There block A, B and C are lying on a smooth horizontal surface,
Question: There block A, B and C are lying on a smooth horizontal surface, as shown in the figure. A and B have equal masses, $\mathrm{m}$ while $\mathrm{C}$ has mass $\mathrm{M}$. Block $\mathrm{A}$ is given an inital speed $v$ towards B due to which it collides with B perfectly inelastically. The combined mass collides with $\mathrm{C}$, also perfectly inelastically $\frac{5}{6}$ th of the initial kinetic energy is lost in whole process. What is value of $\mathrm{M} / \mathrm{m}$ ? (1) 5(2) 2(...
Read More →The average S-F bond energy in
Question: The average S-F bond energy in $\mathrm{kJmol}^{-1}$ of $\mathrm{SF}_{6}$ is (Rounded off to the nearest integer)[Given : The values of standard enthalpy of formation of $\mathrm{SF}_{6}(\mathrm{~g}), \mathrm{S}(\mathrm{g})$ and $\mathrm{F}(\mathrm{g})$ are $-1100,275$ and $80 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively.] Solution: $(309)$ $\mathrm{SF}_{6}(\mathrm{~g}) \longrightarrow \mathrm{S}(\mathrm{g})+6 \mathrm{~F}(\mathrm{~g})$ $\Delta \mathrm{H}_{\text {reaction }}^{\circ}=6 ...
Read More →Let
Question: Let $I_{n}=\int_{1}^{e} x^{19}(\log |x|)^{n} d x$, where $n \in N$. If (20) $\mathrm{I}_{10}=\alpha \mathrm{I}_{9}+\beta \mathrm{I}_{8}$, for natural numbers $\alpha$ and $\beta$, then $\alpha-\beta$ equal to_________. Solution: Let $\overrightarrow{\mathrm{x}}=\lambda \overrightarrow{\mathrm{a}}+\mu \overrightarrow{\mathrm{b}} \quad(\lambda$ and $\mu$ are scalars $)$ $\overrightarrow{\mathrm{x}}=\hat{\mathrm{i}}(2 \lambda+\mu)+\hat{\mathrm{j}}(2 \mu-\lambda)+\hat{\mathrm{k}}(\lambda-\...
Read More →By what number should 1365 be divided to get 31 as quotient and 32 as remainder?
Question: By what number should 1365 be divided to get 31 as quotient and 32 as remainder? Solution: Given: Dividend = 1365, Quotient = 31, Remainder = 32Letthe divisor bex.Dividend = Divisor xQuotient + Remainder $1365=x \times 31+32$ $\Rightarrow \quad 1365-32=31 x$ $\Rightarrow \quad 1333=31 x$ $\Rightarrow \quad x=\frac{1333}{31}=43$ Hence, 1365 should be divided by 43 to get 31 as quotient and 32 as remainder....
Read More →If the integral
Question: If the integral $\int_{0}^{10} \frac{\left[\sin 2 \pi_{\mathrm{x}}\right]}{\mathrm{e}^{\mathrm{x}-[\mathrm{x}]}} \mathrm{d} \mathrm{x}=\alpha \mathrm{e}^{-1}+\beta \mathrm{e}^{-\frac{1}{2}}+\gamma$, where $\alpha, \beta, \gamma$ are integers and $[x]$ denotes the greatest integer less than or equal to $\mathrm{x}$, then the value of $\alpha+\beta+\gamma$ is equal to :(1) 0(2) 20(3) 25(4) 10Correct Option: 1, Solution: Let $I=\int_{0}^{10} \frac{[\sin 2 \pi x]}{e^{x-[x]}} d x=\int_{0}^{...
Read More →A number when divided by 61 gives 27 as quotient and 32 as remainder.
Question: A number when divided by 61 gives 27 as quotient and 32 as remainder.Find the number. Solution: We know, Dividend = Divisor $\times$ Quotient + Remainder Given: Divisor $=61$, Quotient $=27$, Remainder $=32$ Let the Dividend be $x$. $\therefore x=61 \times 27+32$ = 1679Hence, the required number is 1679....
