Express each of the following as the sum or difference of sines and cosines:
Question: Express each of the following as the sum or difference of sines and cosines: (i) 2 sin 3xcosx (ii) 2 cos 3xsin 2x (iii) 2 sin 4xsin 3x (iv) 2 cos 7xcos 3x Solution: (i) 2 sin 3xcosx $=\sin (3 x+x)+\sin (3 x-x) \quad[\because 2 \sin A \cos B=\sin (A+B)+\sin (A-B)]$ $=\sin 4 x+\sin 2 x$ (ii) 2 cos 3xsin 2x $=\sin (3 x+2 x)-\sin (3 x-2 x) \quad[\because 2 \cos A \sin B=\sin (A+B)-\sin (A-B)]$ $=\sin 5 x-\sin x$ (iii) 2 sin 4xsin 3x $=\cos (4 x-3 x)-\cos (4 x+3 x) \quad[\because 2 \sin A \...
Read More →Given below is the bar graph indicating the marks obtained out of 50 in mathematics paper by 100 students.
Question: Given below is the bar graph indicating the marks obtained out of 50 in mathematics paper by 100 students. Read the bar graph and answer the following questions. (i) It is decided to distribute workbooks on mathematics to the students obtaining less than 20 marks, giving one workbook to each of such students. If a workbook costs Rs. 5, what sum is required to buy the workbooks? (ii) Every student belonging to the highest mark group is entitled to get a prize of Rs. 10. How much amount ...
Read More →If cos (A + B) sin (C − D) = cos (A − B) sin (C + D),
Question: If cos (A+B) sin (CD) = cos (AB) sin (C+D), then write the value of tanAtanBtanC. Solution: cos (A+B) sin (CD) = cos (AB) sin (C+D) $\Rightarrow[\cos A \cos B-\sin A \sin B][\sin C \cos D-\cos C \sin D]=[\cos A \cos B+\sin A \sin B][\sin C \cos D+\cos C \sin D]$ Dividing both sides by $\cos A \cos B \cos C \cos D:$ $\frac{[\cos A \cos B-\sin A \sin B][\sin C \cos D-\cos C \sin D]}{\cos A \cos B \cos C \cos D}=\frac{[\cos A \cos B+\sin A \sin B][\sin C \cos D+\cos C \sin D]}{\cos A \cos...
Read More →If cos (A + B) sin (C − D) = cos (A − B) sin (C + D),
Question: If cos (A+B) sin (CD) = cos (AB) sin (C+D), then write the value of tanAtanBtanC. Solution: cos (A+B) sin (CD) = cos (AB) sin (C+D) $\Rightarrow[\cos A \cos B-\sin A \sin B][\sin C \cos D-\cos C \sin D]=[\cos A \cos B+\sin A \sin B][\sin C \cos D+\cos C \sin D]$ Dividing both sides by $\cos A \cos B \cos C \cos D:$ $\frac{[\cos A \cos B-\sin A \sin B][\sin C \cos D-\cos C \sin D]}{\cos A \cos B \cos C \cos D}=\frac{[\cos A \cos B+\sin A \sin B][\sin C \cos D+\cos C \sin D]}{\cos A \cos...
Read More →Read the bar graph shown in the figure and answer the following questions:
Question: Read the bar graph shown in the figure and answer the following questions: (i) What is the information given by the bar graph? (ii) What was the number of commercial banks in 1977? (iii) What is the ratio of the number of commercial banks in 1969 to that in 1980? (iv) State whether true or false: The number of commercial banks in 1983 is less than double the number of commercial banks in 1969. Solution: (i) The bar graph represents the number of commercial banks in India during some pa...
