Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Question: Show that the points (2, 3, 4), (1, 2, 1), (5, 8, 7) are collinear. Solution: The given points are A (2, 3, 4), B ( 1, 2, 1), and C (5, 8, 7). It is known that the direction ratios of line joining the points, (x1,y1,z1) and (x2,y2,z2), are given by,x2x1,y2y1, andz2z1. The direction ratios of AB are (1 2), (2 3), and (1 4) i.e., 3, 5, and 3. The direction ratios of BC are (5 ( 1)), (8 ( 2)), and (7 1) i.e., 6, 10, and 6. It can be seen that the direction ratios of BC are 2 times that of...
Read More →Find the mean of 994, 996, 998, 1000, 1002.
Question: Find the mean of 994, 996, 998, 1000, 1002. Solution: Numbers are 994, 996, 998, 1000, 1002. $\therefore$ Mean $=\frac{\text { Sum of numbers }}{\text { Total numbers }}$ $=\frac{994+996+998+1000+1002}{5}$ $=\frac{4990}{5}=998$ Mean = 998...
Read More →Prove that:
Question: Prove that: $=4 \sin x \cos ^{3} x-4 \sin ^{3} x \cos x=$ RHS Solution: LHS $=\sin 4 x$ $=2 \sin 2 x \cos 2 x \quad(\because \sin 2 \theta=2 \sin \theta \cos \theta)$ Now, using the identities $\sin 2 \alpha=2 \sin \alpha \cos \alpha$ and $\cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha$, we get $\mathrm{LHS}=2(2 \sin x \cos x) \cdot\left(\cos ^{2} x-\sin ^{2} x\right)$ $=4 \sin x \cos ^{3} x-4 \sin ^{3} x \cos x=$ RHS Hence proved....
Read More →If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Question: If a line has the direction ratios 18, 12, 4, then what are its direction cosines? Solution: If a line has direction ratios of 18, 12, and 4, then its direction cosines are $\frac{-18}{\sqrt{(-18)^{2}+(12)^{2}+(-4)^{2}}}, \frac{12}{\sqrt{(-18)^{2}+(12)^{2}+(-4)^{2}}}, \frac{-4}{\sqrt{(-18)^{2}+(12)^{2}+(-4)^{2}}}$ i.e., $\frac{-18}{22}, \frac{12}{22}, \frac{-4}{22}$ $\frac{-9}{11}, \frac{6}{11}, \frac{-2}{11}$ Thus, the direction cosines are $-\frac{9}{11}, \frac{6}{11}$, and $\frac{-2...
Read More →If the heights of 5 persons are 140 cm, 150 cm, 152 cm,
Question: If the heights of 5 persons are 140 cm, 150 cm, 152 cm, 158 cm and 161 cm respectively. Find the mean height. Solution: Given: The heights of 5 persons are 140 cm, 150 cm, 152 cm, 158 cm and 161 cm $\therefore$ Mean Weight $=\frac{\text { Sum of heights }}{\text { Total no. of persons }}$ $=\frac{140+150+152+158+161}{5}$ $=\frac{761}{5}=152.2$...
Read More →Find the direction cosines of a line which makes equal angles with the coordinate axes.
Question: Find the direction cosines of a line which makes equal angles with the coordinate axes. Solution: Let the direction cosines of the line make an anglewith each of the coordinate axes. l= cos,m= cos,n= cos $l^{2}+m^{2}+n^{2}=1$ $\Rightarrow \cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \alpha=1$ $\Rightarrow 3 \cos ^{2} \alpha=1$ $\Rightarrow \cos ^{2} \alpha=\frac{1}{3}$ $\Rightarrow \cos \alpha=\pm \frac{1}{\sqrt{3}}$ Thus, the direction cosines of the line, which is equally inclined to ...
