A train covers a distance of 480 km at a uniform speed.
Question: A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/h less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train. Solution: Let the usual speed of the train bexkm/h. Reduced speed of the train = (x 8) km/hTotal distance to be covered = 480 km Time taken by the train to cover the distance at usual speed $=\frac{480}{x} \mathrm{~h}$ $\left(\right.$ Time $\left.=\frac{\text { Distance }}{\text { Speed }}\right)$ ...
Read More →The following system of linear equations
Question: The following system of linear equations $7 x+6 y-2 z=0$ $3 x+4 y+2 z=0$ $x-2 y-6 z=0$, has (1) infinitely many solutions, $(x, y, z)$ satisfying $y=2 z$.(2) no solution.(3) infinitely many solutions, $(x, y, z)$ satisfying $x=2 z$.(4) only the trivial solution.Correct Option: , 4 Solution: The given system of linear equations $7 x+6 y-2 z=0$...(i) $3 x+4 y+2 z=0$...(ii) $x-2 y-6 z=0$...(iii) Now, determinant of coefficient matrix $\Delta=\left|\begin{array}{ccc}7 6 -2 \\ 3 4 2 \\ 1 -2...
Read More →The correct pair(s) of the ambident nucleophiles is (are):
Question: The correct pair(s) of the ambident nucleophiles is (are): (A) $\mathrm{AgCN} / \mathrm{KCN}$ (B) RCOOAg/RCOOK (C) $\mathrm{AgNO}_{2} / \mathrm{KNO}_{2}$ (D) $\mathrm{AgI} / \mathrm{KI}$(B) and (C) only(A) only(A) and (C) only(B) onlyCorrect Option: , 3 Solution: Ambident nucleophile (A) $\mathrm{KCN} \ \mathrm{AgCN}$ (C) $\mathrm{AgNO}, \ \mathrm{KNO}_{2}$...
Read More →In a certain themodynamical process,
Question: In a certain themodynamical process, the pressure of gas depends on its volume as $\mathrm{kV}^{3}$. The work done when the temperature changes from $100^{\circ} \mathrm{C}$ to $300^{\circ} \mathrm{C}$ will be______ $\mathrm{nR}$, where $\mathrm{n}$ denotes number of moles of a gas. Solution: $(50)$ $P=k v^{3}$ $p v^{-3}=k$ $x=-3$ $w=\frac{n R\left(T_{1}-T_{2}\right)}{x-1}$ $=\frac{n R(100-300)}{-3-1}$ $=\frac{n R(-200)}{-4}$ $=50 n R$...
Read More →Mesityl oxide is a common name of :
Question: Mesityl oxide is a common name of :2,4 -Dimethyl pentan- 3 -one3-Methyl cyclohexane carbaldehyde2 -Methyl cyclohexanone4-Methyl pent-3-en-2-oneCorrect Option: , 4 Solution: IUPAC [4-Methylpent-3-en-2-one]...
Read More →The root mean square speed of molecules of a given mass
Question: The root mean square speed of molecules of a given mass of a gas at $27^{\circ} \mathrm{C}$ and 1 atmosphere pressure is $200 \mathrm{~ms}^{-1}$. The root mean square speed of molecules of the gas at $127^{\circ} \mathrm{C}$ and 2 atmosphere pressure is $\frac{\mathrm{x}}{\sqrt{3}} \mathrm{~ms}^{-1}$. The value of $\mathrm{x}$ will be Solution: $(400)$ $V_{\text {rms }} \sqrt{\frac{3 \mathrm{RT}_{1}}{\mathrm{M}_{0}}}$ $200=\sqrt{\frac{3 R \times 300}{M_{0}}}$ Also, $\frac{x}{\sqrt{3}}=...
Read More →If the matrices
Question: If the matrices $\mathrm{A}=\left[\begin{array}{ccc}1 1 2 \\ 1 3 4 \\ 1 -1 3\end{array}\right], \mathrm{B}=\operatorname{adj} \mathrm{A}$ and $\mathrm{C}=3 \mathrm{~A}$, then $\frac{|\operatorname{adj} \mathrm{B}|}{|\mathrm{C}|}$ is equal to :(1) 8(2) 16(3) 72(4) 2Correct Option: 1 Solution: $|A|=\left|\begin{array}{ccc}1 1 2 \\ 1 3 4 \\ 1 -1 3\end{array}\right|=((9+4)-1(3-4)+2(-1-3))$ $=13+1-8=6$ $|\operatorname{adj} B|=|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(n-1)^{2}}=|A|^{4...
Read More →Which of the following is an aromatic compound?
Question: Which of the following is an aromatic compound?Correct Option: 1 Solution:...
