If A is a matrix of order m × n and B is a matrix such that
Question: If $A$ is a matrix of order $m \times n$ and $B$ is a matrix such that $A B^{T}$ and $B^{T} A$ are both defined, then the order of matrix $B$ is (a) $m \times n$ (b) $n \times n$ (c) $n \times m$ (d) $m \times n$ Disclaimer: option (a) and (d) both are the same. Solution: Since, $A B^{T}$ and $B^{T} A$ are both defined. And, order of $A$ is $m \times n .$ So, Order of $B^{T}$ must be $n \times m$. Thus, order of matrix $B$ is $m \times n$. Hence, the correct option is (d)....
Read More →Two chemical species X and Y combine
Question: Two chemical species $X$ and $Y$ combine together to form a product $P$ which contains both $X$ and $Y$ $X+Y \rightarrow P$ $X$ and $Y$ cannot be broken down into simpler substances by simple chemical reactions. Which of the following concerning the species $X, Y$ and $P$ are correct? (i) $P$ is a compound (ii) $X$ and $Y$ are compounds (iii) $X$ and $Y$ are elements (iv) P has a fixed composition (a) (i), (ii) and (iii) (b) (i), (ii) and (iv) (c) (ii), (iii) and (iv) (d) (i), (iii) an...
Read More →Two substances, A and B were made to react to form a third substance,
Question: Two substances, $A$ and $B$ were made to react to form a third substance, $A_{2} B$ according to the following reaction $2 A+B \rightarrow A_{2} B$ Which of the following statements concerning this reaction are incorrect? (i) The product $\mathrm{A}_{2} \mathrm{~B}$ shows the properties of substances $\mathrm{A}$ and $\mathrm{B}$ (ii) The product will always have a fixed composition (iii) The product so formed cannot be classified as a compound (iv) The product so formed is an element....
Read More →If A and B are two matrices of order 3 × m and 3 × n respectively
Question: If $A$ and $B$ are two matrices of order $3 \times m$ and $3 \times n$ respectively and $m=n$, then the order of $5 A-2 B$ is (a) $m \times 3$ (b) $3 \times 3$ (c) $m \times n$ (d) $3 \times n$ Solution: Since, the order of both $A$ and $B$ are same. i.e., $3 \times m$ or $3 \times n$. Therefore, the order of $5 A-2 B$ is also $3 \times m$ or $3 \times n$. Hence, the correct option is (d)....
Read More →If cos A + cos2 A = 1 then (sin2 A + sin4 A) = ?
Question: If $\cos A+\cos ^{2} A=1$ then $\left(\sin ^{2} A+\sin ^{4} A\right)=?$ (a) $\frac{1}{2}$ (b) 2(c) 1(d) 4 Solution: Given : $\cos A+\cos ^{2} A=1$ $\cos A+\cos ^{2} A=1$ $\Rightarrow \cos A=1-\cos ^{2} A$ $\Rightarrow \cos A=\sin ^{2} A \quad\left(\because \sin ^{2} A+\cos ^{2} A=1\right)$ Now, $\sin ^{2} A+\sin ^{4} A=\sin ^{2} A+\left(\sin ^{2} A\right)^{2}$ $=\cos A+(\cos A)^{2}$ $=\cos A+\cos ^{2} A$ $=1$ Hence, the correct option is (c)....
Read More →Which of the following are chemical changes?
Question: Which of the following are chemical changes? (i) Decaying of wood (ii) Burning of wood (iii) Growth of wood in a tree (iv) Hammering of a nail into a piece of wood. (a) (i) and (ii) (b) (ii) and (iv) (c) (iii) and (iv) (d) (i), (ii) (iiii). Solution: (d) All the three are chemical changes....
Read More →Which of the following are physical changes?
Question: Which of the following are physical changes? (i) Melting of iron metal (ii) Rusting of iron (iii) Bending of an iron rod (iv) Drawing a wire of iron metal (a) (i), (ii) and (iii) (b) (i), (ii) and (iv) (c) (i), (iii) and (iv) (d) (ii), (iii) and (iv). Solution: (c) All the three are examples of physical changes....
Read More →If A is a square matrix such that
Question: If $A$ is a square matrix such that $A^{2}=l$, then $(A-l)^{3}+(A+l)^{3}-7 A$ is equal to (a) $A$ (b) $I-A$ (c) $I+A$ (d) $3 \mathrm{~A}$ Solution: $(A-I)^{3}+(A+I)^{3}-7 A$ $=A^{3}-I^{3}-3 A^{2} I+3 A I^{2}+A^{3}+I^{3}+3 A^{2} I+3 A I^{2}-7 A$ $=2 A^{3}+6 A I^{2}-7 A$ $=2 A \cdot A^{2}+6 A-7 A$ $=2 A \cdot I-A \quad\left(\because A^{2}=I\right)$ $=2 A-A$ $=A$ Hence, the correct option is (a)....