Read More →If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
Question: If a line makes angles $90^{\circ}, 135^{\circ}, 45^{\circ}$ with $x, y$ and $z$-axes respectively, find its direction cosines. Solution: Let direction cosines of the line be $I, m$, and $n$. $l=\cos 90^{\circ}=0$ $m=\cos 135^{\circ}=-\frac{1}{\sqrt{2}}$ $n=\cos 45^{\circ}=\frac{1}{\sqrt{2}}$ Therefore, the direction cosines of the line are $0,-\frac{1}{\sqrt{2}}$, and $\frac{1}{\sqrt{2}}$....
Read More →Prove that:
Question: Prove that: $\sin 4 x=4 \sin x \cos ^{3} x-4 \cos x \sin ^{3} x$ Solution: $\mathrm{LHS}=\sin 4 x$ $=2 \sin 2 x \cos 2 x \quad(\because \sin 2 \theta=2 \sin \theta \cos \theta)$ Now, using the identities $\sin 2 \alpha=2 \sin \alpha \cos \alpha$ and $\cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha$, we get LHS $=2(2 \sin x \cos x) \cdot\left(\cos ^{2} x-\sin ^{2} x\right)$ $=4 \sin x \cos ^{3} x-4 \sin ^{3} x \cos x=$ RHS Hence proved....
Read More →Prove that:
Question: Prove that: $\cos 4 x=1-8 \cos ^{2} x+8 \cos ^{4} x$ Solution: LHS $=\cos 4 x$ $=\cos (2 \times 2 x)$ $=2 \cos ^{2} \times 2 x-1 \quad\left[\because \cos 2 \theta=2 \cos ^{2} \theta-1\right]$ $=2\left(2 \cos ^{2} x-1\right)^{2}-1\left[\because \cos ^{2} 2 \theta=\left(2 \cos ^{2} \theta-1\right)^{2}\right]$ $=2\left(4 \cos ^{4} x-4 \cos ^{2} x+1\right)-1$ $=8 \cos ^{4} x-8 \cos ^{2} x+1$ $=1-8 \cos ^{2} x+8 \cos ^{4} x=$ RHS Hence proved....
Read More →Prove that:
Question: Prove that: $\cos ^{2}\left(\frac{\pi}{4}-x\right)-\sin ^{2}\left(\frac{\pi}{4}-x\right)=\sin 2 x$ Solution: LHS $=\cos ^{2}\left(\frac{\pi}{4}-x\right)-\sin ^{2}\left(\frac{\pi}{4}-x\right)$ $=\cos 2\left(\frac{\pi}{4}-x\right)$ $\left[\because \cos ^{2} \alpha-\sin ^{2} \alpha=\cos 2 \alpha\right]$ $=\cos \left(\frac{\pi}{2}-2 x\right)$ $=\sin 2 x=\mathrm{RHS}$ $\left[\because \cos \left(\frac{\pi}{2}-2 \alpha\right)=\sin 2 \alpha\right]$ Hence proved....
Read More →If θ is the angle between any two vectors
Question: If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b}$, then $|\vec{a} \vec{b}|=|\vec{a} \times \vec{b}|$ when $\theta$ isequal to (A) 0 (B) $\frac{\pi}{4}$ (C) $\frac{\pi}{2}$ (D) $\pi$ Solution: Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$. Then, without loss of generality, $\vec{a}$ and $\vec{b}$ are non-zero vectors, so that $|\vec{a}|$ and $|\vec{b}|$ are positive. $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$ $\Rightarrow|\vec{a}||\...
Read More →If cos θ=35, find the value of sin θ−1tan θ2 tan θ.
Question: If $\cos \theta=\frac{3}{5}$, find the value of $\frac{\sin \theta-\frac{1}{\tan \theta}}{2 \tan \theta}$. Solution: Given: $\cos \theta=\frac{3}{5}$...(1) To find the value of $\frac{\sin \theta-\frac{1}{\tan \theta}}{2 \tan \theta}$ Now, we know the following trigonometric identity $\cos ^{2} \theta+\sin ^{2} \theta=1$ Therefore, by substituting the value of $\cos \theta$ from equation (1), We get, $\left(\frac{3}{5}\right)^{2}+\sin ^{2} \theta=1$ Therefore, $\sin ^{2} \theta=1-\left...