Read More →The system of linear equations
Question: The system of linear equations $\lambda x+2 y+2 z=5$ $2 \lambda x+3 y+5 z=8$ $4 x+\lambda y+6 z=10$ has:(1) no solution when $\lambda=8$(2) a unique solution when $\lambda=-8$(3) no solution when $\lambda=2$(4) infinitely many solutions when $\lambda=2$Correct Option: , 3 Solution: $D=\left|\begin{array}{ccc}\lambda 2 2 \\ 2 \lambda 3 5 \\ 4 \lambda 6\end{array}\right|$ $D=\lambda^{2}+6 \lambda-16$ $D=(\lambda+8)(2-\lambda)$ For no solutions, $D=0$ $\Rightarrow \lambda=-8,2$ when $\lam...
Read More →In chromotography technique,
Question: In chromotography technique, the purification of compound is independent of :Mobility or flow of solvent systemSolubility of the compoundLength of the column or TLC PlatePhysical state of the pure compound OfficialCorrect Option: , 4 Solution: In chromotography technique, the purification of a compound is independent of the physical state of the pure compound....
Read More →Among the following, the aromatic compounds are :
Question: Among the following, the aromatic compounds are : Choose the correct answer from the following options :(A) and (B) only(B) and (C) only(B), (C) and (D) only$(\mathrm{A}),(\mathrm{B})$ and $(\mathrm{C})$ onlyCorrect Option: , 2 Solution: (A) Non-Aromatic (B) Aromatic (C) Aromatic (D) Anti-Aromatic...
Read More →The root mean square speed of molecules of a given mass of a gas
Question: The root mean square speed of molecules of a given mass of a gas at $27^{\circ} \mathrm{C}$ and 1 atmosphere pressure is $200 \mathrm{~ms}^{-1}$. The root mean square speed of molecules of the gas at $127^{\circ} \mathrm{C}$ and 2 atmosphere pressure is $\frac{x}{\sqrt{3}} m s^{-1}$. The value of $x$ will be Solution: (400) $V_{\mathrm{rms}} \sqrt{\frac{3 \mathrm{RT}_{1}}{\mathrm{M}_{0}}}$ $200=\sqrt{\frac{3 R \times 300}{M_{0}}}$ Also, $\frac{x}{\sqrt{3}}=\sqrt{\frac{3 R \times 400}{M...
Read More →If one mole of an ideal gas
Question: If one mole of an ideal gas at $\left(P_{1}, V_{1}\right)$ is allowed to expand reversibly and isothemally $(A$ to B) its pressure is reduced to one-half of the original pressure (see figure). This is followed by a constant volume cooling till its pressure is reduced to one-fourth of the initial value $(\mathrm{B} \rightarrow \mathrm{C})$. Then it is restored to its initial state by a reversible adiabutic compression ( $C$ to A). The net workdone by the gas is equal to : (1) 0$(2)-\fra...
Read More →Prove the following
Question: If $\mathrm{A}=\left(\begin{array}{ll}2 2 \\ 9 4\end{array}\right)$ and $\mathrm{I}=\left(\begin{array}{ll}1 0 \\ 0 1\end{array}\right)$, then $10 \mathrm{~A}^{-1}$ is equal to:(1) $A-4 I$(2) $6 I-A$(3) $A-6 I$(4) $4 I-A$Correct Option: , 3 Solution: Characteristics equation of matrix ' $A$ ' is $|A-\lambda I|=0$ $\left|\begin{array}{cc}2-\lambda 2 \\ 9 4-\lambda\end{array}\right|=0 \Rightarrow \lambda^{2}-6 \lambda-10=0$ $\therefore \quad A^{2}-6 A-10 I=0$ $\Rightarrow A^{-1}\left(A^{...
Read More →While boarding an aeroplane, a passenger got hurt.
Question: While boarding an aeroplane, a passenger got hurt. The pilot showing promptness and concern, made arrangements to hospitalise the injured and so the plane started late by 30 minutes. To reach the destination, 1500 km away, in time, the pilot increased the speed by 100 km/hour.Find the original speed of the plane.Do you appreciate the values shown by the pilot, namely promptness in providing help to the injured and his efforts to reach in time? Solution: Let the original speed of the pl...
Read More →The number of all
Question: The number of all $3 \times 3$ matrices $A$, with enteries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $A A^{T}$ is 3, is________. Solution: Let $\mathrm{A}=\left[a_{\mathrm{ii}}\right]_{3 \times 3}$ It is given that sum of diagonal elements of $\mathrm{AA}^{\mathrm{T}}$ is 3 i.e., $\operatorname{tr}\left(\mathrm{AA}^{\mathrm{T}}\right)=3$ $a_{11}^{2}+a_{12}^{2}+a_{13}^{2}+a_{21}^{2}+\ldots . .+a_{33}^{2}=3$ Possible cases are $\left.\begin{array}{ll}0,0,0,0...
Read More →For which of the following ordered pairs
Question: For which of the following ordered pairs $(\mu, \delta)$, the system of linear equations $x+2 y+3 z=1$ $3 x+4 y+5 z=\mu$ $4 x+4 y+4 z=\delta$ is inconsistent?(1) $(4,3)$(2) $(4,6)$(3) $(1,0)$(4) $(3,4)$Correct Option: 1 Solution: From the given linear equation, we get $D=\left|\begin{array}{lll}1 2 3 \\ 3 4 5 \\ 4 4 4\end{array}\right|\left(R_{3} \rightarrow R_{3}-2 R_{2}+3 R_{3}\right)$ $=\left|\begin{array}{lll}1 2 3 \\ 3 4 5 \\ 0 0 0\end{array}\right|=0$ Now, let $P_{3}=4 x+4 y+4 z-...
