A car is standing 200 m behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and the car has acceleration $4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The car will catch up with the bus after time : (1) $\sqrt { 120 } \mathrm {~s}$ (2) 15 s (3) $\sqrt { 110 } \mathrm {~s}$ (4) $10 \sqrt { 2 } \mathrm {~s}$
A conical pendulum of length $l$ makes an angle $\theta = 45 ^ { \circ }$ with respect to $Z$-axis and moves in a circle in the $X Y$ plane. The radius of the circle is 0.4 m and its center is vertically below $O$. The speed of the pendulum, in its circular path, will be - (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$) (1) $0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (2) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (3) $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (4) $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
The machine as shown has 2 rods of length 1 m connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has roller that rolls along the floor in a slot. As the roller goes back and forth, a 2 kg weight moves up and down. If the roller is moving towards right at a constant speed, the weight moves up with a : (1) Speed which is $\frac { 3 } { 4 }$ th of that of the roller when the weight is 0.4 m above the ground (2) Constant speed (3) Decreasing speed (4) Increasing speed
A circular hole of radius $\frac { R } { 4 }$ is made in a thin uniform disc having mass $M$ and radius $R$, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point $O$ and perpendicular to the plane of the disc is- (1) $\frac { 219 M R ^ { 2 } } { 256 }$ (2) $\frac { 237 M R ^ { 2 } } { 512 }$ (3) $\frac { 197 M R ^ { 2 } } { 256 }$ (4) $\frac { 19 M R ^ { 2 } } { 512 }$
The set of all values of $\lambda$ for which the system of linear equations $$x - 2 y - 2 z = \lambda x$$ $$x + 2 y + z = \lambda y$$ $$- x - y = \lambda z$$ has a non-trivial solution: (1) is an empty set (2) is a singleton (3) contains two elements (4) contains more than two elements
Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is: (1) 12.5 (2) 10 (3) 25 (4) 30
If $f : \mathbb { R } \to \mathbb { R }$ is a differentiable function and $f ( 2 ) = 6$, then $\lim _ { x \to 2 } \int _ { 6 } ^ { f ( x ) } \frac { 2 t \, d t } { ( x - 2 ) }$ is: (1) $2 f ^ { \prime } ( 2 )$ (2) $12 f ^ { \prime } ( 2 )$ (3) $0$ (4) $24 f ^ { \prime } ( 2 )$
The function $f : \mathbb { R } \to \left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$ defined as $f ( x ) = \frac { x } { 1 + x ^ { 2 } }$, is: (1) Surjective but not injective (2) Neither injective nor surjective (3) Invertible (4) Injective but not surjective
The eccentricity of an ellipse whose centre is at the origin is $\frac { 1 } { 2 }$. If one of its directrices is $x = - 4$, then the equation of the normal to it at $\left( 1 , \frac { 3 } { 2 } \right)$ is: (1) $4 x - 2 y = 1$ (2) $4 x + 2 y = 7$ (3) $x + 2 y = 4$ (4) $2 y - x = 2$
If a variable line drawn through the intersection of the lines $\frac { x } { 3 } + \frac { y } { 4 } = 1$ and $\frac { x } { 4 } + \frac { y } { 3 } = 1$, meets the coordinate axes at $A$ and $B$, $( A \neq B )$, then the locus of the midpoint of $AB$ is: (1) $7 x y = 6 ( x + y )$ (2) $4 ( x + y ) ^ { 2 } - 28 ( x + y ) + 49 = 0$ (3) $6 x y = 7 ( x + y )$ (4) $14 ( x + y ) ^ { 2 } - 97 ( x + y ) + 168 = 0$
If two different numbers are taken from the set $\{ 0 , 1 , 2 , 3 , \ldots , 10 \}$; then the probability that their sum as well as absolute difference are both multiple of 4, is: (1) $\frac { 6 } { 55 }$ (2) $\frac { 12 } { 55 }$ (3) $\frac { 14 } { 45 }$ (4) $\frac { 7 } { 55 }$
A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is: (1) $\frac { 6 } { 25 }$ (2) 6 (3) 4 (4) $\frac { 12 } { 5 }$
The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is: (1) 25 (2) 30 (3) 35 (4) 40
Let $I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ^ { 3 } ) ^ { n } d x$, where $n \in \mathbb { N }$. Then $\frac { 3 n + 1 } { 3 n } \cdot \frac { I _ { n + 1 } } { I _ { n } }$ is equal to: (1) 1 (2) $\frac { n } { n + 1 }$ (3) $\frac { n + 1 } { n }$ (4) $\frac { 3 n + 1 } { 3 n - 2 }$
Let $a , b , c \in \mathbb { R }$. If $f ( x ) = a x ^ { 2 } + b x + c$ is such that $a + b + c = 3$ and $f ( x + y ) = f ( x ) + f ( y ) + x y$, $\forall x , y \in \mathbb { R }$, then $\sum _ { n = 1 } ^ { 10 } f ( n )$ is equal to: (1) 330 (2) 165 (3) 190 (4) 255
For any three positive real numbers $a$, $b$ and $c$, $9 ( 25 a ^ { 2 } + b ^ { 2 } ) + 25 ( c ^ { 2 } - 3 a c ) = 15 b ( 3 a + c )$. Then: (1) $b$, $c$ and $a$ are in G.P. (2) $b$, $c$ and $a$ are in A.P. (3) $a$, $b$ and $c$ are in A.P. (4) $a$, $b$ and $c$ are in G.P.
If the image of the point $P ( 1 , - 2 , 3 )$ in the plane $2 x + 3 y - 4 z + 22 = 0$ measured parallel to the line $\frac { x } { 1 } = \frac { y } { 4 } = \frac { z } { 5 }$ is $Q$, then $P Q$ is equal to: (1) $3 \sqrt { 5 }$ (2) $2 \sqrt { 42 }$ (3) $\sqrt { 42 }$ (4) $6 \sqrt { 5 }$
Let $\vec { u } = \hat { i } + \hat { j }$, $\vec { v } = \hat { i } - \hat { j }$ and $\vec { w } = \hat { i } + 2 \hat { j } + 3 \hat { k }$. If $\hat { n }$ is a unit vector such that $\vec { u } \cdot \hat { n } = 0$ and $\vec { v } \cdot \hat { n } = 0$, then $| \vec { w } \cdot \hat { n } |$ is equal to: (1) 0 (2) 1 (3) 2 (4) 3
The number of real values of $\lambda$ for which the system of linear equations $$2 x + 4 y - \lambda z = 0$$ $$4 x + \lambda y + 2 z = 0$$ $$\lambda x + 2 y + 2 z = 0$$ has infinitely many solutions, is: (1) 0 (2) 1 (3) 2 (4) 3