Two vertical poles of heights,
Question: Two vertical poles of heights, $20 \mathrm{~m}$ and $80 \mathrm{~m}$ stand apart on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is :(1) 15(2) 18(3) 12(4) 16Correct Option: , 4 Solution: Equations of lines $\mathrm{OB}$ and $\mathrm{AC}$ are respectively $y=\frac{80}{x_{1}} x$....(i) $\frac{x}{x_{1}}+\frac{y}{20}=1$..(ii) $\because$ equations (i) and (ii) intersect ...
Read More →For an elementary chemical reaction,
Question: For an elementary chemical reaction, $A_{2} \underset{\mathrm{k}_{-1}}{\stackrel{\mathrm{k}_{1}}{\rightleftharpoons}} 2 A$, the expression for $\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}$ is:$\mathrm{k}_{1}\left[\mathrm{~A}_{2}\right]-\mathrm{k}_{-1}[\mathrm{~A}]^{2}$$2 \mathrm{k}_{1}\left[\mathrm{~A}_{2}\right]-\mathrm{k}_{-1}[\mathrm{~A}]^{2}$$\mathrm{k}_{1}\left[\mathrm{~A}_{2}\right]+\mathrm{k}_{-1}[\mathrm{~A}]^{2}$$2 \mathrm{k}_{1}\left[\mathrm{~A}_{2}\right]-2 \mathrm{k}_{-1}[\m...
Read More →Consider the given plots for a reaction obeying Arrhenius equation
Question: Consider the given plots for a reaction obeying Arrhenius equation $\left(\begin{array}{lll}0 \left.{ }^{\circ} \mathrm{C}\mathrm{T}300{ }^{\circ} \mathrm{C}\right) \text { : }\end{array}\right.$ ( $\mathrm{K}$ and $\mathrm{E}_{\mathrm{a}}$ are rate constant and activation energy, respectively) Choose the correct option:I is right but II is wrongBoth I and II are correctI is wrong but II is rightBoth I and II are wrong.Correct Option: , 2 Solution: From Arrhenius equation, $K=A e^{-E_{...
Read More →The angle of elevation of the summit of a mountain
Question: The angle of elevation of the summit of a mountain from a point on the ground is $45^{\circ}$. After climbing up on $\mathrm{km}$ towards the summit at an inclination of $30^{\circ}$ from the ground, the angle of elevation of the summit is found to be $60^{\circ}$. Then the height (in $\mathrm{km}$ ) of the summit from the ground is:(1) $\frac{\sqrt{3}-1}{\sqrt{3}+1}$(2) $\frac{\sqrt{3}+1}{\sqrt{3}-1}$(3) $\frac{1}{\sqrt{3}-1}$(4) $\frac{1}{\sqrt{3}+1}$Correct Option: , 3 Solution: $\b...
Read More →Which of the following are the roots of
Question: Which of the following are the roots of $3 x^{2}+2 x-1=0 ?$ (i) $-1$ (ii) $\frac{1}{3}$ (iii) $-\frac{1}{2}$ Solution: The given equation is $\left(3 \mathrm{x}^{2}+2 \mathrm{x}-1=0\right)$. (i) $\mathrm{x}=(-1)$ L.H .S. $=\mathrm{x}^{2}+2 \mathrm{x}-1$ $=3 \times(-1)^{2}+2 \times(-1)-1$ $=3-2-1$ $=0$ $=$ R.H.S. Thus, $(-1)$ is a root of $\left(3 x^{2}+2 x-1=0\right)$. (ii) On substituting $\mathrm{x}=\frac{1}{3}$ in the given equation, we get: L.H.S. $=3 \mathrm{x}^{2}+2 \mathrm{x}-1$...
Read More →The angle of elevation of the top of a hill from
Question: The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $45^{\circ}$. After walking a distance of 80 meters towards the top, up a slope inclined at an angle of $30^{\circ}$ to the horizontal plane, the angle of elevation of the top of the hill becomes $75^{\circ}$. Then the height of the hill (in meters) is_____________. Solution: Let height $(A B)=h \mathrm{~m}, C D=x \mathrm{~m}$ and $E D=y \mathrm{~m}$ In r...
