Let f
Question: Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $f(x)=\left\{\begin{array}{rr}x^{5} \sin \left(\frac{1}{x}\right)+5 x^{2}, x0 \\ 0, x=0 \\ x^{5} \cos \left(\frac{1}{x}\right)+\lambda x^{2}, x0\end{array}\right.$ The value of $\lambda$ for which $f^{\prime \prime}(0)$ exists, is_________. Solution: $f^{\prime}(x)=\left\{\begin{array}{cc}5 x^{4} \cdot \sin \left(\frac{1}{x}\right)-x^{3} \cos \left(\frac{1}{x}\right)+10 x, x0 \\ 0, x=0 \\ 5 x^{4} \cos \left(\frac{1}{x}\right)+x^{...
Read More →On a particular day, in a city, 240 vehicles of various types going past a crossing during a time interval were observed, as shown:
Question: On a particular day, in a city, 240 vehicles of various types going past a crossing during a time interval were observed, as shown: Out of these vehicles, one is chosen at random. What is the probability that the given vehicle is a two-wheeler? Solution: Total number of vehicles going past the crossing = 240Number of two-wheelers = 84 LetE be the event that the selected vehicle is a two-wheeler. Then, required probability $=P(E)=\frac{84}{240}=0.35$...
Read More →The bond order and the magnetic characteristics of
Question: The bond order and the magnetic characteristics of $\mathrm{CN}^{-}$are:$2 \frac{1}{2}$, diamagnetic3 , diamagnetic3 , paramagnetic$2 \frac{1}{2}$, paramagneticCorrect Option: , 2 Solution: Total number of electrons in $\mathrm{CN}^{-}=6+7+1=14$ $\therefore$ Molecular orbital distribution $\therefore$ Bond order $=\frac{10-4}{2}=3$ $\mathrm{CN}^{-}$is diamagnetic because all electrons are paired....
Read More →Let f(x)=x
Question: Let $f(x)=x .\left[\frac{x}{2}\right]$, for $-10x10$, where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to_________. Solution: We know $[x]$ discontinuous for $x \in Z$ $f(x)=x\left[\frac{x}{2}\right]$ may be discontinuous where $\frac{x}{2}$ is an integer. So, points of discontinuity are, $x=\pm 2, \pm 4, \pm 6, \pm 8$ and 0 but at $x=0$ $\lim _{x \rightarrow 0^{+}} f(x)=0=f(0)=\lim _{x \rightarrow 0^{-}} f(x)$ So, $f(x)$ wil...
Read More →Two bodies of the same mass are moving with the same speed,
Question: Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial speed. The angle between the initial velocities of the two bodies (in degree) is______ Solution: Momentum conservation along $x$ direction, $2 m v_{0} \cos \theta=2 m \frac{v_{0}}{2} \Rightarrow \cos \theta=\frac{1}{2}$ or $\theta=60^{\circ}$ Hence angle between the i...
Read More →The percentages of marks obtained by a student in six unit tests are given below.
Question: The percentages of marks obtained by a student in six unit tests are given below. A unit test is selected at random. What is the probability that the student gets more than 60% marks in the test? Solution: Total number of unit tests = 6Number of tests in which the student scored more than 60% marks = 2 LetE be the event that he got more than 60% marks in the unit tests.Then, required probability $=P(E)=\frac{\text { Number of unit tests in which he got more than } 60 \% \text { marks }...
Read More →If the function
Question: If the function $f(x)\left\{\begin{array}{ll}k_{1}(x-\pi)^{2}-1, x \leq \pi \\ k_{2} \cos x, x\pi\end{array}\right.$ is twice differentiable, then the ordered pair $\left(k_{1}, k_{2}\right)$ is equal to:(1) $\left(\frac{1}{2}, 1\right)$(2) $(1,0)$(3) $\left(\frac{1}{2},-1\right)$(4) $(1,1)$Correct Option: 1 Solution: $f(x)$ is differentiable then, $f(x)$ is also continuous. $\therefore \lim _{x \rightarrow \pi^{+}} f(x)=\lim _{x \rightarrow \pi^{-}} f(x)=f(\pi)$ $\Rightarrow-1=-K_{2} ...
