Solve the following
Question: For the reaction $\mathrm{A}(\mathrm{g}) \rightleftharpoons \mathrm{B}(\mathrm{g})$ at $495 \mathrm{~K}$, $\Delta_{\mathrm{r}} \mathrm{G}^{\mathrm{o}}=-9.478 \mathrm{~kJ} \mathrm{~mol}^{-1}$ If we start the reaction in a closed container at $495 \mathrm{~K}$ with 22 millimoles of $\mathrm{A}$, the amount of $\mathrm{B}$ is the equilibrium mixture is....millimoles. (Round off to the Nearest Integer). $\left[\mathrm{R}=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} ; \ell \ln 10=2...
Read More →To know the opinion of the students about the subject Sanskrit, a survey of 200 students was conducted.
Question: To know the opinion of the students about the subject Sanskrit, a survey of 200 students was conducted. The data is recorded as under. What is the probability that a student chosen at random does not like it? (a) $\frac{13}{27}$ (b) $\frac{27}{40}$ (c) $\frac{13}{40}$ (d) $\frac{27}{13}$ Solution: Total number of students surveyed = 200Number of students who does not like the subject Sanskrit = 65 $\therefore \mathrm{P}($ Student chosen at random does not like the subject Sanskrit) $)=...
Read More →is continuous at $x=0$, then the ordered pair $(p, q)$ is equal to:
Question: If $f(x)=\left\{\begin{array}{cc}\frac{\sin (p+1) x+\sin x}{x} , x0 \\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}} , x0\end{array}\right.$ is continuous at $x=0$, then the ordered pair $(p, q)$ is equal to: (1) $\left(-\frac{3}{2},-\frac{1}{2}\right)$(2) $\left(-\frac{1}{2}, \frac{3}{2}\right)$(3) $\left(-\frac{3}{2}, \frac{1}{2}\right)$(4) $\left(\frac{5}{2}, \frac{1}{2}\right)$Correct Option: , 3 Solution: $f(x)=\left\{\begin{array}{ll}\frac{\sin (\mathrm{p}+1) \mathrm{x}+\sin \mathrm{...
Read More →80 bulbs are selected at random from a lot and their lifetime in hours is recorded as under.
Question: 80 bulbs are selected at random from a lot and their lifetime in hours is recorded as under. One bulb is selected at random from the lot. What is the probability that the selected bulb has a life more than 500 hours? (a) $\frac{27}{40}$ (b) $\frac{29}{40}$ (c) $\frac{5}{16}$ (d) $\frac{11}{40}$ Solution: (b) $\frac{29}{40}$ Explanation:Total number of bulbs in the lot = 80Number of bulbs with life time of more than 500 hours = (23 + 25 + 10) = 58 LetEbe the eventthat the chosen bulb's ...
Read More →Let
Question: Let $\mathrm{f}=\mathbf{R} \rightarrow \mathbf{R}$ be differentiable at $\mathrm{c} \in \mathbf{R}$ and $\mathrm{f}(\mathrm{c})=0$. If $g(x)=|f(x)|$, then at $x=c, g$ is :(1) not differentiable if $f^{\prime}(c)=0$(2) differentiable if $f^{\prime \prime}(c) \neq 0$(3) differentiable if $\mathrm{f}^{\prime \prime}(\mathrm{c})=0$(4) not differentiableCorrect Option: , 3 Solution: $g^{\prime}(c)=\lim _{x \rightarrow c} \frac{g(x) \quad g(c)}{x \quad c}$ $\Rightarrow g^{\prime}(c)=\lim _{x...
Read More →If the function
Question: If the function $f(x)=\left\{\begin{array}{l}a|\pi-x|+1, x \leq 5 \\ b|x-\pi|+3, x5\end{array}\right.$ is continuous at $x=5$, then the value of $a-b$ is:(1) $\frac{2}{\pi+5}$(2) $\frac{-2}{\pi+5}$(3) $\frac{2}{\pi-5}$(4) $\frac{2}{5-\pi}$Correct Option: , 4 Solution: R.H.L. $\lim _{x \rightarrow 5^{+}} b|(x-\pi)|+3=(5-\pi) b+3$ $f(5)=$ L.H.L. $\lim _{x \rightarrow 5^{-}} a|(\pi-x)|+1=a(5-\pi)+1$ $\because$ function is continuous at $x=5$ $\therefore \mathrm{LHL}=\mathrm{RHL}$ $(5-\pi)...