Read More →Find λ and μ if
Question: Find $\lambda$ and $\mu$ if $(2 \hat{i}+6 \hat{j}+27 \hat{k}) \times(\hat{i}+\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}$ Solution: $(2 \hat{i}+6 \hat{j}+27 \hat{k}) \times(\hat{i}+\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}$ $\Rightarrow\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 2 6 27 \\ 1 \lambda \mu\end{array}\right|=0 \hat{i}+0 \hat{j}+0 \hat{k}$ $\Rightarrow \hat{i}(6 \mu-27 \lambda)-\hat{j}(2 \mu-27)+\hat{k}(2 \lambda-6)=0 \hat{i}+0 \hat{j}+0 \hat{k}$ On comparin...
Read More →If cos θ=1213, show that sin θ (1 − tan θ)=35156.
Question: If $\cos \theta=\frac{12}{13}$, show that $\sin \theta(1-\tan \theta)=\frac{35}{156}$ Solution: Given: $\cos \theta=\frac{12}{13}$....(1) To show that $\sin \theta(1-\tan \theta)=\frac{35}{156}$ Now, we know that $\cos \theta=\frac{\text { Base side adjacent to } \angle \theta}{\text { Hypotenuse }}$....(2) Therefore, by comparing equation (1) and (2) We get, Base side adjacent to $\angle \theta=12$ And Hypotenuse = 13 Therefore from above figure Base side $B C=12$ Hypotenuse $A C=13$ ...
Read More →Write the value of
Question: Write the value of $\frac{\sin A+\sin 3 A}{\cos A+\cos 3 A}$. Solution: $\frac{\sin A+\sin 3 A}{\cos A+\cos 3 A}$ $=\frac{2 \sin \left(\frac{A+3 A}{2}\right) \cos \left(\frac{A-3 A}{2}\right)}{2 \cos \left(\frac{A+3 A}{2}\right) \cos \left(\frac{A-3 A}{2}\right)}$ $\left[\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right.$, and $\left.\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $=\frac{\sin 2 A \cos...
Read More →Study the bar graph representing the number of persons in various age groups in a town shown in figure Observe the bar graph and answer the following questions:
Question: Study the bar graph representing the number of persons in various age groups in a town shown in figure Observe the bar graph and answer the following questions: (i) What is the percentage of the youngest age-group persons over those in the oldest age group? (ii) What is the total population of the town? (iii) What is the number of persons in the age-group 60 65? (iv) How many persons are more in the age-group 10 15 than in the age group 30 35? (v) What is the age-group of exactly 1200 ...
Read More →If sin 2A = λ sin 2B,
Question: If $\sin 2 A=\lambda \sin 2 B$, then write the value of $\frac{\lambda+1}{\lambda-1}$. Solution: Given: sin 2A= sin 2B $\Rightarrow \frac{\sin 2 A}{\sin 2 B}=\lambda$ $\Rightarrow \frac{\sin 2 A+\sin 2 B}{\sin 2 A-\sin 2 B}=\frac{\lambda+1}{\lambda-1}$ $\Rightarrow \frac{2 \sin \left(\frac{2 A+2 B}{2}\right) \cos \left(\frac{2 A-2 B}{2}\right)}{2 \sin \left(\frac{2 A-2 B}{2}\right) \cos \left(\frac{2 A+2 B}{2}\right)}=\frac{\lambda+1}{\lambda-1}$ $\left[\because \sin A+\sin B=2 \sin \l...
Read More →Write the value of sin
Question: Write the value of $\sin \frac{\pi}{15} \sin \frac{4 \pi}{15} \sin \frac{3 \pi}{10}$ Solution: $\frac{\pi}{15}=12^{\circ}, \frac{4 \pi}{15}=48^{\circ}, \frac{3 \pi}{10}=54^{\circ}$ $\sin 12^{\circ} \sin 48^{\circ} \sin 54^{\circ}$ $=\frac{1}{2}\left[2 \sin 12^{\circ} \sin 48^{\circ}\right] \sin 54^{\circ}$ $=\frac{1}{2}\left[\cos \left(12^{\circ}-48^{\circ}\right)-\cos \left(12^{\circ}+48^{\circ}\right)\right] \sin 54^{\circ}$ $=\frac{1}{2}\left[\cos \left(-36^{\circ}\right)-\cos 60^{\...