Read More →In a survey of 200 ladies, it was found that 142 like coffee, while 58 dislike it.
Question: In a survey of 200 ladies, it was found that 142 like coffee, while 58 dislike it.Find the probability that a lady chosen at random(i) likes coffee(ii) dislikes coffee Solution: Total number of ladies = 200Number of ladies who like coffee = 142 Number of ladies who dislike coffee = 58 LetE1andE2be the events that the selected lady likes and dislikes coffee, respectively.Then, (i) $P($ selected lady likes coffee $)=P\left(E_{1}\right)=\frac{\text { Number of ladies who like coffee }}{\t...
Read More →The compound that has the largest
Question: The compound that has the largest $\mathrm{H}-\mathrm{M}-\mathrm{H}$ bond angle $(\mathrm{M}=\mathrm{N}, \mathrm{S}, \mathrm{C})$, is :$\mathrm{H}_{2} \mathrm{O}$$\mathrm{NH}_{3}$$\mathrm{H}_{2} \mathrm{~S}$$\mathrm{CH}_{4}$Correct Option: Solution: $\mathrm{H}_{2} \mathrm{O}-104.5^{\circ}\left(\mathrm{sp}^{3}\right.$ with 2 lone pair at $\left.\mathrm{O}\right)$ $\mathrm{NH}_{3}-107^{\circ}\left(s p^{3}\right.$ with 1 lone pair at $\left.\mathrm{N}\right)$ $\mathrm{CH}_{4}-109.5^{\cir...
Read More →A dice is thrown 300 times and the outcomes are noted as given below.
Question: A dice is thrown 300 times and the outcomes are noted as given below. When a dice is thrown at random, what is the probability of getting a(i) 3?(ii) 6?(iii) 5?(iv) 1? Solution: Total number of throws = 300In a random throw of a dice, let E1, E2, E3, E4, be the events of getting 3, 6, 5 and 1, respectively. Then (i) $P($ getting 3$)=P\left(E_{1}\right)=\frac{\text { Number of times } 3 \text { appears }}{\text { Total number of trials }}=\frac{54}{300}=0.18$ (ii) $P$ (getting 6 ) $=P\l...
Read More →A thin rod of mass 0.9 kg and length
Question: A thin rod of mass $0.9 \mathrm{~kg}$ and length $1 \mathrm{~m}$ is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of move $0.1 \mathrm{~kg}$ moving in a straight line with velocity $80 \mathrm{~m} / \mathrm{s}$ hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in $\mathrm{rad} / \mathrm{s}$ ) of the rod immediately after the collision will be _____ Solution: Using principal of conservation of ang...
Read More →The structure of
Question: The structure of $\mathrm{PCl}_{5}$ in the solid state is:tetrahedral $\left[\mathrm{PCl}_{4}\right]^{+}$and octahedral $\left[\mathrm{PCl}_{6}\right]^{-}$square planar $\left[\mathrm{PCl}_{4}\right]^{+}$and octahedral $\left[\mathrm{PCl}_{6}\right]^{-}$square pyramıdaltrigonal bipyramidalCorrect Option: 1 Solution:...
Read More →The function
Question: The function $f(x)=\left\{\begin{array}{l}\frac{\pi}{4}+\tan ^{-1} x,|x| \leq 1 \\ \frac{1}{2}(|x|-1),|x|1\end{array}\right.$ is : (1) continuous on $\mathbf{R}-\{1\}$ and differentiable on $\mathbf{R}-\{-1,1\} .$(2) both continuous and differentiable on $\mathbf{R}-\{1\}$.(3) continuous on $\mathbf{R}-\{-1\}$ and differentiable on $\mathbf{R}-\{-1,1\} .$(4) both continuous and differentiable on $\mathbf{R}-\{-1\}$.Correct Option: 1 Solution: $f(x)= \begin{cases}\frac{-x-1}{2}, x-1 \\ ...
