If A is an m × n matrix and B is n × p matrix does AB exist? If yes, write its order.
Question: If $A$ is an $m \times n$ matrix and $B$ is $n \times p$ matrix does $A B$ exist? If yes, write its order. Solution: Given: Order of $A=m \times n$ Order of $B=n \times p$ Since the number of columns in $A$ are equal to the number of rows in $B$, i.e. $n, A B$ exists. Order of $A B=$ Number of rows in $A \times$ Number of columns in $B$ $=m \times p$...
Read More →Prove that
Question: If $\sin (A-B)=\frac{1}{2}$ and $\cos (A+B)=\frac{1}{2}, 0^{\circ}(A+B) \leq 90^{\circ}$ and $AB$ then values of $A$ and $B$ are (a) (60, 30)(b) (60, 15)(c) (45, 15)(d) (60, 25) Solution: As we know that, $\sin 30^{\circ}=\frac{1}{2}$ Thus, if $\sin (A-B)=\frac{1}{2}$ $\Rightarrow A-B=30^{\circ} \quad \ldots(1)$ and $\cos 60^{\circ}=\frac{1}{2}$ Thus, if $\cos (A+B)=\frac{1}{2}$ $\Rightarrow A+B=60^{\circ} \quad \ldots(2)$ Solving (1) and (2), we get $A=45^{\circ}$ and $B=15^{\circ}$ H...
Read More →Why does the skin of your fingers shrink when you wash clothes for a long time?
Question: Why does the skin of your fingers shrink when you wash clothes for a long time? Solution: Clothes are washed with soap or detergent solution. This solution is hypertonic as composed to osmotic concentration of our skin cells. The washing solution, therefore, causes exosmosis in the skin cells that come in contact with it for some time. Because of it, the skin over the fingers shrinks while washing clothes for a long time....
Read More →Find x, if
Question: Findx, if (i) $\left(\frac{1}{4}\right)^{-4} \times\left(\frac{1}{4}\right)^{-8}=\left(\frac{1}{4}\right)^{-4 x}$ (ii) $\left(\frac{-1}{2}\right)^{-19} \times\left(\frac{-1}{2}\right)^{8}=\left(\frac{-1}{2}\right)^{-2 x+1}$ (iii) $\left(\frac{3}{2}\right)^{-3} \times\left(\frac{3}{2}\right)^{5}=\left(\frac{3}{2}\right)^{2 x+1}$ (iv) $\left(\frac{2}{5}\right)^{-3} \times\left(\frac{2}{5}\right)^{15}=\left(\frac{2}{5}\right)^{2+3 x}$ (v) $\left(\frac{5}{4}\right)^{-x} \div\left(\frac{5}{...
Read More →Do you agree that "A cell is a building unit of an organism"?
Question: Do you agree that "A cell is a building unit of an organism"? Solution: Yes. Cell is a building unit of every living organism as every living being is made up of one or more cells. In unicellular or acellular organisms, the single cell performs all the functions of life. In multicellular organisms all the cells have a similar basic structure and perform similar basic life activities. However, they become specialised to form components of different structures that perform different func...
Read More →Find x, if
Question: Findx, if (i) $\left(\frac{1}{4}\right)^{-4} \times\left(\frac{1}{4}\right)^{-8}=\left(\frac{1}{4}\right)^{-4 x}$ (ii) $\left(\frac{-1}{2}\right)^{-19} \times\left(\frac{-1}{2}\right)^{8}=\left(\frac{-1}{2}\right)^{-2 x+1}$ (iii) $\left(\frac{3}{2}\right)^{-3} \times\left(\frac{3}{2}\right)^{5}=\left(\frac{3}{2}\right)^{2 x+1}$ (iv) $\left(\frac{2}{5}\right)^{-3} \times\left(\frac{2}{5}\right)^{15}=\left(\frac{2}{5}\right)^{2+3 x}$ (v) $\left(\frac{5}{4}\right)^{-x} \div\left(\frac{5}{...
Read More →If A is a non-singular matrix,
Question: If $A$ is a non-singular matrix, then $\left(A^{\top}\right)^{-1}=$_______ Solution: It is given that, $A$ is a non-singular matrix. $\therefore|A| \neq 0$ $\Rightarrow\left|A^{\top}\right| \neq 0 \quad \quad\left(|A|=\left|A^{\top}\right|\right)$ Now. $A A^{-1}=I_{n}=A^{-1} A$ $\Rightarrow\left(A A^{-1}\right)^{T}=\left(I_{n}\right)^{T}=\left(A^{-1} A\right)^{T}$ $\Rightarrow\left(A^{-1}\right)^{T} A^{T}=I_{n}=A^{T}\left(A^{-1}\right)^{T}$ $\Rightarrow\left(A^{T}\right)^{-1}=\left(A^{...
Read More →Why are lysosomes known as “suicide bags” of a cell ?
