Which condition out of the following will increase the evaporation of water?
Question: Which condition out of the following will increase the evaporation of water? (a) Increase in temperature of water (b) Decrease in temperature of water (c) Less exposed surface area of water (d) Adding common salt to water. Solution: (a) Increase in temperature always helps in evaporation of a liquid....
Read More →The boiling points of diethyl ether,
Question: The boiling points of diethyl ether, acetone and -butyl alcohol are $35^{\circ} \mathrm{C}, 56^{\circ} \mathrm{C}$ and $118^{\circ} \mathrm{C}$ respectively. Which one of the following correctly represents their boiling points in kelvin scale? (a) 306 K, 329 K, 391 K (b) $308 \mathrm{~K}, 329 \mathrm{~K}, 392 \mathrm{~K}$ (c) $308 \mathrm{~K}, 329 \mathrm{~K}, 391 \mathrm{~K}$ (d) $329 \mathrm{~K}, 392 \mathrm{~K}, 308 \mathrm{~K}$. Solution: (c) It is the correct representation of the...
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Question: If $\sin \theta=\frac{1}{2}$ then $\cot \theta=$ ? (a) $\frac{\sqrt{3}}{2}$ (b) 1 (c) $\sqrt{3}$ (d) $\frac{1}{\sqrt{3}}$ Solution: Given : $\sin \theta=\frac{1}{2}$ Since, $\sin \theta=\frac{P}{H}$ $\Rightarrow P=1$ and $H=2$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 1^{2}+B^{2}=2^{2}$ $\Rightarrow B^{2}=4-1$ $\Rightarrow B^{2}=3$ $\Rightarrow B=\sqrt{3}$ Therefore, $\cot \theta=\frac{B}{P}=\frac{\sqrt{3}}{1}$ Hence, the correct option is (c)....
Read More →Choose the correct statement from the following:
Question: Choose the correct statement from the following: (a) conversion of solid into vapours without passing through the liquid state is called vapourisation. (b) conversion of vapours into solid without passing through the liquid state is called sublimation. (c) conversion of vapours into solid without passing through the liquid state is called freezing. (d) conversion of solid into liquid is called sublimation. Solution: (c) It is the correct statement....
Read More →On converting 25°C, 38°C and 66°C to kelvin scale,
Question: On converting 25C, 38C and 66C to kelvin scale, the correct sequence of temperature will be (a)298 K, 311 K and 339 K (b)298 K, 300 K and 338 K (c)273 K, 278 K and 543 K (d)298 K, 310 K and 338 K. Solution: (a)It is the correct sequence of temperature obtained by adding 273 K to the given temperature....
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Question: If $A=\left[\begin{array}{rr}\cos \theta -\sin \theta \\ \sin \theta \cos \theta\end{array}\right]$, then $A^{T}+A=I_{2}$, if (a) $\theta=n \pi, n \in \mathrm{Z}$ (b) $\theta=(2 n+1) \frac{\pi}{2}, n \in Z$ (c) $\theta=2 n \pi+\frac{\pi}{3}, n \in Z$ (d) none of these Solution: (c) $\theta=2 n \pi+\frac{\pi}{3}, n \in Z$ Here, $A=\left[\begin{array}{cc}\cos \theta -\sin \theta \\ \sin \theta \cos \theta\end{array}\right]$ $\Rightarrow A^{T}=\left[\begin{array}{cc}\cos \theta \sin \thet...
Read More →A few substances are arranged in the increasing
Question: A few substances are arranged in the increasing order of forces of attraction between their particles. Which one of the following represents a correct arrangement ? (a)Water, air, wind (b)Air, sugar, oil (c)Oxygen, water, sugar (d)Salt, juice, air Solution: (c) It is the correct sequence....
