Following are the ages (in years) of 360 patients, getting medical treatment in a hospital:
Question: Following are the ages (in years) of 360 patients, getting medical treatment in a hospital: One of the patients is selected at random.What is the probability that his age is(i) 30 years or more but less than 40 years?(ii) 50 years or more but less than 70 years?(iii) 10 years or more but less than 40 years?(iv) 10 years or more?(v) less than 10 years? Solution: Total number of patients = 360(i) Number of patients whose age is 30 years or more but less than 40 years = 60 Let E1 be the e...
Read More →During the change of
Question: During the change of $\mathrm{O}_{2}$ to $\mathrm{O}_{2}^{-}$, the incoming electron goes to the orbital :$\pi 2 \mathrm{p}_{\mathrm{y}}$$\sigma^{*} 2 p_{z}$$\pi^{*} 2 p_{x}$$\pi 2 \mathrm{p}_{\mathrm{x}}$Correct Option: , 3 Solution: Electronic configuration of $\mathrm{O}_{2}$ is $\sigma 1 s^{2} \sigma * 1 s^{2} \sigma 2 s^{2} \sigma * 2 s^{2} \sigma 2 p_{z}^{2} \pi 2 p_{x}^{2}$ $=\pi 2 p_{y}^{2} \pi^{*} 2 p_{x}^{1}=\pi * 2 p_{y}^{1}$ When an electron is added in $\mathrm{O}_{2}$ to ...
Read More →If f(x)=
Question: If $f(x)=\left\{\begin{array}{cc}\frac{\sin (a+2) x+\sin x}{x} ; x0 \\ b ; x=0 \\ \frac{\left(x+3 x^{2}\right)^{1 / 3}-x^{1 / 3}}{x^{4 / 3}} ; x0\end{array}\right.$ is continuous at $x=0$, then $a+2 b$ is equal to: (1) 1(2) $-1$(3) 0(4) $-2$Correct Option: , 3 Solution: $\mathrm{LHL}=\lim _{x \rightarrow 0} \frac{\sin (a+2) x+\sin x}{x}$ $=\lim _{x \rightarrow 0}\left(\frac{\sin (a+2) x}{(a+2) x}\right)(a+2)+\lim _{x \rightarrow 0} \frac{\sin x}{x}=a+3$ $f(0)=b$ RHL $=\lim _{h \rightar...
Read More →The coordinates of centre of mass of a uniform flag shaped lamina
Question: The coordinates of centre of mass of a uniform flag shaped lamina (thin flat plale) of mass $4 \mathrm{~kg}$. (The coordinates of the same are shown in figure) are: (1) $(1.25 \mathrm{~m}, 1.50 \mathrm{~m})$(2) $(0.75 \mathrm{~m}, 1.75 \mathrm{~m})$(3) $(0.75 \mathrm{~m}, 0.75 \mathrm{~m})$(4) $(1 \mathrm{~m}, 1.75 \mathrm{~m})$Correct Option: , 2 Solution: (2) For given Lamina $x y$ $(1.5,2.5)$ $m_{1}=1, C_{1}=$ $m_{2}=3, C_{2}=$ $(0.5,1.5)$ $X_{c m}=\frac{m_{1} x_{1}+m_{2} x_{2}}{m_{...
Read More →Let f be any function continuous on
Question: Let $f$ be any function continuous on $[a, b]$ and twice differentiable on $(a, b)$. If for all $x \in(a, b), f^{\prime}(x)0$ and $f^{\prime \prime}(x)0$, then for any $c \in(a, b), \frac{f(c)-f(a)}{f(b)-f(c)}$ is greater than:(1) $\frac{b+a}{b-a}$(2) 1(3) $\frac{b-c}{c-a}$(4) $\frac{c-a}{b-c}$Correct Option: , 4 Solution: Since, function $f(x)$ is twice differentiable and continuous in $x \in[a, b]$. Then, by LMVT for $x \in[a, c]$ $\frac{f(c)-f(a)}{c-a}=f^{\prime}(\alpha), \alpha \in...
Read More →The table given below shows the ages of 75 teachers in a school.