Read More →Show that
Question: Show that $(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})=2(\vec{a} \times \vec{b})$ Solution: $(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})$ $=(\vec{a}-\vec{b}) \times \vec{a}+(\vec{a}-\vec{b}) \times \vec{b} \quad$ [By distributivity of vector product over addition] $=\vec{a} \times \vec{a}-\vec{b} \times \vec{a}+\vec{a} \times \vec{b}-\vec{b} \times \vec{b} \quad$ [Again, by distributivity of vector product over addition] $=\overrightarrow{0}+\vec{a} \times \vec{b}+\vec{a} \times \vec{b}-\o...
Read More →Read the bar graph shown in the figure and answer the following questions:
Question: Read the bar graph shown in the figure and answer the following questions: (i) What is the information given by the bar graph? (ii) How many tickets of Assam State Lottery were sold by the agent? (iii) Of which state, were the maximum number of tickets sold? (iv) State whether true or false. The maximum number of tickets sold is three times the minimum number of tickets sold. (v) Of which state were the minimum numbers of tickets sold? Solution: (i) The bar graph represents the number ...
Read More →If A + B =
Question: If $A+B=\frac{\pi}{3}$ and $\cos A+\cos B=1$, then find the value of $\cos \frac{A-B}{2}$. Solution: Given: $A+B=\frac{\pi}{3}$ and cosA+ cosB= 1 $\Rightarrow 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)=1 \quad\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $\Rightarrow 2 \cos \left(\frac{\pi}{6}\right) \cos \left(\frac{A-B}{2}\right)=1$ $\left[\because A+B=\frac{\pi}{3}\right]$ $\Rightarrow 2 \times \frac{\sq...
Read More →If A + B =
Question: If $A+B=\frac{\pi}{3}$ and $\cos A+\cos B=1$, then find the value of $\cos \frac{A-B}{2}$. Solution: Given: $A+B=\frac{\pi}{3}$ and cosA+ cosB= 1 $\Rightarrow 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)=1 \quad\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $\Rightarrow 2 \cos \left(\frac{\pi}{6}\right) \cos \left(\frac{A-B}{2}\right)=1$ $\left[\because A+B=\frac{\pi}{3}\right]$ $\Rightarrow 2 \times \frac{\sq...
Read More →The population of Delhi State in different census years is as given below:
Question: The population of Delhi State in different census years is as given below: Represent the above information with the help of a bar graph. Solution: While drawing a bar graph, we keep in mind that: The width of the bars should be uniform throughout. The gap between any two bars should be uniform throughout. Bars may be either horizontal or vertical. To represent the given data by a vertical bar graph, we first draw horizontal and vertical axes. Let us consider that the horizontal and ver...
Read More →If sec θ=135, show that 2 sin θ−3 cos θ4 sin θ− 9 cos θ=3.
Question: If $\sec \theta=\frac{13}{5}$, show that $\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}=3$. Solution: Given: $\sec \theta=\frac{13}{5}$ To show that $\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}=3$ Now, we know that $\cos \theta=\frac{1}{\sec \theta}$ Therefore, $\cos \theta=\frac{1}{\frac{13}{5}}$ Therefore, $\cos \theta=\frac{5}{13}$...(1) Now, we know that $\cos \theta=\frac{\text { Base side adjacent to } \angle \theta}{\text { Hypotenuse }}$......
Read More →Write the value of the expression
Question: Write the value of the expression $\frac{1-4 \sin 10^{\circ} \sin 70^{\circ}}{2 \sin 10^{\circ}}$. Solution: $\frac{1-4 \sin 10^{\circ} \sin 70^{\circ}}{2 \sin 10^{\circ}}$ $=\frac{1-2\left[2 \sin 10^{\circ} \sin 70^{\circ}\right]}{2 \sin 10^{\circ}}$ $=\frac{1-2\left[\cos \left(10^{\circ}-70^{\circ}\right)-\cos \left(10^{\circ}+70^{\circ}\right)\right]}{2 \sin 10^{\circ}}$ $=\frac{1-2\left[\cos \left(-60^{\circ}\right)-\cos 80^{\circ}\right]}{2 \sin 10^{\circ}}$ $=\frac{1-2\left[\cos ...