Read More →Three coins are tossed 200 times and we get
Question: Three coins are tossed 200 times and we getthree heads: 39 times; two heads: 58 times;one head: 67 times; 0 head: 36 times.When three coins are tossed at random, what is the probability of getting(i) 3 heads?(ii) 1 head?(iii) 0 head?(iv) 2 heads? Solution: Total number of tosses = 200Number of times 3 heads appear = 39Number of times 2 heads appear = 58Number of times 1 head appears = 67Number of times 0 head appears = 36 In a random toss of three coins, letE1,E2,E3andE4 be the events ...
Read More →The potential energy curve for the
Question: The potential energy curve for the $\mathrm{H}_{2}$ molecule as a function of internuclear distance is :Correct Option: , 2 Solution: When two H-atoms come closer then initially due to attraction P.E. is -ve, which decreases more as atoms come closer and after reacting to a minimum value as repulsion starts dominating. So, P.E. increases then....
Read More →Two coins are tossed 400 times and we get
Question: Two coins are tossed 400 times and we gettwo heads: 112 times; one head: 160 times; 0 head: 128 times.When two coins are tossed at random, what is the probability of getting(i) 2 heads?(ii) 1 head?(iii) 0 head? Solution: Total number of tosses = 400Number of times 2 heads appear = 112Number of times 1 head appears = 160Number of times 0 head appears = 128 In a random toss of two coins, letE1,E2,E3be the events of getting 2 heads, 1 head and 0 head, respectively. Then, (i) $P($ getting ...
Read More →If a function $f(x)$ defined by
Question: If a function f(x) defined by $f(x)=\left\{\begin{array}{l}a e^{x}+b e^{-x},-1 \leq x1 \\ c x^{2}, 1 \leq x \leq 3 \\ a x^{2}+2 c x, 3x \leq 4\end{array}\right.$ be continuous for some $a, b, c \in \mathbf{R}$ and $f^{\prime}(0)+f^{\prime}(2)=e$, then the value of $a$ is :(1) $\frac{1}{e^{2}-3 e+13}$(2) $\frac{e}{e^{2}-3 e-13}$(3) $\frac{e}{e^{2}+3 e+13}$(4) $\frac{e}{e^{2}-3 e+13}$Correct Option: , 4 Solution: Since, function $f(x)$ is continuous at $x=1,3$ $\therefore f(1)=f\left(1^{...
Read More →A coin is tossed 500 times and we get
Question: A coin is tossed 500 times and we gethead: 285 times, tail: 215 times.When a coin is tossed at random, what is the probability of getting(i) a head?(ii) a tail? Solution: Total number of tosses = 500Number of heads = 285Number of tails = 215(i) Let E be the event of getting a head. $P($ getting a head $)=P(E)=\frac{\text { Number of heads coming up }}{\text { Total number of trials }}=\frac{285}{500}=0.57$ (ii) Let F be the event of getting a tail. $P$ (getting a tail) $=P(F)=\frac{\te...
Read More →A particle of mass
Question: A particle of mass $200 \mathrm{MeV} / \mathrm{c}^{2}$ collides with a hydrogen atom at rest. Soon after the collision the particle comes to rest, and the atom recoils and goes to its first excited state. The initial kinetic energy of the particle (in eV) is $\frac{\mathrm{N}}{4}$ The value of $\mathrm{N}$ is (Given the mass of the hydrogen atom to be $1 \mathrm{GeV} / \mathrm{c}^{2}$ ) Solution: From linear momentum conservation, $L_{i}=L_{f}$ $m V+0=0+5 m V^{\prime} \Rightarrow V^{\p...
