Prove that
Question: Prove that $(\vec{a}+\vec{b}) \cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2}$, if and only if $\vec{a}, \vec{b}$ are perpendicular, given $\vec{a} \neq \overrightarrow{0}, \vec{b} \neq \overrightarrow{0}$. Solution: $(\vec{a}+\vec{b}) \cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+|\vec{b}|^{2}$ $\Leftrightarrow \vec{a} \cdot \vec{a}+\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{a}+\vec{b} \cdot \vec{b}=|\vec{a}|^{2}+|\vec{b}|^{2}$ [Distributivity of scalar products over addition] $\Leftrightarro...
Read More →Prove that:
Question: Prove that: $\sin ^{2} \frac{\pi}{8}+\sin ^{2} \frac{3 \pi}{8}+\sin ^{2} \frac{5 \pi}{8}+\sin ^{2} \frac{7 \pi}{8}=2$ Solution: $\mathrm{LHS}=\sin ^{2} \frac{\pi}{8}+\sin ^{2} \frac{3 \pi}{8}+\sin ^{2} \frac{5 \pi}{8}+\sin ^{2} \frac{7 \pi}{8}$ $=\sin ^{2}\left(\frac{\pi}{2}-\frac{3 \pi}{8}\right)+\sin ^{2}\left(\frac{\pi}{2}-\frac{\pi}{8}\right)+\sin ^{2} \frac{5 \pi}{8}+\sin ^{2} \frac{7 \pi}{8}$ $=\cos ^{2} \frac{3 \pi}{8}+\sin ^{2} \frac{\pi}{8}+\sin ^{2}\left(\pi-\frac{3 \pi}{8}\r...
Read More →Prove that:
Question: Prove that: $\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2} \frac{5 \pi}{8}+\cos ^{2} \frac{7 \pi}{8}=2$ Solution: $\mathrm{LHS}=\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2} \frac{5 \pi}{8}+\cos ^{2} \frac{7 \pi}{8}$ $=\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2}\left(\pi-\frac{3 \pi}{8}\right)+\cos ^{2}\left(\pi-\frac{\pi}{8}\right)$ $=\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\left\{-\cos \left(\frac{3 \pi}{8}\right)\right\}^{2}+\lef...
Read More →If are mutually perpendicular vectors of equal magnitudes, show that the vector
Question: If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{a}+\vec{b}+\vec{c}$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$. Solution: Since $\vec{a}, \vec{b}$, and $\vec{c}$ are mutually perpendicular vectors, we have $\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}=\vec{c} \cdot \vec{a}=0$ It is given that: $|\vec{a}|=|\vec{b}|=|\vec{c}|$ Let vector $\vec{a}+\vec{b}+\vec{c}$ be inclined to $\vec{a}, \vec{b}$, and $\vec{c}$...
Read More →Prove that:
Question: Prove that: $\frac{\cos x}{1-\sin x}=\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)$ Solution: $\mathrm{LHS}=\frac{\cos x}{1-\sin x}$ $=\frac{\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}}{\sin ^{2} \frac{x}{2}+\cos ^{2} \frac{x}{2}-2 \sin \frac{x}{2} \times \cos \frac{x}{2}} \quad\left[\because \cos x=\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}, \sin x=2 \sin \frac{x}{2} \cos \frac{x}{2}\right.$ and $\left.\sin ^{2} \frac{x}{2}+\cos ^{2} \frac{x}{2}=1\right]$ $=\frac{\left(\cos \frac{x}{2}...
Read More →Prove that:
Question: Prove that: $\frac{\cos 2 x}{1+\sin 2 x}=\tan \left(\frac{\pi}{4}-x\right)$ Solution: $\mathrm{LHS}=\frac{\cos 2 x}{1+\sin 2 x}$ $=\frac{\cos ^{2} x-\sin ^{2} x}{\sin ^{2} x+\cos ^{2} x+2 \sin x \times \cos x} \quad\left[\because \cos 2 x=\cos ^{2} x-\sin ^{2} x\right.$ and $\left.\sin ^{2} x+\cos ^{2} x=1\right]$ $=\frac{(\cos x-\sin x)(\cos x+\sin x)}{(\cos x+\sin x)^{2}} \quad\left[\because a^{2}-b^{2}=(a+b)(a-b)\right]$ $=\frac{\cos x-\sin x}{\cos x+\sin x}$ On dividing the numerat...
Read More →Prove that:
Question: Prove that: $\frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x$ Solution: $\mathrm{LHS}=\frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}$ $=\frac{\sin x+\sin 2 x}{\cos x+(1+\cos 2 x)}$ $=\frac{\sin x+2 \sin x \cos x}{\cos x+2 \cos ^{2} x} \quad\left[\because \sin 2 x=2 \sin x \cos x\right.$ and $\left.2 \cos ^{2} x=1+\cos 2 x\right]$ $=\frac{\sin x(1+2 \cos x)}{\cos x(1+2 \cos x)}$ $=\tan x=\mathrm{RHS}$ Hence proved....