Read More →Which of the following are homogeneous mixtures in nature?
Question: Which of the following are homogeneous mixtures in nature? (i) ice (ii) Wood (iii) soil (iv) air. (a) (i) and (iii) (b) (ii) and (iv) (c) (i) and (iv) (d) (iii) and (iv). Solution: (c) Both ice and air are homogeneous mixtures....
Read More →If sec θ + tan θ + 1 = 0 then (sec θ – tan θ) = ?
Question: If sec + tan + 1 = 0 then (sec tan ) = ?(a) 1(b) 1(c) 0(d) 2 Solution: Given: $\sec \theta+\tan \theta+1=0$ $\sec \theta+\tan \theta+1=0$ $\Rightarrow \sec \theta+\tan \theta=-1$ Multiplying and dividing LHS by $\sec \theta-\tan \theta$, we get $\Rightarrow(\sec \theta+\tan \theta) \times\left(\frac{\sec \theta-\tan \theta}{\sec \theta-\tan \theta}\right)=-1$ $\Rightarrow\left(\frac{\sec ^{2} \theta-\tan ^{2} \theta}{\sec \theta-\tan \theta}\right)=-1$ $\Rightarrow\left(\frac{1+\tan ^{...
Read More →Tincture of iodine has antiseptic properties.
Question: Tincture of iodine has antiseptic properties. This solution is made by dissolving (a) iodine in benzene (b) iodine in ether (c) iodine in water (d) iodine in alcohol. Solution: (d) Tincture of iodine is a dilute solution of iodine formed by dissolving iodine in ethyl alcohol also called alcohol....
Read More →A mixture of sulphur and carbon disulphide is
Question: A mixture of sulphur and carbon disulphide is (a) heterogeneous and shows Tyndall effect (b) homogeneous and shows Tyndall effect (c) heterogeneous and does not show Tyndall effect (d) homogeneous and does not show Tyndall effect. Solution: (d) Sulphur dissolves in carbon disulphide to form a solution which is homogeneous in nature. It does not show any Tyndall effect....
Read More →Rusting of an article made up of iron is called
Question: Rusting of an article made up of iron is called (a) corrosion and it is a physical as well as chemical change (b) dissolution and it is a physical change (c) corrosion and it is a chemical change (d) dissolution and it is a chemical change. Solution: (c) corrosion and it is a chemical change....
Read More →If 3cos θ = 2 then (2sec2 θ + 2tan2 θ – 7) = ?
Question: If 3cos = 2 then (2sec2 + 2tan2 7) = ?(a) 0(b) 1(c) 3(4) 4 Solution: Given : $3 \cos \theta=2$ $\Rightarrow \cos \theta=\frac{2}{3}$ Since, $\cos \theta=\frac{B}{H}$ $\Rightarrow B=2$ and $H=3$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow P^{2}+2^{2}=3^{2}$ $\Rightarrow P^{2}=9-4$ $\Rightarrow P^{2}=5$ $\Rightarrow P=\sqrt{5}$ Therefore, $\sec \theta=\frac{H}{B}=\frac{3}{2}$ $\tan \theta=\frac{P}{B}=\frac{\sqrt{5}}{2}$ Now, $\left(2 \sec ^{2} \theta+2 \tan ^{2} \theta-7\r...
Read More →solve this matrices
Question: If $\left[\begin{array}{cc}2 x+y 4 x \\ 5 x-7 4 x\end{array}\right]=\left[\begin{array}{cc}7 7 y-13 \\ y x+6\end{array}\right]$, then the value of $x$ and $y$ is (a) $x=3, y=1$ (b) $x=2, y=3$ (c) $x=2, y=4$ (d) $x=3, y=3$ Solution: $\left[\begin{array}{cc}2 x+y 4 x \\ 5 x-7 4 x\end{array}\right]=\left[\begin{array}{cc}7 7 y-13 \\ y x+6\end{array}\right]$ Corresponding elements of equal matrices are equal. $\therefore 4 x=x+6 \quad$ and $\quad 2 x+y=7$ $\Rightarrow 3 x=6 \quad$ and $\qu...
Read More →Which of the following statements are true for pure substances?
Question: Which of the following statements are true for pure substances? (i) Pure substances contain only one kind of particles (ii) Pure substances may be compounds or mixtures (iii) Pure substances have the same composition throughout (iv) Pure substances can be exemplified by all elements other than nickel. (a) (i) and (ii) (b) (i) and (iii) (c) (iii) and (iv) (d) (ii) and (iii). Solution: (b) Both these statements are correct....
Read More →The number of possible matrices of order 3 × 3 with each entry 2 or 0 is
Question: The number of possible matrices of order 3 3 with each entry 2 or 0 is (a) 9 (b) 27 (c) 81 (d) none of these Solution: In a matrix of order $3 \times 3$, there are 9 elements. Each element of the matrix have two options (either 2 or 0 ). Hence, total number of possible matrices are $2^{9}$. Hence, the correct option is (d)....
