If tan θ=ab, prove that a sin θ+b cos θa sin θ+b cos θ=a2−b2a2+b2.
Question: If $\tan \theta=\frac{a}{b}$, prove that $\frac{a \sin \theta+b \cos \theta}{a \sin \theta+b \cos \theta}=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$. Solution: Given: $\tan \theta=\frac{a}{b}$ Now, we know that $\tan \theta=\frac{\sin \theta}{\cos \theta}$ Therefore equation (1) becomes $\frac{\sin \theta}{\cos \theta}=\frac{a}{b}$....(2) Now, multiplying by $\frac{a}{b}$ on both sides of equation (2) We get, $\frac{a}{b} \times \frac{\sin \theta}{\cos \theta}=\frac{a}{b} \times \frac{a}{b}$ The...
Read More →Given below is a cumulative frequency distribution table showing ages of the people living in a locality:
Question: Given below is a cumulative frequency distribution table showing ages of the people living in a locality: Prepare a frequency distribution table. Solution:...
Read More →If (cos α + cos β)
Question: If $(\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}=\lambda \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$, write the value of $\lambda$ Solution: $(\cos \alpha+\cos \beta)^{2}+(\sin \alpha+\sin \beta)^{2}=\lambda \cos ^{2}\left(\frac{\alpha-\beta}{2}\right)$ Consider LHS: (cos + cos )2+ (sin + sin)2 $=\left[2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)\right]^{2}+\left[2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\be...
Read More →Find a unit vector perpendicular to each of the vector
Question: Find a unit vector perpendicular to each of the vector $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$, where $\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$. Solution: We have, $\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}$ $\therefore \vec{a}+\vec{b}=4 \hat{i}+4 \hat{j}, \vec{a}-\vec{b}=2 \hat{i}+4 \hat{k}$ $(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 4 4 0 \\ 2 0 4\end{array...
Read More →The following cumulative frequency distribution table shows the daily electricity consumption (in KW) of 40 factories in an industrial state.
Question: The following cumulative frequency distribution table shows the daily electricity consumption (in KW) of 40 factories in an industrial state. (1) Represent this as a frequency distribution table. (2) Prepare a cumulative frequency table. Solution: (1) (2)...
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Question: Find $|\vec{a} \times \vec{b}|$, if $\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}$ and $\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}$ Solution: We have, $\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}$ and $\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}$ $\vec{a} \times \vec{b}=\left|\begin{array}{rrc}\hat{i} \hat{j} \hat{k} \\ 1 -7 7 \\ 3 -2 2\end{array}\right|$ $=\hat{i}(-14+14)-\hat{j}(2-21)+\hat{k}(-2+21)=19 \hat{j}+19 \hat{k}$ $\therefore|\vec{a} \times \vec{b}|=\sqrt{(19)^{2}+(19)^{2}}=\sqrt{2 \times(19)^{2}}=19 \s...
Read More →Ifis a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λis unit vector if
Question: If $\vec{a}$ is a nonzero vector of magnitude ' $a$ ' and $\lambda$ a nonzero scalar, then $\lambda \vec{a}$ is unit vector if (A) $\lambda=1$ (B) $\lambda=-1$ (C) $a=|\lambda|$ (D) $a=\frac{1}{|\lambda|}$ Solution: Vector $\lambda \vec{a}$ is a unit vector if $|\lambda \vec{a}|=1$. Now, $|\lambda \vec{a}|=1$ $\Rightarrow|\lambda||\vec{a}|=1$ $\Rightarrow|\vec{a}|=\frac{1}{|\lambda|}$ $[\lambda \neq 0]$ $\Rightarrow a=\frac{1}{|\lambda|}$ $[|\vec{a}|=a]$ Hence, vector $\lambda \vec{a}$...
Read More →If 3 cot θ = 2, find the value of 4 sin θ−3 cos θ2 sin θ+6 cos θ.
Question: If $3 \cot \theta=2$, find the value of $\frac{4 \sin \theta-3 \cos \theta}{2 \sin \theta+6 \cos \theta}$. Solution: Given: $3 \cot \theta=2$ Therefore, $\cot \theta=\frac{2}{3}$..(1) Now, we know that $\cot \theta=\frac{\cos \theta}{\sin \theta}$ Therefore equation (1) becomes $\frac{\cos \theta}{\sin \theta}=\frac{2}{3}$...(2) Now, by applying Invertendo to equation (2) We get, $\frac{\sin \theta}{\cos \theta}=\frac{3}{2}$ (3) Now, multiplying by $\frac{4}{3}$ on both sides We get, $...
Read More →Given below are the cumulative frequencies showing the weights of 685 students of a school.
Question: Given below are the cumulative frequencies showing the weights of 685 students of a school. Prepare a frequency distribution table. Solution:...
