(cos 0° + sin 30° + sin 45°) (sin 90° + cos 60° – cos 45°) = ?
Question: (cos 0 + sin 30 + sin 45) (sin 90 + cos 60 cos 45) = ? (a) $\frac{7}{4}$ (b) $\frac{5}{6}$ (c) $\frac{3}{5}$ (d) $\frac{5}{8}$ Solution: As we know that, $\cos 0^{\circ}=1$ $\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ $\sin 30^{\circ}=\frac{1}{2}$ $\sin 90^{\circ}=1$ $\cos 60^{\circ}=\frac{1}{2}$ $\sin 45^{\circ}=\frac{1}{\sqrt{2}}$ By substituting these values, we get $\left(\cos 0^{\circ}+\sin 30^{\circ}+\sin 45^{\circ}\right)\left(\sin 90^{\circ}+\cos 60^{\circ}-\cos 45^{\circ}\right)=\left...
Read More →In applying one or more row operations while finding
Question: In applying one or more row operations while finding $A^{-1}$ by elementary row operations, we obtain all zeroes in one or more row, then $A^{-1}$ Solution: Let $A$ be a square matrix. In order to find the inverse of matrix $A$ using elementary row operations, we write $A=I A$. Now, perform a sequence of elementary row operations successively on $A$ on the LHS and the pre-factor $I$ on RHS, till we get $I=B A$. Here, $B$ is the inverse of of matrix $A$. However, in applying one or more...
Read More →In the atom of an element X, 6
Question: In the atom of an element $X, 6$ electrons are present in the outermost shell. If it acquires noble gas configuration by accepting requisite number of electrons, then what would be the charge on the ion so formed? Solution: With six (6) electrons in the outermost shell, the atom of the element needs two (2) electrons more to have eight (8) electrons in the outermost shell. The ion $\left(x^{2-}\right)$ has $-2$ units charge....
Read More →Write down the electron distribution of chlorine atom.
Question: Write down the electron distribution of chlorine atom. How many electrons are there in the $L$ shell ? (Atomic number of chlorine is 17 ). Solution: The electronic distribution in chlorine $(Z=17)$ is $K(2), L(8)$ and $M(7)$. The $L$-shell has eight electrons....
Read More →One electron is present in the outermost
Question: One electron is present in the outermost shell of the atom of an element X. What would be the nature and value of charge on the ion formed if this electron is removed from the outermost shell? Solution: Since the atom of the element $X$ has lost one electron from the outermost shell, it has +1 unit charge....
Read More →Simplify:
Question: Simplify: (i) $\left\{4^{-1} \times 3^{-1}\right\}^{2}$ (ii) $\left\{5^{-1} \div 6^{-1}\right\}^{3}$ (iii) $\left(2^{-1}+3^{-1}\right)^{-1}$ (iv) $\left\{3^{-1} \times 4^{-1}\right\}^{-1} \times 5^{-1}$ (v) $\left(4^{-1}-5^{-1}\right) \div 3^{-1}$ Solution: $(i)\left(4^{-1} \times 3^{-1}\right)^{2}=\left(\frac{1}{4} \times \frac{1}{3}\right)^{2} \quad \ldots\left(a^{-1}=1 / a\right)$ $=\left(\frac{1}{12}\right)^{2}$ $=\frac{(1)^{2}}{(12)^{2}} \quad \cdots\left((a / b)^{n}=\left(a^{n}\r...
Read More →If A and B are symmetric matrices of the same order, then
Question: IfAandBare symmetric matrices of the same order, then (i) $A B-B A$ is a_________ (ii) $B A-2 B A$ is a________ Solution: It is given that, $A$ and $B$ are symmetric matrices of the same order. $\therefore A^{T}=A$ and $B^{T}=B$ (i) $(A B-B A)^{T}$ $=(A B)^{T}-(B A)^{T} \quad\left[(X+Y)^{T}=X^{T}+Y^{T}\right]$ $=B^{T} A^{T}-A^{T} B^{T} \quad\left[(X Y)^{T}=Y^{T} X^{T}\right]$ $=B A-A B$ [Using (1)] $=-(A B-B A)$ Since $(A B-B A)^{T}=-(A B-B A)$, so the matrix $A B-B A$ is a skew-symmet...