Read More →If a unit vector makes an angles
Question: If a unit vector $\vec{a}$ makes an angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$, then find $\theta$ and hence, the compounds of $\vec{a}$. Solution: Let unit vector $\vec{a}$ have $\left(a_{1}, a_{2}, a_{3}\right)$ components. $\Rightarrow \vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$ Since $\vec{a}$ is a unit vector, $|\vec{a}|=1$. Also, it is given that $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{...
Read More →The following table shows the number of Maruti cars sold by five dealers in a particular month:
Question: The following table shows the number of Maruti cars sold by five dealers in a particular month: Represent the above information by a pictograph. Solution: The given information can be represented using a pictograph in the following manner:...
Read More →If cos A = m cos B,
Question: If $\cos A=m \cos B$, then write the value of $\cot \frac{A+B}{2} \cot \frac{A-B}{2} .$ Solution: Given: $\cos A=m \cos B$ $\Rightarrow \frac{\cos A}{\cos B}=\frac{m}{1}$ $\Rightarrow \frac{\cos A+\cos B}{\cos A-\cos B}=\frac{m+1}{m-1}$ $\Rightarrow \frac{2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)}{-2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)}=\frac{m+1}{m-1}$ $\left[\because \cos A+\cos B=2 \cos \left(\frac{A-B}{2}\right) \cos \left(\frac{A...
Read More →The following table shows the daily production of T.V. sets in an industry for 7 days of a week.
Question: The following table shows the daily production of T.V. sets in an industry for 7 days of a week. Represent the above information by a pictograph. Solution: The given information can be represented using a pictograph in the following manner:...
Read More →If sin A + sin B = α and cos A + cos B = β,
Question: If $\sin A+\sin B=\alpha$ and $\cos A+\cos B=\beta$, then write the value of $\tan \left(\frac{A+B}{2}\right)$ Solution: Given: sinA+ sinB= .....(i) cosA+ cosB= .....(ii) Dividing (i) by (ii): $\Rightarrow \frac{\sin A+\sin B}{\cos A+\cos B}=\frac{\alpha}{\beta}$ $\Rightarrow \frac{2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)}{2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)}=\frac{\alpha}{\beta}$ $\left[\because \sin A+\sin B=2 \sin \left(\frac{A+...
Read More →If sin A + sin B = α and cos A + cos B = β,
Question: If $\sin A+\sin B=\alpha$ and $\cos A+\cos B=\beta$, then write the value of $\tan \left(\frac{A+B}{2}\right)$ Solution: Given: sinA+ sinB= .....(i) cosA+ cosB= .....(ii) Dividing (i) by (ii):...
Read More →Write the value of sin
Question: Write the value of $\sin \frac{\pi}{12} \sin \frac{5 \pi}{12}$. Solution: $\sin \frac{\pi}{12} \sin \frac{5 \pi}{12}$ $=\frac{1}{2} \times 2\left(\sin \frac{\pi}{12}\right)\left(\sin \frac{5 \pi}{12}\right)$ $=\frac{1}{2}\left[\cos \left(\frac{\pi}{12}-\frac{5 \pi}{12}\right)-\cos \left(\frac{\pi}{12}+\frac{5 \pi}{12}\right)\right]$ $[\because 2 \sin A \sin B=\cos (A-B)-\cos (A+B)]$ $=\frac{1}{2}\left[\cos \left(-\frac{\pi}{3}\right)-\cos \frac{\pi}{2}\right]$ $=\frac{1}{2}\left(\frac{...
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