Question: Why are lysosomes known as suicide bags of a cell ? Solution: Lysosomes contain digestive enzymes against all types of organic materials. If their covering membrane breaks as it happens during injury to cell, the digestive enzymes will spill over the cell contents and digest the same. As lysosomes are organelles which on bursting can kill the cells possessing them, they are called suicide bags....
Read More →If sin θ = cos θ, 0 ≤ θ ≤ 90° then θ = ?
Question: If sin = cos , 0 90 then = ?(a) 30(b) 45(c) 60(d) 90 Solution: Given : $\sin \theta=\cos \theta$ $\Rightarrow \sin \theta=\sin \left(90^{\circ}-\theta\right) \quad\left(\because \sin \left(90^{\circ}-\theta\right)=\cos \theta\right)$ $\Rightarrow \theta=90^{\circ}-\theta$ $\Rightarrow 2 \theta=90^{\circ}$ $\Rightarrow \theta=45^{\circ}$ Hence, the correct option is (b)....
Read More →If tan (3x + 30°) = 1 then x = ?
Question: If tan (3x+ 30) = 1 thenx= ?(a) 20(b) 15(c) 10(d) 5 Solution: As we know that, $\tan 45^{\circ}=1$ Thus, if $\tan \left(3 x+30^{\circ}\right)=1$ $\Rightarrow 3 x+30^{\circ}=45^{\circ}$ $\Rightarrow 3 x=45^{\circ}-30^{\circ}$ $\Rightarrow 3 x=15^{\circ}$ $\Rightarrow x=5^{\circ}$ Hence, the correct option is (d)....
Read More →In the Gold foil experiment of Geiger and Marsden,
Question: In the Gold foil experiment of Geiger and Marsden, that paved the way for Rutherfords model of an atom, $1.00 \%$ of the a-particles were found to deflect at angles $50^{\circ}$. If one mole of a-particles were bombarded on the gold foil, compute the number of a-particles that would deflect at angles less than $50^{\circ}$. Solution: Percentage $(\%)$ of particles deflected at an angle more than $50^{\circ}=1 \%$ Percentage $(\%)$ of a-particles deflected at an angle less than $50^{\ci...
Read More →The negative of a matrix is obtained by
Question: The negative of a matrix is obtained by multiplying it by________ Solution: Let $A$ be any matrix and $k$ be any scalar. Then $(-k) A=-(k A)$ Putting $k=1$, we get $(-1) A=-(1 A)$ $\Rightarrow(-1) A=-A$ Thus, the negative of a matrix $A$ is obtained by multiplying $A$ by $-1$. The negative of a matrix is obtained by multiplying it by $-1$...
Read More →By what number should
Question: By what number should $\left(\frac{5}{3}\right)^{-2}$ be multiplied so that the product may be $\left(\frac{7}{3}\right)^{-1} ?$ Solution: Expressing as a positive exponent, we have: $\left(\frac{5}{3}\right)^{-2}=\frac{1}{(5 / 3)^{2}} \quad \cdots\left(a^{-1}=1 / a\right)$ $=\frac{1}{25 / 9} \quad \ldots\left((a / b)^{n}=\left(a^{n}\right) /\left(b^{n}\right)\right)$ $=\frac{9}{25}$ and (7/3)1= 3/7. --- (a1= 1/a) We have to find a numberxsuch that $\frac{9}{25} \times x=\frac{3}{7}$ M...
Read More →If x tan 45° cos 60° = sin 60°cot 60° then x = ?
Question: Ifxtan 45 cos 60 = sin 60cot 60 thenx= ? (a) 1 (b) $\frac{1}{2}$ (c) $\frac{1}{\sqrt{2}}$ (d) $\sqrt{3}$ Solution: As we know that, $\tan 45^{\circ}=1$ $\cos 60^{\circ}=\frac{1}{2}$ $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ $\cot 60^{\circ}=\frac{1}{\sqrt{3}}$ By substituting these values, we get $x \tan 45^{\circ} \cos 60^{\circ}=\sin 60^{\circ} \cot 60^{\circ}$ $\Rightarrow x(1)\left(\frac{1}{2}\right)=\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{\sqrt{3}}\right)$ $\Rightarrow \frac{x}{2...
Read More →Show diagramatically the electronic
Question: Show diagramatically the electronic distribution in a sodium atom and a sodium ion and also give their atomic number. Solution: The atomic number of sodium (Na) is 11 . The electronic distribution in the atom is $K(2) L(8)$ and $L(1)$. Sodium ion (Na^t) is formed by removal of one electron from the atom. Its electronic distribution is $K(2)$ and $L(8)$. This means that $N a^{+}$ion has the elecronic configuration of Ne atom which is inert gas atom. The electronic distribution of the at...
Read More →What are the postulates of Bohr's model of an atom?
Question: What are the postulates of Bohr's model of an atom? Solution: The main postulates of the theory are listed: 1. In the extra nuclear portion of an atom, the electrons revolve in well defined circular paths known as orbits. 2. These circular orbits are also known as energy levels or energy shells. 3. These have been designated as $K, L, M, N, O, \ldots$ (or as $1,2,3,4,5, \ldots$ ) based on the energy present. 4. The order of the energy of these energy shells is: $KLMN0\ldots .$ or $1234...
