Read the bar graph given in figure and answer the following questions:
Question: Read the bar graph given in figure and answer the following questions: (i) What is the information given by the bar graph? (ii) What is the number of families having 6 members? (iii) How many members per family are there in the maximum number of families? Also tell the number of such families. (iv) What are the number of members per family for which the number of families are equal? Also, tell the number of such families? Solution: (i) The bar graph represents the number of families wi...
Read More →If tan θ=17√, show that cosec2 θ−sec2 θcosec2 θ+sec2 θ=34
Question: If $\tan \theta=\frac{1}{\sqrt{7}}$, show that $\frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\sec ^{2} \theta}=\frac{3}{4}$ Solution: Given: $\tan \theta=\frac{1}{\sqrt{7}}$....(1) To show that $\frac{\operatorname{cosec}^{2} \theta-\sec ^{2} \theta}{\operatorname{cosec}^{2} \theta+\sec ^{2} \theta}=\frac{3}{4}$ Now, we know that Since $\tan \theta=\frac{\text { Perpendicular side opposite to } \angle \theta}{\text { Base side adjacent to } \a...
Read More →Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
Question: Write down a unit vector in XY-plane, making an angle of 30 with the positive direction ofx-axis. Solution: If $\vec{r}$ is a unit vector in the $\mathrm{XY}$-plane, then $\vec{r}=\cos \theta \hat{i}+\sin \theta \hat{j}$. Here,is the angle made by the unit vector with the positive direction of thex-axis. Therefore, for= 30: $\vec{r}=\cos 30^{\circ} \hat{i}+\sin 30^{\circ} \hat{j}=\frac{\sqrt{3}}{2} \hat{i}+\frac{1}{2} \hat{j}$ Hence, the required unit vector is $\frac{\sqrt{3}}{2} \hat...
Read More →Read the bar graph given in figure and answer the following questions:
Question: Read the bar graph given in figure and answer the following questions: (i) What information is given by the bar graph? (ii) What was the expenditure on health and family planning in the year 1982-83? (iii) In which year is the increase in expenditure maximum over the expenditure in the previous year? What is the maximum increase? Solution: (i) The bar graph represents the expenditure (in 100 Crores of rupees) on health and family planning during the Sixth Five Year Plan in India. (ii) ...
Read More →Area of a rectangle having vertices A, B, C, and D with position vectors
Question: Area of a rectangle having vertices $A, B, C$, and $D$ with position vectors $-\hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}$ and $-\hat{i}-\frac{1}{2} \hat{j}+4 \hat{k}$ respectively is (A) $\frac{1}{2}$ (B) 1 (C) 2 (D) 4 Solution: The position vectors of vertices A, B, C, and D of rectangle ABCD are given as: $\overrightarrow{\mathrm{OA}}=-\hat{i}+\frac{1}{2} \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{OB}}=\hat{i}+...
Read More →Read the graph given in figure and answer the following question
Question: Read the graph given in figure and answer the following question (i) What information is given by the bar given? (ii) In which years the areas under the sugarcane crop were the maximum and the minimum? (iii) State whether true or false: The area under sugarcane crop in the year 1982 83 is three times that of the year 1950 51. Solution: (i) The bar graph represents the areas (in lath hectares) under sugarcane crop during different years in India. (ii) It is seen from the bar graph that ...
Read More →Read the bar graph given in figure and answer the following questions:
Question: Read the bar graph given in figure and answer the following questions: (i) What information does it give? (ii) In which part the expenditure on education is maximum in 1980? (iii) In which part the expenditure has gone up from 1980 to 1990? (iv) In which part the gap between 1980 and 1990 is maximum? Solution: (i) The bar graph represents the public expenditure on education in different countries and sub continents in the years 1980 and 1990. (ii) The expenditure on education in Africa...
Read More →Let the vectors
Question: Let the vectors $\vec{a}$ and $\vec{b}$ be such that $|\vec{a}|=3$ and $|\vec{b}|=\frac{\sqrt{2}}{3}$, then $\vec{a} \times \vec{b}$ is a unit vector, if the angle between $\vec{a}$ and $\vec{b}$ is (A) $\frac{\pi}{6}$ (B) $\frac{\pi}{4}$ (C) $\frac{\pi}{3}$ (D) $\frac{\pi}{2}$ Solution: It is given that $|\vec{a}|=3$ and $|\vec{b}|=\frac{\sqrt{2}}{3}$. We know that $\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to both $...
Read More →Read the bar graph given in and answer the following questions:
Question: Read the bar graph given in and answer the following questions: (i) What information is given by the bar graph? (ii) What was the crop-production of rice in 1970 71? (iii) What is the difference between the maximum and minimum production of rice? Solution: (i)The bar graph represents the production of the rice crop in India in different years. (ii)According to the height of the 3rd bar from the left, the crop-production of rice in 1970 71 is 42.5 lakh tonnes. (iii) The maximum product ...
