As shown in the figure, a bob of mass $m$ is tied to a massless string whose other end portion is wound on a fly wheel (disc) of radius $r$ and mass $m$. When released from rest the bob starts falling vertically. When it has covered a distance of $h$, the angular speed of the wheel will be: (1) $\frac { 1 } { \mathrm { r } } \sqrt { \frac { 4 \mathrm { gh } } { 3 } }$ (2) $r \sqrt { \frac { 3 } { 2 g h } }$ (3) $\frac { 1 } { \mathrm { r } } \sqrt { \frac { 2 \mathrm { gh } } { 3 } }$ (4) $r \sqrt { \frac { 3 } { 4 g h } }$
The radius of gyration of a uniform rod of length $l$, about an axis passing through a point $\frac { l } { 4 }$ away from the centre of the rod, and perpendicular to it, is: (1) $\frac { 1 } { 4 } l$ (2) $\frac { 1 } { 8 } l$ (3) $\sqrt { \frac { 7 } { 48 } } l$ (4) $\sqrt { \frac { 3 } { 8 } } l$
A satellite of mass $M$ is launched vertically upwards with an initial speed $u$ from the surface of the earth. After it reaches height $R$ ($R =$ radius of the earth), it ejects a rocket of mass $\frac { M } { 10 }$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ($G$ is the gravitational constant; $M _ { e }$ is the mass of the earth): (1) $\frac { M } { 20 } \left( u ^ { 2 } + \frac { 113 } { 200 } \frac { G M _ { e } } { R } \right)$ (2) $5 M \left( u ^ { 2 } - \frac { 119 } { 200 } \frac { G M _ { e } } { R } \right)$ (3) $\frac { 3 M } { 8 } \left( u + \sqrt { \frac { 5 G M _ { e } } { 6 R } } \right) ^ { 2 }$ (4) $\frac { M } { 20 } \left( u - \sqrt { \frac { 2 G M _ { e } } { 3 R } } \right) ^ { 2 }$
A particle ($\mathrm { m } = 1 \mathrm {~kg}$) slides down a frictionless track (AOC) starting from rest at a point $A$ (height 2 m). After reaching $C$, the particle continues to move freely in air as a projectile. When it reaching its highest point P (height 1 m), the kinetic energy of the particle (in J) is: (Figure drawn is schematic and not to scale; take $g = 10 \mathrm {~ms} ^ { - 2 }$) $\_\_\_\_$.
Let $\alpha$ and $\beta$ be two real roots of the equation $(k + 1) \tan ^ { 2 } x - \sqrt { 2 } \cdot \lambda \tan x = (1 - k)$, where $k (\neq -1)$ and $\lambda$ are real numbers. If $\tan ^ { 2 } (\alpha + \beta) = 50$, then a value of $\lambda$ is (1) $10 \sqrt { 2 }$ (2) 10 (3) 5 (4) $5 \sqrt { 2 }$
If $\operatorname { Re } \left( \frac { z - 1 } { 2 z + i } \right) = 1$, where $z = x + i y$, then the point $(x, y)$ lies on a (1) circle whose centre is at $\left( - \frac { 1 } { 2 } , - \frac { 3 } { 2 } \right)$ (2) straight line whose slope is $- \frac { 2 } { 3 }$ (3) straight line whose slope is $\frac { 3 } { 2 }$ (4) circle whose diameter is $\frac { \sqrt { 5 } } { 2 }$
Total number of 6-digit numbers in which only and all the five digits $1, 3, 5, 7$ and 9 appears, is (1) $\frac { 1 } { 2 } (6!)$ (2) $6!$ (3) $5 ^ { 6 }$ (4) $\frac { 5 } { 2 } (6!)$
Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers is $- \frac { 1 } { 2 }$, then the greatest number amongst them is (1) 27 (2) 7 (3) $\frac { 21 } { 2 }$ (4) 16
If $y = m x + 4$ is a tangent to both the parabolas, $y ^ { 2 } = 4 x$ and $x ^ { 2 } = 2 b y$, then $b$ is equal to (1) $-32$ (2) $-64$ (3) $-128$ (4) 128
If the distance between the foci of an ellipse is 6 and the distance between its directrix is 12, then the length of its latus rectum is (1) $\sqrt { 3 }$ (2) $3 \sqrt { 2 }$ (3) $\frac { 3 } { \sqrt { 2 } }$ (4) $2 \sqrt { 3 }$
If the system of linear equations $$\begin{aligned}
& 2x + 2ay + az = 0 \\
& 2x + 3by + bz = 0 \\
& 2x + 4cy + cz = 0
\end{aligned}$$ where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then (1) $\frac { 1 } { a } , \frac { 1 } { b } , \frac { 1 } { c }$ are in $A.P$. (2) $a, b, c$ are in $G.P$. (3) $a + b + c = 0$ (4) $a, b, c$ are in $A.P$.
