 14 x + 12 = 0$ (4) $x ^ { 2 } + 10 x + 16 = 0$
Q62. Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $( \bar { z } ) ^ { 2 } + | z | = 0 , z \in \mathrm { C }$. Then $4 \left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$ is equal to : (1) 6 (2) 8 (3) 2 (4) 4
Q63. There are 5 points $P _ { 1 } , P _ { 2 } , P _ { 3 } , P _ { 4 } , P _ { 5 }$ on the side $A B$, excluding $A$ and $B$, of a triangle $A B C$. Similarly there are 6 points $\mathrm { P } _ { 6 } , \mathrm { P } _ { 7 } , \ldots , \mathrm { P } _ { 11 }$ on the side BC and 7 points $\mathrm { P } _ { 12 } , \mathrm { P } _ { 13 } , \ldots , \mathrm { P } _ { 18 }$ on the side $C A$ of the triangle. The number of triangles, that can be formed using the points $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \ldots , \mathrm { P } _ { 18 }$ as vertices, is : (1) 776 (2) 796 (3) 751 (4) 771
Q64. Let the first three terms $2 , p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7 ^ { \text {th } } , 8 ^ { \text {th } }$ and $13 ^ { \text {th } }$ terms of an A.P. If the $5 ^ { \text {th } }$ term of the G.P. is the $n ^ { \text {th } }$ term of the A.P., then $n$ is equal to: (1) 163 (2) 151 (3) 177 (4) 169
Q66. The vertices of a triangle are $\mathrm { A } ( - 1,3 ) , \mathrm { B } ( - 2,2 )$ and $\mathrm { C } ( 3 , - 1 )$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is : (1) $x + y + ( 2 - \sqrt { 2 } ) = 0$ (2) $- x + y - ( 2 - \sqrt { 2 } ) = 0$ (3) $x + y - ( 2 - \sqrt { 2 } ) = 0$ (4) $x - y - ( 2 + \sqrt { 2 } ) = 0$
Q67. A square is inscribed in the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 6 y + 30 = 0$. One side of this square is parallel to $y = x + 3$. If $\left( x _ { i } , y _ { i } \right)$ are the vertices of the square, then $\boldsymbol { \Sigma } \left( x _ { i } ^ { 2 } + y _ { i } ^ { 2 } \right)$ is equal to: (1) 148 (2) 152 (3) 160 (4) 156
Q68. Let $\alpha , \beta \in \mathbf { R }$. Let the mean and the variance of 6 observations $- 3,4,7 , - 6 , \alpha , \beta$ be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is : (1) $\frac { 13 } { 3 }$ (2) $\frac { 16 } { 3 }$ (3) $\frac { 11 } { 3 }$ (4) $\frac { 14 } { 3 }$
Q72. Let the sum of the maximum and the minimum values of the function $f ( x ) = \frac { 2 x ^ { 2 } - 3 x + 8 } { 2 x ^ { 2 } + 3 x + 8 }$ be $\frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$. Then $\mathrm { m } + \mathrm { n }$ is equal to : (1) 195 (2) 201 (3) 217 (4) 182 Q73. Let $f : \mathbf { R } \rightarrow \mathbf { R }$ be a function given by $f ( x ) = \left\{ \begin{array} { l l } \frac { 1 - \cos 2 x } { x ^ { 2 } } , & x < 0 \\ \alpha , & x = 0 \\ \frac { \beta \sqrt { 1 - \cos x } } { x } , & x > 0 \end{array} \right.$, where $\alpha , \beta \in \mathbf { R }$. If $f$ is continuous at $x = 0$, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to : (1) 3 (2) 12 (3) 48 (4) 6
Q74. Let $f ( x ) = x ^ { 5 } + 2 \mathrm { e } ^ { x / 4 }$ for all $x \in \mathbf { R }$. Consider a function $g ( x )$ such that $( g \circ f ) ( x ) = x$ for all $x \in \mathbf { R }$. Then the value of $8 g ^ { \prime } ( 2 )$ is : (1) 2 (2) 8 (3) 4 (4) 16