Read More →Find the area of the parallelogram whose adjacent sides are determined by the vector
Question: Find the area of the parallelogram whose adjacent sides are determined by the vector $\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$. Solution: The area of the parallelogram whose adjacent sides are $\vec{a}$ and $\vec{b}$ is $|\vec{a} \times \vec{b}|$. Adjacent sides are given as: $\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$ $\therefore \vec{a} \times \vec{b}=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 1 -1 3 \\ 2 -...
Read More →The bar graph shown in figure represents the circulation of newspapers in 10 languages.
Question: The bar graph shown in figure represents the circulation of newspapers in 10 languages. Study the bar graph and answer the following questions: (i) What is the total number of newspapers published in Hindi, English, Urdu, Punjabi, and Bengali? (ii) What percent is the number of newspapers published in Hindi of the total number of newspapers? (iii) Find the excess of the number of newspapers published in English over those published in Urdu. (iv) Name two pairs of languages which publis...
Read More →Read the following bar graph and answer the following questions:
Question: Read the following bar graph and answer the following questions: (i) What information is given by the bar graph? (ii) What was the production of cement in the year 1980 81? (iii) What are the minimum and maximum production of cement and corresponding years? Solution: (i) The bar graph represents the industrial production of cement in different years in India. (ii) According to the height of the 6th bar from the left. the production of cement in the year 1980 81 was 186 lakh tonnes. (ii...
Read More →Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
Question: Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) andC (1, 5, 5). Solution: The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and C (1, 5, 5). The adjacent sides $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{BC}}$ of $\triangle \mathrm{ABC}$ are given as: $\overrightarrow{\mathrm{AB}}=(2-1) \hat{i}+(3-1) \hat{j}+(5-2) \hat{k}=\hat{i}+2 \hat{j}+3 \hat{k}$ $\overrightarrow{\mathrm{BC}}=(1-2) \hat{i}+(5-3) \hat{j}+(5-5) \hat{k}=-\hat{i}+2...
Read More →Prove that
Question: Prove that $4 \cos x \cos \left(\frac{\pi}{3}+x\right) \cos \left(\frac{\pi}{3}-x\right)=\cos 3 x .$ Solution: $\mathrm{LHS}=4 \cos x \cos \left(\frac{\pi}{3}+x\right) \cos \left(\frac{\pi}{3}-x\right)$ $=2 \cos x\left[2 \cos \left(\frac{\pi}{3}+x\right) \cos \left(\frac{\pi}{3}-x\right)\right]$ $=2 \cos x\left[\cos \left(\frac{\pi}{3}+x+\frac{\pi}{3}-x\right)+\cos \left(\frac{\pi}{3}+x-\frac{\pi}{3}+2 x\right)\right]$ $[\because 2 \cos A \cos B=\cos (A+B)+\cos (A-B)]$ $=2 \cos x\left[...
Read More →If either
Question: If either $\vec{a}=\overrightarrow{0}$ or $\vec{b}=\overrightarrow{0}$, then $\vec{a} \times \vec{b}=\overrightarrow{0}$. Is the converse true? Justify your answer with an example. Solution: Take any parallel non-zero vectors so that $\vec{a} \times \vec{b}=\overrightarrow{0}$. Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=4 \hat{i}+6 \hat{j}+8 \hat{k}$ Then, $\vec{a} \times \vec{b}=\left|\begin{array}{ccc}\hat{i} \hat{j} \hat{k} \\ 2 3 4 \\ 4 6 8\end{array}\right|=\hat{i}(24-24)...
Read More →Show that :
Question: Show that : (i) $\sin 50^{\circ} \cos 85^{\circ}=\frac{1-\sqrt{2} \sin 35^{\circ}}{2 \sqrt{2}}$ (ii) $\sin 25^{\circ} \cos 115^{\circ}=\frac{1}{2}\left(\sin 140^{\circ}-1\right)$ Solution: (i) $\mathrm{LHS}=2 \sin 50^{\circ} \cos 85^{\circ}$ $=\frac{\sin \left(50^{\circ}+85^{\circ}\right)+\sin \left(50^{\circ}-85^{\circ}\right)}{2}$ $\left[\because \sin A \cos B=\frac{1}{2}\{\sin (A+B)+\sin (A-B)\}\right]$ $=\frac{\sin 135^{\circ}+\sin \left(-35^{\circ}\right)}{2}$ $=\frac{\sin 135^{\c...
Read More →If cot θ=13√, show that 1−cos2 θ2−sin2 θ=35.