Read More →On the basis of kinetic theory of gases, the gas exerts pressure because its molecules:
Question: On the basis of kinetic theory of gases, the gas exerts pressure because its molecules:(1) suffer change in momentum when impinge on the walls of container:(2) continuously stick to the walls of container.(3) continuously lose their energy till it reaches wall.(4) are attracted by the walls of container.Correct Option: 3, Solution: (1) On the basis of kinetic theory of gases, the gas pressure is due to the molecules suffering change in momentum when impinge on the walls of container....
Read More →A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed
Question: A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed which is 20 km per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck. Solution: Let the first speed of the truck bexkm/h. $\therefore$ Time taken to cover $150 \mathrm{~km}=\frac{150}{x} \mathrm{~h} \quad$ (Time $\left.=\frac{\text { Distance }}{\text { Speed }}\right)$ New speed of the truck =(x+ 20) km/h $...
Read More →If the system of linear equations,
Question: If the system of linear equations, $x+y+z=6$ $x+2 y+3 z=10$ $3 x+2 y+\lambda z=\mu$ has more than two solutions, then $\mu-\lambda^{2}$ is equal to_________. Solution: $x+y+z=6$.....(i) $x+2 y+3 z=10$...(ii) $3 x+2 y+\lambda z=\mu$...(iii) From (i) and (ii), If $z=0 \Rightarrow x+y=6$ and $x+2 y=10$ $\Rightarrow y=4, x=2$ $(2,4,0)$ If $y=0 \Rightarrow x+z=6$ and $x+3 z=10$ $\Rightarrow \quad z=2$ and $x=4$ $(4,0,2)$ So, $3 x+2 y+\lambda z=\mu$, must pass through $(2,4,0)$ and $(4,0,2)$...
Read More →Water samples with BOD values of 4ppm and 18 ppm,
Question: Water samples with BOD values of $4 \mathrm{ppm}$ and $18 \mathrm{ppm}$, respectively, are :Clean and CleanHighly polluted and CleanClean and Highly pollutedHighly polluted and Highly pollutedCorrect Option: , 2 Solution: Clean water would have BOD value $5 \mathrm{ppm}$ whereas highly polluted water could have BOD value $\geq 17$ $\mathrm{ppm}$....
Read More →n mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes -
Question: $\mathrm{n}$ mole of a perfect gas undergoes a cyclic process $\mathrm{ABCA}$ (see figure) consisting of the following processes - $\mathrm{A} \rightarrow \mathrm{B}$ : Isothermal expansion at temperature $\mathrm{T}$ so that the volume is doubled from $\mathrm{V}_{1}$ to $\mathrm{V}_{2}=2 \mathrm{~V}_{1}$ and pressure changes from $\mathrm{P}_{1}$ to $\mathrm{P}_{2}$. $\mathrm{B} \rightarrow \mathrm{C}$ : Isobaric compression at pressure $\mathrm{P}_{2}$ to initial volume $\mathrm{V}_...
Read More →Solve the following
Question: Let $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ be two $3 \times 3$ real matrices such that $b_{i j}=(3)^{(i+j-2)} a_{i j}$, where $i, j=1,2,3$. If the determinant of $B$ is 81 , then the determinant of $A$ is:(1) $1 / 3$(2) 3(3) $1 / 81$(4) $1 / 9$Correct Option: , 4 Solution: It is given that $|B|=81$ $\therefore \quad|B|=\left|\begin{array}{lll}b_{11} b_{12} b_{13} \\ b_{21} b_{22} b_{23} \\ b_{31} b_{32} b_{33}\end{array}\right|=\left|\begin{array}{lll}3^{0} a_{11} 3^{1} ...
Read More →The molecule that has minimum/no role in the formation
Question: The molecule that has minimum/no role in the formation of photochemical smog, is: $\mathrm{N}_{2}$$\mathrm{CH}_{2}=\mathrm{O}$$\mathrm{O}_{3}$NOCorrect Option: 1 Solution: $\mathrm{NO}, \mathrm{O}_{3}$ and $\mathrm{HCHO}$ are responsible for the formation of photochemical smog while $\mathrm{N}_{2}$ has no role in photochemical smog....
Read More →The upper stratosphere consisting of the ozone layer protects us from the sun's
Question: The upper stratosphere consisting of the ozone layer protects us from the sun's radiation that falls in the wavelength region$200-315 \mathrm{~nm}$$400-550 \mathrm{~nm}$$0.8-1.5 \mathrm{~nm}$$600-750 \mathrm{~nm}$Correct Option: 1 Solution: Wavelength range lies in $200 \mathrm{~nm}-315 \mathrm{~nm}$ (U.V light)...
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