Read More →Two coins are tossed 1000 times and the outcomes are recorded as given below:
Question: Two coins are tossed 1000 times and the outcomes are recorded as given below: Now, if two coins are tossed at random, what is the probability of getting at most one head? (a) $\frac{3}{4}$ (b) $\frac{4}{5}$ (c) $\frac{1}{4}$ (d) $\frac{1}{5}$ Solution: Total number of times two coins are tossed = 1000Number of times of getting at most one head = Number of times of getting 0 heads + Number of times of getting 1 head = 250 + 550 = 800 $\therefore P($ Getting at most one head $)=\frac{\te...
Read More →As shown in fig.
Question: As shown in fig. when a spherical cavity (centred at $O$ ) of radius 1 is cut out of a uniform sphere of radius $R$ (centred at $C$ ), the centre of mass of remaining (shaded) part of sphere is at $G$, i.e on the surface of the cavity. $R$ can be determined by the equation: (1) $\left(R^{2}+R+1\right)(2-R)=1$(2) $\left(R^{2}-R-1\right)(2-R)=1$(3) $\left(R^{2}-R+1\right)(2-\mathrm{R})=1$(4) $\left(R^{2}+R-1\right)(2-R)=1$Correct Option: Solution: (1) Mass of sphere = volume of sphere $x...
Read More →if f(x) = [x]
Question: If $f(x)=[x]-\left[\frac{x}{4}\right], x \in \mathrm{R}$, where $[x]$ denotes the greatest integer function, then: (1) $f$ is continuous at $x=4$.(2) $\lim _{x \rightarrow 4+} f(x)$ exists but $\lim _{x \rightarrow 4-} f(x)$ does not exist.(3) Both $\lim _{x \rightarrow 4-} f(x)$ and $\lim _{x \rightarrow 4+} f(x)$ exist but are not equal.(4) $\lim _{x \rightarrow 4-} f(x)$ exists but $\lim _{x \rightarrow 4+} f(x)$ does not exist.Correct Option: 1 Solution: L.H.L. $\lim _{x \rightarro...
Read More →In a survey of 364 children aged 19 – 36 months, it was found that 91 liked to eat potato chips.
Question: In a survey of 364 children aged 19 36 months, it was found that 91 liked to eat potato chips. If a child is selected at random, the probability that he / she does not like to eat potato chips is (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) $\frac{3}{4}$ (d) $\frac{4}{5}$ Solution: Total number of children surveyed = 364Number of children who liked to eat potato chips = 91Number of children who do not liked to eat potato chips = 364 91 = 273 $\therefore P($ Child does not like to eat potato...
Read More →80 bulbs are selected at random from a lot and their lifetime is hours is recorded as under.
Question: 80 bulbs are selected at random from a lot and their lifetime is hours is recorded as under. One bulb is selected at random from the lot. What is the probability that its life is 1150?(a) $\frac{1}{80}$ (b) $\frac{7}{16}$ (c) 1(d) 0 Solution: (d) 0Maximum lifetime a bulb has is 1100 hours. There is no bulb with lifetime 1150 hours....
Read More →Let f(x)=15-|x-10| ; x in R.
Question: Let $f(x)=15-|x-10| ; x \in R$. Then the set of all values of $x$, at which the function, $g(x)=f(f(x))$ is not differentiable, is:(1) $\{5,10,15\}$(2) $\{10,15\}$(3) $\{5,10,15,20\}$(4) $\{10\}$Correct Option: 1 Solution: Since, $f(x)=15-|(10-x)|$ $\therefore \mathrm{g}(\mathrm{x})=\mathrm{f}(\mathrm{f}(\mathrm{x}))=15-|10-[15-|10-\mathrm{x}|]|$ $=15-|| 10-x|-5|$ $\therefore$ Then, the points where function $g(x)$ is Nondifferentiable are $10-x=0$ and $|10-x|=5$ $\Rightarrow x=10$ and...