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Question: If $\sec \theta=\frac{25}{7}$ then $\sin \theta=$ ? (a) $\frac{7}{25}$ (b) $\frac{24}{25}$ (c) $\frac{7}{24}$ (d) $\frac{24}{7}$ Solution: Given: $\sec \theta=\frac{25}{7}$ Since, $\sec \theta=\frac{H}{B}$ $\Rightarrow B=7$ and $H=25$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow P^{2}+(7)^{2}=(25)^{2}$ $\Rightarrow P^{2}=625-49$ $\Rightarrow P^{2}=576$ $\Rightarrow P=24$ Therefore, $\sin \theta=\frac{P}{H}=\frac{24}{25}$ Hence, the correct option is (b)....
Read More →During summer, water kept in an earthen pot becomes
Question: During summer, water kept in an earthen pot becomes cool because of the phenomenon of (a)diffusion (b)transpiration (c)osmosis (d)evaporation. Solution: (d) The evaporation of a liquid is always accompanied by the lowering of temperature and results in cooling....
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Question: If $A=\left[a_{i j}\right]$ is a square matrix of even order such that $a_{i j}=i^{2}-j^{2}$, then (a) $A$ is a skew-symmetric matrix and $|A|=0$ (b) $A$ is symmetric matrix and $|A|$ is a square (c) $A$ is symmetric matrix and $|A|=0$ (d) none of these. Solution: (d) none of these Given: $A$ is a square matrix of even order. Let $A=\left[\begin{array}{ll}a_{11} a_{12} \\ a_{21} a_{22}\end{array}\right]$ $\Rightarrow A=\left[\begin{array}{cc}0 -3 \\ 3 0\end{array}\right] \quad\left[\be...
Read More →If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined.
Question: If $A$ is $3 \times 4$ matrix and $B$ is a matrix such that $A^{\prime} B$ and $B A^{\prime}$ are both defined. Then, $B$ is of the type (a) $3 \times 4$ (b) $3 \times 3$ (c) $4 \times 4$ (d) $4 \times 3$ Solution: (a) $3 \times 4$ The order of $A$ is $3 \times 4$. So, the order of $A$ ' is $4 \times 3$. Now, both $A^{\prime} B$ and $B A^{\prime}$ are defined. So, the number of columns in $A^{\prime}$ should be equal to the number of rows in $B$ for $A^{\prime} B$. Also, the number of ...
Read More →The property to flow is unique to fluids.
Question: The property to flow is unique to fluids. Which one of the following statements is correct ? (a)Only gases behave like fluids (b)Gases and solids behave like fluids (c)Gases and liquids behave like fluids (d)Only liquids are fluids. Solution: (c)The gases as well as liquids can behave as fluids. There is hardly any fluidity in solids as the particles are very closely packed...
Read More →If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined.
Question: If $A$ is $3 \times 4$ matrix and $B$ is a matrix such that $A^{\prime} B$ and $B A^{\prime}$ are both defined. Then, $B$ is of the type (a) $3 \times 4$ (b) $3 \times 3$ (c) $4 \times 4$ (d) $4 \times 3$ Solution: (a) $3 \times 4$ The order of $A$ is $3 \times 4$. So, the order of $A$ ' is $4 \times 3$. Now, bothA'BandBA'A'BandBA'are defined. So, the number of columns inA'should be equal to the number of rows inBforA'B.Also, the number of columns inBshould be equal to number of rows i...
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Question: If $\operatorname{cosec} \theta=\sqrt{10}$ then $\sec \theta=?$ (a) $\frac{1}{\sqrt{10}}$ (b) $\frac{2}{\sqrt{10}}$ (c) $\frac{3}{\sqrt{10}}$ (d) $\frac{\sqrt{10}}{3}$ Solution: Given : $\operatorname{cosec} \theta=\frac{\sqrt{10}}{1}$ Since, $\operatorname{cosec} \theta=\frac{H}{P}$ $\Rightarrow P=1$ and $H=\sqrt{10}$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 1^{2}+B^{2}=(\sqrt{10})^{2}$ $\Rightarrow B^{2}=10-1$ $\Rightarrow B^{2}=9$ $\Rightarrow B=3$ Therefore, $\sec...