Read More →The intermolecular potential energy for the molecules A
Question: The intermolecular potential energy for the molecules $A$, $B, C$ and $D$ given below suggests that : A-D has the shortest bond lengthA-A has the largest bond enthalpy$\mathrm{D}$ is more electronegative than other atomsA-B has the stiffest bondCorrect Option: , 4 Solution: A-B bond has highest intermolecular potential energy among the given molecules. Hence, it is strongest bond and has maximum bond enthalpy....
Read More →Let f
Question: Let $f: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{c}2 \sin \left(-\frac{2 x}{2}\right), \text { if } x-1 \\ \left|a x^{2}+x+b\right|, \text { if }-1 \leq x \leq 1 \\ \sin (\pi x) \quad \text { if } x1\end{array}\right.$ If $\mathrm{f}(\mathrm{x})$ is continuous on $\mathrm{R}$, then a $+\mathrm{b}$ equals :(1) 3(2) - 1(3) -3(4) 1Correct Option: , 2 Solution: If $f$ is continuous at $x=-1$, then $f\left(-1^{-}\right)=f(-1)$ $\Rightarrow 2=|a-1+b|$ $\Rightarrow|a+b-1|=2 \...
Read More →The median of the data arranged in ascending order 8, 9, 12, 18, (x + 2), (x + 4), 30, 31, 34, 39 is 24. The value of x is
Question: The median of the data arranged in ascending order 8, 9, 12, 18, (x+ 2), (x+ 4), 30, 31, 34, 39 is 24. The value ofxis(a) 22(b) 21(c) 20(d) 24 Solution: (b) 21The given data is in ascending order.Here,nis 10, which is an even number.Thus, we have: Median $=$ Mean of $\left(\frac{n}{2}\right)$ th $\\left(\frac{n}{2}+1\right)$ th observations $=\frac{1}{2}$ (5 th observation $+6$ th observation) $=\frac{1}{2}(x+2+x+4)=(x+3)$ $=24$ Also, $x+3=24$ $\Rightarrow x=21$...
Read More →A function f is defined on [-3,3] as
Question: A function $f$ is defined on $[-3,3]$ as $f(x)=\left\{\begin{array}{cc}\min \left\{|x|, 2-x^{2}\right\}, -2 \leq x \leq 2 \\ {[|x|]} , 2|x| \leq 3\end{array}\right.$ where $[\mathrm{x}]$ denotes the greatest integer $\leq \mathrm{x}$. The number of points, where $\mathrm{f}$ is not differentiable in $(-3,3)$ is Solution: Points of non-differentiability in $(-3,3)$ are at $x=-2,-1,0,1,2$. i.e. 5 points....
Read More →Blocks of masses m, 2 m, 4 m and 8 m are arranged in a line on a frictionless floor.
Question: Blocks of masses $m, 2 m, 4 m$ and $8 m$ are arranged in a line on a frictionless floor. Another block of mass $m$, moving with speed $v$ along the same line (see figure) collides with mass $m$ in perfectly inelastic manner. All the subsequent collisions are also perfectly inelastic. By the time the last block of mass $8 m$ starts moving the total energy loss is $p \%$ of the original energy. Value of ' $p$ ' is close to : (1) 77(2) 94(3) 37(4) 87Correct Option: , 2 Solution: (2) Accor...
Read More →Mode of the data 15, 17, 15, 19, 14, 18, 15, 14, 16, 15, 14, 20, 19, 14, 15 is
Question: Mode of the data 15, 17, 15, 19, 14, 18, 15, 14, 16, 15, 14, 20, 19, 14, 15 is(a) 14(b) 15(c) 16(d) 17 Solution: (b) 15Here, 14 occurs 4 times, 15 occurs 5 times, 16 occurs 1 time, 18 occurs 1 time, 19 occurs 1 time and 20 occurs 1 time. Therefore, the mode, which is the most occurring item, is 15....
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