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Question: The matrix $A=\left[\begin{array}{ccc}0 -5 8 \\ 5 0 12 \\ -8 -12 0\end{array}\right]$ is a (a) diagonal matrix (b) symmetric matrix (c) skew-symmetric matrix (d) scalar matrix Solution: Given: $A=\left[\begin{array}{ccc}0 -5 8 \\ 5 0 12 \\ -8 -12 0\end{array}\right]$ $A^{T}=\left[\begin{array}{ccc}0 -5 8 \\ 5 0 12 \\ -8 -12 0\end{array}\right]^{T}$ $=\left[\begin{array}{ccc}0 5 -8 \\ -5 0 -12 \\ 8 12 0\end{array}\right]$ $=-1\left[\begin{array}{ccc}0 -5 8 \\ 5 0 12 \\ -8 -12 0\end{arra...
Read More →Give an example each for the mixture having the following characteristics.
Question: Give an example each for the mixture having the following characteristics. Suggest a suitable method to separate the components of these mixtures 1. A volatile and a non-volatile component. 2. Two volatile components with appreciable difference in boiling points. 3. Two immiscible liquids. 4. One of the components changes directly from solid to gaseous state. 5. Two or more coloured constituents soluble in same solvent. Solution: 1. A mixture of naphthalene (volatile) and common salt (...
Read More →An element is sonorous and highly ductile.
Question: An element is sonorous and highly ductile. Under which category would you classify this element? What other characteristics do you expect the element to possess? Solution: The element is a metal. It is expected to possess the important characteristics of the metals. Semi-Metals or Metalloids There are a few elements which possess the characteristics of both metals and non-metals. These are actually border-line elements and are known as semi-metals. Semi-metals are also called metalloid...
Read More →What are the favourable qualities given to
Question: What are the favourable qualities given to gold when it is alloyed with copper or silver for the purpose of making ornaments? Solution: Pure gold (24 carats) is very soft and cannot be used for making ornaments. It is alloyed with a small quantity of copper or silver and becomes hard (22 carats). It can be drawn into wires (becomes ductile) or beaten into fine sheets (becomes malleable). It can, therefore, be used for making ornaments....
Read More →You are given two samples of water labelled as
Question: You are given two samples of water labelled as $A$ and ' $B$ '. Sample $A$ boils at $100^{\circ} C$ and sample ' $B$ ' boils at $102^{\circ} \mathrm{C}$. Which sample of water will not freeze at $0^{\circ} \mathrm{C}$ ? Comment. Solution: The standard boiling point temperature of water is $100^{\circ} \mathrm{C}$. Therefore, the sample v/ith boiling point $100^{\circ} \mathrm{C}$ represents the pure form of water. The sample with boiling point $102^{\circ} \mathrm{C}$ is impure. Please...
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Question: If $3 x=\operatorname{cosec} \theta$ and $\frac{3}{x}=\cot \theta$, find the value of $3\left(x^{2}-\frac{1}{x^{2}}\right)$. Solution: $3\left(x^{2}-\frac{1}{x^{2}}\right)$ $=\frac{9}{3}\left(x^{2}-\frac{1}{x^{2}}\right)$ $=\frac{1}{3}\left(9 x^{2}-\frac{9}{x^{2}}\right)$ $=\frac{1}{3}\left[(3 x)^{2}-\left(\frac{3}{x}\right)^{2}\right]$ $=\frac{1}{3}\left[(\operatorname{cosec} \theta)^{2}-(\cot \theta)^{2}\right]$ $=\frac{1}{3}\left(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta\righ...
Read More →Solve the following equations
Question: If $A=\left[\begin{array}{ccc}2 -1 3 \\ -4 5 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2 3 \\ 4 -2 \\ 1 5\end{array}\right]$, then (a) only $A B$ is defined (b) only $B A$ is defined (c) $A B$ and $B A$ both are defined (d) $A B$ and $B A$ both are not defined Solution: Given: $A=\left[\begin{array}{ccc}2 -1 3 \\ -4 5 1\end{array}\right]$ and $B=\left[\begin{array}{cc}2 3 \\ 4 -2 \\ 1 5\end{array}\right]$ Order of $A$ is $2 \times 3$ and order of $B$ is $3 \times 2$. Therefore...
Read More →Name the process associated with the following:
Question: Name the process associated with the following: 1. Dry ice is kept at room temperature and under one atmospheric pressure. 2. A drop of ink placed on the surface of water contained in a glass spreads throughout the water. 3. A potassium permanganate crystal is in a beaker and water is poured into the beaker with stirring. 4. An acetone bottle is left open and the bottle, becomes empty. 5. Milk is churned to separate cream from it. . 6. Settling of sand when a mixture of sand and water ...
