The stepwise formation of
Question: The stepwise formation of $\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right]^{2+}\right.$ is given below: $\mathrm{Cu}^{2+}+\mathrm{NH}_{3} \stackrel{\mathrm{K}_{1}}{\Longrightarrow}\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2+}$$\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{2}\right]^{2+}+\mathrm{NH}_{3} \stackrel{\mathrm{K}_{2}}{\Longrightarrow}\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{2}\right]^{2+}$$\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{2}\right]^{2+}+\mat...
Read More →A body of mass
Question: A body of mass $m_{1}$ moving with an unknown velocity of $v_{1} \hat{i}$, undergoes a collinear collision with a body of mass $m_{2}$ moving with a velocity $v_{2} \hat{i}$. After collision, $m_{1}$ and $m_{2}$ move with velocities of $v_{3} \hat{i}$ and $v_{4} \hat{i}$, respectively. If $m_{2}=0.5 \mathrm{~m}_{1}$ and $v_{3}=0.5 v_{1}$, then $v_{1}$ is :(1) $v_{4}-\frac{v_{2}}{2}$(2) $v_{4}-v_{2}$(3) $v_{4}-\frac{v_{2}}{4}$(4) $v_{4}+v_{2}$Correct Option: , 2 Solution: $m_{1} v_{1}+m...
Read More →In 50 throws of a dice, the outcomes were noted as shown below:
Question: In 50 throws of a dice, the outcomes were noted as shown below: A dice is thrown at random. What is the probability of getting an even number? (a) $\frac{12}{25}$ (b) $\frac{3}{50}$ (C) $\frac{1}{8}$ (d) $\frac{1}{2}$ Solution: (a) $\frac{12}{25}$ Explanation:Total number of trials = 50Let E be the event of getting an even number.Then, E contains 2, 4 and 6, i.e. 3 even numbers. $\therefore P($ getting an even number $)=P(E)=\frac{\text { Number of times even number } s \text { appear ...
Read More →Let S be the set of all points in
Question: Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function $f(x)=\min \{\sin x, \cos x\}$ is not differentiable. Then $S$ is a subset of which of the following?(1) $\left\{-\frac{\pi}{4}, 0, \frac{\pi}{4}\right\}$(2) $\left\{-\frac{3 \pi}{4},-\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{\pi}{4}\right\}$(3) $\left\{-\frac{\pi}{2},-\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}\right\}$(4) $\left\{-\frac{3 \pi}{4},-\frac{\pi}{2}, \frac{\pi}{2}, \frac{3 \pi}{4}\right\}$Correct Option: ...
Read More →Four particles A, B, C and D with masses
Question: Four particles A, B, C and D with masses $m_{\text {A }}=\mathrm{m}, m_{\text {B }}=2 \mathrm{~m}$, $m_{\mathrm{C}}=3 \mathrm{~m}$ and $m_{\mathrm{D}}=4 \mathrm{~m}$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is: (1) $\frac{a}{5}(\hat{i}-\hat{j})$(2) $a(\hat{i}+\hat{j})$(3) Zero(4) $\frac{a}{5}(\hat{i}+\hat{j})$Correct Option: 1 Solution: (1) Acceleration of centre of mass ...
Read More →Let K be the set of all real values
Question: Let $\mathrm{K}$ be the set of all real values of $x$ where the function $f(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ is not differentiable. Then the set $K$ is equal to :(1) $\phi$ (an emptyset)(2) $\{\pi\}$(3) $\{0\}$(4) $\{0, \pi\}$Correct Option: 1 Solution: $f(x)=\sin |x|-|x|+2(x-\pi) \cos |x|$ There are two cases, Case (1), $x0$ $f(x)=\sin x-x+2(x-\pi) \cos x$ $f^{\prime}(x)=\cos x-1+2(1-0) \cos x-2 \sin (x-\pi)$ $f^{\prime}(x)=3 \cos x-2(x-\pi) \sin x-1$ Then, function $f(x)$ is differ...
