A charged particle carrying charge
Question: A charged particle carrying charge $1 \mu \mathrm{C}$ is moving with velocity $(2 \hat{i}+3 \hat{j}+4 \hat{k}) \mathrm{ms}^{-1}$. If an external magnetic field of $(5 \hat{i}+3 \hat{j}-6 \hat{k}) \times 10^{-3} \mathrm{~T}$ exists in the region where the particle is moving then the force on the particle is $\vec{F} \times 10^{-9}$ N. The vector $\vec{F}$ is :(1) $-0.30 \hat{i}+0.32 \hat{j}-0.09 \hat{k}$(2) $-30 \hat{i}+32 \hat{j}-9 \hat{k}$(3) $-300 \hat{i}+320 \hat{j}-90 \hat{k}$(4) $...
Read More →Let A and B be two events such that the
Question: Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $A$ or $B$ occurs is $\frac{1}{2}$, then the probability of both of them occur together is:(1) $0.02$(2) $0.20$(3) $0.01$(4) $0.10$Correct Option: , 4 Solution: $P($ exactly one $)=\frac{2}{5}$ $\Rightarrow \quad P(A)+P(B)-2 P(A \cap B)=\frac{2}{5}$ $P(A$ or $B)=P(A \cup B)=\frac{1}{2}$ $\Rightarrow P(A)+P(B)-P(A \cap B)=\frac{1}{2}$ $\therefore \quad P(A \c...
Read More →The decreasing order of reactivity of the following
Question: The decreasing order of reactivity of the following compounds towards nucleophilic substitution $\left(\mathrm{S}_{\mathrm{N}} 2\right)$ is : (II) $$ (III) $$ (I) $$ (IV)(II) $$ (III) $$ (IV) $$ (I)(III) $$ (II) $$ (IV) $$ (I)(IV) $$ (II) $$ (III) $$ (I)Correct Option: , 2 Solution: $\mathrm{S}_{\mathrm{N}} 2$ reactions depend upon $-\mathrm{I}$ and $-\mathrm{M}$ effect on substrate. On increasing - I and -M effect, rate of $\mathrm{S}_{\mathrm{N}} 2$ reaction will increase....
Read More →Let A and B be two independent events such that
Question: Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then, which of the following is TRUE ?(1) $P(A / B)=\frac{2}{3}$(2) $P\left(A / B^{\prime}\right)=\frac{1}{3}$(3) $P\left(A^{\prime} / B^{\prime}\right)=\frac{1}{3}$(4) $P(A /(A \cup B))=\frac{1}{4}$Correct Option: , 2 Solution: $\mathrm{A}$ and $\mathrm{B}$ are independent events. So, $P\left(\frac{A}{B^{\prime}}\right)=\frac{P\left(A \cap B^{\prime}\right)}{P\left(B^{\prime}\right)}=\frac{\...
Read More →Very-Short-Answer Questions
Question: Very-Short-Answer Questions Find the values of $k$ for which the quadratic equation $9 x^{2}-3 k x+k=0$ has equal roots. Solution: It is given that the quadratic equation $9 x^{2}-3 k x+k=0$ has equal roots. $\therefore D=0$ $\Rightarrow(-3 k)^{2}-4 \times 9 \times k=0$ $\Rightarrow 9 k^{2}-36 k=0$ $\Rightarrow 9 k(k-4)=0$ $\Rightarrow k=0$ or $k-4=0$ $\Rightarrow k=0$ or $k=4$ Hence, 0 and 4 are the required values ofk....
Read More →Magnitude of magnetic field (in SI units)
Question: Magnitude of magnetic field (in SI units) at the centre of a hexagonal shape coil of side $10 \mathrm{~cm}, 50$ turns and carrying current $I$ (Ampere) in units of $\frac{\mu_{0} I}{\pi}$ is :(1) $250 \sqrt{3}$(2) $50 \sqrt{3}$(3) $500 \sqrt{3}$(4) $5 \sqrt{3}$Correct Option: , 3 Solution: Magnetic field due to one side of hexagon $B=\frac{\mu_{0} I}{4 \pi \frac{\sqrt{3} a}{2}}\left(\sin 30^{\circ}+\sin 30^{\circ}\right)$ $\Rightarrow B=\frac{\mu_{0} I}{2 \sqrt{3} a}\left(\frac{1}{2}+\...
Read More →In a workshop, there are five machines and
Question: In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$. If the probability that at most two machines will be out of service on the same day is $\left(\frac{3}{4}\right)^{3} k$, then $k$ is equal to:(1) $\frac{17}{8}$(2) $\frac{17}{4}$(3) $\frac{17}{2}$(4) 4Correct Option: 1 Solution: Required probability $=$ when no machine has fault $+$ when only one machine has fault $+$ when only two machines have fault. $={ }^{5}...
Read More →The product formed in the first step of the reaction of
Question: The product formed in the first step of the reaction of with excess $\mathrm{Mg} / \mathrm{Et}_{2} \mathrm{O}\left(\mathrm{Et}=\mathrm{C}_{2} \mathrm{H}_{5}\right)$ is :Correct Option: , 3 Solution:...