Question: The table given below shows the ages of 75 teachers in a school. A teacher from this school is chosen at random. What is the probability that the selected teacher is(i) 40 or more than 40 years old?(ii) of an age lying between 30 39 years (including both)?(iii) 18 years or more and 49 years or less?(iv) 18 years or more old?(v) above 60 years of age?Note Here 18 29 means 18 or more but less than or equal to 29. Solution: Total number of teachers = 75(i) Number of teachers who are 40 or...
Read More →Among the following species, the diamagnetic molecule is:
Question: Among the following species, the diamagnetic molecule is:NOCO$\mathrm{B}_{2}$$\mathrm{O}_{2}$Correct Option: , 2 Solution: The molecules with no unpaired electrons are diamagnetic. Since $\mathrm{CO}$ has no unpaired electron. Hence $\mathrm{CO}$ is diamagnetic....
Read More →Let f be any function continuous on
Question: Let $f$ be any function continuous on $[a, b]$ and twice differentiable on $(a, b)$. If for all $x \in(a, b), f^{\prime}(x)0$ and $f^{\prime \prime}(x)0$, then for any $c \in(a, b), \frac{f(c)-f(a)}{f(b)-f(c)}$ is greater than:(1) $\frac{b+a}{b-a}$(2) 1(3) $\frac{b-c}{c-a}$(4) $\frac{c-a}{b-c}$Correct Option: , 4 Solution: Since, function $f(x)$ is twice differentiable and continuous in $x \in[a, b]$. Then, by LMVT for $x \in[a, c]$ $\frac{f(c)-f(a)}{c-a}=f^{\prime}(\alpha), \alpha \in...
Read More →If the function
Question: If the function $f$ defined on $\left(-\frac{1}{3}, \frac{1}{3}\right)$ by $f(x)=\left\{\begin{array}{l}\frac{1}{x} \log _{e}\left(\frac{1+3 x}{1-2 x}\right), \text { when } x \neq 0 \\ k, \text { when } x=0\end{array}\right.$ is continuous, then $k$ is equal to_____. Solution: $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}\left(\frac{1}{x} \ln \left(\frac{1+3 x}{1-2 x}\right)\right)$ $=\lim _{x \rightarrow 0}\left(\frac{\ln (1+3 x)}{x}-\frac{\ln (1-2 x)}{x}\right)$ $=\lim _{x \...
Read More →Among the following, the molecule expected to be stabilized by anion formation is:
Question: Among the following, the molecule expected to be stabilized by anion formation is: $\mathrm{C}_{2}, \mathrm{O}_{2}, \mathrm{NO}, \mathrm{F}_{2}$$\mathrm{C}_{2}$$\mathrm{F}_{2}$NO$\mathrm{O}_{2}$Correct Option: 1 Solution: Configuration of $C_{2}$ $=\sigma 1 s^{2} \sigma * 1 s^{2} \sigma 2 s^{2} \sigma * 2 s^{2} \pi 2 p_{x}^{2}=\pi 2 p_{y}^{2}$ Configuration of $\mathrm{C}_{2}^{-}$ $=\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} \sig...
Read More →The table given below shows the marks obtained by 30 students in a test.
Question: The table given below shows the marks obtained by 30 students in a test. Out of these students, one is chosen at random. What is the probability that the marks of the chosen student(i) are 30 or less?(ii) are 31 or more?(iii) lie in the interval 21 30? Solution: Total number of students = 30(i) Number of students whose marks are 30 or less = 7 + 10 + 6 = 23 $\therefore P$ (Marks of the chosen student are 30 or less) $=\frac{\text { Number of students whose marks are } 30 \text { or les...
Read More →het S be the set of points where the function,
Question: het $S$ be the set of points where the function, $(x)=|2-| x-3||, x \in \boldsymbol{R}$, is not differentiable. hen $\sum_{x \in S} f(f(x))$ is equal to Solution: $f(x)$ is non differentiable at $x=1,3,5$ $[\because|x-3|$ is not differentiable at $x=3]$ $\Sigma f(f(x))=f(f(1)+f(f(3))+f(f(5))$ $=1+1+1=3$...
Read More →An organisation selected 2400 families at random and surveyed them to determine a relationship between the income level and the number of vehicles in a family.