Read More →Three particles of masses
Question: Three particles of masses $50 \mathrm{~g}, 100 \mathrm{~g}$ and $150 \mathrm{~g}$ are placed at the vertices of an equilateral triangle of side $1 \mathrm{~m}$ (as shown in the figure). The $(x, y)$ coordinates of the centre of mass will be : (1) $\left(\frac{\sqrt{3}}{4} m, \frac{5}{12} m\right)$(2) $\left(\frac{7}{12} m, \frac{\sqrt{3}}{8} m\right)$(3) $\left(\frac{7}{12} m, \frac{\sqrt{3}}{4} m\right)$(4) $\left(\frac{\sqrt{3}}{8} m, \frac{7}{12} m\right)$Correct Option: , 3 Solutio...
Read More →What do you mean by Euclid's division algorithm.
Question: What do you mean by Euclid's division algorithm. Solution: Euclid's division algorithm states that for any two positive integersaandb,there exist unique integersqandr,such thata = bq + r,where 0rb....
Read More →Let f
Question: Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $\mathrm{f}(\mathrm{x})=\mathrm{e}^{-\mathrm{x}} \sin \mathrm{x}$. If $\mathrm{F}:[0,1] \rightarrow \mathrm{R}$ is a differentiable function such that $\mathrm{F}(\mathrm{x})=\int_{0}^{\mathrm{x}} \mathrm{f}(\mathrm{t}) \mathrm{dt}$, then the value of $\int_{0}^{1}\left(\mathrm{~F}^{\prime}(\mathrm{x})+\mathrm{f}(\mathrm{x})\right) \mathrm{e}^{\mathrm{x}} \mathrm{dx}$ lies in the interval(1) $\left[\frac{327}{360}, \frac...
Read More →Two particles, of masses M and 2 M,
Question: Two particles, of masses $\mathrm{M}$ and $2 \mathrm{M}$, moving, as shown, with speeds of $10 \mathrm{~m} / \mathrm{s}$ and $5 \mathrm{~m} / \mathrm{s}$, collide elastically at the origin. After the collision, they move along the indicated directions with speeds $\mathrm{v}_{1}$ and $\mathrm{v}_{2}$, respectively. The values of $v_{1}$ and $v_{2}$ are nearly : (1) $6.5 \mathrm{~m} / \mathrm{s}$ and $6.3 \mathrm{~m} / \mathrm{s}$(2) $3.2 \mathrm{~m} / \mathrm{s}$ and $6.3 \mathrm{~m} /...
Read More →If $[.]$ represents the greatest integer function,
Question: If $[\cdot]$ represents the greatest integer function, then the value of $\left|\int_{0}^{\sqrt{\frac{\pi}{2}}}\left[\left[x^{2}\right]-\cos x\right] d x\right|$ is________ Solution: $I=\int_{0}^{\sqrt{\pi / 2}}\left(\left[x^{2}\right]+[-\cos x]\right) d x$ $=\int_{0}^{1} 0 d x+\int_{1}^{\sqrt{\pi / 2}} d x+\int_{0}^{\sqrt{\pi / 2}}(-1) d x$ $=\sqrt{\frac{\pi}{2}}-1-\sqrt{\frac{\pi}{2}}=-1$ $\Rightarrow|I|=1$...
Read More →Consider the integral
Question: Consider the integral $I=\int_{0}^{10} \frac{[x] e^{[x]}}{e^{x-1}} d x$ where $[\mathrm{x}]$ denotes the greatest integer less than or equal to $\mathrm{x}$. Then the value of $\mathrm{I}$ is equal to: (1) $9(\mathrm{e}-1)$(2) $45(\mathrm{e}+1)$(3) $45(\mathrm{e}-1)$(4) $9(\mathrm{e}+1)$Correct Option: , 3 Solution: $\mathrm{I}=\int_{0}^{10}[\mathrm{x}] \cdot \mathrm{e}^{[\mathrm{x}]-\mathrm{x}+1}$ $\mathrm{I}=\int_{0}^{1} 0 \mathrm{~d} \mathrm{x}+\int_{1}^{2} 1 \cdot \mathrm{e}^{2-\ma...
Read More →A homogeneous ideal gaseous reaction
Question: A homogeneous ideal gaseous reaction $\mathrm{AB}_{2(\mathrm{~g})} \rightleftharpoons \mathrm{A}_{(\mathrm{g})}+2 \mathrm{~B}_{(\mathrm{g})}$ is carried out in a 25 litre flask at $27^{\circ} \mathrm{C}$. The initial amount of $\mathrm{AB}_{2}$ was 1 mole and the equilibrium pressure was $1.9$ atm. The value of $K_{p}$ is $x \times 10^{-2}$. The value of $x$ is ______________$\quad\left[R=0.08206 \mathrm{dm}^{3} \mathrm{~atm} K^{-1} \mathrm{~mol}^{-1}\right]$ Solution: (74)...