Read More →Solve this
Question: $\left(\cos ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \sec ^{2} 60^{\circ}+\frac{1}{2} \cos ^{2} 90^{\circ}-2 \tan ^{2} 60^{\circ}\right)=?$ (a) $\frac{81}{8}$ (b) $\frac{83}{8}$ (C) $\frac{73}{8}$ (d) $\frac{75}{8}$ Solution: As we know that, $\cos 30^{\circ}=\frac{\sqrt{3}}{2}$ $\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ $\sec 60^{\circ}=2$ $\cos 90^{\circ}=0$ $\tan 60^{\circ}=\sqrt{3}$ By substituting these values, we get $\left(\cos ^{2} 30^{\circ} \cos ^{2} 45^{\circ}+4 \sec ^{2} 60^{\circ}...
Read More →Find out the valency of
Question: Find out the valency of the atoms represented by the Fig. (a) and (b). Solution: (a) The electronic configuration of the atom is $2,8,8$. It has completely filled $K, L, M$ shells. Its valency is zero. The atom belongs to the element Argon (Ar). (b) The electronic configuration of the atom is 2,7 . It has seven electrons in the valence shell. Its valency is $(8-7)$ equal to one. The atom belongs to the element fluorine (F)....
Read More →Why did Rutherford select a gold foil in his a-scattering experiment?
Question: Why did Rutherford select a gold foil in his a-scattering experiment? Solution: Gold is a highly malleable. It could be converted into very fine foils or leaves. Moreover, the nucleus of the element is very heavy. It could not be displaced by the impact of fast moving a-particles....
Read More →Express each of the following as a rational number in the form
Question: Express each of the following as a rational number in the form $\frac{p}{q}$ : (i) 61 (ii) (7)1 (iii) $\left(\frac{1}{4}\right)^{-1}$ (iv) $(-4)^{-1} \times\left(\frac{-3}{2}\right)^{-1}$ (v) $\left(\frac{3}{5}\right)^{-1} \times\left(\frac{5}{2}\right)^{-1}$ Solution: (i) $6^{-1}=\frac{1}{6} \quad \ldots\left(a^{-1}=1 / a\right)$ (ii) $(-7)^{-1}=\frac{1}{-7} \quad \cdots\left(a^{-1}=1 / a\right)$ $=\frac{-1}{7}$ (iii) $\left(\frac{1}{4}\right)^{-1}=\frac{1}{1 / 4} \quad \cdots\left(a^...
Read More →(3cos260° + 2cot230° – 5sin245°) = ?
Question: (3cos260 + 2cot230 5sin245) = ? (a) 1(b) 4 (c) $\frac{17}{4}$ (d) $\frac{13}{6}$ Solution: As we know that, $\sin 45^{\circ}=\frac{1}{\sqrt{2}}$ $\cos 60^{\circ}=\frac{1}{2}$ $\cot 30^{\circ}=\sqrt{3}$ By substituting these values, we get $\left(3 \cos ^{2} 60^{\circ}+2 \cot ^{2} 30^{\circ}-5 \sin ^{2} 45^{\circ}\right)=3\left(\frac{1}{2}\right)^{2}+2(\sqrt{3})^{2}-5\left(\frac{1}{\sqrt{2}}\right)^{2}$ $=3\left(\frac{1}{4}\right)+2(3)-5\left(\frac{1}{2}\right)$ $=\frac{3}{4}+6-\frac{5}...
Read More →Prove the following
Question: Will ${ }^{35} \mathrm{Cl}$ and ${ }^{37} \mathrm{Cl}$ have different valencies? Justify your answer. Solution: The atomic number $(Z)$ of the element $(C l)$ is $17 .$ Its electronic distribution is $2,8,7 .$ It has valency $(8-7)=1$ which is the same for both the species, which act as isotopes....