Question: An organisation selected 2400 families at random and surveyed them to determine a relationship between the income level and the number of vehicles in a family. The information gathered is listed in the table below: Suppose a family is chosen at random. Find the probability that the family chosen is(i) earning ₹ 25000 ₹ 30000 per month and owning exactly 2 vehicles.(ii) earning ₹ 40000 or more per month and owning exactly 1 vehicle.(iii) earning less than ₹ 25000 per month and not ownin...
Read More →Three point particles of masses
Question: Three point particles of masses $1.0 \mathrm{~kg}, 1.5 \mathrm{~kg}$ and $2.5 \mathrm{~kg}$ are placed at three corners of a right angle triangle of sides $4.0 \mathrm{~cm}, 3.0 \mathrm{~cm}$ and $5.0 \mathrm{~cm}$ as shown in the figure. The center of mass of the system is at a point: (1) $0.6 \mathrm{~cm}$ right and $2.0 \mathrm{~cm}$ above $1 \mathrm{~kg}$ mass(2) $1.5 \mathrm{~cm}$ right and $1.2 \mathrm{~cm}$ above $1 \mathrm{~kg}$ mass(3) $2.0 \mathrm{~cm}$ right and $0.9 \mathrm...
Read More →The correct statement about
Question: The correct statement about $\mathrm{ICl}_{5}$ and $\mathrm{ICl}_{4}^{-}$is :both are is isostructural.$\mathrm{ICl}_{5}$ is trigonal bipyramidal and $\mathrm{ICl}_{4}^{-}$is tetrahedral,$\mathrm{ICl}_{5}$ is square pyramidal and $\mathrm{ICl}_{4}^{-}$is tetrahedral.$\mathrm{ICl}_{5}$ is square pyramidal and $\mathrm{ICl}_{4}^{-}$is square planar.Correct Option: , 4 Solution: $\mathrm{ICl}_{5}$ is $s p^{3} d^{2}$ hybridised $(5 b p, 1 l p)$ $\mathrm{ICl}_{4}^{-}$is $s p^{3} d^{2}$ hybr...
Read More →In a circket match, a batsman hits a boundary 6 times out of 30 balls he plays
Question: In a circket match, a batsman hits a boundary 6 times out of 30 balls he plays. Find the probability that he did not hit a boundary. Solution: Number of balls played by the batsman = 30Number of balls in which he hits boundaries = 6 Number of balls in which he did not hit a boundary = 30 6 = 24 $P($ Batsman did not hit a boundary $)=\frac{\text { Number of balls in which he did not hit a boundary }}{\text { Number of balls played by the batsman }}=\frac{24}{30}=\frac{4}{5}$ Thus, the p...
Read More →12 Packets of salt, each marked 2 kg, actually contained the following weights (in kg) of salt:
Question: 12 Packets of salt, each marked 2 kg, actually contained the following weights (in kg) of salt:1.950, 2.020, 2.060, 1.980, 2.030, 1.970,2.040, 1.990, 1.985, 2.025, 2.000, 1.980.Out of these packets, one packet is chosen at random.What is the probability that the chosen packet contains more than 2 kg of salt? Solution: Total number of salt packets = 12Number of packets which contains more than 2 kg of salt = 5 $\therefore P($ Chosen packet contains more than $2 \mathrm{~kg}$ of salt $)=...
Read More →The centre of mass of a solid hemisphere of radius
Question: The centre of mass of a solid hemisphere of radius $8 \mathrm{~cm}$ is $x \mathrm{~cm}$ from the centre of the flat surface. Then value of $x$ is ______ Solution: (3) Centre of mass of solid hemisphere of radius $R$ lies at a distance $\frac{3 R}{8}$ above the centre of flat side of hemisphere. $\therefore h_{\mathrm{cm}}=\frac{3 R}{8}=\frac{3 \times 8}{8}=3 \mathrm{~cm}$...
Read More →Let the function,
Question: 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 $f$, $f^{\prime}(-1)+f(0)$ lies in the interval:(1) $(-\infty, 20]$(2) $[-3,11]$(3) $(-\infty, 11]$(4) $[-6,20]$Correct Option: 1 Solution: From, LMVT for $x \in[-7,-1]$ $\frac{f(-1)-f(-7)}{(-1+7)} \leq 2 \Rightarrow \frac{f(-1)+3}{6} \leq 2 \Rightarrow f(-1) \leq 9$ From, LMVT for $x \in[-7,0]$...