Read More →In a medical examination of students of a class, the following blood groups are recorded:
Question: In a medical examination of students of a class, the following blood groups are recorded: From this class, a student is chosen at random. What is the probability that the chosen student has blood group AB? (a) $\frac{13}{20}$ (b) $\frac{3}{8}$ (C) $\frac{1}{5}$ (d) $\frac{11}{40}$ Solution: (c) $\frac{1}{5}$ Explanation:Total number of students = 40Number of students with blood group AB = 8 LetE be the eventthat the selected student's blood group is AB. $\therefore$ Required probabilit...
Read More →Two pi and half sigma bonds are present in:
Question: Two pi and half sigma bonds are present in:$\mathrm{O}_{2}^{+}$$\mathrm{N}_{2}$$\mathrm{O}_{2}$$\mathrm{N}_{2}^{+}$Correct Option: , 4 Solution: $\mathrm{N}_{2}^{+}=13 e^{-}$ $=\sigma 1 s^{2} \sigma^{*} 1 s^{2} \sigma 2 s^{2} \sigma^{*} 2 s^{2} \pi 2 p_{x}^{2}$ $=\pi 2 p_{y}^{2} \sigma 2 p_{z}^{1}$ B.O. $=\frac{\begin{array}{c}\text { Bonding } \\ \text { electrons }\end{array}-\begin{array}{c}\text { Antibonding } \\ \text { electrons }\end{array}}{2}$ B.O. $=\frac{9-4}{2}=2.5=2 \pi$ ...
Read More →If the function
Question: If the function $\mathrm{f}$ defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}\frac{\sqrt{2} \cos \mathrm{x}-1}{\cot \mathrm{x}-1}, \mathrm{x} \neq \frac{\pi}{4} \\ \mathrm{k}, \mathrm{x}=\frac{\pi}{4}\end{array}\right.$ is continuous, then $\mathrm{k}$ is equal to:(1) 2(2) $\frac{1}{2}$(3) 1(4) $\frac{1}{\sqrt{2}}$Correct Option: , 2 Solution: Since, $f(x)$ is continuous, then $\lim _{x \rightarrow \frac{\pi}{4}} f(x)=f\left(\fr...
Read More →A body A, of mass
Question: A body $A$, of mass $m=0.1 \mathrm{~kg}$ has an initial velocity of $3 \hat{i} \mathrm{~ms}^{-1}$. It collides elastically with another body, $B$ of the same mass which has an initial velocity of $5 \hat{j} \mathrm{~ms}^{-1}$. After collision, $A$ moves with a velocity $\vec{v}=4(\hat{i}+\hat{j})$. The energy of $B$ after collision is written as $\frac{x}{10} J$. The value of $x$ is Solution: (1) For elastic collision $K E_{\mathrm{i}}=K E_{f}$ $\frac{1}{2} m \times 25+\frac{1}{2} \tim...
Read More →In a sample survey of 645 people, it was found that 516 people have a high school certificate.
Question: In a sample survey of 645 people, it was found that 516 people have a high school certificate. If a person is chosen at random, what is the probability that he / she has a high school certificate? (a) $\frac{1}{2}$ (b) $\frac{3}{5}$ (c) $\frac{7}{10}$ (d) $\frac{4}{5}$ Solution: Total number of people surveyed = 645Number of people who have a high school certificate = 516 $\therefore P(P e r s o n$ has a high school certificate $)=\frac{\text { Number of people who have a high school c...
Read More →In which of the following processes,
Question: In which of the following processes, the bond order has increased and paramagnetic character has changed to diamagnetic?$\mathrm{NO} \rightarrow \mathrm{NO}^{+}$$\mathrm{N}_{2} \rightarrow \mathrm{N}_{2}^{+}$$\mathrm{O}_{2} \rightarrow \mathrm{O}_{2}^{+}$$\mathrm{O}_{2} \rightarrow \mathrm{O}_{2}^{2-}$Correct Option: Solution: In case of $\mathrm{NO}$ (paramagnetic) $\rightarrow \mathrm{NO}^{+}$(diamagnetic) the bond order has increased from $2.5$ to 3 . For other cases:...
Read More →Fill in the blanks.