Read More →Ap unbiased coin is tossed 5 times.
Question: Ap unbiased coin is tossed 5 times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3,4,5$, otherwise $X$ takes the value $-1$. Then the expected value of $X$, is:(1) $\frac{3}{16}$(2) $\frac{1}{8}$(3) $-\frac{3}{16}$(4) $-\frac{1}{8}$Correct Option: , 2 Solution: $k=$ No. of times head occur consecutively Now expectation $=\sum x P(k)=(-1) \times \frac{1}{32}+(-1) \times \frac{12}{32}+(-1) \times \frac{11}{32}$ $+3 \times \frac{5}{3...
Read More →Very-Short-Answer Questions
Question: Very-Short-Answer Questions Find the values of $k$ so that the quadratic equation $x^{2}-4 k x+k=0$ has equal roots. Solution: It is given that the quadratic equation $x^{2}-4 k x+k=0$ has equal roots. $\therefore D=0$ $\Rightarrow(-4 k)^{2}-4 \times 1 \times k=0$ $\Rightarrow 16 k^{2}-4 k=0$ $\Rightarrow 4 k(4 k-1)=0$ $\Rightarrow k=0$ or $4 k-1=0$ $\Rightarrow k=0$ or $k=\frac{1}{4}$ Hence, 0 and $\frac{1}{4}$ are the required values of $k$....
Read More →An elliptical loop having resistance R, of semi major axis a,
Question: An elliptical loop having resistance $R$, of semi major axis $a$, and semi minor axis $b$ is placed in a magnetic field as shown in the figure. If the loop is rotated about the $x$-axis with angular frequency $\omega$, the average power loss in the loop due to Joule heating is : (1) $\frac{\pi^{2} a^{2} b^{2} B^{2} \omega^{2}}{2 R}$(2) zero(3) $\frac{\pi a b B \omega}{R}$(4) $\frac{\pi^{2} a^{2} b^{2} B^{2} \omega^{2}}{R}$Correct Option: 1 Solution: (1) As we know, emf $\varepsilon=N A...
Read More →The above reaction requires which of the following
Question: The above reaction requires which of the following reaction conditions? $573 \mathrm{~K}, \mathrm{Cu}, 300 \mathrm{~atm}$$623 \mathrm{~K}, \mathrm{Cu}, 300 \mathrm{~atm}$$573 \mathrm{~K}, 300 \mathrm{~atm}$$623 \mathrm{~K}, 300 \mathrm{~atm}$Correct Option: , 4 Solution: Temperature $=623 \mathrm{~K}$ Pressure $=300 \mathrm{~atm}$...
Read More →Very-Short-Answer Questions
Question: Very-Short-Answer Questions If the roots of the quadratic equation $p x(x-2)+6=0$ are equal, find the value of $p$. Solution: It is given that the roots of the quadratic equation $p x^{2}-2 p x+6=0$ are equal. $\therefore D=0$ $\Rightarrow(-2 p)^{2}-4 \times p \times 6=0$ $\Rightarrow 4 p^{2}-24 p=0$ $\Rightarrow 4 p(p-6)=0$ $\Rightarrow p=0$ or $p-6=0$ $\Rightarrow p=0$ or $p=6$ Forp= 0, we get 6 = 0, which is not true.p 0Hence, the value ofpis 6....
Read More →The probabilities of three events
Question: The probabilities of three events $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are given by $P(A)=0.6, P(B)=0.4$ and $P(C)=0.5$. If $P(A \cup B)=0.8$, $\mathrm{P}(\mathrm{A} \cap \mathrm{C})=0.3, \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0.2, \mathrm{P}(\mathrm{B} \cap \mathrm{C})=\beta$ and $\mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=\alpha$, where $0.85 \leq \alpha \leq 0.95$, then $\beta$ lies in the interval:(1) $[0.35,0.36]$(2) $[0.25,0.35]$(3) $[0.20,0.25]$(...
Read More →Very-Short-Answer Questions
Question: Very-Short-Answer Questions If one root of the quadratic equation $3 x^{2}-10 x+k=0$ is reciprocal of the other, find the value of $k$. Solution: Let $\alpha$ and $\beta$ be the roots of the equation $3 x^{2}-10 x+k=0$. $\therefore \alpha=\frac{1}{\beta}$ (Given) $\Rightarrow \alpha \beta=1$ $\Rightarrow \frac{k}{3}=1 \quad$ (Product of the roots $=\frac{c}{a}$ ) $\Rightarrow k=3$ Hence, the value ofkis 3....