Read More →Write any two observations to show that atoms are divisible.
Question: Write any two observations to show that atoms are divisible. Solution: The discovery of electrons and protons led to the opinion that the atoms are divisible. Even neutrons were also discovered at a later stage....
Read More →An element has one proton,
Question: An element has one proton, one electron and no neutron. Name the element. How will you represent it? Solution: The element is called protium or hydrogen $(\mathrm{H})$. It can be represented as ${ }^{1}{ }_{1} \mathrm{H}$....
Read More →(sin230° – sec260° + 4cot245°) = ?
Question: (sin230 sec260 + 4cot245) = ? (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 →Evaluate:
Question: Evaluate: (i) 52 (ii) (3)2 (iii) $\left(\frac{1}{3}\right)^{-4}$ (iv) $\left(\frac{-1}{2}\right)^{-1}$ Solution: (i) $5^{-2}=\frac{1}{5^{2}} \quad \quad \cdots\left(a^{-n}=1 /\left(a^{n}\right)\right)$ $=\frac{1}{25}$ $(\mathrm{ii})(-3)^{2}=\frac{1}{3^{2}} \quad \ldots\left(a^{-n}=1 /\left(a^{n}\right)\right)$ $=\frac{1}{9}$ (iii) $\left(\frac{1}{3}\right)^{-4}=\frac{1}{(1 / 3)^{4}} \quad \cdots\left(a^{-n}=1 /\left(a^{n}\right)\right)$ $=\frac{1}{1 / 81}$ = 81 $($ iv $)\left(\frac{-1}...
Read More →Atomic models have been improved over the years.
Question: Atomic models have been improved over the years. Arrange the following atomic models in the order of their chronological order: (i) Rutherford's atomic model (ii) Thomson's atomic model (iii) Bohr's atomic model (a) (i), (ii) and (iii) (b) (ii), (iii) and (i) (c) (ii), (i) and (iii) (d) (iii), (ii) and (i). Solution: (c). It is the correct order....
Read More →Which of the following statements is always correct?
Question: Which of the following statements is always correct? (a) An atom has equal number of electrons and protons. (b) An atom has equal number of electrons and neutrons. (c) An atom has equal number of protons and neutrons. (d) An atom has equal number of electrons, protons and neutrons. Solution: (a) The number of electrons or protons in an atom are always equal....
Read More →Which of the following do not represent
Question: Which of the following do not represent Bohr's model of an atom correctly? (a) (i) and (ii) (b) (ii) and (iii) (c) (ii) and (iv) (d) (i) and (iv). Solution: (c) The first shell (K) cannot have more than two electrons in it....
Read More →Write each of the following in exponential form:
Question: Write each of the following in exponential form: (i) $\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1}$ (ii) $\left(\frac{2}{5}\right)^{-2} \times\left(\frac{2}{5}\right)^{-2} \times\left(\frac{2}{5}\right)^{-2}$ Solution: (i) $\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1} \times\left(\frac{3}{2}\right)^{-1}=\left(\frac{3}{2}\right)^{-1+(-1)+(...
Read More →The electronic distribution in an aluminium atom is :
Question: The electronic distribution in an aluminium atom is : (a) $2,8,3$ (b) $2,8,2$ (c) $8,2,3$ (d) $2,3,8$ Solution: (d) $2,3,8$...
Read More →An atom with 3 protons and 4 neutrons
Question: An atom with 3 protons and 4 neutrons will have a valency of : (a) 3 (b) 7 (c) 1 (d) $4 .$ Solution: (c) The atom has also 3 electrons with electronic configuration 2,1 . It has valency of 1 ....
Read More →The first model of an atom was given by.
Question: The first model of an atom was given by. (a) N. Bohr (b) E. Goldstein (c) Rutherford (d) J.J. Thomson. Solution: (d) J.J. Thomson....
Read More →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)....
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