Read More →For the reaction,
Question: For the reaction, $2 \mathrm{~A}+\mathrm{B} \rightarrow$ products, when the concentrations of $A$ and $B$ both were doubled, the rate of the reaction increased from $0.3 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$ to $2.4 \mathrm{~mol}$ $\mathrm{L}^{-1} \mathrm{~s}^{-1}$. When the concentration of A alone is doubled, the rate increased from $0.3$ and $\mathrm{L}^{-1} \mathrm{~s}^{-1}$ to $0.6 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$. Which one of the following statements i...
Read More →Let AD and BC
Question: Let $\mathrm{AD}$ and $\mathrm{BC}$ be two vertical poles at $\mathrm{A}$ and $\mathrm{B}$ respectively on a horizontal ground. If $A D=8 \mathrm{~m}, B C=11 \mathrm{~m}$ and $A B=10$ $\mathrm{m}$; then the distance (in meters) of a point $\mathrm{M}$ on $\mathrm{AB}$ from the point $\mathrm{A}$ such that $\mathrm{MD}^{2}+\mathrm{MC}^{2}$ is minimum is_____________. Solution: Let $A M=x \mathrm{~m}$ $\therefore(M D)^{2}+(M C)^{2}=64+x^{2}+121+(10-x)^{2}=f(x)$ (say) $f^{\prime}(x)=2 x-2...
Read More →The following results were obtained during kinetic studies of the reaction;
Question: The following results were obtained during kinetic studies of the reaction; The time (in minutes) required to consume half of $\mathrm{A}$ is:5101100Correct Option: 1 Solution: From experiment 1 and II, it is observed that order of reaction w.r.t. (3) is zero. From experiment II and III, $\alpha$ can be calculated as: $\frac{1.386 \times 10^{-2}}{6.93 \times 10^{-3}}=\left(\frac{0.2}{0.1}\right)^{\alpha}$ $\therefore \quad \alpha=1$ Now, $\quad$ Rate $=\mathrm{K}[\mathrm{A}]^{1}$ or, $...
Read More →The electric fields of two plane electromagnetic plane waves in vacuum are given by
Question: The electric fields of two plane electromagnetic plane waves in vacuum are given by $\overrightarrow{\mathrm{E}}_{1}=\mathrm{E}_{0} \hat{j} \cos (\omega t-k x)$ and $\overrightarrow{\mathrm{E}}_{2}=\mathrm{E}_{0} \hat{k} \cos (\omega t-k y)$ At $t=0$, a particle of charge $q$ is at origin with a velocity $\vec{v}=0.8 c \hat{j}$ ( $c$ is the speed of light in vacuum). The instantaneous force experienced by the particle is:(1) $\mathrm{E}_{0} q(0.8 \hat{i}-\hat{j}+0.4 \hat{k})$(2) $\math...
Read More →The angle of elevation of a cloud C
Question: The angle of elevation of a cloud $C$ from a point $P, 200 \mathrm{~m}$ above a still lake is $30^{\circ}$. If the angle of depression of the image of $C$ in the lake from the point $P$ is $60^{\circ}$, then $P C$ (in $\mathrm{m}$ ) is equal to:(1) 100(2) $200 \sqrt{3}$(3) 400(4) $400 \sqrt{3}$Correct Option: , 3 Solution: Here in $\triangle P C D$, $\sin 30^{\circ}=\frac{h}{P C} \Rightarrow P C=2 h$......$\ldots$ (i) $\tan 30^{\circ}=\frac{h}{x} \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}...
Read More →In the following reaction
Question: In the following reaction $: x \mathrm{~A} \rightarrow y \mathrm{~B}$ $\log _{10}\left[-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\right]=\log _{10}\left[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\right]+0.3010$ 'A' and 'B' respectively can be:$n$-Butane and Iso-butane$\mathrm{C}_{2} \mathrm{H}_{2}$ and $\mathrm{C}_{6} \mathrm{H}_{6}$$\mathrm{C}_{2} \mathrm{H}_{4}$ and $\mathrm{C}_{4} \mathrm{H}_{8}$$\mathrm{N}_{2} \mathrm{O}_{4}$ and $\mathrm{NO}_{2}$Correct Option: , 3 Solution: $x A...
Read More →Which of the following are quadratic equation in x ?