Read More →The following table shows the blood groups of 40 students of a class.
Question: The following table shows the blood groups of 40 students of a class. One student of the class is chosen at random. What is the probability that the chosen student's blood group is(i) O?(ii) AB? Solution: Total number of students = 4 (i) Number of students with blood group O = 14 Let E1 be the eventthat the selected student's blood group is O. $\therefore$ Required probability $=P\left(E_{1}\right)=\frac{14}{40}=0.35$ (ii)Number of students with blood group AB = 6 Let E2 be the eventth...
Read More →Among the following molecules/ions,
Question: Among the following molecules/ions, $\mathrm{C}_{2}^{2-}, \mathrm{N}_{2}^{2-}, \mathrm{O}_{2}^{2-}, \mathrm{O}_{2}$ Which one is diamagnetic and has the shortest bond length?$\mathrm{O}_{2}$$\mathrm{N}_{2}^{2-}$$\mathrm{O}_{2}^{2-}$$\mathrm{C}_{2}^{2-}$Correct Option: , 4 Solution: Bond length $\propto \frac{1}{\text { Bond order }}$ and diamagnetic species has no unpaired electron in their molecular orbitals....
Read More →Particle A of mass
Question: Particle $A$ of mass $m_{1}$ moving with velocity $(\sqrt{3} \hat{i}+\hat{j}) \mathrm{ms}^{-1}$ collides with another particle $\mathrm{B}$ of mass $\mathrm{m}_{2}$ which is at rest initially. Let $\vec{V}_{1}$ and $\vec{V}_{2}$ be the velocities of particles A and $\mathrm{B}$ after collision respectively. If $\mathrm{m}_{1}=2 \mathrm{~m}_{2}$ and after collision $\overrightarrow{\mathrm{V}}_{1}=(\hat{i}+\sqrt{3} \hat{j}) \mathrm{ms}^{-1}$, the angle between $\overrightarrow{\mathrm{V...
Read More →Let f:
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=\max \left\{x, x^{2}\right\}$. Let $\mathrm{S}$ denote the set of all points in $\mathrm{R}$, where $f$ is not differentiable. Then: (1) $\{0,1\}$(2) $\{0\}$(3) $\phi$ (an empty set)(4) $\{1\}$Correct Option: 1 Solution: $f(x)=\max \cdot\left\{x, x^{2}\right\}$ $\Rightarrow f(x)=\left\{\begin{array}{cc}x^{2}, x0 \\ x, 0 \leq x1 \\ x^{2}, x \geq 1\end{array}\right.$ $\therefore f(x)$ is not differentiable at $x=0,...
Read More →On one page of a telephone directory, there are 200 phone numbers.
Question: On one page of a telephone directory, there are 200 phone numbers. The frequency distribution of their units digits is given below: One of the numbers is chosen at random from the page. What is the probability that the units digit of the chosen number is(i) 5?(ii) 8? Solution: Total phone numbers on the directory page = 200(i) Number of numbers with units digit 5 = 24 Let E1 be the eventthat the units digit of selected number is 5. $\therefore$ Required probability $=P\left(E_{1}\right...
Read More →If the magnetic moment of a dioxygen species is
Question: If the magnetic moment of a dioxygen species is $1.73 \mathrm{~B} . \mathrm{M}$, it may be:$\mathrm{O}_{2}^{-}$or $\mathrm{O}_{2}^{+}$$\mathrm{O}_{2}$ or $\mathrm{O}_{2}^{+}$$\mathrm{O}_{2}$ or $\mathrm{O}_{2}^{-}$$\mathrm{O}_{2}, \mathrm{O}_{2}^{-}$or $\mathrm{O}_{2}^{+}$Correct Option: 1 Solution: $\mu=\sqrt{n(n+2)}$ B.M. $1.73=\sqrt{n(n+2)}$ $n=1$ $\mathrm{O}_{2}^{+}=\sigma 1 s^{2} \sigma^{*} 1 s^{2} \sigma 2 s^{2} \sigma^{*} 2 s^{2} \sigma 2 p_{z}^{2} \pi 2 p_{x}^{2}$ $=\pi 2 p_{y}...
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