Mass of one atom of oxygen is:

Question: Mass of one atom of oxygen is: (a) $\frac{16}{6.023 \times 10^{23}} \mathrm{~g}$ (b) $\frac{32}{6.023 \times 10^{23}} \mathrm{~g}$ (c) $\frac{1}{6.023 \times 10^{23}} \mathrm{~g}$ (d) $8 \mathrm{~g}$. Solution: (a) $\frac{16}{6.023 \times 10^{23}} \mathrm{~g}$...

If A is an m × n matrix and B is a matrix such that both AB and BA

Question: If $A$ is an $m \times n$ matrix and $B$ is a matrix such that both $A B$ and $B A$ are defined, then the order of $B$ is______ Solution: Let $X=\left[x_{i j}\right]_{m \times n}$ and $Y=\left[y_{i j}\right]_{p \times q}$ be two matrices of order $m \times n$ and $p \times q$. The multiplication of matrices $X$ and $Y$ is defined if number of columns of $X$ is same as the number of rows of $Y$ i.e. $n=p$. Also, $X Y$ is a matrix of order $m \times q$. It is given that, $A$ is an $m \ti...

Which of the following contains maximum number of molecules?

Question: Which of the following contains maximum number of molecules? (a) $1 \mathrm{~g} \mathrm{CO}_{2}$ (b) $1 \mathrm{~g} \mathrm{~N}_{2}$ (c) $1 \mathrm{~g} \mathrm{H}_{2}$ (d) $1 \mathrm{~g} \mathrm{CH}_{4}$. Solution: (c). (a) $\frac{(1 \mathrm{~g})}{(44 \mathrm{~g})} \times \mathrm{N}_{\mathrm{A}}=0.023 \times \mathrm{N}_{\mathrm{A}}$ (b) $\frac{(1 \mathrm{~g})}{(28 \mathrm{~g})} \times \mathrm{N}_{\mathrm{A}}=0.036 \times \mathrm{N}_{\mathrm{A}}$ (c) $\frac{(1 g)}{(2 g)} \times N_{A}=0....

Which of the following has maximum number of atoms?

Question: Which of the following has maximum number of atoms? (a) $18 \mathrm{~g}$ of $\mathrm{H}_{2} \mathrm{O}$ (b) $18 \mathrm{~g}$ of $\mathrm{O}_{2}$ (c) $18 \mathrm{~g}$ of $\mathrm{CO}_{2}$ (d) $18 \mathrm{~g}$ of $\mathrm{CH}_{4}$ Solution: (d) No. of atoms = (a) $\frac{(18 \mathrm{~g}) \times 3}{(18 \mathrm{~g})} \times \mathrm{N}_{\mathrm{A}}=3 \mathrm{~N}_{\mathrm{A}}$ (b) $\frac{(18 \mathrm{~g}) \times 2}{(32 \mathrm{~g})} \times \mathrm{N}_{\mathrm{A}}=1.23 \mathrm{~N}_{\mathrm{A}}$...

Show that:

Question: Show that: (i) $\frac{1-\sin 60^{\circ}}{\cos 60^{\circ}}=\frac{\tan 60^{\circ}-1}{\tan 60^{\circ}+1}$ (ii) $\frac{\cos 30^{\circ}+\sin 60^{\circ}}{1+\sin 30^{\circ}+\cos 60^{\circ}}=\cos 30^{\circ}$ Solution: (i) $\mathrm{LHS}=\frac{1-\sin 60^{\circ}}{\cos 60^{\circ}}=\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\left(\frac{2 \sqrt{3}}{2}\right)}{\frac{1}{2}}=\left(\frac{2-\sqrt{3}}{2}\right) \times 2=2-\sqrt{3}$ RHS $=\frac{\tan 60^{\circ}-1}{\tan 60^{\circ}+1}=\frac{\sqrt{3}-1}{\s...

Which of the following would weigh the maximum?

Question: Which of the following would weigh the maximum? (a) $0.2$ mole of sucrose $\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)$ (b) 2 moles of $\mathrm{CO}_{2}$ (c) 2 moles of $\mathrm{CaCO}_{3}$ (d) 10 moles of $\mathrm{H}_{2} \mathrm{O}$. Solution: (c). (a) $0.2 \times 342 \mathrm{~g}=68.4 \mathrm{~g}$ (b) $2 \times 44 \mathrm{~g}=88 \mathrm{~g}$ (c) $2 \times 100 \mathrm{~g}=200 \mathrm{~g}$ (d) $10 \times 18 \mathrm{~g}=180 \mathrm{~g}$....

