If A is a skew-symmetric matrix,
Question: If $A$ is a skew-symmetric matrix, then $k A$ is a_____ ( $k$ is any scalar). Solution: It is given that, $A$ is a skew-symmetric matrix. $\therefore A^{T}=-A$ ...(1) Now, $(k A)^{T}$ $=k\left(A^{T}\right)$ $=k(-A)$ [Using (1)] $=-k A$ Since $(k A)^{T}=-k A$, so the matrix $k A$ ( $k$ is any scalar) is a skew-symmetric matrix. If $A$ is a skew-symmetric matrix, then $k A$ is a ( $k$ is any scalar)....
Read More →Prove that
Question: $\left(\sin ^{2} 30^{\circ}-\sec ^{2} 60^{\circ}+4 \cot ^{2} 45^{\circ}\right)=?$ (a) 4(b) 2(c) 1 (d) $\frac{1}{4}$ Solution: As we know that, $\sin 30^{\circ}=\frac{1}{2}$ $\sec 60^{\circ}=2$ $\cot 45^{\circ}=1$ By substituting these values, we get $\left(\sin ^{2} 30^{\circ}-\sec ^{2} 60^{\circ}+4 \cot ^{2} 45^{\circ}\right)=\left(\frac{1}{2}\right)^{2}-(2)^{2}+4(1)^{2}$ $=\frac{1}{4}-4+4$ $=\frac{1}{4}$ Hence, the correct option is (d)....
Read More →If A is a symmetric matrix,
Question: If $A$ is a symmetric matrix, then $A^{3}$ is a______ matrix. Solution: It is given that, $A$ is symmetric matrix. $\therefore A^{T}=A$ .....(1) Now, $\left(A^{3}\right)^{T}$ $=\left(A^{T}\right)^{3} \quad\left[\left(A^{n}\right)^{T}=\left(A^{T}\right)^{n}, \forall n \in \mathrm{N}\right]$ $=A^{3}$ [Using (1)] Since $\left(A^{3}\right)^{T}=A^{3}$, so the matrix $A^{3}$ is symmetric. If $A$ is a symmetric matrix, then $A^{3}$ is a symmetric matrix...
Read More →Solve this
Question: $\left(\sec ^{2} 60^{\circ}-1\right)=?$ (a) 0(b) 2(c) 3(d) 4 Solution: As we know that, $\sec 60^{\circ}=2$ By substituting the value, we get $\left(\sec ^{2} 60^{\circ}-1\right)=(2)^{2}-1$ $=4-1$ $=3$ Hence, the correct option is (c)....
Read More →If A is a skew-symmetric matrix,
Question: If $A$ is a skew-symmetric matrix, then $A^{2}$ is a________ matrix. Solution: It is given that, $A$ is a skew-symmetric matrix. $\therefore A^{T}=-A$ Now, $\left(A^{2}\right)^{T}$ $=(A A)^{T}$ $=A^{T} A^{T}$ [For any matrices $X, Y,(X Y)^{\top}=Y^{\top} X^{\top}$ ] $=(-A)(-A)$ [Using (1)] Since $\left(A^{2}\right)^{T}=A^{2}$, so the matrix $A^{2}$ is a symmetric matrix. If $A$ is a skew-symmetric matrix, then $A^{2}$ is a symmetric matrix...
Read More →Elements with valency 1 are :
Question: Elements with valency 1 are : (a) always metals (b) always metalloids (c) either metals or non-metals (d) always non-metals. Solution: (c) Elements with valency 1 may be metals (+1 valency) or non-metals ( $-1$ valency)...
Read More →In a sample of methyl ethanoate
Question: In a sample of methyl ethanoate $\left(\mathrm{CH}_{3} \mathrm{COOCH}_{3}\right)$ the two oxygen atoms have the same number of electrons but different number of neutrons. Which of the following is the correct reason for it? (a) One of the oxygen atoms has gained electrons (b) One of the oxygen atoms has gained two neutrons (c) The two oxygen atoms are isotopes (d) The two oxygen atoms are isobars. Solution: (c) The two oxygen atoms are the isotopes. These are the different atoms of the...