Question: If $\cot \theta=\frac{1}{\sqrt{3}}$, show that $\frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta}=\frac{3}{5}$. Solution: Given: $\cot \theta=\frac{1}{\sqrt{3}}$....(1) To show that $\frac{1-\cos ^{2} \theta}{2-\sin ^{2} \theta}=\frac{3}{5}$ Now, we know that $\cot \theta=\frac{1}{\tan \theta}$ Since $\tan \theta=\frac{\text { Perpendicular side opposite to } \angle \theta}{\text { Base } \text { side } \text { adjacent to } \angle \theta}$ Therefore, $\cot \theta=\frac{1}{\frac{\text { Pe...
Read More →Let the vectors given as
Question: Let the vectors $\vec{a}, \vec{b}, \vec{c}$ given as $a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}, b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}, c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}$. Then show that $=\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$ Solution: We have, $\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}, \vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}, \vec{c}=c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}$ $(\vec{b}+\vec{c})=\left(b_{1}+c_{1}\...
Read More →Prove that:
Question: Prove that: (i) $2 \sin \frac{5 \pi}{12} \sin \frac{\pi}{12}=\frac{1}{2}$ (ii) $2 \cos \frac{5 \pi}{12} \cos \frac{\pi}{12}=\frac{1}{2}$ (iii) $2 \sin \frac{5 \pi}{12} \cos \frac{\pi}{12}=\frac{\sqrt{3}+2}{2}$ Solution: (i) LHS $=2\left(\sin \frac{5 \pi}{12}\right)\left(\sin \frac{\pi}{12}\right)$ $=\cos \left(\frac{5 \pi}{12}-\frac{\pi}{12}\right)-\cos \left(\frac{5 \pi}{12}+\frac{\pi}{12}\right) \quad[\because 2 \sin A \sin B=\cos (A-B)-\cos (A+B)]$ $=\cos \frac{\pi}{3}-\cos \frac{\p...
Read More →Prove that:
Question: Prove that: (i) $2 \sin \frac{5 \pi}{12} \sin \frac{\pi}{12}=\frac{1}{2}$ (ii) $2 \cos \frac{5 \pi}{12} \cos \frac{\pi}{12}=\frac{1}{2}$ (iii) $2 \sin \frac{5 \pi}{12} \cos \frac{\pi}{12}=\frac{\sqrt{3}+2}{2}$ Solution: (i) LHS $=2\left(\sin \frac{5 \pi}{12}\right)\left(\sin \frac{\pi}{12}\right)$ $=\cos \left(\frac{5 \pi}{12}-\frac{\pi}{12}\right)-\cos \left(\frac{5 \pi}{12}+\frac{\pi}{12}\right) \quad[\because 2 \sin A \sin B=\cos (A-B)-\cos (A+B)]$ $=\cos \frac{\pi}{3}-\cos \frac{\p...
Read More →The following bar graph represents the heights (in cm) of 50 students of Class XI of a particular school.
Question: The following bar graph represents the heights (in cm) of 50 students of Class XI of a particular school. Study the graph and answer the following questions: (i) What percentage of the total number of students have their heights more than 149 cm? (ii) How many students in the class are in the range of maximum height of the class? (iii) The school wants to provide a particular type of tonic to each student below the height of 150 cm to improve his height. If the cost of the tonic for ea...
Read More →Read the following bar graph and answer the following questions:
Question: Read the following bar graph and answer the following questions: (i) What information is given by the bar graph? (ii) Which state is the largest producer of rice? (iii) Which state is the largest producer of wheat? (iv) Which state has total production of rice and wheat at its maximum? (v) Which state has total production of wheat and rice at its minimum? Solution: (i) The bar graph represents the production of rice and wheat in different states of India. (ii) According to the height o...
Read More →Given that
Question: Given that $\vec{a} \cdot \vec{b}=0$ and $\vec{a} \times \vec{b}=\overrightarrow{0}$. What can you conclude about the vectors $\vec{a}$ and $\vec{b} ?$ Solution: $\vec{a} \cdot \vec{b}=0$ Then, (i) Either $|\vec{a}|=0$ or $|\vec{b}|=0$, or $\vec{a} \perp \vec{b}$ (in case $\vec{a}$ and $\vec{b}$ are non-zero) $\vec{a} \times \vec{b}=0$ (ii) Either $|\vec{a}|=0$ or $|\vec{b}|=0$, or $\vec{a} \| \vec{b}$ (in case $\vec{a}$ and $\vec{b}$ are non-zero) But, $\vec{a}$ and $\vec{b}$ cannot b...
Read More →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 →Read the following bar graph and answer the following questions:
Question: Read the following bar graph and answer the following questions: (i) What is the information given by the bar graph? (ii) State each of the following whether true or false. (a) The number of government companies in 1957 is that of 1982 is 1:9. (b) The number of government companies has decreased over the year 1957 to 1983. Solution: (i) The bar graph represents the number of government companies in India during some years. (ii) (a) The number of companies in 1957 was 50 and the number ...
Read More →