Read More →The teacher instructed three students A',B'
Question: The teacher instructed three students $A^{\prime}, B^{\prime}$ and ' $C$ ' respectively to prepare a $50 \%$ (mass by volume) solution of sodium hydroxide $(\mathrm{NaOH})$. 'A' dissolved $50 \mathrm{~g}$ of $\mathrm{NaOH}$ in $100 \mathrm{~mL}$ of water. ' $\mathrm{B}$ ' dissolved $50 \mathrm{~g}$ of $\mathrm{NaOH}$ in $100 \mathrm{~g}$ of water while ' $\mathrm{C}$ ' dissolved $50 \mathrm{~g}$ of $\mathrm{NaOH}$ in water to make $100 \mathrm{~mL}$ of solution. Which one of them has m...
Read More →If A and B are square matrices of the same order,
Question: If $A$ and $B$ are square matrices of the same order, then $(A+B)(A-B)$ is equal to (a) $A^{2}-B^{2}$ (b) $A^{2}-B A-A B-B^{2}$ (c) $A^{2}-B^{2}+B A-A B$ (d) $A^{2}-B A+B^{2}+A B$ Solution: $(A+B)(A-B)=A^{2}-A B+B A-B^{2}$ Hence, the correct option is (c)....
Read More →Classify the following as physical
Question: Classify the following as physical or chemical properties (CBSE 2012) 1. The composition of a sample of steel is : $98 \%$ iron. $1-5 \%$ carbon and $0-5 \%$ other elements. 2. Zinc dissolves in hydrochloric acid with the evolution of hydrogen gas. 3. Metallic sodium is soft enough to be cut with a knife. 4. Most metal oxides form alkalis on interacting with water. Solution: 1. Physical 2. Chemical 3. Physical 4. Chemical....
Read More →Solve the following equations for
Question: If $A=\frac{1}{\pi}\left[\begin{array}{cc}\sin ^{-1}(\pi \mathrm{x}) \tan ^{-1}\left(\frac{x}{\pi}\right) \\ \sin ^{-1}\left(\frac{x}{\pi}\right) \cot ^{-1}(\pi \mathrm{x})\end{array}\right], B=\frac{1}{\pi}\left[\begin{array}{cc}-\cos ^{-1}(\pi x) \tan ^{-1}\left(\frac{x}{\pi}\right) \\ \sin ^{-1}\left(\frac{x}{\pi}\right) -\tan ^{-1}(\pi x)\end{array}\right]$, then $A-B$ is equal to (a) $I$ (b) 0 (c) 21 (d) $\frac{1}{2} I$ Disclaimer: There is a misprint in the question. $\operatorna...
Read More →Smoke and fog both are aerosols. In what way are they different?
Question: Smoke and fog both are aerosols. In what way are they different? Solution: Air is the dispersion medium in both the cases. However, dispersed phases are not the same. In smoke, carbon particles act as the dispersed phase while water drops represent the dispersed phase in fog....
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Question: If $\sqrt{3} \tan \theta=3 \sin \theta$ then $\left(\sin ^{2} \theta-\cos ^{2} \theta\right)=?$ (a) $\frac{1}{3}$ (b) $\frac{1}{\sqrt{3}}$ (c) $\sqrt{3}$ (d) $\frac{2}{\sqrt{3}}$ Solution: Given : $\sqrt{3} \tan \theta=3 \sin \theta$ $\sqrt{3} \tan \theta=3 \sin \theta$ $\Rightarrow \sqrt{3} \frac{\sin \theta}{\cos \theta}=3 \sin \theta$ $\Rightarrow \frac{\sqrt{3} \sin \theta}{\cos \theta}-3 \sin \theta=0$ $\Rightarrow \frac{\sqrt{3} \sin \theta-3 \sin \theta \cos \theta}{\cos \theta}...
Read More →Explain why particles of a colloidal solution
Question: Explain why particles of a colloidal solution do not settle down when left undisturbed while in the case of a suspension, they do. Solution: In a colloidal solution, the particle size is smaller as compared to particle size in a suspension. The effect of gravity on these particles is less than in a suspension. Therefore, the particles in a suspension settle under the influence of gravity while they remain dispersed in a colloidal solution....
Read More →What would you observe when
Question: What would you observe when (a) a saturated solution of potassium chloride prepared at $60^{\circ} \mathrm{C}$ is allowed to cool to room temperature. (b) an aqueous sugar solution is heated to dryness (c) a mixture of iron filings and sulphur powder is heated strongly. Solution: (a) Upon cooling, crystals of potassium chloride would separate. (b) Initially, entire water slowly gets removed. Upon further heating, sugar gets charred and burning smell will be noticed. (c) Iron sulphide i...