Read More →The outcomes of 65 throws of a dice were noted as shown below
Question: The outcomes of 65 throws of a dice were noted as shown below A dice is thrown at random. What is the probability of getting a prime number? (a) $\frac{3}{35}$ (b) $\frac{3}{5}$ (c) $\frac{31}{65}$ (d) $\frac{36}{65}$ Solution: (c) $\frac{31}{65}$ Explanation:Total number of throws = 65Let E be the event of getting a prime number.Then, E contains 2, 3 and 5, i.e. three numbers. $\therefore P($ getting a prime number $)=P(E)=\frac{\text { Number of times prime number s occur }}{\text { ...
Read More →The gas phase reaction
Question: The gas phase reaction $2 \mathrm{~A}(\mathrm{~g}) \rightleftharpoons \mathrm{A}_{2}(\mathrm{~g})$ at $400 \mathrm{~K}$ has $\Delta \mathrm{G}^{\circ}=+25.2 \mathrm{~kJ} \mathrm{~mol}^{-1}$. The equilibrium constant $\mathrm{K}_{\mathrm{C}}$ for this reaction is off to the Nearest integer) [ Use : $R=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \ln 10=2.3$$\left.\log _{10} 2=0.30,1 \mathrm{~atm}=1 \mathrm{bar}\right]$ $[$ antilog $(-0.3)=0.501]$ Solution: (166) Using formula $\...
Read More →A particle of mass m is projected with a speed u from the
Question: A particle of mass $m$ is projected with a speed $u$ from the ground at an angle $\theta=\frac{\pi}{3}$ w.r.t. horizontal (x-axis). When it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity $u \hat{i}$. The horizontal distance covered by the combined mass before reaching the ground is:(1) $\frac{3 \sqrt{3}}{8} \frac{u^{2}}{g}$(2) $\frac{3 \sqrt{2}}{4} \frac{u^{2}}{g}$(3) $\frac{5}{8} \frac{u^{2}}{g}$(4) $2 \sqrt{2} ...
Read More →A bag contains 5 red, 8 black and 7 white balls.
Question: A bag contains 5 red, 8 black and 7 white balls. One ball is chosen at random. What is the probability that the chosen ball is black? (a) $\frac{2}{3}$ (b) $\frac{2}{5}$ (c) $\frac{3}{5}$ (d) $\frac{1}{3}$ Solution: (b) $\frac{2}{5}$ Explanation:Total number of balls in the bag = 5 + 8 + 7 = 20Number of black balls = 8 LetE be the eventthat the chosen ball is black. $\therefore$ Required probability $=P(E)=\frac{8}{20}=\frac{2}{5}$...
Read More →Let f(x)=
Question: Let $f(x)=\left\{\begin{array}{cc}-1, -2 \leq x0 \\ x^{2}-1, 0 \leq x \leq 2\end{array}\right.$ and $g(x)=|f(x)|+f(|x|)$. Then, in the interval $(-2,2), g$ is : (1) differentiable at all points(2) not continuous(3) not differentiable at two points(4) not differentiable at one pointCorrect Option: , 4 Solution: $f(x)=\left\{\begin{array}{cc}-1, -2 \leq x0 \\ x^{2}-1, 0 \leq x \leq 2\end{array}\right.$ Then, $f(|x|)=\left\{\begin{array}{cc}-1, -2 \leq|x|0 \\ |x|^{2}-1, 0 \leq|x| \leq 2\e...
Read More →The gas phase reaction
Question: The gas phase reaction $2 \mathrm{~A}(\mathrm{~g}) \rightleftharpoons \mathrm{A}_{2}(\mathrm{~g})$ at $400 \mathrm{~K}$ has $\Delta \mathrm{G}^{\circ}=+25.2 \mathrm{~kJ} \mathrm{~mol}^{-1}$. The equilibrium constant $\mathrm{K}_{\mathrm{C}}$ for this reaction is off to the Nearest integer) [ Use : $R=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \ln 10=2.3$$\left.\log _{10} 2=0.30,1 \mathrm{~atm}=1 \mathrm{bar}\right]$ $[$ antilog $(-0.3)=0.501]$ Solution: (166) Using formula $\...
Read More →A bag contains 16 cards bearing numbers 1, 2, 3, ..., 16 respectively. One card is chosen at random.