Read More →The number of books in different shelves of a library is as follows:
Question: The number of books in different shelves of a library is as follows: 30, 32, 28, 24, 20, 25, 38, 37, 40, 45, 16, 20 19, 24, 27, 30, 32, 34, 35, 42, 27, 28, 19, 34 38, 39, 42, 29, 24, 27, 22, 29, 31, 19, 27, 25 28, 23, 24, 32, 34, 18, 27, 25, 37, 31, 24, 23 43, 32, 28, 31, 24, 23, 26, 36, 32, 29, 28, 21. Prepare a cumulative frequency distribution table using 45 49 as the last class-interval. Solution: The minimum number of bookshelves is 16 and maximum number of bookshelves is 45 Range...
Read More →Show that the vectors
Question: Show that the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$ form the vertices of a right angled triangle. Solution: Let vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$ be position vectors of points $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$ respectively. i.e., $\overrightarrow{\mathrm{OA}}=2 \hat{i}-\hat{j}+\hat{k}, \overrightarrow{\mathrm{OB}}=\hat{i}-3 \hat{j}-5 \hat{k}$ and $\...
Read More →The value of 2 cos
Question: The value of $2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13}+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13}$ is ______________ Solution: $2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13}+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13}$ Using identity:- 2 cosxcosy= cos (x + y) + cos (x y) $=\cos \left(\frac{9 \pi}{13}+\frac{\pi}{13}\right)+\cos \left(\frac{9 \pi}{13}-\frac{\pi}{13}\right)+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13}$ $=\cos \left(\frac{10 \pi}{13}\right)+\cos \left(\frac{8 \pi}{13}\rig...
Read More →The value of 2 cos
Question: The value of $2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13}+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13}$ is ______________ Solution: $2 \cos \frac{\pi}{13} \cos \frac{9 \pi}{13}+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13}$ Using identity:- 2 cosxcosy= cos (x + y) + cos (x y) $=\cos \left(\frac{9 \pi}{13}+\frac{\pi}{13}\right)+\cos \left(\frac{9 \pi}{13}-\frac{\pi}{13}\right)+\cos \frac{3 \pi}{13}+\cos \frac{5 \pi}{13}$ $=\cos \left(\frac{10 \pi}{13}\right)+\cos \left(\frac{8 \pi}{13}\rig...
Read More →The water bills (in rupees) of 32 houses in a certain street for the period 1.198 to 31.398 are given below:
Question: The water bills (in rupees) of 32 houses in a certain street for the period 1.198 to 31.398 are given below: 56, 43, 32, 38, 56, 24, 68, 85, 52, 47, 35, 58, 63, 74, 27, 84, 69, 35, 44, 75, 55, 30, 54, 65, 45, 67, 95, 72, 43, 65, 35, 69. Tabulate the data and present the data as a cumulative frequency table using 70-79 as one of the class intervals. Solution: The minimum bill is Rs 24 The maximum bill is Rs 95 Range = Maximum bill-Minimum bill = 95 - 24 = 71 Given class interval is 70-7...
Read More →The value of sin 50° – sin 70° + sin 10°
Question: The value of sin 50 sin 70 + sin 10 is ___________. Solution: sin 50 sin 70 + sin 10 $=2 \sin \left(\frac{50^{\circ}-70^{\circ}}{2}\right) \cos \left(\frac{50^{\circ}+70^{\circ}}{2}\right)+\sin 10^{\circ}$ using identity $\sin a-\sin b=2 \cos \left(\frac{a+b}{2}\right) \sin \left(\frac{a-b}{2}\right)$ $=2 \sin \left(-10^{\circ}\right) \cos \left(\frac{120^{\circ}}{2}\right)+\sin 10^{\circ}$ $=-2 \sin 10^{\circ} \cos 60^{\circ}+\sin 10^{\circ}$ $(\because \sin (-\theta)=-\sin \theta)$ $...
Read More →Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.
Question: Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, 1) are collinear. Solution: The given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, 1). $\therefore \overrightarrow{\mathrm{AB}}=(2-1) \hat{i}+(6-2) \hat{j}+(3-7) \hat{k}=\hat{i}+4 \hat{j}-4 \hat{k}$ $\overrightarrow{\mathrm{BC}}=(3-2) \hat{i}+(10-6) \hat{j}+(-1-3) \hat{k}=\hat{i}+4 \hat{j}-4 \hat{k}$ $\overrightarrow{\mathrm{AC}}=(3-1) \hat{i}+(10-2) \hat{j}+(-1-7) \hat{k}=2 \hat{i}+8 \hat{j}-8 \hat{k}$ $|\overrightarrow...
Read More →The value of sin 50° – sin 70° + sin 10°
Question: The value of sin 50 sin 70 + sin 10 is ___________. Solution: sin 50 sin 70 + sin 10 $=2 \sin \left(\frac{50^{\circ}-70^{\circ}}{2}\right) \cos \left(\frac{50^{\circ}+70^{\circ}}{2}\right)+\sin 10^{\circ}$ using identity $\sin a-\sin b=2 \cos \left(\frac{a+b}{2}\right) \sin \left(\frac{a-b}{2}\right)$ $=2 \sin \left(-10^{\circ}\right) \cos \left(\frac{120^{\circ}}{2}\right)+\sin 10^{\circ}$ $=-2 \sin 10^{\circ} \cos 60^{\circ}+\sin 10^{\circ}$ $(\because \sin (-\theta)=-\sin \theta)$ $...