Solve this

Question: If $A=\left[\begin{array}{cc}x 1 \\ -1 -x\end{array}\right]$ satisfies the equation $A^{2}=0$, then $x=$______ Solution: The given matrix is $A=\left[\begin{array}{cc}x 1 \\ -1 -x\end{array}\right]$. $A^{2}=0$ $\therefore\left[\begin{array}{cc}x 1 \\ -1 -x\end{array}\right]\left[\begin{array}{cc}x 1 \\ -1 -x\end{array}\right]=\left[\begin{array}{ll}0 0 \\ 0 0\end{array}\right]$ $\Rightarrow\left[\begin{array}{cc}x^{2}-1 0 \\ 0 x^{2}-1\end{array}\right]=\left[\begin{array}{ll}0 0 \\ 0 0...

The chemical symbol for sodium is:

Question: The chemical symbol for sodium is: (a) So (b) $S d$ (c) NA (d) $\mathrm{Na}$. Solution: (d) $\mathrm{Na}$....

The chemical symbol for nitrogen gas is :

Question: The chemical symbol for nitrogen gas is : (a) $\mathrm{Ni}$ (b) $\mathrm{N}_{2}$ (c) $\mathrm{N}^{+}$ (d) $N$ Solution: (b) $N_{2}$...

Which of the following statements is not true about atoms?

Question: Which of the following statements is not true about atoms? (a) Atoms are not able to exist independently (b) Atoms are the basic units from which molecules and ions are formed (c) Atoms are always neutral in nature (d) Atoms aggregate in large numbers to form the matter that we can see, feel or touch. Solution: (a) Atoms of inert gas elements can exist independently. However, atoms of all other elements cannot exist independently....

Solve this

Question: If $A=\left[\begin{array}{cc}\frac{1}{3} 2 \\ 0 2 x-3\end{array}\right], B=\left[\begin{array}{cc}3 6 \\ 0 -1\end{array}\right]$ and $A B=l$, then $x=$______ Solution: The given matrices are $A=\left[\begin{array}{cc}\frac{1}{3} 2 \\ 0 2 x-3\end{array}\right]$ and $B=\left[\begin{array}{cc}3 6 \\ 0 -1\end{array}\right]$ $A B=I$ (Given) $\Rightarrow\left[\begin{array}{cc}\frac{1}{3} 2 \\ 0 2 x-3\end{array}\right]\left[\begin{array}{cc}3 6 \\ 0 -1\end{array}\right]=\left[\begin{array}{ll...

Question: If $A=\left[\begin{array}{cc}\frac{1}{3} 2 \\ 0 2 x-3\end{array}\right], B=\left[\begin{array}{cc}3 6 \\ 0 -1\end{array}\right]$ and $A B=l$, then $x=$______ Solution: The given matrices are $A=\left[\begin{array}{cc}\frac{1}{3} 2 \\ 0 2 x-3\end{array}\right]$ and $B=\left[\begin{array}{cc}3 6 \\ 0 -1\end{array}\right]$ $A B=I$ (Given) $\Rightarrow\left[\begin{array}{cc}\frac{1}{3} 2 \\ 0 2 x-3\end{array}\right]\left[\begin{array}{cc}3 6 \\ 0 -1\end{array}\right]=\left[\begin{array}{ll...

(cos 0° + cos 45° + sin 30°) (sin 90° – cos 45° + cos 60°)

Question: (cos 0 + cos 45 + sin 30) (sin 90 cos 45 + cos 60) Solution: As we know that, $\cos 0^{\circ}=1$ $\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ $\sin 30^{\circ}=\frac{1}{2}$ $\sin 90^{\circ}=1$ $\cos 60^{\circ}=\frac{1}{2}$ By substituting these values, we get $\left(\cos 0^{\circ}+\cos 45^{\circ}+\sin 30^{\circ}\right)\left(\sin 90^{\circ}-\cos 45^{\circ}+\cos 60^{\circ}\right)=\left(1+\frac{1}{\sqrt{2}}+\frac{1}{2}\right)\left(1-\frac{1}{\sqrt{2}}+\frac{1}{2}\right)$ $=\left(\left(1+\frac{1}{2...