Read More →A wire carrying current I is bent in the shape A B C D E F A as shown,
Question: A wire carrying current $I$ is bent in the shape $A B C D E F A$ as shown, where rectangle $A B C D A$ and $A D E F A$ are perpendicular to each other. If the sides of the rectangles are of lengths $a$ and $b$, then the magnitude and direction of magnetic moment of the loop $A B C D E F A$ is : (1) $a b I$, along $\left(\frac{\hat{j}}{\sqrt{2}}+\frac{\hat{k}}{\sqrt{2}}\right)$(2) $\sqrt{2} a b I$, along $\left(\frac{\hat{j}}{\sqrt{2}}+\frac{\hat{k}}{\sqrt{2}}\right)$(3) $\sqrt{2} a b I...
Read More →Out of 11 consecutive natural numbers
Question: Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is:(1) $\frac{15}{101}$(2) $\frac{5}{101}$(3) $\frac{5}{33}$(4) $\frac{10}{99}$Correct Option: , 3 Solution: For an A.P. $2 b=a+c$ (even), so both $a$ and $c$ even numbers or odd numbers from given numbers and $b$ number will be fixed automatically. Required probability $=\frac{{ }^{6} C_{2}+{ }^{5} C_{2}}{{ }^{1...
Read More →Ammonolysis of Alkyl halides followed by the treatment with
Question: Ammonolysis of Alkyl halides followed by the treatment with $\mathrm{NaOH}$ solution can be used to prepare primary, secondary and tertiary amines. The purpose of $\mathrm{NaOH}$ in the reaction is:to remove basic impuritiesto activate $\mathrm{NH}_{3}$ used in the reactionto remove acidic impuritiesto increase the reactivity of alkyl halideCorrect Option: , 3 Solution:...
Read More →Very-Short-Answer Questions
Question: Very-Short-Answer Questions If one zero of the polynomial $x^{2}-4 x+1$ is $(2+\sqrt{3})$, write the other zero. Solution: Let the other zero of the given polynomial be $\alpha$. Now, Sum of the zeroes of the given polynomial $=\frac{-(-4)}{1}=4$ $\therefore \alpha+(2+\sqrt{3})=4$ $\Rightarrow \alpha=4-2-\sqrt{3}=2-\sqrt{3}$ Hence, the other zero of the given polynomial is $(2-\sqrt{3})$....
Read More →In a bombing attack, there is 50 % chance
Question: In a bombing attack, there is $50 \%$ chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least $99 \%$ chance of completely destroying the target, is________. Solution: Let ' $n$ ' bombs are required, then $1-{ }^{n} C_{1} \cdot\left(\frac{1}{2}\right)^{1}\left(\frac{1}{2}\right)^{n-1}-{ }^{n} C_{0}\left(\frac{1}{2}\right)^{0}\left(\frac{1}...
Read More →Very-Short-Answer Questions
Question: Very-Short-Answer Questions If 1 is a root of the equation $a y^{2}+a y+3=0$ and $y^{2}+y+b=0$ then find the value of $a b$. Solution: It is given that $y=1$ is a root of the equation $a y^{2}+a y+3=0$. $\therefore a \times(1)^{2}+a \times 1+3=0$ $\Rightarrow a+a+3=0$ $\Rightarrow 2 a+3=0$ $\Rightarrow a=-\frac{3}{2}$ Also, $y=1$ is a root of the equation $y^{2}+y+b=0$. $\therefore(1)^{2}+1+b=0$ $\Rightarrow 1+1+b=0$ $\Rightarrow b+2=0$ $\Rightarrow b=-2$ $\therefore a b=\left(-\frac{3...
Read More →Solve the following
Question: The product "A" and "B" formed in above reactions areCorrect Option: , 3 Solution:...
Read More →Four fair dice are thrown independently
Question: Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is______. Solution: Probability of getting at least two 3 's or 5 's in one trial $={ }^{4} C_{2}\left(\frac{2}{6}\right)^{2}\left(\frac{4}{6}\right)^{2}+{ }^{4} C_{3}\left(\frac{2}{6}\right)^{3}\left(\frac{4}{6}\right)+{ }^{4} C_{4}\left(\frac{2}{6}\right)^{4}$ $=\frac{33}{3^{4}}=\frac{11}{27}$ $E(x)=n p=27\left(\frac{11}{27}\right)=11 .$...
Read More →The figure shows a region of length ' l ' with a uniform magnetic field of 0.3 T
Question: The figure shows a region of length ' $l$ ' with a uniform magnetic field of $0.3 \mathrm{~T}$ in it and a proton entering the region with velocity $4 \times 10^{5} \mathrm{~ms}^{-1}$ making an angle $60^{\circ}$ with the field. If the proton completes 10 revolution by the time it cross the region shown, ' $l$ ' is close to (mass of proton $=1.67 \times 10^{-27} \mathrm{~kg}$, charge of the proton $=1.6 \times 10^{-19} \mathrm{C}$ ) (1) $0.11 \mathrm{~m}$(2) $0.88 \mathrm{~m}$(3) $0.44...
Read More →In a game two players A and B take turns
Question: In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is :(1) $\frac{5}{31}$(2) $\frac{31}{61}$(3) $\frac{5}{6}$(4) $\frac{30}{61}...
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