The system of linear equations
Question: The system of linear equations $\lambda x+2 y+2 z=5$ $2 \lambda x+3 y+5 z=8$ $4 x+\lambda y+6 z=10$ hasinfinitely many solutions when $\lambda=2$a unique solution when $\lambda=-8$no solution when $\lambda=8$no solution when $\lambda=2$Correct Option: , 4 Solution:...
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Question: Let $f:(1,3) \rightarrow \mathrm{R}$ be a function defined by $f(\mathrm{x})=\frac{\mathrm{x}[\mathrm{x}]}{1+\mathrm{x}^{2}}$, where $[\mathrm{x}]$ denotes the greatest integer $\leq \mathrm{x}$. Then the range of $f$ is$\left(\frac{3}{5}, \frac{4}{5}\right)$$\left(\frac{2}{5}, \frac{3}{5}\right] \cup\left(\frac{3}{4}, \frac{4}{5}\right)$$\left(\frac{2}{5}, \frac{4}{5}\right]$$\left(\frac{2}{5}, \frac{1}{2}\right) \cup\left(\frac{3}{5}, \frac{4}{5}\right]$Correct Option: , 4 Solution:...
Read More →If the 10^th term of an A.P. is
Question: If the $10^{\text {th }}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text {th }}$ term is $\frac{1}{10}$, then the sum of its first 200 terms is$50 \frac{1}{4}$$100 \frac{1}{2}$50100Correct Option: , 2 Solution:...
Read More →Which of the following statements is a tautology?
Question: Which of the following statements is a tautology?$\sim(\mathrm{p} \vee \sim \mathrm{q}) \rightarrow \mathrm{p} \vee \mathrm{q}$$\sim(\mathrm{p} \wedge \sim \mathrm{q}) \rightarrow \mathrm{p} \vee \mathrm{q}$$\sim(\mathrm{p} \vee \sim \mathrm{q}) \rightarrow \mathrm{p} \wedge \mathrm{q}$$\mathrm{p} \vee(\sim \mathrm{q}) \rightarrow \mathrm{p} \wedge \mathrm{q}$Correct Option: 1 Solution: $\sim(\mathrm{p} \vee \sim \mathrm{q}) \rightarrow \mathrm{p} \vee \mathrm{q}$ $(\sim \mathrm{p} \we...
Read More →Let S be the set of all functions
Question: Let $\mathrm{S}$ be the set of all functions $f:[0,1] \rightarrow \mathrm{R}$, which are continuous on $[0,1]$ and differentiable on $(0,1)$. Then for every $f$ in $\mathrm{S}$, there exists a $\mathrm{c} \in(0,1)$, depending on $f$, such that$|f(\mathrm{c})-f(1)|(1-\mathrm{c})\left|f^{\prime}(\mathrm{c})\right|$$|f(\mathrm{c})-f(1)|\left|f^{\prime}(\mathrm{c})\right|$$|f(\mathrm{c})+f(1)|(1+\mathrm{c})\left|f^{\prime}(\mathrm{c})\right|$$\frac{f(1)-f(\mathrm{c})}{1-\mathrm{c}}=f^{\pri...
Read More →If a line,
Question: If a line, $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent to the circle, $(x-3)^{2}+y^{2}=1$ and it is perpendicular to a line $L_{1}$, where $L_{1}$ is the tangent to the circle, $x^{2}+y^{2}=1$ at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$, then$c^{2}-6 c+7=0$$\mathrm{c}^{2}+6 \mathrm{c}+7=0$$\mathrm{c}^{2}+7 \mathrm{c}+6=0$$\mathrm{c}^{2}-7 \mathrm{c}+6=0$Correct Option: , 2 Solution:...
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Question: If $I=\int_{1}^{2} \frac{d x}{\sqrt{2 x^{3}-9 x^{2}+12 x+4}}$, then :$\frac{1}{9}\mathrm{I}^{2}\frac{1}{8}$$\frac{1}{16}\mathrm{I}^{2}\frac{1}{9}$$\frac{1}{6}\mathrm{I}^{2}\frac{1}{2}$$\frac{1}{8}\mathrm{I}^{2}\frac{1}{4}$Correct Option: 1 Solution:...
Read More →The length of the perpendicular from the origin,
Question: The length of the perpendicular from the origin, on the normal to the curve, $x^{2}+2 x y-3 y^{2}=0$ at the point $(2,2)$ is$4 \sqrt{2}$$2 \sqrt{2}$2$\sqrt{2}$Correct Option: , 2 Solution:...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region $\left\{(x, y) \in R^{2}: x^{2} \leq y \leq 3-2 x\right\}$, is$\frac{29}{3}$$\frac{31}{3}$$\frac{34}{3}$$\frac{32}{3}$Correct Option: , 4 Solution:...