Read More →Let f:
Question: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x)+f(x+1)=2$, for all $x \in \mathbb{R}$. If $\mathrm{I}_{1}=\int_{0}^{8} f(\mathrm{x}) \mathrm{d} \mathrm{x}$ and $\mathrm{I}_{2}=\int_{-1}^{3} f(\mathrm{x}) \mathrm{d} \mathrm{x}$, then the value of $\mathrm{I}_{1}+2 \mathrm{I}_{2}$ is equal to______. Solution: $f(x)+f(x+1)=2$ $\Rightarrow f(x)$ is periodic with period $=2$ $\mathrm{I}_{1}=\int_{0}^{8} f(\mathrm{x}) \mathrm{d} \mathrm{x}=4 \int_{0}^{2} f...
Read More →A wedge of mass M=4 m lies on a frictionless plane.
Question: A wedge of mass $M=4 \mathrm{~m}$ lies on a frictionless plane. $A$ particle of mass $m$ approaches the wedge with speed $v$. There is no friction between the particle and the plane or between the particle and the wedge. The maximum height climbed by the particle on the wedge is given by:(1) $\frac{v^{2}}{g}$(2) $\frac{2 v^{2}}{7 g}$(3) $\frac{2 v^{2}}{5 g}$(4) $\frac{v^{2}}{2 g}$Correct Option: , 3 Solution: (3) $m v=(m+M) V^{\prime}$ or $v=\frac{m v}{m+M}=\frac{m v}{m+4 m}=\frac{v}{5...
Read More →The ionization enthalpy of
Question: The ionization enthalpy of $\mathrm{Na}^{+}$formation from $\mathrm{Na}_{(g)}$ is $495.8 \mathrm{~kJ} \mathrm{~mol}^{-1}$, while the electron gain enthalpy of Br is $-325.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Given the lattice enthalpy of $\mathrm{NaBr}$ is $-728.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$. The energy for the formation of $\mathrm{NaBr}$ ionic solid is (-)____________ $\times 10^{-1} \mathrm{~kJ} \mathrm{~mol}^{-1}$ Solution: $(\mathbf{5 5 7 6})$...
Read More →The ionization enthalpy of
Question: The ionization enthalpy of $\mathrm{Na}^{+}$formation from $\mathrm{Na}_{(g)}$ is $495.8 \mathrm{~kJ} \mathrm{~mol}^{-1}$, while the electron gain enthalpy of Br is $-s 25.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Given the lattice enthalpy of $\mathrm{NaBr}$ is $-728.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$. The energy for the formation of $\mathrm{NaBr}$ ionic solid is $(-) ____________\quad \times 10^{-1} \mathrm{~kJ} \mathrm{~mol}^{-1}$ Solution: $(\mathbf{5 5 7 6})$...
Read More →Let f:(0,2)
Question: Let $f:(0,2) \rightarrow \mathbb{R}$ be defined as $f(\mathrm{x})=\log _{2}\left(1+\tan \left(\frac{\pi \mathrm{x}}{4}\right)\right)$ Then, $\lim _{n \rightarrow \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\ldots+f(1)\right)$ is equal to_________. Solution: $\mathrm{E}=2 \lim _{\mathrm{n} \rightarrow \infty} \sum_{\mathrm{r}-1}^{\mathrm{n}} \frac{1}{\mathrm{n}} \mathrm{f}\left(\frac{\mathrm{r}}{\mathrm{n}}\right)$ $\mathrm{E}=\frac{2}{\ell_{\mathrm{n} 2...
Read More →A particle of mass ' m ' is moving with speed ' 2 v '
Question: A particle of mass ' $m$ ' is moving with speed ' $2 v$ ' and collides with a mass ' $2 m$ ' moving with speed ' $v$ ' in the same direction. After collision, the first mass is stopped completely while the second one splits into two particles each of mass ' $m$ ', which move at angle $45^{\circ}$ with respect to the original direction. The speed of each of the moving particle will be:(1) $\sqrt{2} v$(2) $2 \sqrt{2} v$(3) $v /(2 \sqrt{2})$(4) $v / \sqrt{2}$Correct Option: , 2 Solution: ...