Read More →Solve this
Question: If $4 \cot \theta=3$ then $\left(\frac{\sin \theta-\cos \theta}{\sin \theta+\cos \theta}\right)=?$ (a) $\frac{3}{7}$ (b) $\frac{2}{7}$ (c) $\frac{1}{7}$ (d) 0 Solution: Given: $4 \cot \theta=3$ $\Rightarrow \cot \theta=\frac{3}{4}$ Since, $\cot \theta=\frac{B}{P}$ $\Rightarrow P=4$ and $B=3$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 4^{2}+3^{2}=H^{2}$ $\Rightarrow H^{2}=16+9$ $\Rightarrow H^{2}=25$ $\Rightarrow H=5$ Therefore, $\sin \theta=\frac{P}{H}=\frac{4}{5}$ $\co...
Read More →Solve this
Question: The matrix $A=\left[\begin{array}{lll}0 0 4 \\ 0 4 0 \\ 4 0 0\end{array}\right]$ is a (a) square matrix (b) diagonal matrix (c) unit matrix (d) none of these Solution: Given: $A=\left[\begin{array}{lll}0 0 4 \\ 0 4 0 \\ 4 0 0\end{array}\right]$ Since, number of rows is equal to number of columns.Therefore,Ais a square matrix.Hence, the correct option is (a)....
Read More →Why does the temperature of a substance remain
Question: Why does the temperature of a substance remain constant during its melting point or boiling point? Solution: As long as a substance has not started either melting or boiling, the heat energy supplied increases the kinetic energy of the particles. Therefore, the temperature of the substance increases. Once it starts melting or boiling, the further heat energy is absorbed either as latent heat of fusion or as latent heat of vaporisation. This does not cause any increase in kinetic energy...
Read More →If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to
Question: If $A=\left[a_{i j}\right]$ is a scalar matrix of order $n \times n$ such that $a_{i j}=k$, for all $i$, then trace of $A$ is equal to (a) $n k$ (b) $n+k$ (c) $\frac{n}{k}$ (d) none of these Solution: (a) $n k$ Here, $A=\left[a_{i j}\right]_{n \times n}$ Trace of $A$, i. e. $\operatorname{tr}(A)=\sum_{i=1}^{n} a_{i j}=a_{11}+a_{22}+\ldots+a_{n n}$ $=k+k+k+\ldots n$ times $=k n$...
Read More →Comment on the following statements :
Question: Comment on the following statements : 1. Evaporation produces cooling 2. Rate of evaporation of an aqueous solution decreases with increase in humidity. 3. Sponge though compressible is a solid. Solution: 1. As a liquid changes into vapours during evaporation, it needs some energy which it takes from the surroundings. Since the temperature of the surroundings gets lowered, cooling is therefore, caused. 2. Increase in humidity results in increase in moisture in the atmosphere. As a resu...
Read More →Tarun got an invitation to attend a party.
Question: Tarun got an invitation to attend a party. On coming to his place, he found that both his shirt and pent were wet. What step he would take to dry them quickly? (C.B.S.E. 2013) Solution: The wet clothes can be dried in a number of ways. 1. By spreading them in air under sun so that the water may evaporate. 2. By spreading them under a fan in a room. 3. By ironing the clothes. As a result, the moisture present will escape as steam / vapours and the clothes will soon become dry....
Read More →It is a hot summer day,
Question: It is a hot summer day, Priyanshi and Ali are wearing cotton and nylon clothes respectively. Who do you think would be more comfortable and why? Solution: Priyanshi would be more comfortable than Ali since she is wearing cotton clothes. Being porous in nature, cotton clothes would absorb moisture from the body. As it evaporates, it results in cooling giving comfort to the body. Since Ali is wearing nylon clothes which have hardly any pores, he would feel less comfortable....
Read More →If 4tan θ = 3 then (cos2 θ – sin2 θ) = ?
Question: If 4tan = 3 then (cos2 sin2) = ? (a) $\frac{4}{25}$ (b) $\frac{7}{25}$ (c) 1 (d) $\frac{11}{25}$ Solution: Given : $4 \tan \theta=3$ $\Rightarrow \tan \theta=\frac{3}{4}$ Since, $\tan \theta=\frac{P}{B}$ $\Rightarrow P=3$ and $B=4$ Using Pythagoras theorem, $P^{2}+B^{2}=H^{2}$ $\Rightarrow 3^{2}+4^{2}=H^{2}$ $\Rightarrow H^{2}=9+16$ $\Rightarrow H^{2}=25$ $\Rightarrow H=5$ Therefore, $\sin \theta=\frac{P}{H}=\frac{3}{5}$ $\cos \theta=\frac{B}{H}=\frac{4}{5}$ $\cos ^{2} \theta-\sin ^{2}...
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