Read More →Find the value of x for which
Question: Find the value ofxfor which(i)xtan 45 cot 60 = sin 30 cosec 60 (ii) $2 \operatorname{cosec}^{2} 30^{\circ}+x \sin ^{2} 60^{\circ}-\frac{3}{4} \tan ^{2} 30^{\circ}=10$ Solution: (i) As we know that, $\tan 45^{\circ}=1$ $\cot 60^{\circ}=\frac{1}{\sqrt{3}}$ $\sin 30^{\circ}=\frac{1}{2}$ $\operatorname{cosec} 60^{\circ}=\frac{2}{\sqrt{3}}$ On substituting these values, we get $x \tan 45^{\circ} \cot 60^{\circ}=\sin 30^{\circ} \operatorname{cosec} 60^{\circ}$ $\Rightarrow x(1)\left(\frac{1}...
Read More →identify the mg2+ ion from the given
Question: Identify the $\mathrm{Mg}^{2+}$ ion from the given figure where $\mathrm{n}$ and $\mathrm{p}$ represent the number of neutrons and protons respectively Solution: (d) Since the ion has 10 electrons, it is $\mathrm{Mg}^{2+}$ ion....
Read More →The ion of an element has 3 positive charges.
Question: The ion of an element has 3 positive charges. Mass, number of the atom is 27 and the number of neutrons is 14 . What is the number of electrons in the ion? (a) 13 (b) 10 (c) 14 (d) $16 .$ Solution: (b) The ion has 13 protons $(27-14)$ and 10 electrons. It is $\mathrm{Al}^{3+}$ ion....
Read More →If A is symmetric matrix,
Question: If $A$ is symmetric matrix, then $B^{\top} A B$ is Solution: It is given that, $A$ is symmetric matrix. $\therefore A^{T}=A$ ...(1) Now, $\left(B^{T} A B\right)^{T}$ $\left(B^{T} A B\right)^{T}$ $=B^{T} A^{T}\left(B^{T}\right)^{T} \quad\left[\right.$ For any matrices $\left.X, Y, Z,(X Y Z)^{\top}=Z^{\top} Y^{\top} X^{\top}\right]$ $=B^{T} A B$ [Using (1)] Since $\left(B^{T} A B\right)^{T}=B^{T} A B$, so the matrix $B^{\top} A B$ is symmetric. If $A$ is symmetric matrix, then $B^{\top} ...
Read More →If A is symmetric matrix,
Question: If $A$ is symmetric matrix, then $B^{\top} A B$ is Solution: It is given that, $A$ is symmetric matrix. $\therefore A^{T}=A$ ...(1) Now, $\left(B^{T} A B\right)^{T}$ $\left(B^{T} A B\right)^{T}$ $=B^{T} A^{T}\left(B^{T}\right)^{T} \quad\left[\right.$ For any matrices $\left.X, Y, Z,(X Y Z)^{\top}=Z^{\top} Y^{\top} X^{\top}\right]$ $=B^{T} A B$ [Using (1)] Since $\left(B^{T} A B\right)^{T}=B^{T} A B$, so the matrix $B^{\top} A B$ is symmetric. If $A$ is symmetric matrix, then $B^{\top} ...
Read More →Rutherford's a-particle scattering experiment showed that
Question: Rutherford's a-particle scattering experiment showed that (i) electrons have negative charge (ii) the mass and positive charge of the atom is concentrated in the nucleus (iii) neutron exists in the nucleus (iv) most of the space in atom is empty Which of the above statements are correct? (a) (i) and (iii) (b) (ii) and (iv) (c) (i) and (iv) (d) (iii) and (iv). Solution: (b) The statements (ii) and (iv) are correct....
Read More →In the Thomson's model of atom,
Question: In the Thomson's model of atom, which of the following statements are correct? (i) The mass of the atom is assumed to be uniformly distributed over the atom (ii) The positive charge is assumed to be uniformly distributed over the atom (iii) The electrons are uniformly distributed in the positively charged sphere (iv) The electrons attract each other to stabilise the atom (a) (i), (ii) and (iii) (b) (i) and (iii) (c) (i) and (iv) (d) (i), (iii) and (iv). Solution: (a) Except for the sta...
Read More →If cosec (A + B) = 1 and cosec(A – B) = 2, 0°
Question: If cosec (A+B) = 1 and cosec(A B) = 2, 0 (A + B) 90 andABthen find the values of :(i) sinAcosB+ cosAsinB (ii) $\frac{\tan A-\tan B}{1+\tan A \tan B}$ Solution: As we know that, $\operatorname{cosec} 90^{\circ}=1$ Thus, if $\operatorname{cosec}(A+B)=1$ $\Rightarrow A+B=90^{\circ} \quad \ldots(1)$ and $\operatorname{cosec} 30^{\circ}=2$ Thus, if $\operatorname{cosec}(A-B)=2$ $\Rightarrow A-B=30^{\circ} \quad \ldots(2)$ Solving $(1)$ and $(2)$, we get $A=60^{\circ}$ and $B=30^{\circ} \qua...