Question: A bag contains 16 cards bearing numbers 1, 2, 3, ..., 16 respectively. One card is chosen at random. What is the probability that the chosen card bears a number divisible by 3? (a) $\frac{3}{16}$ (b) $\frac{5}{16}$ (c) $\frac{11}{16}$ (d) $\frac{13}{16}$ Solution: (b) $\frac{5}{16}$ Explanation:Total number of cards in the bag = 16Numbers on the cards that are divisible by 3 are 3, 6, 9, 12 and 15. Number of cards with numbers divisible by 3 = 5 LetE be the eventthat the chosen card b...
Read More →A rod of length L has non-uniform linear mass density
Question: A rod of length $L$ has non-uniform linear mass density given by $\rho(x)=a+b\left(\frac{x}{\mathrm{~L}}\right)^{2}$, where $a$ and $b$ are constants and $0 \leq x \leq \mathrm{L}$. The value of $x$ for the centre of mass of the rod is at:(1) $\frac{3}{2}\left(\frac{a+b}{2 a+b}\right) \mathrm{L}$(2) $\frac{3}{4}\left(\frac{2 a+b}{3 a+b}\right) \mathrm{L}$(3) $\frac{4}{3}\left(\frac{a+b}{2 a+3 b}\right) \mathrm{L}$(4) $\frac{3}{2}\left(\frac{2 a+b}{3 a+b}\right) \mathrm{L}$Correct Optio...
Read More →In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays.
Question: In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays. What is the probability that in a given delivery, the ball does not hit the boundary? (a) $\frac{1}{4}$ (b) $\frac{1}{5}$ (C) $\frac{4}{5}$ (d) $\frac{3}{4}$ Solution: (C) $\frac{4}{5}$ Explanation:Total number of balls faced = 30Number of times the ball hits the boundary = 6Number of times the ball does not hit the boundary = (30 6 )= 24 LetEbe the eventthat theball does not hit the boundary. Then, $P(E)=\...
Read More →For the reaction
Question: For the reaction $\mathrm{C}_{2} \mathrm{H}_{6} \rightarrow \mathrm{C}_{2} \mathrm{H}_{4}+\mathrm{H}_{2}$ the reaction enthalpy $\Delta_{\mathrm{r}} \mathrm{H}=$ $\mathrm{kJmol}^{-1}$ (Round off to the Nearest Integer). [Given : Bond enthalpies in $\mathrm{kJ} \mathrm{mol}^{-1}: \mathrm{C}-\mathrm{C}$ : $347, \mathrm{C}=\mathrm{C}: 611 ; \mathrm{C}-\mathrm{H}: 414, \mathrm{H}-\mathrm{H}: 436]$ Solution: (128) $\Delta_{\mathrm{r}} \mathrm{H}=\left[\epsilon_{\mathrm{C}-\mathrm{C}}+2 \eps...
Read More →Let f
Question: Let $f:(-1,1) \rightarrow \mathbf{R}$ be a function defined by $f(x)=\max$ $\left\{-|x|,-\sqrt{1-x^{2}}\right\} .$ If $\mathrm{K}$ be the set of all points at which $f$ is not differentiable, then $\mathrm{K}$ has exactly: (1) five elements(2) one element(3) three elements(4) two elementsCorrect Option: , 3 Solution: Consider the function $f(x)=\max \left\{-|x|,-\sqrt{1-x^{2}}\right\}$ Now, the graph of the function From the graph, it is clear that $f(x)$ is not differentiable at $x=0,...
Read More →It is given that the probability of winning a game is 0.7.
Question: It is given that the probability of winning a game is 0.7. What is the probability of losing the game?(a) 0.8(b) 0.3(c) 0.35(d) 0.15 Solution: (b) 0.3Explanation:Let E be the event of winning the game. Then,P(E) = 0.7P(not E) =P(losing the game) = 1P(E) ⇒ 10.7 = 0.3...
Read More →A coin is tossed 60 times and the tail appears 35 times. What is the probability of getting a head?
Question: A coin is tossed 60 times and the tail appears 35 times. What is the probability of getting a head? (a) $\frac{7}{12}$ (b) $\frac{12}{7}$ (c) $\frac{5}{12}$ (d) $\frac{1}{25}$ Solution: (C) $\frac{5}{12}$ Explanation:Total number of trials = 60Number of times tail appears = 35 Number of times head appears = 6035 = 25 LetE be the event of getting a head. $\therefore P($ getting a head $)=P(E)=\frac{\text { Number of times head appears }}{\text { Total number of trials }}=\frac{25}{60}=\...