Read More →If 1 + cos 2x + cos 4x + cos 6x = k cos x cos 2x cos 3x,
Question: If 1 + cos 2x+ cos 4x+ cos 6x=kcosxcos 2xcos 3x, thenk= ____________. Solution: Given 1 + cos 2x+ cos 4x+ cos 6x=kcosxcos 2xcos 3x Consider, L.H.S $1+\cos 2 x+\cos 4 x+\cos 6 x$$=(1+\cos 2 x)+(\cos 4 x+\cos 6 x)$ $=2 \cos ^{2} x+2 \cos \left(\frac{4 x+6 x}{2}\right) \cos \left(\frac{4 x-6 x}{2}\right)$ using identities : $-1+\cos 2 \theta=2 \cos ^{2} \theta$ and $\cos a+\cos b=2 \cos \left(\frac{a+b}{2}\right) \cos \left(\frac{a-b}{2}\right)$ $=2 \cos ^{2} x+2 \cos 5 x \cos (-x)$ $=2 \...
Read More →If 1 + cos 2x + cos 4x + cos 6x = k cos x cos 2x cos 3x,
Question: If 1 + cos 2x+ cos 4x+ cos 6x=kcosxcos 2xcos 3x, thenk= ____________. Solution: Given 1 + cos 2x+ cos 4x+ cos 6x=kcosxcos 2xcos 3x Consider, L.H.S $1+\cos 2 x+\cos 4 x+\cos 6 x$$=(1+\cos 2 x)+(\cos 4 x+\cos 6 x)$ $=2 \cos ^{2} x+2 \cos \left(\frac{4 x+6 x}{2}\right) \cos \left(\frac{4 x-6 x}{2}\right)$ using identities : $-1+\cos 2 \theta=2 \cos ^{2} \theta$ and $\cos a+\cos b=2 \cos \left(\frac{a+b}{2}\right) \cos \left(\frac{a-b}{2}\right)$ $=2 \cos ^{2} x+2 \cos 5 x \cos (-x)$ $=2 \...
Read More →Following are the ages of 360 patients getting medical treatment in a hospital on a day.
Question: Following are the ages of 360 patients getting medical treatment in a hospital on a day. Construct a cumulative frequency table. Solution:...
Read More →If 3 tan θ = 4, find the value of 4 cos θ−sin θ2 cos θ+sin θ.
Question: If $3 \tan \theta=4$, find the value of $\frac{4 \cos \theta-\sin \theta}{2 \cos \theta+\sin \theta}$. Solution: Given: $3 \tan \theta=4$ Therefore, $\tan \theta=\frac{4}{3}$...(1) Now, we know that $\tan \theta=\frac{\sin \theta}{\cos \theta}$ Therefore equation (1) becomes $\frac{\sin \theta}{\cos \theta}=\frac{4}{3}$....(2) Now, by applying Invertendo to equation (2) We get, $\frac{\cos \theta}{\sin \theta}=\frac{3}{4}$ Now, multiplying by 4 on both sides We get, $4 \times \frac{\co...
Read More →The marks scored by 55 students in a test are given below:
Question: The marks scored by 55 students in a test are given below: Prepare a cumulative frequency table Solution:...
Read More →If tan (A + B) = p, tan (A – B) = q,
Question: If tan (A+B) =p, tan (AB) =q, then the value of tan 2Ain terms ofpandqis ___________. Solution: Given (A+B) =p tan (AB) =q $\tan 2 A=\tan [A+A]$ $=\tan [(A+B)+(A-B)]$ $=\frac{\tan (A+B)+\tan (A-B)}{1-\tan (A+B) \tan (A-B)} \quad\left[\right.$ using identity $\left.\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}\right]$ $\tan 2 A=\frac{p+q}{1-p q}$...
Read More →If the vertices A, B, C of a triangle ABC are
Question: If the vertices $A, B, C$ of a triangle $A B C$ are $(1,2,3),(-1,0,0),(0,1,2)$, respectively, then find $\angle A B C$. [ $\angle A B C$ is the angle between the vectors $\overrightarrow{B A}$ and $\overrightarrow{B C}$ ] Solution: The vertices of ΔABC are given as A (1, 2, 3), B (1, 0, 0), and C (0, 1, 2). Also, it is given that $\angle \mathrm{ABC}$ is the angle between the vectors $\overrightarrow{\mathrm{BA}}$ and $\overrightarrow{\mathrm{BC}}$. $\overrightarrow{\mathrm{BA}}=\{1-(-...
Read More →Explain the difference between a frequency distribution and a cumulative frequency distribution.
Question: Explain the difference between a frequency distribution and a cumulative frequency distribution. Solution: Frequency table or frequency distribution is a method to represent raw data in the form from which one can easily understand the information contained in a raw data, where as a table which plays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution....
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