Question: Which of the following are quadratic equation inx? (i) $x^{2}-x+3=0$ (ii) $2 x^{2}+\frac{5}{2} x-\sqrt{3}=0$ (iii) $\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0$ (iv) $\frac{1}{3} x^{2}+\frac{1}{5} x-2=0$ (v) $x^{2}-3 x-\sqrt{x}+4=0$ (vi) $x-\frac{6}{x}=3$ (vii) $x+\frac{2}{x}=x^{2}$ (viii) $x^{2}-\frac{1}{x^{2}}=5$ (ix) $(x+2)^{3}=x^{3}-8$ (x) $(2 x+3)(3 x+2)=6(x-1)(x-2)$ (xi) $\left(x+\frac{1}{x}\right)^{2}=2\left(x+\frac{1}{x}\right)+3$ Solution: i) $\left(x^{2}-x+3\right)$ is a quadratic polyno...
Read More →Two vertical poles A B=15
Question: Two vertical poles $A B=15 \mathrm{~m}$ and $C D=10 \mathrm{~m}$ are standing apart on a horizontal ground with points $A$ and $C$ on the ground. If $P$ is the point of intersection of $B C$ and $A D$, then the height of $P($ in $\mathrm{m})$ above the line $A C$ is :(1) $20 / 3$(2) 5(3) $10 / 3$(4) 6Correct Option: , 4 Solution: Let $P E \perp A C$ and $\frac{A E}{E C}=\frac{m}{n}$ $\because \Delta A E P \sim \Delta A C D, \frac{m}{P E}=\frac{m+n}{10}$ $\Rightarrow P E=\frac{10 m}{m+n...
Read More →For the reaction of
Question: For the reaction of $\mathrm{H}_{2}$ with $\mathrm{I}_{2}$, the rate constant is $2.5 \times 10^{-4} \mathrm{dm}^{3} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}$ at $327^{\circ} \mathrm{C}$ and $1.0 \mathrm{dm}^{3} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}$ at $527^{\circ} \mathrm{C}$. The activation energy for the reaction, in $\mathrm{kJ} \mathrm{mol}^{-1}$ is : $\left(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right)$1661507259Correct Option: 1 Solution: $K=e^{-\frac{E_{a}}{...
Read More →A plane electromagnetic wave of frequency 25 GHz is propagating in vacuum along the z-direction.
Question: A plane electromagnetic wave of frequency $25 \mathrm{GHz}$ is propagating in vacuum along the $z$-direction. At a particular point in space and time, the magnetic field is given by $\vec{B}=5 \times 10^{-8} \hat{j} T$. The corresponding electric field $\vec{E}$ is (speed of light $c=3 \times 10^{8} \mathrm{~ms}^{-1}$ )(1) $1.66 \times 10^{-16} \hat{i} \mathrm{~V} / \mathrm{m}$(2) $-1.66 \times 10^{-16} \hat{i} \mathrm{~V} / \mathrm{m}$(3) $-15 \hat{i} \mathrm{~V} / \mathrm{m}$(4) $15 ...
Read More →A bacterial infection in an internal wound grows as
Question: A bacterial infection in an internal wound grows as $\mathrm{N}^{\prime}(\mathrm{t})=$ $\mathrm{N}_{0} \exp (\mathrm{t})$, where the time $\mathrm{t}$ is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down as $\frac{\mathrm{d} \mathrm{N}}{\mathrm{dt}}=-5 \mathrm{~N}^{2}$. What will be the plot of $\frac{\mathrm{N}_{0}}{\mathrm{~N}} \mathrm{vs}$. t after 1 hour?Correct Option: , 3 Solution: When drug is...
Read More →A bacterial infection in an internal wound grows as
Question: A bacterial infection in an internal wound grows as $\mathrm{N}^{\prime}(\mathrm{t})=$ $\mathrm{N}_{0} \exp (\mathrm{t})$, where the time $\mathrm{t}$ is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down as $\frac{\mathrm{d} \mathrm{N}}{\mathrm{dt}}=-5 \mathrm{~N}^{2}$. What will be the plot of $\frac{\mathrm{N}_{0}}{\mathrm{~N}} \mathrm{vs}$. t after 1 hour?Correct Option: Solution: When drug is adm...