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Question: Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b} \times \vec{c}=\vec{b} \times \vec{a}$ and $\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}=0$, then $\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{b}}$ is equal to$\frac{1}{2}$$-1$$-\frac{1}{2}$$-\frac{3}{2}$Correct Option: , 3 Sol...
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Question: Figure shows a uniform rod of length $30 \mathrm{~cm}$ having a mass of $3.0 \mathrm{~kg}$. The strings shown in the figure are pulled by constant forces of $20 \mathrm{~N}$ and $32 \mathrm{~N}$. Find the force exerted by the $20 \mathrm{~cm}$ part of the rod on the $10 \mathrm{~cm}$ part. All the surfaces are smooth and the strings and the pulleys are light. Solution: $\frac{m}{l}=\frac{3}{30 \times 10^{-2}}=10 \mathrm{~kg} / \mathrm{m}$ For $0.1 \mathrm{~m}-m_{A}=10 \times 0.10=1 \ma...
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Question: Consider the Atwood machine of the previous problem. The larger mass is stopped for a moment $2.0$ $s$ after the system is set into motion. Find the time elapsed before the string is tight again. Solution: Use law of kinematics $v=u+a t$ $\mathrm{V}=0+3.26 \times 2$ $=6.52 \mathrm{~m} / \mathrm{s}$ $\mathrm{m}_{2}$ is moving down with $\mathrm{v}$ $m_{1}$ is moving down with $v$ at $\mathrm{t}=25, \mathrm{~m}_{2}$ stops $\mathrm{m}_{1}$ moves upward and reaches $\mathrm{v}=0$ $v=0, u=6...
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Question: In a simple Atwood machine, two unequal masses $\mathrm{m} 1$, and $\mathrm{m} 2$ are connected by a string going over a clamped light smooth pulley. In a typical arrangement $\mathrm{m} 1=300 \mathrm{~g}$ and $\mathrm{m} 2=600 \mathrm{~g}$. The system is released from rest. (a) Find the distance travelled by the first block in the first two seconds. (b) Find the tension in the string. (c) Find the force exerted by the clamp on the pulley. Solution: $\mathrm{T}-\mathrm{m}_{1} \mathrm{~...
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Question: A force $\vec{F}=\vec{v} \times \vec{A}$ is exerted on a particle in addition to the force of gravity, where $\vec{v}$ is the velocity of the particle and $\vec{A}$ is a constant vector in the horizontal direction. With what minimum speed a particle of mass $m$ be projected so that it continues to move undeflected with a constant velocity? Solution: $\vec{F}=\vec{v} \times \vec{A}$ $\vec{A}=A \hat{\imath}$ $\left|\begin{array}{llr}\hat{\imath} \hat{\jmath} \hat{k} \\ V_{x} V_{y} V_{z} ...
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Question: An empty plastic box of mass $m$ is found to accelerate up at the rate of $g / 6$ when placed deep inside water. How much sand should be put inside the box so that it may accelerate down at the rate of $g$ / 6 ? Solution: $\frac{m g}{6}=-(m g-B)$ $(m+\propto) \frac{g}{6}=(m+\propto) g-B$ B force is constant $\frac{m g}{6}+m g=(\mathrm{m}+\propto) g-(\mathrm{m}+\propto) \frac{g}{6}$ $=\frac{m g}{6}+m g=m g+\propto g-\frac{\mathrm{mg}}{6}-\frac{\propto g}{6}$ $\Rightarrow 2 \frac{m g}{6}...
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Question: The force of buoyancy exerted by the atmosphere on a balloon is $B$ in the upward direction and remains constant. The force of air resistance on the balloon acts opposite to the direction of velocity and is proportional to it. The balloon carries a mass $M$ and is found to fall down near the earth's surface with a constant velocity v. How much mass should be removed from the balloon so that it may rise with a constant velocity v? Solution: $\mathrm{B}+\mathrm{bv}=\mathrm{mg}$ $B=b v+(m...
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Question: Suppose the ceiling in the previous problem is that of an elevator which is going up with an acceleration of $2.0 \mathrm{~m} / \mathrm{s}^{2}$. Find the elongations. Solution: $-k x=2\left(a_{0}+g\right)$ $=2(2+g)$ $x=\frac{-2(2+g)}{k}=\frac{-2(2+9.8)}{100}=-0.236 \mathrm{~m}$ $-k x^{\prime}=3\left(a_{0}+g\right)$ $=3(2+g)$ $x^{\prime}=\frac{-3(2+g)}{k}=\frac{-3(2+9.8)}{100}=-0.36 m$ $x^{\prime}-x=0.36-0.24 m=0.12 m$...