Read More →If cos θ=35, find the value of sin θ−1tan θ2 tan θ.
Question: If $\cos \theta=\frac{3}{5}$, find the value of $\frac{\sin \theta-\frac{1}{\tan \theta}}{2 \tan \theta}$. Solution: Given: $\cos \theta=\frac{3}{5}$...(1) To find the value of $\frac{\sin \theta-\frac{1}{\tan \theta}}{2 \tan \theta}$ Now, we know the following trigonometric identity $\cos ^{2} \theta+\sin ^{2} \theta=1$ Therefore, by substituting the value of $\cos \theta$ from equation (1), We get, $\left(\frac{3}{5}\right)^{2}+\sin ^{2} \theta=1$ Therefore, $\sin ^{2} \theta=1-\left...
Read More →The expenditure (in 10 crores of rupees) on health by the Government of India during the various five-year plans is shown below:
Question: The expenditure (in 10 crores of rupees) on health by the Government of India during the various five-year plans is shown below: . Construct a bar graph to represent the above data. 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 expenditures on health in 10 Crores rupees respectively. We have to draw 6 bars of different lengths given in the table...
Read More →Prove that:
Question: Prove that: $(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x=0$ Solution: LHS $=(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x$ Using the identities $\sin C+\sin D=2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$ and $\cos C-\cos D=-2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}$, we get LHS $=\left(2 \sin \frac{3 x+x}{2} \times \cos \frac{3 x-x}{2} \times \sin x\right)+\left(-2 \sin \frac{3 x+x}{2} \times \sin \frac{3 x-x}{2}\right) \cos x$ $=(2 \sin 2 x \times \cos x \times \sin x)-(2 \sin 2 x...
Read More →The value of
Question: The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ is (A) 0 (B) $-1$ (C) 1 (D) 3 Solution: $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ $=\hat{i} \cdot \hat{i}+\hat{j} \cdot(-\hat{j})+\hat{k} \cdot \hat{k}$ $=1-\hat{j} \cdot \hat{j}+1$ $=1-1+1$ $=1$ The correct answer is C....
Read More →Prove that:
Question: Prove that: $\cos ^{3} 2 x+3 \cos 2 x=4\left(\cos ^{6} x-\sin ^{6} x\right)$ Solution: RHS $=4\left(\cos ^{6} x-\sin ^{6} x\right)$ $=4\left[\left(\cos ^{2} x\right)^{3}-\left(\sin ^{2} x\right)^{3}\right]$ Using the identity $a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)$, we get $=4\left(\cos ^{2} x-\sin ^{2} x\right)\left(\cos ^{4} x+\sin ^{4} x+\sin ^{2} x \cos ^{2} x\right)$ $=4\left(\cos ^{2} x-\sin ^{2} x\right)\left(\cos ^{4} x+\sin ^{4} x+\sin ^{2} x \cos ^{2} x\right)$ $=4 \c...
Read More →The production of oil (in lakh tonnes) in some of the refineries in India during 1982 was given below:
Question: The production of oil (in lakh tonnes) in some of the refineries in India during 1982 was given below: Construct a bar graph to represent the above data so that the bars are drawn horizontally. Solution: To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let u consider that the vertical and horizontal axes represent the refineries and the production of oil in la] tonnes respectively. We have to draw 5 bars of different lengths given in the ...
Read More →Let and be two unit vectors andθ is the angle between them.
Question: Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a}+\vec{b}$ is a unit vector if (A) $\theta=\frac{\pi}{4}$ (B) $\theta=\frac{\pi}{3}$ (C) $\theta=\frac{\pi}{2}$ (D) $\theta=\frac{2 \pi}{3}$ Solution: Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ be the angle between them. Then, $|\vec{a}|=|\vec{b}|=1$ Now, $\vec{a}+\vec{b}$ is a unit vector if $|\vec{a}+\vec{b}|=1$. $|\vec{a}+\vec{b}|=1$ $\Rightarrow(\vec{a}+\vec{b})^{2}...