Read More →While diluting a solution of salt in water,
Question: While diluting a solution of salt in water, a student by mistake added acetone (boiling point $56^{\circ} \mathrm{C}$ ). What technique can be employed to get back acetone? Justify your choice. Solution: Acetone can be recovered by carrying out the distillation in a distillation flask. Its vapours will rise since it is volatile, get condensed and collected in a receiver. Salt being non-volatile in nature remains in the flask....
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Question: If matrix $A=\left[a_{i j}\right]_{2 \times 2}$, where $a_{i j}=\left\{\begin{array}{ll}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{array}\right.$, then $A^{2}$ is equal to (a) $I$ (b) $A$ (c) 0 (d) $-1$ Solution: Given: $a_{i j}= \begin{cases}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{cases}$ $\therefore a_{11}=0$ $a_{12}=1$ $a_{21}=1$ $a_{22}=0$ Therefore, matrix $A=\left[\begin{array}{ll}0 1 \\ 1 0\end{array}\right]$ $A^{2}=\left[\begin{array}{ll}0 1 \\ 1 0\end{arra...
Read More →If matrix A=[aij]2×2,
Question: If matrix $A=\left[a_{i j}\right]_{2 \times 2}$, where $a_{i j}=\left\{\begin{array}{ll}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{array}\right.$, then $A^{2}$ is equal to (a) $I$ (b) $A$ (c) 0 (d) $-1$ Solution: Given: $a_{i j}= \begin{cases}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{cases}$ $\therefore a_{11}=0$ $a_{12}=1$ $a_{21}=1$ $a_{22}=0$ Therefore, matrix $A=\left[\begin{array}{ll}0 1 \\ 1 0\end{array}\right]$ $A^{2}=\left[\begin{array}{ll}0 1 \\ 1 0\end{arra...
Read More →Sea-water can be classified as homogeneous
Question: Sea-water can be classified as homogeneous as well as heterogeneous mixture. Comment. Solution: The sample of sea water from mid stream is transparent and has certain salts dissolved in water. It is therefore, homogeneous., However, the sample collected from near the seashore contains mud and many suspended particles. It is therefore, heterogeneous in nature....
Read More →Salt can be recovered from its solution
Question: Salt can be recovered from its solution by evaporation. Suggest some other technique for the same? Solution: Crystallisation technique can also be used. For example, both these can be used to separate salts like sodium chloride (common salt) and potasium nitrate (nitre) from their aqueous solutions....
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Question: If $\sin \theta=\frac{\sqrt{3}}{2}$ then $(\operatorname{cosec} \theta+\cot \theta)=?$ (a) $(2+\sqrt{3})$ (b) $2 \sqrt{3}$ (c) $\sqrt{2}$ (d) $\sqrt{3}$ Solution: Given : $\sin \theta=\frac{\sqrt{3}}{2}$ Since, $\sin \theta=\frac{P}{H}$ $\Rightarrow P=\sqrt{3}$ and $H=2$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow(\sqrt{3})^{2}+B^{2}=2^{2}$ $\Rightarrow B^{2}=4-3$ $\Rightarrow B^{2}=1$ $\Rightarrow B=1$ Therefore, $\operatorname{cosec} \theta=\frac{H}{P}=\frac{2}{\sqrt{3...
Read More →Which of the tubes (a) and (b) will be more
Question: Which of the tubes (a) and (b) will be more effective as a condenser in the distillation apparatus? Solution: Condenser (a) has a number of bends or obstructions. It will provide more opportunity to vapours to condense and act as a better condenser than (b) which has no such obstructions....
Read More →If A and B are matrices of the same order,
Question: If $A$ and $B$ are matrices of the same order, then $A B^{T}-B A^{T}$ is a (a) skew-symmetric matrix (b) : matrix (c) unit matrix (d) symmetric matrix Solution: $\left(A B^{T}-B A^{T}\right)^{T}=\left(A B^{T}\right)^{T}-\left(B A^{T}\right)^{T}$ $=B A^{T}-A B^{T}$ $=-\left(A B^{T}-B A^{T}\right)$ Therefore, $A B^{T}-B A^{\top}$ is a skew-symmetric matrix. Hence, the correct option is (a). Disclaimer: There is a misprint in the question. It should be $B A^{\top}$ instead of $B^{\top} A$...
Read More →Suggest separation technique(s) one would need
Question: Suggest separation technique(s) one would need to employ to separate the following mixtures. 1. Mercury and water 2. Potassium chloride and ammonium chloride 3. Common salt, water and sand 4. Kerosene oil, water and salt. Solution: 1. Separation can be done with a separating funnel. Mercury being heavy forms lower layer. 2. Sublimation can be used. Ammonium chloide sublimes. 3. Filtration followed by crystallisation techniques can be used here. Sand can be removed as an insoluble resid...
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