Read More →Which of the following are true for an element?
Question: Which of the following are true for an element? (i) Atomic number = number of protons + number of electrons (ii) Mass number = number of protons + number of neutrons (iii) Atomic Mass = number of protons + number of neutrons (iv) Atomic number = number of protons = number of electrons (a) (i) and (ii) (b) (i) and (iii) (c) (ii) and (iii) (d) (ii) and (iv). Solution: (d) The statements (it) and (iv) are both correct....
Read More →Which of the following statements about Rutherford's model of atom are correct ?
Question: Which of the following statements about Rutherford's model of atom are correct ? (i) Considered the nucleus as positively charged (ii) Established that the a-particles are four times as heavy as a hydrogen atom (iii) Can be compared to solar system (iv) Was in agreement with Thomson's model (a) (i) and (iii) (b) (ii) and (iii) (c) (i) and (iv) (d) only (i). Solution: (a) The statements (i) and (iii) are both correct....
Read More →Daltons atomic theory successfully explained:
Question: Daltons atomic theory successfully explained: (i) Law of conservation of mass (ii) Law of constant composition (iii)Law of radioactivity (iv) Law of multiple proportion (a) (i), (ii) and (iii) (b) (i), (ii) and (iv) (c) (ii), (iii) and (iv) (d) (i), (ii) and (iv). Solution: (d) Except for the law of radioactivity, Dalton's atomic theory explained all other laws which have been listed....
Read More →If A and B are symmetric matrices of the same order, then AB is symmetric iff
Question: If $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff _________ Solution: It is given that,AandB are symmetric matrices of the same order. $\therefore A^{T}=A$ and $B^{T}=B$ ....(1) Now, $A B$ is symmetric if $(A B)^{T}=A B$ $\Rightarrow B^{T} A^{T}=A B$ $\Rightarrow B A=A B$ [Using (1)] Thus, if $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff $A B=B A$. If $A$ and $B$ are symmetric matrices of the same order, then $A B$ ...
Read More →The number of electrons in an element X
Question: The number of electrons in an element $X$ is 15 and the number of neutrons is 16 . Which of the following is the correct representation of the element? (a) ${ }_{15}^{31} \mathrm{X}$ (b) ${ }_{16}^{31} \mathrm{X}$ (c) ${ }_{15}^{16} \mathrm{X}$ (d) ${ }_{16}^{15} \mathrm{X}$. Solution: (a) Mass no. $=$ no. of electrons (protons) $+$ no. of neutrons $=15+16=31$....
Read More →If A and B are symmetric matrices of the same order, then AB is symmetric iff
Question: If $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff _________ Solution: It is given that,AandB are symmetric matrices of the same order. $\therefore A^{T}=A$ and $B^{T}=B$ ....(1) Now, $A B$ is symmetric if $(A B)^{T}=A B$ $\Rightarrow B^{T} A^{T}=A B$ $\Rightarrow B A=A B$ [Using (1)] Thus, if $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff $A B=B A$. If $A$ and $B$ are symmetric matrices of the same order, then $A B$ ...
Read More →Rutherfords alpha (a) particles scattering
Question: Rutherfords alpha (a) particles scattering experiment resulted in to discovery of: (a) Electron (b) Proton (c) Nucleus in the atom (d) Atomic mass. Solution: (c) Nucleus in the atom...
Read More →Which of the following correctly represents
Question: Which of the following correctly represents the electonic distribution in the Mg atom? (a) $3,8,1$ (b) $2,8,2$ (c) $1,8,3$ (d) $8,2,2$ Solution: (b) $2,8,2$...
Read More →Addition of matrices is defined if order of the matrices is
Question: Addition of matrices is defined if order of the matrices is __________ Solution: Two or more matrices can be added if they are of same order. Addition of matrices is defined if order of the matrices is same...
Read More →Addition of matrices is defined if order of the matrices is
Question: Addition of matrices is defined if order of the matrices is __________ Solution: Two or more matrices can be added if they are of same order. Addition of matrices is defined if order of the matrices is same...
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