Read More →Consider the reaction
Question: Consider the reaction $\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}_{2}(\mathrm{~g})$ The temperature at which $\mathrm{K}_{C}=20.4$ and $\mathrm{K}_{\mathrm{p}}=600.1$, is___________ $\mathrm{K}$. (Round off to the Nearest Integer). [Assume all gases are ideal and $\mathrm{R}=0.0831 \mathrm{~L}$ bar $\mathrm{K}^{-1} \mathrm{~mol}^{-1}$ ] Solution: (354) $\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}_{2}(\mathrm{~g}) ; \Delta...
Read More →Let f(x)
Question: Let $f(x)=\left\{\begin{array}{cc}\max \left\{|x|, x^{2}\right\} |x| \leq 2 \\ 8-2|x|, 2|x| \leq 4\end{array}\right.$ Let $\mathrm{S}$ be the set of points in the interval $(-4,4)$ at which $f$ is not differentiable. Then $\mathrm{S}$ :(1) is an empty set(2) equals $\{-2,-1,0,1,2\}$(3) equals $\{-2,-1,1,2\}$(4) equals $\{-2,2\}$Correct Option: , 2 Solution: Given $f(x)=\left\{\begin{array}{cc}\max \left\{|x|, x^{2}\right\} |x| \leq 2 \\ 8-2|x| 2|\mathrm{x}| \leq 4\end{array}\right.$ $\...
Read More →A body A of mass $m$ is moving in a circular orbit of radius
Question: A body A of mass $m$ is moving in a circular orbit of radius $\mathrm{R}$ about a planet. Another body $\mathrm{B}$ of mass $\frac{\mathrm{m}}{2}$ collides with A with a velocity which is half $\left(\frac{\vec{v}}{2}\right)$ the instantaneous velocity $\vec{v}$ or A. The collision is completely inelastic. Then, the combined body:(1) continues to move in a circular orbit(2) Escapes from the Planet's Gravitational field(3) Falls vertically downwards towards the planet(4) starts moving i...
Read More →The standard enthalpies of formation of
Question: The standard enthalpies of formation of $\mathrm{Al}_{2} \mathrm{O}_{3}$ and $\mathrm{CaO}$ are $-1675 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $-635 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively. For the reaction $3 \mathrm{CaO}+2 \mathrm{Al} \rightarrow 3 \mathrm{Ca}+\mathrm{Al}_{2} \mathrm{O}_{3}$ the standard reaction enthalpy $\Delta_{\mathrm{r}} \mathrm{H}^{0=}$ __________ kJ. (Round off to the Nearest Integer). Solution: (230) Given reaction: $3 \mathrm{CaO}+\mathrm{Al} \rightarrow ...
Read More →be a function defined as
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function defined as $f(x)=\left\{\begin{array}{ccc}5, \text { if } x \leq 1 \\ \mathrm{a}+\mathrm{b} x, \text { if } 1x3 \\ \mathrm{~b}+5 x, \text { if } 3 \leq x5 \\ 30, \text { if } x \geq 5\end{array}\right.$ Then, $f$ is :(1) continuous if $a=5$ and $b=5$(2) continuous if $a=-5$ and $b=10$(3) continous if $\mathrm{a}=0$ and $\mathrm{b}=5$(4) not continuous for any values of $a$ and $b$Correct Option: , 4 Solution: Let $f(x)$ is contin...
Read More →Two particles of equal mass m have respective initial velocities
Question: Two particles of equal mass $m$ have respective initial velocities $u \hat{i}$ and $u\left(\frac{\hat{i}+\hat{j}}{2}\right)$. They collide completely inelastically. The energy lost in the process is:(1) $\frac{1}{3} \mathrm{mu}^{2}$(2) $\frac{1}{8} \mathrm{mu}^{2}$(3) $\frac{3}{4} \mathrm{mu}^{2}$(4) $\sqrt{\frac{2}{3}} \mathrm{mu}^{2}$Correct Option: , 2 Solution: $m u+\frac{m u}{2}=2 m v_{x}^{\prime} \Rightarrow V_{x}^{\prime}=\frac{3 u}{4}$ $\mathrm{y}$-direction $0+\frac{m u}{2}=2 ...
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