Read More →Matrix multiplication is______ over matrix addition.
Question: Matrix multiplication is over matrix addition. Solution: Matrix multiplication is distributive over matrix addition. Let $A, B$ and $C$ be three matrices. Then (i) $A(B+C)=A B+A C$, whenever both sides of equality are defined (ii) $(A+B) C=A C+B C$, whenever both sides of equality are defined Matrix multiplication is distributive over matrix addition....
Read More →By what number should
Question: By what number should (15)1be divided so that the quotient may be equal to (5)1? Solution: Expressing in fractional form, we get: (15)1= 1/15,--- (a1= 1/a) and (5)1= 1/5 --- (a1= 1/a) We have to find a numberxsuch that $-\frac{1}{15} \div x=-\frac{1}{5}$ Solving this equation, we get: $-\frac{1}{15} \times \frac{1}{x}=-\frac{1}{5}$ $-\frac{1}{15}=-\frac{x}{5}$ $\frac{-5}{-15}=x$ $\therefore x=\frac{1}{3}$ Hence, (15)1should be divided by 1/3 to obtain (5)1....
Read More →In what way is the Rutherfords atomic
Question: In what way is the Rutherfords atomic model different from Thomsons atomic model? Solution: According to Thomsons model, an atom may be regarded as a positively charged sphere containing protons in which the negatively charged protons are supposed to be studied or embedded. He gave no clue about the nucleus and extranuclear portion. This was given for the first time by Rutherfords model atom with the help of a-ray scattering experiment....
Read More →Enlist the conclusions drawn by
Question: Enlist the conclusions drawn by Rutherford from his a-ray scattering experiment. Solution: He made the following conclusions from the experiment. 1. As most of the alpha particles passed through undeflected, this means that they did not come across any obstruction in their path. Thus, most of the space in an atom is expected to be empty. 2. As a few alpha particles suffered minor deflections and a very few major deflections, this means that these must have met with some obstructions in...
Read More →Prove that
Question: If $2 \cos 3 \theta=1$ then $\theta=$ ? (a) 10(b) 15(c) 20(d) 30 Solution: As we know that, $\cos 60^{\circ}=\frac{\sqrt{3}}{2}$ Thus, if $2 \cos 3 \theta=1$ $\Rightarrow \cos 3 \theta=\frac{1}{2}$ $\Rightarrow 3 \theta=60^{\circ}$ $\Rightarrow \theta=20^{\circ}$ Hence, the correct option is (c)....
Read More →Enlist the conclusions drawn by
Question: Enlist the conclusions drawn by Rutherford from his a-ray scattering experiment. Solution: He made the following conclusions from the experiment. 1. As most of the alpha particles passed through undeflected, this means that they did not come across any obstruction in their path. Thus, most of the space in an atom is expected to be empty. 2. As a few alpha particles suffered minor deflections and a very few major deflections, this means that these must have met with some obstructions in...
Read More →matrix is both symmetric and skew-symmetric matrix.
Question: __________matrix is both symmetric and skew-symmetric matrix. Solution: Let $A=\left[a_{i j}\right]$ be a matrix which is both symmetric and skew-symmetric.\ $A$ is a symmetric matrix. $\therefore a_{i j}=a_{j i}$, for all $i, j$ ....(1) Also, $A$ is a skew-symmetric matrix. $\therefore a_{i j}=-a_{j i}$, for all $i, j$ .....(2) From (1) and (2), we have $a_{i j}=-a_{i j}$, for all $i, j$ $\Rightarrow 2 a_{i j}=0$, for all $i, j$ $\Rightarrow a_{i j}=0$, for all $i, j$ $\therefore$ The...
Read More →By what number should
Question: By what number should $\left(\frac{1}{2}\right)^{-1}$ be multiplied so that the product may be equal to $\left(\frac{-4}{7}\right)^{-1} ?$ Solution: Expressing in fractional form, we get: (1/2)1= 2, --- (a1= 1/a) and (4/7)1= 7/4 --- (a1= 1/a) We have to find a numberxsuch that $2 x=-\frac{7}{4}$ Dividing both sides by 2, we get: $x=-\frac{7}{8}$ Hence, (1/2)1should be multiplied by 7/8 to obtain (4/7)1....
Read More →The ratio of the radii of hydrogen atom and its
Question: The ratio of the radii of hydrogen atom and its nucleus is $10^{5}$. Assuming the atom and the nucleus to be spherical, (i) what will be the ratio of their sizes? (ii) If atom is represented by planet earth ' $\mathrm{Re}^{\prime}=6.4 \times 10^{6} \mathrm{~m}$, estimate the radius of the nucleus. Solution: (i) For a sphere, the volume $=\frac{4}{3} \pi r^{3}$ Let ' $R$ ' be the radius of the atom and ' $r$ ' be that of the nucleus. According to available information $\mathrm{R}=10^{5}...
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