Read More →A man is observing, from the top of a tower,
Question: A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point $\mathrm{A}$, with uniform speed. At that point, angle of depression of the boat with the man's eye is $30^{\circ}$ (Ignore man's height). After sailing for 20 seconds towards the base of the tower (which is at the level of water), the boat has reached a point $\mathrm{B}$, where the angle of depression is $45^{\circ}$. Then the time taken (in seconds) by the boat from B to reach the bas...
Read More →The electric field of a plane electromagnetic wave is given by
Question: The electric field of a plane electromagnetic wave is given by $\vec{E}=E_{0} \frac{\hat{i}+\hat{j}}{\sqrt{2}} \cos (k z+\omega t)$ At $t=0$, a positively charged particle is at the point $(x, y, z)=\left(0,0, \frac{\pi}{k}\right)$. If its instantaneous velocity at $(t=0)$ is $v_{0} \hat{k}$, the force acting on it due to the wave is:(1) parallel to $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$(2) zero(3) antiparallel to $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$(4) parallel to $\hat{k}$Correct Option: ,...
Read More →The angle of elevation of a jet plane from a
Question: The angle of elevation of a jet plane from a point $\mathrm{A}$ on the ground is $60^{\circ}$. After a flight of 20 seconds at the speed of $432 \mathrm{~km}$ / hour, the angle of elevation changes to $30^{\circ}$. If the jet plane is flying at a constant height, then its height is:(1) $1200 \sqrt{3} \mathrm{~m}$(2) $1800 \sqrt{3} \mathrm{~m}$(3) $3600 \sqrt{3} \mathrm{~m}$(4) $2400 \sqrt{3} \mathrm{~m}$Correct Option: 1 Solution: $v=432 \times \frac{1000}{60 \times 60} \mathrm{~m} / \...
Read More →Two vertical poles are 150
Question: Two vertical poles are $150 \mathrm{~m}$ apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is:(1) 25(2) $20 \sqrt{3}$(3) 30(4) $25 \sqrt{3}$Correct Option: , 4 Solution: $\tan \theta=\frac{\mathrm{h}}{75}=\frac{75}{3 \mathrm{~h}}$ $\Rightarrow \mathrm{h}^{2}=\frac{(75)^{2}}{3}$ $\mathrm{h}=25 \s...
Read More →A pole stands vertically inside a triangular park ABC
Question: A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi}{3}$. If the radius of the circumcircle ot $\triangle \mathrm{ABC}$ is 2 , then the height of the pole is equal to:(1) $\frac{2 \sqrt{3}}{3}$(2) $2 \sqrt{3}$(3) $\sqrt{3}$(4) $\frac{1}{\sqrt{3}}$Correct Option: , 2 Solution: Let $\mathrm{PD}=\mathrm{h}, \mathrm{R}=2$ As angle of elevation of top of pole from $A, B, C$ are equal So $\mathrm{...
Read More →A particle of mass $m$ and charge q has an initial velocity
Question: A particle of mass $m$ and charge $q$ has an initial velocity $\vec{v}=v_{0} \hat{j}$. If an electric field $\vec{E}=E_{0} \vec{i}$ and magnetic field $\vec{B}=B_{0} \hat{i}$ act on the particle, its speed will double after a time:(1) $\frac{2 m v_{0}}{q E_{0}}$(2) $\frac{3 m v_{0}}{q E_{0}}$(3) $\frac{\sqrt{3} m v_{0}}{q E_{0}}$(4) $\frac{\sqrt{2} m v_{0}}{q E_{0}}$Correct Option: , 3 Solution: (3) In the $x$ direction $F_{x}=q E$ $\Rightarrow \quad m a_{x}=q E$ $\Rightarrow \quad a_{...
Read More →If the magnetic field in a plane electromagnetic wave
Question: If the magnetic field in a plane electromagnetic wave is given by $\vec{B}=3 \times 10^{-8} \sin \left(1.6 \times 10^{3} x+48 \times 10^{10} \mathrm{t}\right) \hat{j} \mathrm{~T}$, then what will be expression for electric field?(1) $\vec{E}=\left(60 \sin \left(1.6 \times 10^{3} x+48 \times 10^{10} \mathrm{t}\right) \hat{k} \mathrm{v} / \mathrm{m}\right)$(2) $\vec{E}=\left(9 \sin \left(1.6 \times 10^{3} x+48 \times 10^{10} \mathrm{t}\right) \hat{k} \mathrm{v} / \mathrm{m}\right)$(3) $\...
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