Let the function, $f : [-7, 0] \rightarrow R$ be continuous on $[-7, 0]$ and differentiable on $(-7, 0)$. If $f(-7) = -3$ and $f ^ { \prime } (x) \leq 2$ for all $x \in (-7, 0)$, then for all such functions $f$, $f(-1) + f(0)$ lies in the interval (1) $(-\infty, 20]$ (2) $[-3, 11]$ (3) $(-\infty, 11]$ (4) $[-6, 20]$
If $f(a + b + 1 - x) = f(x)$, for all $x$, where $a$ and $b$ are fixed positive real numbers, then $\frac { 1 } { a + b } \int _ { a } ^ { b } x (f(x) + f(x + 1)) d x$ is equal to (1) $\int _ { a - 1 } ^ { b - 1 } f(x + 1) d x$ (2) $\int _ { a - 1 } ^ { b - 1 } f(x) d x$ (3) $\int _ { a + 1 } ^ { b + 1 } f(x) d x$ (4) $\int _ { a + 1 } ^ { b + 1 } f(x + 1) d x$
The area of the region (in sq. units), enclosed by the circle $x ^ { 2 } + y ^ { 2 } = 2$ which is not common to the region bounded by the parabola $y ^ { 2 } = x$ and the straight line $y = x$, is (1) $\frac { 1 } { 6 } (24 \pi - 1)$ (2) $\frac { 1 } { 3 } (6 \pi - 1)$ (3) $\frac { 1 } { 3 } (12 \pi - 1)$ (4) $\frac { 1 } { 6 } (12 \pi - 1)$
If $y = y(x)$ is the solution of the differential equation, $e ^ { y } \left( \frac { d y } { d x } - 1 \right) = e ^ { x }$ such that $y(0) = 0$, then $y(1)$ is equal to (1) $1 + \log _ { e } 2$ (2) $2 + \log _ { e } 2$ (3) $2e$ (4) $\log _ { e } 2$
A vector $\vec { a } = \alpha \hat { i } + 2 \hat { j } + \beta \hat { k }$ $(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec { b } = \hat { i } + \hat { j }$ and $\vec { c } = \hat { i } - \hat { j } + 4 \hat { k }$. If $\vec { a }$ bisects the angle between $\vec { b }$ and $\vec { c }$, then (1) $\vec { a } \cdot \hat { i } + 3 = 0$ (2) $\vec { a } \cdot \hat { i } + 1 = 0$ (3) $\vec { a } \cdot \widehat { k } + 2 = 0$ (4) $\vec { a } \cdot \widehat { k } + 4 = 0$
Let $P$ be a plane passing through the points $(2,1,0)$, $(4,1,1)$ and $(5,0,1)$ and $R$ be any point $(2,1,6)$. Then the image of $R$ in the plane $P$ is (1) $(6,5,2)$ (2) $(6,5,-2)$ (3) $(4,3,2)$ (4) $(3,4,-2)$
An unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k = 3, 4, 5$, otherwise $X$ takes the value $-1$. Then the expected value of $X$, is (1) $\frac { 3 } { 16 }$ (2) $\frac { 1 } { 8 }$ (3) $- \frac { 3 } { 16 }$ (4) $- \frac { 1 } { 8 }$
If the sum of the coefficients of all even powers of $x$ in the product $\left(1 + x + x ^ { 2 } + \ldots + x ^ { 2n} \right) \left(1 - x + x ^ { 2 } - x ^ { 3 } + \ldots + x ^ { 2n } \right)$ is 61, then $n$ is equal to
Let $A(1,0)$, $B(6,2)$ and $C \left( \frac { 3 } { 2 } , 6 \right)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle $ABC$ such that the triangles $APC$, $APB$ and $BPC$ have equal areas, then the length of the line segment $PQ$, where $Q$ is the point $\left( - \frac { 7 } { 6 } , - \frac { 1 } { 3 } \right)$, is
If the variance of the first $n$ natural numbers is 10 and the variance of the first $m$ even natural numbers is 16, then the value of $m + n$ is equal to
Let $S$ be the set of points where the function, $f(x) = |2 - |x - 3||$, $x \in R$, is not differentiable. Then $\sum _ { x \in S } f(f(x))$ is equal to