Read More →The following table gives the quantity of goods (in crores of rupees)
Question: The following table gives the quantity of goods (in crores of rupees) Represent this information with the help of a bar graph. Explain through the bar graph if the quantity of goods carried by the Indian Railways in 1965-66 is more than double the quantity of goods carried in the year 1950 51. Solution: To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the horizontal and vertical axes represent the years and the quanti...
Read More →Prove that:
Question: Prove that: $\frac{1-\cos 2 x+\sin 2 x}{1+\cos 2 x+\sin 2 \mathrm{x}}=\tan x$ Solution: LHS $=\frac{1-\cos 2 x+\sin 2 x}{1+\cos 2 x+\sin 2 x}$ $=\frac{2 \sin ^{2} x+\sin 2 x}{2 \cos ^{2} x+\sin 2 x} \quad\left[\because 2 \sin ^{2} x=1-\cos 2 x\right.$ and $\left.2 \cos ^{2} x=1+\cos 2 x\right]$ $=\frac{2 \sin ^{2} x+2 \sin x \cos x}{2 \cos ^{2} x+2 \sin x \cos x} \quad(\because \sin 2 x=2 \sin x \cos x)$ $=\frac{2 \sin x(\sin x+\cos x)}{2 \cos x(\cos x+\sin x)}$ $=\tan \theta=\mathrm{R...
Read More →Prove that:
Question: Prove that: $\sqrt{2+\sqrt{2+2 \cos 4 x}}=2 \cos x$ Solution: LHS $=\sqrt{2+\sqrt{2+2 \cos 4 x}}$ $=\sqrt{2+\sqrt{2(1+\cos 4 x)}}$ $=\sqrt{2+\sqrt{2 \times 2 \cos ^{2} 2 x}} \quad\left(\because 2 \cos ^{2} 2 x=1+\cos 4 x\right)$ $=\sqrt{2+2 \cos 2 x}$ $=\sqrt{2(1+\cos 2 x)}$ $=\sqrt{2.2 \cos ^{2} x} \quad\left(\because 2 \cos ^{2} x=1+\cos 2 x\right)$ $=2 \cos x=\mathrm{RHS}$ Hence proved....
Read More →The scalar product of the vector
Question: The scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is equal to one. Find the value of $\lambda$. Solution: $(2 \hat{i}+4 \hat{j}-5 \hat{k})+(\lambda \hat{i}+2 \hat{j}+3 \hat{k})$ $=(2+\lambda) \hat{i}+6 \hat{j}-2 \hat{k}$ Therefore, unit vector along $(2 \hat{i}+4 \hat{j}-5 \hat{k})+(\lambda \hat{i}+2 \hat{j}+3 \hat{k})$ is given as: $\frac{(2+\lambda) \hat{i}+6...
Read More →The following data gives the value (in crores of rupees) of the Indian export of cotton textiles for different years:
Question: The following data gives the value (in crores of rupees) of the Indian export of cotton textiles for different years: Represent the above data with the help of a bar graph. Indicate with the help of a bar graph the year in which the rate of increase in exports is maximum over the preceding year. Solution: To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the horizontal and vertical axes represent the years and the valu...
Read More →Prove that:
Question: Prove that: $\frac{\sin 2 x}{1+\cos 2 x}=\tan x$ Solution: $\mathrm{LHS}=\frac{\sin 2 x}{1+\cos 2 x}$ $=\frac{2 \sin x \times \cos x}{1+2 \cos ^{2} x-1} \quad\left[\because \sin 2 x=2 \sin x \times \cos x\right.$ and $\left.\cos 2 x=2 \cos ^{2} x-1\right]$ $=\frac{2 \sin x \times \cos x}{2 \cos x \times \cos x}$ $=\frac{\sin x}{\cos x}$ $=\tan x=\mathrm{RHS}$ Hence proved....
Read More →Prove that:
Question: Prove that: $\frac{\sin 2 x}{1-\cos 2 x}=\cot x$ Solution: LHS $=\frac{\sin 2 x}{1-\cos 2 x}$ $=\frac{2 \sin x \times \cos x}{2 \sin ^{2} x} \quad\left(\because \sin 2 x, 1-\cos 2 x=2 \sin ^{2} x\right)$ $=\frac{2 \sin x \times \cos x}{2 \sin x \times \sin x}$ $=\frac{\cos x}{\sin x}$ $=\cot x=\mathrm{RHS}$ Hence proved....
Read More →Let
Question: Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$. Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d}=15$. Solution: Let $\vec{d}=d_{1} \hat{i}+d_{2} \hat{j}+d_{3} \hat{k}$ Since $\vec{d}$ is perpendicular to both $\vec{a}$ and $\vec{b}$, we have: $\vec{d} \cdot \vec{a}=0$ $\Rightarrow d_{1}+4 d_{2}+2 d_{3}=0$ ...(1) And, $\vec{d} \cdot \vec{b}=0$ $\Rightarro...