Read More →Solve the following
Question: For the reaction $\mathrm{A}_{(g)} \rightarrow \mathrm{B}_{(\mathrm{g})}$, the value of the equilibrium constant at $300 \mathrm{~K}$ and $1 \mathrm{~atm}$ is equal to $100.0$. The value of $\Delta_{r} G$ for the reaction at $300 \mathrm{~K}$ and 1 atm in $\mathrm{Jmol}^{-1}$ is $-\mathrm{xR}$, where $\mathrm{x}$ is__________________. (Rounded off to the nearest integer) $\left[\mathrm{R}=8.31 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right.$ and $\left.\ln 10=2.3\right]$ Solutio...
Read More →A body of mass
Question: A body of mass $2 \mathrm{~kg}$ makes an elastic collision with a second body at rest and continues to move in the original direction but with one fourth of its original speed. What is the mass of the second body?(1) $1.0 \mathrm{~kg}$(2) $1.5 \mathrm{~kg}$(3) $1.8 \mathrm{~kg}$(4) $1.2 \mathrm{~kg}$Correct Option: , 4 Solution: (4) For head on elastic collision we have $\mathrm{V}_{1}=\frac{\left(\mathrm{m}_{1}-\mathrm{m}_{2}\right) \mathrm{u}_{1}}{\mathrm{~m}_{1}+\mathrm{m}_{2}}+\fra...
Read More →Solve the following
Question: At $1990 \mathrm{~K}$ and 1 atrm pressure, there are equal number of $\mathrm{Cl}_{2}$ molecules and $\mathrm{Cl}$ atoms in the reaction mixture. The value of $\mathrm{K}_{\mathrm{p}}$ for the reaction $\mathrm{Cl}_{2(\mathrm{~g})}=2 \mathrm{Cl}_{(\mathrm{g})}$ under the above conditions is $\mathrm{x} \times 10^{-1}$. The value of $x$ is ____________________. (Rounded off to the nearest integer) Solution: (5) $\mathrm{Cl}_{2} \rightleftharpoons 2 \mathrm{Cl}$ $\mathrm{K}_{\mathrm{p}}=...
Read More →A uniform rectangular thin sheet
Question: A uniform rectangular thin sheet $\mathrm{ABCD}$ of mass $\mathrm{M}$ has length a and breadth $b$, as shown in the figure. If the shaded portion HBGO is cut-off, the coordinates of the centre of mass of the remaining portion will be : (1) $\left(\frac{3 a}{4}, \frac{3 b}{4}\right)$(2) $\left(\frac{5 a}{3}, \frac{5 b}{3}\right)$(3) $\left(\frac{2 a}{3}, \frac{2 b}{3}\right)$(4) $\left(\frac{5 a}{12}, \frac{5 b}{12}\right)$Correct Option: , 4 Solution: (4) With respect to point $\theta$...
Read More →Two coins are tossed simultaneously 600 times to get 2 heads : 234 times, 1 head : 206 times, 0 head : 160 times.
Question: Two coins are tossed simultaneously 600 times to get 2 heads : 234 times, 1 head : 206 times, 0 head : 160 times.If two coins are tossed at random, what is the probability of getting at least one head? (a) $\frac{103}{300}$ (b) $\frac{39}{100}$ (C) $\frac{11}{15}$ (d) $\frac{4}{15}$ Solution: Number of times two coins are tossed simultaneously = 600 Number of times of getting at least one head = Number of times of getting 1 head + Number of times of getting 2 heads = 206 + 234 = 440 $\...
Read More →The table given below shows the month of birth of 36 students of a class.
Question: The table given below shows the month of birth of 36 students of a class. A student is chosen at random from the class. What is the probability that the chosen student was born in October? (a) $\frac{1}{3}$ (b) $\frac{2}{3}$ (c) $\frac{1}{4}$ (d) $\frac{1}{12}$ Solution: (d) $\frac{1}{12}$ Explanation:Total number of students = 36Number of students born in October = 3 LetEbe the event that the chosen student was born in October. Then, $P(E)=\frac{\text { Number of students born in Octo...
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