Read More →Seema visited a Natural Gas Compressing Unit
Question: Seema visited a Natural Gas Compressing Unit and found that the gas can be liquefied under specific conditions of temperature and pressure. While sharing her experience with friends, she got confused. Help her to identify the correct set of conditions (a)Low temperature, low pressure (b)High temperature, low pressure (c)Low temperature, high pressure (d)High temperature, high pressure. Solution: (c)Both low temperature and high pressure bring the gas molecules closer and help in its li...
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Question: If $A=\left[\begin{array}{ll}5 x \\ y 0\end{array}\right]$ and $A=A^{T}$, then (a) $x=0, y=5$ (b) $x+y=5$ (c) $x=y$ (d) none of these Solution: (c) $x=y$ Here, $A=\left[\begin{array}{ll}5 x \\ y 0\end{array}\right]$ $A^{T}=\left[\begin{array}{ll}5 y \\ x 0\end{array}\right]$ Now, $A=A^{T}$ The corresponding elements of two equal matrices are equal. $\therefore\left[\begin{array}{ll}5 x \\ y 0\end{array}\right]=\left[\begin{array}{ll}5 y \\ x 0\end{array}\right]$ $\Rightarrow x=y$...
Read More →Which one of the following sets of phenomena
Question: Which one of the following sets of phenomena would increase on raising the temperature? (a)Diffusion, evaporation, compression of gases (b)Evaporation, compression of gases, solubility (c)Evaporation, diffusion, expansion of gases (d)Evaporation, solubility, diffusion, compression of gases. Solution: (c)All the three phenomenon would increase upon increasing the temperature....
Read More →If A and B are symmetric matrices, then ABA is
Question: If $A$ and $B$ are symmetric matrices, then $A B A$ is (a) symmetric matrix (b) skew-symmetric matrix (c) diagonal matrix (d) scalar matrix Solution: (a) symmetric matrix Since $A$ and $B$ are symmetric matrices, we get $A=A^{\prime}$ and $B=B^{\prime}$ $(A B A)^{\prime}=(B A)^{\prime}(A)^{\prime}$ $=A^{\prime} B^{\prime} A^{\prime}$ $=A B A \quad\left[\because A=A^{\prime}\right.$ and $\left.B=B^{\prime}\right]$ Since $(A B A)^{\prime}=A B A, A B A$ is a symmetric matrix....
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Question: If $\tan \theta=\sqrt{3}$ then $\sec \theta=$ ? (a) 2 (b) $\frac{1}{2}$ (c) $\frac{\sqrt{3}}{2}$ (d) $\frac{2}{\sqrt{3}}$ Solution: Given : $\tan \theta=\frac{\sqrt{3}}{1}$ Since, $\tan \theta=\frac{P}{B}$ $\Rightarrow P=\sqrt{3}$ and $B=1$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow(\sqrt{3})^{2}+1^{2}=H^{2}$ $\Rightarrow H^{2}=3+1$ $\Rightarrow H^{2}=4$ $\Rightarrow H=2$ Therefore, $\sec \theta=\frac{H}{B}=\frac{2}{1}=2$ Hence, the correct option is (a)....
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Question: If $\tan \theta=\frac{8}{15}$ then $\operatorname{cosec} \theta=?$ (a) $\frac{15}{17}$ (b) $\frac{17}{15}$ (c) $\frac{17}{8}$ (d) $\frac{8}{17}$ Solution: Given : $\tan \theta=\frac{8}{15}$ Since, $\tan \theta=\frac{P}{B}$ $\Rightarrow P=8$ and $B=15$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 8^{2}+(15)^{2}=H^{2}$ $\Rightarrow H^{2}=64+225$ $\Rightarrow H^{2}=289$ $\Rightarrow H=17$ Therefore, $\operatorname{cosec} \theta=\frac{H}{P}=\frac{17}{8}$ Hence, the correct op...