Which of the following correctly represents 360g of water?

Question: Which of the following correctly represents $360 \mathrm{~g}$ of water? (i) 2 moles of $\mathrm{H}_{2} \mathrm{O}$ (ii) 20 moles of water (iii) $6.022 \times 10^{23}$ molecules of water (iv) $1.2044 \times 10^{25}$ molecules of water (a) (i) (b) (i) and (iv) (c) (ii) and (iii) (d) (ii) and (iv). Mass of water Solution: (d) (ii) $\frac{\text { Mass of water }}{\text { Molar mass of water }}$ $=\frac{(360 \mathrm{~g})}{\left(18 \mathrm{~g} \mathrm{~mol}^{-1}\right)}=20 \mathrm{~mol}$ (iv...

If a, b are positive integers such that a < b and [a b]

Question: If $a, b$ are positive integers such that $ab$ and $[a b]\left[\begin{array}{l}a \\ b\end{array}\right]=25$, then $(a, b)=$_____ Solution: $\left[\begin{array}{ll}a b\end{array}\right]\left[\begin{array}{l}a \\ b\end{array}\right]=[25]$ $\Rightarrow\left[a^{2}+b^{2}\right]=[25]$ $\Rightarrow a^{2}+b^{2}=25$ Now, $3^{2}+4^{2}=5^{2}=25$ $\therefore a=3$ and $b=4 \quad(ab)$ Thus, the ordered pair $(a, b)$ is $(3,4)$. If $a, b$ are positive integers such that $ab$ and $[a b]\left[\begin{ar...

cot230° − 2cos230° solve this

Question: $\cot ^{2} 30^{\circ}-2 \cos ^{2} 30^{\circ}-\frac{3}{4} \sec ^{2} 45^{\circ}+\frac{1}{4} \operatorname{cosec}^{2} 30^{\circ}$ Solution: On substituting the values of various T-ratios, we get: $\cot ^{2} 30^{\circ}-2 \cos ^{2} 30^{\circ}-\frac{3}{4} \sec ^{2} 45^{\circ}+\frac{1}{4} \operatorname{cosec}^{2} 30^{\circ}$ $=(\sqrt{3})^{2}-2 \times\left(\frac{\sqrt{3}}{2}\right)^{2}-\frac{3}{4} \times(\sqrt{2})^{2}+\frac{1}{4} \times(2)^{2}$ $=3-2 \times \frac{3}{4}-\frac{3}{4} \times 2+\fr...

Prove that

Question: $\sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}+\frac{1}{8} \cot ^{2} 60^{\circ}$ Solution: As we know that, $\sin 30^{\circ}=\frac{1}{2}$ $\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ $\tan 30^{\circ}=\frac{1}{\sqrt{3}}$ $\sin 90^{\circ}=1$ $\cot 60^{\circ}=\frac{1}{\sqrt{3}}$ By substituting these values, we get $\sin ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \tan ^{2} 30^{\circ}+\frac{1}{2} \sin ^{2} 90^{\circ}+\frac{1}{8} \cot ^{2} 60^{\circ}=...

Solve the following equations

Question: If $A=\left[\begin{array}{ccc}-1 2 3 x \\ 2 y 4 -1 \\ 6 5 0\end{array}\right]$ is a symmetric matrix, then the value of $2 x+y$ is______ Solution: A matrix $X$ is a symmetric matrix if $X^{T}=X$ It is given that, the matrix $A=\left[\begin{array}{ccc}-1 2 3 x \\ 2 y 4 -1 \\ 6 5 0\end{array}\right]$ is a symmetric matrix.$\therefore A^{T}=A$ $\therefore A^{T}=A$ $\Rightarrow\left[\begin{array}{ccc}-1 2 3 x \\ 2 y 4 -1 \\ 6 5 0\end{array}\right]^{T}=\left[\begin{array}{ccc}-1 2 3 x \\ 2 ...