Read More →Prove that:
Question: Prove that: $\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}=\tan x$ Solution: $\mathrm{LHS}=\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}$ $=\sqrt{\frac{2 \sin ^{2} x}{2 \cos ^{2} x}} \quad\left[\because 1-\cos 2 x=2 \sin ^{2} x\right.$ and $\left.1+\cos 2 x=2 \cos ^{2} x\right]$ $=\frac{\sin x}{\cos x}$ $=\tan x=\mathrm{RHS}$ Hence proved....
Read More →The investments (in ten crores of rupees) of Life Insurance Corporation of India in different sectors is given below:
Question: The investments (in ten crores of rupees) of Life Insurance Corporation of India in different sectors is given below: Represent the above data with the help of a bar graph. Solution: To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the horizontal and vertical axes represent the sectors and the investment in ten Crores of rupees respectively. We have to draw 6 bars of different lengths given in the table. At first, we ...
Read More →Show that the direction cosines of a vector equally inclined to the axes
Question: Show that the direction cosines of a vector equally inclined to the axes $O X, O Y$ and $O Z$ are $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$. Solution: Let a vector be equally inclined to axes OX, OY, and OZ at angle. Then, the direction cosines of the vector are cos, cos, and cos. Now, $\cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha=1$ $\Rightarrow 3 \cos ^{2} \alpha=1$ $\Rightarrow \cos \alpha=\frac{1}{\sqrt{3}}$ Hence, the direction cosines of the vector which ...
Read More →The income and expenditure for 5 years of a family is given in the following data:
Question: The income and expenditure for 5 years of a family is given in the following data: Represent the above data by a bar graph. Solution: To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the horizontal and vertical axes represent the years and the income or expenditure in thousand rupees respectively. We have to draw 5 bars for each income and expenditure side by side of different lengths given in the table. At first, we ...
Read More →The two adjacent sides of a parallelogram are
Question: The two adjacent sides of a parallelogram are $2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\hat{i}-2 \hat{j}-3 \hat{k}$. Find the unit vector parallel to its diagonal. Also, find its area. Solution: Adjacent sides of a parallelogram are given as: $\vec{a}=2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}-3 \hat{k}$ Then, the diagonal of a parallelogram is given by $\vec{a}+\vec{b}$. $\vec{a}+\vec{b}=(2+1) \hat{i}+(-4-2) \hat{j}+(5-3) \hat{k}=3 \hat{i}-6 \hat{j}+2 \hat{k}$ Thus, the...
Read More →The following data gives the demand estimates of the Government of India,
Question: The following data gives the demand estimates of the Government of India, Department of Electronics for the personnel in the Computer sector during the Eighth Plan period (1990 95): Represent the data with the help of a bar graph. Indicate with the help of the bar graph the course where the estimated requirement is least. Solution: To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the horizontal and vertical axes repre...
Read More →If cos θ=513, find the value of sin2 θ−cos2 θ2 sin θ cos θ×1tan2 θ.
Question: If $\cos \theta=\frac{5}{13}$, find the value of $\frac{\sin ^{2} \theta-\cos ^{2} \theta}{2 \sin \theta \cos \theta} \times \frac{1}{\tan ^{2} \theta}$ Solution: Given: $\cos \theta=\frac{5}{13}$ To Find: The value of expression $\frac{\sin ^{2} \theta-\cos ^{2} \theta}{2 \sin \theta \cos \theta} \times \frac{1}{\tan ^{2} \theta}$ Now, we know that $\cos \theta=\frac{\text { Base side adjacent to } \angle \theta}{\text { Hypotenuse }}$ (2) Now when we compare equation (1) and (2) We g...
Read More →The following data gives the amount of manure (in thousand tones) manufactured by a company during some years:
Question: The following data gives the amount of manure (in thousand tones) manufactured by a company during some years: (i) Represent the above data with the help of a bar graph. (ii) Indicate with the help of the bar graph the year in which the amount of manufactured by the company was maximum. (iii) Choose the correct alternative: The consecutive years during which there was the maximum decrease in manure production are: (a) 1994 and 1995 (b) 1992 and 1993 (c) 1996 and 1997 (d) 1995 and 1996 ...
Read More →Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
Question: Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $(2 \vec{a}+\vec{b})$ and $(\vec{a}-3 \vec{b})$ externally in the ratio $1: 2$. Also, show that $P$ is the mid point of the line segment $R Q$. Solution: It is given that $\overrightarrow{\mathrm{OP}}=2 \vec{a}+\vec{b}, \overrightarrow{\mathrm{OQ}}=\vec{a}-3 \vec{b}$. It is given that point R divides a line segment joining two points P and Q externally in the ratio 1...
Read More →The following data gives the production of food grains (In thousand tonnes) for some years:
Question: The following data gives the production of food grains (In thousand tonnes) for some years: Represent the above data with the help of a bar graph. Solution: To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the horizontal and vertical axes represent the years and the production of food grains in thousand tonnes respectively. We have to draw 6 bars of different lengths given in the table. At first, we mark 6 points in t...
Read More →