Q75. Let $f ( x ) = \left\{ \begin{array} { l l } - 2 , & - 2 \leq x \leq 0 \\ x - 2 , & 0 < x \leq 2 \end{array} \right.$ and $h ( x ) = f ( | x | ) + | f ( x ) |$. Then $\int _ { - 2 } ^ { 2 } h ( x ) \mathrm { d } x$ is equal to : (1) 1 (2) 6 (3) 4 (4) 2
Q76. One of the points of intersection of the curves $y = 1 + 3 x - 2 x ^ { 2 }$ and $y = \frac { 1 } { x }$ is $\left( \frac { 1 } { 2 } , 2 \right)$. Let the area of the region enclosed by these curves be $\frac { 1 } { 24 } ( l \sqrt { 5 } + \mathrm { m } ) - \mathrm { n } \log _ { \mathrm { e } } ( 1 + \sqrt { 5 } )$, where $l , \mathrm {~m} , \mathrm { n } \in \mathbf { N }$. Then $l + \mathrm { m } + \mathrm { n }$ is equal to (1) 29 (2) 31 (3) 30 (4) 32
Q79. Let the point, on the line passing through the points $P ( 1 , - 2,3 )$ and $Q ( 5 , - 4,7 )$, farther from the origin and at distance of 9 units from the point P , be $( \alpha , \beta , \gamma )$. Then $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ is equal to : (1) 165 (2) 160 (3) 155 (4) 150
Q80. Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is : (1) $\frac { 5 } { 18 }$ (2) $\frac { 5 } { 16 }$ (3) $\frac { 4 } { 17 }$ (4) $\frac { 7 } { 18 }$
Q82. Let the length of the focal chord PQ of the parabola $y ^ { 2 } = 12 x$ be 15 units. If the distance of PQ from the origin is p , then $10 \mathrm { p } ^ { 2 }$ is equal to $\_\_\_\_$
Q83. Let $A$ be a square matrix of order 2 such that $| A | = 2$ and the sum of its diagonal elements is - 3 . If the points $( x , y )$ satisfying $\mathrm { A } ^ { 2 } + x \mathrm {~A} + y \mathrm { I } = \mathrm { O }$ lie on a hyperbola, whose length of semi major axis is $x$ and semi minor axis is $y$, eccentricity is e and the length of the latus rectum is $l$, then $81 \left( e ^ { 4 } + l ^ { 2 } \right)$ is equal to
Q85. In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$
Q86. Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] = 3 \left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right]$. Then the maximum value of $\operatorname { det } ( \mathrm { A } )$ is $\_\_\_\_$
Q87. If $\int _ { 0 } ^ { \frac { \pi } { 4 } } \frac { \sin ^ { 2 } x } { 1 + \sin x \cos x } \mathrm {~d} x = \frac { 1 } { \mathrm { a } } \log _ { \mathrm { e } } \left( \frac { \mathrm { a } } { 3 } \right) + \frac { \pi } { \mathrm { b } \sqrt { 3 } }$, where $\mathrm { a } , \mathrm { b } \in \mathbf { N }$, then $\mathrm { a } + \mathrm { b }$ is equal to $\_\_\_\_$
Q88
First order differential equations (integrating factor)View
Q88. Let the solution $y = y ( x )$ of the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } - y = 1 + 4 \sin x$ satisfy $y ( \pi ) = 1$. Then $y \left( \frac { \pi } { 2 } \right) + 10$ is equal to $\_\_\_\_$
Q89. Let ABC be a triangle of area $15 \sqrt { 2 }$ and the vectors $\overrightarrow { \mathrm { AB } } = \hat { i } + 2 \hat { j } - 7 \hat { k } , \overrightarrow { \mathrm { BC } } = \mathrm { a } \hat { i } + \mathrm { b } \hat { j } + \mathrm { ck }$ and $\overrightarrow { \mathrm { AC } } = 6 \hat { i } + \mathrm { d } \hat { j } - 2 \hat { k } , \mathrm {~d} > 0$. Then the square of the length of the largest side of the triangle ABC is $\_\_\_\_$