Read More →If A is a square matrix, then AA is a
Question: If $A$ is a square matrix, then $A A$ is a (a) skew-symmetric matrix (b) symmetric matrix (c) diagonal matrix (d) none of these Solution: (d) none of theseGiven: Ais a square matrix. Let $A=\left[\begin{array}{ll}1 2 \\ 1 0\end{array}\right]$ $\Rightarrow A A=\left[\begin{array}{ll}1 2 \\ 1 0\end{array}\right]\left[\begin{array}{ll}1 2 \\ 1 0\end{array}\right]=\left[\begin{array}{ll}3 2 \\ 1 2\end{array}\right]$...
Read More →If ∠A and ∠B are acute angles such that tanA = tanB,
Question: If $\angle \mathrm{A}$ and $\angle \mathrm{B}$ are acute angles such that $\tan \mathrm{A}=\tan \mathrm{B}$, the prove that $\angle \mathrm{A}=\angle \mathrm{B}$. Solution: In $\triangle \mathrm{ABC}, \angle \mathrm{C}=90^{\circ}$, $\tan \mathrm{A}=\frac{\mathrm{BC}}{\mathrm{AC}}$ and $\tan \mathrm{B}=\frac{\mathrm{AC}}{\mathrm{BC}}$ As, $\tan \mathrm{A}=\tan \mathrm{B}$ $\Rightarrow \frac{\mathrm{BC}}{\mathrm{AC}}=\frac{\mathrm{AC}}{\mathrm{BC}}$ $\Rightarrow \mathrm{BC}^{2}=\mathrm{A...
Read More →Solve the following equations for
Question: The matrix $\left[\begin{array}{rrr}0 5 -7 \\ -5 0 11 \\ 7 -11 0\end{array}\right]$ is (a) a skew-symmetric matrix (b) a symmetric matrix (c) a diagonal matrix (d) an uppertriangular matrix Solution: (a) a skew-symmetric matrix Here, $A=\left[\begin{array}{ccc}0 5 -7 \\ -5 0 11 \\ 7 -11 0\end{array}\right]$ $\Rightarrow A^{T}=\left[\begin{array}{ccc}0 -5 7 \\ 5 0 -11 \\ -7 11 0\end{array}\right]$ $\Rightarrow A^{T}=-\left[\begin{array}{ccc}0 5 -7 \\ -5 0 11 \\ 7 -11 0\end{array}\right]...
Read More →If ∠A and ∠B are acute angles such that sinA
Question: If $\angle A$ and $\angle B$ are acute angles such that $\sin A=\sin B$, then prove that $\angle A=\angle B$. Solution: $\ln \triangle \mathrm{ABC}, \angle \mathrm{C}=90^{\circ}$ $\sin \mathrm{A}=\frac{\mathrm{BC}}{\mathrm{AB}}$ and $\sin B=\frac{A C}{A B}$ As, sinA = sinB $\Rightarrow \frac{\mathrm{BC}}{\mathrm{AB}}=\frac{\mathrm{AC}}{\mathrm{AB}}$ ⇒BC = ACSo,A =B (Angles opposite to equal sides are equal)...
Read More →If a matrix A is both symmetric and skew-symmetric, then
Question: If a matrix $A$ is both symmetric and skew-symmetric, then (a) $A$ is a diagonal matrix (b) $A$ is a zero matrix (c) $A$ is a scalar matrix (d) $A$ is a square matrix Solution: (b) $A$ is a zero matrix Let $A=\left[a_{i j}\right]$ be a matrix which is both symmetric and skew-symmetric. If $A=\left[a_{i j}\right]$ is a symmetric matrix, then $a_{i j}=a_{j i}$ for all $\mathrm{i}, \mathrm{j}$ ....(1) If $A=\left[a_{i j}\right]$ is a skew-symmetric matrix, then $a_{i j}=-a_{j i}$ for all ...
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