Question: Fill in the blanks.(i) Probability of an impossible event = ........ .(ii) Probability of a sure event = ........ .(iii) LetEbe an event. Then, P(notE) = ......... .(iv)P(E) +P(notE) = ........ .(v) ....... P(E) ....... . Solution: (i) Probability of an impossible event = 0 (ii) Probability of a sure event = 1 (iii) LetEbe an event. Then, P(notE) = 1 P(E) .(iv)P(E) +P(notE) = 1 (v) 0 P(E) 1...
Read More →Let f:[-1,3]
Question: Let $f:[-1,3] \rightarrow \mathrm{R}$ be defined as $f(x)= \begin{cases}|x|+[x], -1 \leq \mathrm{x}1 \\ x+|x|, 1 \leq x2 \\ x+[x], 2 \leq x \leq 3\end{cases}$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at :(1) only one point(2) only two points(3) only three points(4) four or more pointsCorrect Option: , 3 Solution: Given function is, $f(\mathrm{x})= \begin{cases}|x|+[x], -1 \leq x1 \\ x+|x|, 1 \leq x2 \\ x+[x], 2 \leq x \leq 3\end{cas...
Read More →It is known that a box of 800 electric bulbs contains 36 defective bulbs.
Question: It is known that a box of 800 electric bulbs contains 36 defective bulbs.One bulb is taken at random out of the box. What is the probability that the bulb chosen is non-defective? Solution: Total number of electric bulbs in the box = 800Number of defective electricbulbs in the box = 36 Number of non-defective electric bulbs in the box = 800 36 = 764 $P($ Bulb chosen is non-defective $)=\frac{\text { Number of non defective electric bulbs in the box }}{\text { Total number of electric b...
Read More →The marks obtained by 90 students of a school in mathematics out of 100 are given as under:
Question: The marks obtained by 90 students of a school in mathematics out of 100 are given as under: From these students, a student is chosen at random.What is the probability that the chosen student(i) gets 20% or less marks?(ii) gets 60% or more marks? Solution: Total number of students = 90(i) Number of students who gets 20% or less marks = Number of students who gets 20 or less marks = 7 $\therefore \mathrm{P}($ Student gets $20 \%$ or less marks $)=\frac{\text { Number of students who gets...
Read More →According to molecular orbital theory,
Question: According to molecular orbital theory, which of the following is true with respect to $\mathrm{Li}_{2}^{+}$and $\mathrm{Li}_{2}^{-}$?$\mathrm{Li}_{2}^{+}$is unstable and $\mathrm{Li}_{2}^{-}$is stable$\mathrm{Li}_{2}^{+}$is stable and $\mathrm{Li}_{2}^{-}$is unstableBoth are stableBoth are unstableCorrect Option: , 3 Solution: Electronic configuratios of $\mathrm{Li}_{2}^{+}$and $\mathrm{Li}_{2}^{-}$: $\mathrm{Li}_{2}^{+}: \sigma 1 \mathrm{~s}^{2} \sigma^{*} 1 \mathrm{~s}^{2} \sigma 2 ...
Read More →Consider a uniform rod of mass
Question: Consider $a$ uniform rod of mass $M=4 \mathrm{~m}$ and length $l$ pivoted about its centre. $A$ mass $m$ moving with velocity $v$ making angle $\theta=\frac{\pi}{4}$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is:(1) $\frac{3}{7 \sqrt{2}} \frac{v}{l}$(2) $\frac{3}{7} \frac{v}{l}$(3) $\frac{3 \sqrt{2}}{7} \frac{v}{l}$(4) $\frac{4}{7} \frac{v}{l}$Correct Option: 3, Solution: About point $\math...
Read More →Let [t] denote the greatest integer
Question: Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=\mathrm{A}$. Then the function, $f(x)=\left[x^{2}\right] \sin (\pi x)$ is discontinuous, when $x$ is equal to : (1) $\sqrt{\mathrm{A}+1}$(2) $\sqrt{\mathrm{A}+5}$(3) $\sqrt{\mathrm{A}+21}$(4) $\sqrt{\mathrm{A}}$Correct Option: 1 Solution: $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=A \Rightarrow \lim _{x \rightarrow 0} x\left[\frac{4}{x}-\left\{\frac{4}{x}\right\}\right]=A$ $\Ri...
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