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Question: A block of $2 \mathrm{~kg}$ is suspended from the ceiling through a mass less spring of spring constant $\mathrm{k}=100$ $\mathrm{N} / \mathrm{m}$. What is the elongation of the spring? If another $1 \mathrm{~kg}$ is added to the block, what would be the further elongation? Solution: $1^{\text {st }}$ Case: mq=-kx $x=\frac{-m g}{k}=\frac{-2 \times 9 .}{100}$ $=-0.196 m=-0.2 m$ 2nd $^{\text {nd }}$ Case: $m=3 k g$ $x^{\prime}=\frac{-3 \times 9.8}{100}=-0.294 m=-0.3 m$ $\left|x^{\prime}-...
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Question: Find the reading of the spring balance shown in figure. The elevator is going up with an acceleration of $g / 10$, the pulley and the string are light and the pulley is smooth. Solution: $\mathrm{T}-1.5 \mathrm{~g}-\frac{\frac{1.5 \mathrm{~g}}{10}}{10}=1.5 \mathrm{a} \mathrm{T}-3 \mathrm{~g}-\frac{3 \mathrm{~g}}{10}=-3 \mathrm{a}$ $\mathrm{T}=1.5 \mathrm{~g}^{\frac{1.5 \mathrm{~g}}{10}}+1.5 \mathrm{a} \mathrm{T}=3 \mathrm{~g}^{\frac{3 \mathrm{~g}}{10}}-3 \mathrm{a}$ $1.5 g^{\frac{1.5 g...
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Question: A person is standing on a weighing machine placed on the floor of an elevator. The elevator starts going up with some acceleration, moves with uniform velocity for a while and finally decelerates to stop. The maximum and the minimum weights recorded are $72 \mathrm{~kg}$ and $60 \mathrm{~kg}$. Assuming that the magnitudes of the acceleration and the deceleration are the same, find (a) the true weight of the person and $(b)$ the magnitude of the acceleration. Take $\mathrm{g}=9.9 \mathr...
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Question: A pendulum bob of mass $50 \mathrm{~g}$ is suspended from the ceiling of an elevator. Find the tension in the string if the elevator (a) goes up with acceleration $1.2 \mathrm{~m} / \mathrm{s}^{2}$, (b) goes up with deceleration $1.2 \mathrm{~m} / \mathrm{s}^{2}$, (c) goes up with uniform velocity, (d) goes down with acceleration $1.2 \mathrm{~m} / \mathrm{s}^{2}$, (e) goes down with deceleration $1.2 \mathrm{~m} / \mathrm{s}^{2}$ and ( $f$ ) goes down with uniform velocity. Solution: ...
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Question: The elevator shown in figure is descending with an acceleration of $2 \mathrm{~m} / \mathrm{s}^{2}$. The mass of the block $A$ is $0.5 \mathrm{~kg}$. What force is exerted by the block $A$ on the block B? Solution: $\mathrm{R}=\mathrm{m}_{\mathrm{A}} \mathrm{a}_{0}-0.5 \mathrm{~g}$ $=0.5(2-10)=-4 \mathrm{~N}$ (-ve sign $=$ force acts opposite to our convention)...
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Question: A small block B is placed on another block $A$ of mass $5 \mathrm{~kg}$ and length $20 \mathrm{~cm}$. Initially the block $B$ is near the right end of block $A$. A constant horizontal force of $10 \mathrm{~N}$ is applied to the block $A$. All the surfaces are assumed frictionless. Find the time elapsed before the block $B$ separates from $A$. Solution: $5 a=10$ $a=2 m / s^{2}$ $x-x_{0}=u t+\frac{1}{2} a t^{2}$ $20 \times 10^{-2}=\frac{2 \times t^{2}}{2}$ $(u=0)$ $t=\sqrt{0.2}=0.45 s$...
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Question: Both the springs shown in figure are unstretched. If the block is displaced by a distance $\mathrm{x}$ and released, what will be the initial acceleration? Solution: $-k_{1} x-\left(k_{2} x\right)=m a$ $=a=\frac{-\left(k_{1}+k_{2}\right) x}{m}$...
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Question: A particle of mass $0.3 \mathrm{~kg}$ is subjected to a force $F=-\mathrm{kx}$ with $\mathrm{k}=15 \mathrm{~N} / \mathrm{m}$. What will be its initial acceleration if it is released from a point $x=20 \mathrm{~cm}$ ? Solution: $F=-k x=m a$ $=a=\frac{-k x}{m}=\frac{-15 \times 20 \times 10^{-2}}{0.3}=-10 \mathrm{~m} / \mathrm{s}^{2}$ (-ve sign $=^{a}$ is opposite to displacement)...
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