Read More →Prove that:
Question: Prove that: $1+\cos ^{2} 2 x=2\left(\cos ^{4} x+\sin ^{4} x\right)$ Solution: $\mathrm{LHS}=1+\cos ^{2} 2 x$ Using the identity $\cos 2 x=\cos ^{2} x-s \operatorname{in}^{2} x$, we get LHS $=1+\left(\cos ^{2} x-\sin ^{2} x\right)^{2}$ $=1+\cos ^{4} x+\sin ^{4} x-2 \cos ^{2} x \sin ^{2} x$ $=\left(\cos ^{2} x+\sin ^{2} x\right)^{2}+\cos ^{4} x+\sin ^{4} x-2 \cos ^{2} x \sin ^{2} x \quad\left[\because \cos ^{2} x+\sin ^{2} x=1\right]$ $=\cos ^{4} x+\sin ^{4} x+2 \cos ^{2} x \sin ^{2} x+\...
Read More →Prove that:
Question: Prove that: $\sin ^{2}\left(\frac{\pi}{8}+\frac{x}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{x}{2}\right)=\frac{1}{\sqrt{2}} \sin x$ Solution: $\mathrm{LHS}=\sin ^{2}\left(\frac{\pi}{8}+\frac{x}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{x}{2}\right)$ $=\frac{1}{2}\left\{1-\cos 2\left(\frac{\pi}{8}+\frac{x}{2}\right)\right\}-\frac{1}{2}\left\{1-\cos 2\left(\frac{\pi}{8}-\frac{x}{2}\right)\right\}$ $=\frac{1}{2}\left\{\cos \left(\frac{\pi}{4}-x\right)-\cos \left(\frac{\pi}{4}+x\rig...
Read More →If θ is the angle between two vectors
Question: If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a} \cdot \vec{b} \geq 0$ only when (A) $0\theta\frac{\pi}{2}$ (B) $0 \leq \theta \leq \frac{\pi}{2}$ (C) $0\theta\pi$ (D) $0 \leq \theta \leq \pi$ Solution: Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$. Then, without loss of generality, $\vec{a}$ and $\vec{b}$ are non-zero vectors so that $|\vec{a}|$ and $|\vec{b}|$ are positive. It is known that $\vec{a} \cdot \vec{b}=|\vec{a}||\v...
Read More →If tan θ=1213, find the value of 2 sin θ cos θcos2 θ−sin2 θ
Question: If $\tan \theta=\frac{12}{13}$, find the value of $\frac{2 \sin \theta \cos \theta}{\cos ^{2} \theta-\sin ^{2} \theta}$ Solution: Given: $\tan \theta=\frac{12}{13}$ To find the value of $\frac{2 \sin \theta \cos \theta}{\cos ^{2} \theta-\sin ^{2} \theta}$ Now, we know the following trigonometric identity $\operatorname{cosec}^{2} \theta=1+\tan ^{2} \theta$ Therefore, by substituting the value of $\tan \theta$ from equation (1), We get, $\operatorname{cosec}^{2} \theta=1+\left(\frac{12}...
Read More →Prove that:
Question: Prove that: $\left(\cos \alpha+\cos \beta^{2}\right)+(\sin \alpha+\sin \beta)^{2}=4 \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$ Solution: LHS $=(\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}$ $=\cos ^{2} \alpha+\cos ^{2} \beta+2 \cos \alpha \cos \beta+\sin ^{2} \alpha+\sin ^{2} \beta+2 \sin \alpha \sin \beta$ $=\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right)+\left(\cos ^{2} \beta+\sin ^{2} \beta\right)+2(\cos \alpha \cos \beta+\sin \alpha \sin \beta)$ $=1+1+2 \cos (\alpha-\...
Read More →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...
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