Calculate the mass of sodium sulphate

Question: Calculate the mass of sodium sulphate required to prepare its $20 \%$ (mass percent) solution in $100 \mathrm{~g}$ of water? Solution: Mass percent $=\frac{\text { Mass of solute }}{\text { Mass of solute }+\text { Mass of water }} \times 100$ $20=\frac{x}{(x+100)} \times 100$ $20 x+2000=100 x$ $80 x=2000$ or $x=\frac{2000}{80}=25 \mathrm{~g}$...

Arun has prepared 0301% (by mass) solution of sodium chloride in water.

Question: Arun has prepared $0301 \%$ (by mass) solution of sodium chloride in water. Which of the following correctly represents the composition of the solution? (a) $1.00 \mathrm{~g}$ of $\mathrm{NaCl}+100 \mathrm{~g}$ of water (b) $0.11 \mathrm{~g}$ of $\mathrm{NaCl}+100 \mathrm{~g}$ of water (c) $0.01 \mathrm{~g}$ of $\mathrm{NaCl}+99.99 \mathrm{~g}$ of water (d) $0.10 \mathrm{~g}$ of $\mathrm{NaCl}+99.90 \mathrm{~g}$ of water. Solution: By definition $0.01 \%$ (by mass) solution means that ...

You are provided with a mixture containing sand,

Question: You are provided with a mixture containing sand, iron filings, ammonium chloride and sodium chloride. Describe the procedure you would use to separate these constituents from the mixture? Solution: Separate the constituents from a mixture containing ammonium chloride, sand and iron-filings. Place the mixture on a paper or petridish. Move a magnet over it. Iron filings get attached to the magnet. Scrap off iron filings from the magnet and collect them separately. Transfer the remaining ...

If P and Q are two matrices of orders 3 × n and n × p respectively

Question: If $P$ and $Q$ are two matrices of orders $3 \times n$ and $n \times p$ respectively then the order of the matrix $P Q$ is________ Solution: Let $X=\left[x_{i j}\right]_{m \times n}$ and $Y=\left[y_{i j}\right]_{p \times q}$ be two matrices of order $m \times n$ and $p \times q$. The multiplication of matrices $X$ and $Y$ is defined if number of columns of $X$ is same as the number of rows of $Y$ i.e. $n=p$. Also, $X Y$ is a matrix of order $m \times q$. It is given that, $P$ and $Q$ a...

2 cos2 60° + 3 sin2 45° − 3 sin2 30° + 2 cos2 90°

Question: 2 cos260 + 3 sin245 3 sin230 + 2 cos290 Solution: On substituting the values of variousT-ratios, we get:2 cos260o+ 3 sin245o 3 sin230o+ 2 cos290o $=2 \times\left(\frac{1}{2}\right)^{2}+3 \times\left(\frac{1}{\sqrt{2}}\right)^{2}-3 \times\left(\frac{1}{2}\right)^{2}+2 \times(0)^{2}$ $=2 \times \frac{1}{4}+3 \times \frac{1}{2}-3 \times \frac{1}{4}+0$ $=\left(\frac{1}{2}+\frac{3}{2}-\frac{3}{4}\right)=\left(\frac{2+6-3}{4}\right)=\frac{5}{4}$...

During an experiment the students were asked to prepare

Question: During an experiment the students were asked to prepare a $10 \%$ (Mass/Mass) solution of sugar in water. Ramesh dissolved $10 \mathrm{~g}$ of sugar in $100 \mathrm{~g}$ of water while Sarika prepared it by dissolving $10 \mathrm{~g}$ of sugar in water to make $100 \mathrm{~g}$ of the solution. (a) Are the two solutions of the same concentration (b) Compare the mass $\%$ of the two solutions. Solution: (a) No, the two solutions have different concentrations. (b) Mass $\%$ of solution p...

Classify each of the following, as a physical or a chemical change. Give reasons.

Question: Classify each of the following, as a physical or a chemical change. Give reasons. 1. Drying of a shirt in the sun. 2. Rising of hot air over a radiator. 3. Burning of kerosene in a lantern. 4. hange in the colour of black tea on adding lemon juice to it. 5. Churning of milk cream to get butter. Solution: 1. It is a physical change. Moisture or water drops present on the shirt will evaporate or change into vapours. 2. It is a physical change. On cooling, the level of air over the radiat...