The following data shows the average age of men in various countries in a certain year.
Question: The following data shows the average age of men in various countries in a certain year. Represent the above information 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 countries and the average age of mens respectively. We have to draw 6 bars of different lengths given in the table. At first, we mark 6 points in the horizontal axis at equal dist...
Read More →Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear,
Question: Show that the points $A(1,-2,-8), B(5,0,-2)$ and $C(11,3,7)$ are collinear, and find the ratio in which $B$ divides $A C$. Solution: The given points are A (1, 2, 8), B (5, 0, 2), and C (11, 3, 7). $\therefore \overrightarrow{\mathrm{AB}}=(5-1) \hat{i}+(0+2) \hat{j}+(-2+8) \hat{k}=4 \hat{i}+2 \hat{j}+6 \hat{k}$ $\overrightarrow{\mathrm{BC}}=(11-5) \hat{i}+(3-0) \hat{j}+(7+2) \hat{k}=6 \hat{i}+3 \hat{j}+9 \hat{k}$ $\overrightarrow{\mathrm{AC}}=(11-1) \hat{i}+(3+2) \hat{j}+(7+8) \hat{k}=...
Read More →The following table shows the interest paid by a company (in lakhs):
Question: The following table shows the interest paid by a company (in lakhs): Draw the bar graph to represent the above information. 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 interests in lakhs of rupees respectively. We have to draw 5 bars of different lengths given in the table. At first, we mark 5 points in the horizontal axis at equal distances a...
Read More →If α + β =
Question: If $\alpha+\beta=\frac{\pi}{2}$, show that the maximum value of $\cos \alpha \cos \beta$ is $\frac{1}{2}$. Solution: $\frac{\pi}{2}=90^{\circ}$ Let $x=\cos \alpha \cos \beta$ $\Rightarrow x=\frac{1}{2}[2 \cos \alpha \cos \beta]$ $\Rightarrow x=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]$ $\Rightarrow x=\frac{1}{2}\left[\cos (\alpha-\beta)+\cos 90^{\circ}\right]$ $\Rightarrow x=\frac{1}{2} \cos (\alpha-\beta)$ Now, $-1 \leq \cos (\alpha-\beta) \leq 1$ $\Rightarrow-\frac{1}{2} \...
Read More →If sec θ=54, find the value of sin θ−2 cos θtan θ−cot θ.
Question: If $\sec \theta=\frac{5}{4}$, find the value of $\frac{\sin \theta-2 \cos \theta}{\tan \theta-\cot \theta}$. Solution: Given: $\sec \theta=\frac{5}{4}$....(1) To find the value of $\frac{\sin \theta-2 \cos \theta}{\tan \theta-\cot \theta}$ Now we know that $\sec \theta=\frac{1}{\cos \theta}$ Therefore, $\cos \theta=\frac{1}{\sec \theta}$ Therefore from equation (1) $\cos \theta=\frac{1}{\frac{5}{4}}$ $\cos \theta=\frac{4}{5}$.....(2) Also, we know that $\cos ^{2} \theta+\sin ^{2} \thet...
Read More →The following data gives the amount of loans (in crores of rupees) disbursed by a bank during some years:
Question: The following data gives the amount of loans (in crores of rupees) disbursed by a bank during some years: (i) Represent the above data with the help of a bar graph. (ii) With the help of the bar graph, indicate the year in which amount of loan is not increased over that of 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 amount ...
Read More →if the
Question: If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=\hat{i}-2 \hat{j}+\hat{k}$, find a unit vector parallel to the vector $2 \vec{a}-\vec{b}+3 \vec{c}$. Solution: We have, $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=\hat{i}-2 \hat{j}+\hat{k}$ $2 \vec{a}-\vec{b}+3 \vec{c}=2(\hat{i}+\hat{j}+\hat{k})-(2 \hat{i}-\hat{j}+3 \hat{k})+3(\hat{i}-2 \hat{j}+\hat{k})$ $=2 \hat{i}+2 \hat{j}+2 \hat{k}-2 \hat{i}+\hat{j}-3 \hat...
Read More →if the
Question: If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=\hat{i}-2 \hat{j}+\hat{k}$, find a unit vector parallel to the vector $2 \vec{a}-\vec{b}+3 \vec{c}$. Solution: We have, $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}-\hat{j}+3 \hat{k}$ and $\vec{c}=\hat{i}-2 \hat{j}+\hat{k}$ $2 \vec{a}-\vec{b}+3 \vec{c}=2(\hat{i}+\hat{j}+\hat{k})-(2 \hat{i}-\hat{j}+3 \hat{k})+3(\hat{i}-2 \hat{j}+\hat{k})$ $=2 \hat{i}+2 \hat{j}+2 \hat{k}-2 \hat{i}+\hat{j}-3 \hat...
Read More →Prove that tan x tan
Question: Prove that $\tan x \tan \left(\frac{\pi}{3}-x\right) \tan \left(\frac{\pi}{3}+x\right)=\tan 3 x$ Solution: $\frac{\pi}{3}=60^{\circ}$ LHS $=\tan x \tan \left(60^{\circ}-x\right) \tan \left(60^{\circ}+x\right)$ $=\frac{\sin x \sin \left(60^{\circ}-x\right) \sin \left(60^{\circ}+x\right)}{\cos x \cos \left(60^{\circ}-x\right) \cos \left(60^{\circ}+x\right)}$ $=\frac{\sin x\left(\sin ^{2} 60^{\circ}-\sin ^{2} x\right)}{\cos x\left(\cos ^{2} 60^{\circ}-\sin ^{2} x\right)}$ $=\frac{\sin x\l...
Read More →Show that:
Question: Show that: (i) sinAsin (BC) + sinBsin (CA) + sinCsin (AB) = 0 (ii) sin (BC) cos (AD) + sin (CA) cos (BD) + sin (AB) cos (CD) = 0 Solution: (i) Consider LHS : $\sin A \sin (B-C)+\sin B \sin (C-A)+\sin C \sin (A-B)$ $=\frac{1}{2}[2 \sin A \sin (B-C)]+\frac{1}{2}[2 \sin B \sin (C-A)]+\frac{1}{2}[2 \sin C \sin (A-B)]$ $=\frac{1}{2}[\cos \{A-(B-C)\}-\cos \{A+(B-C)\}]+\frac{1}{2}[\cos \{B-(C-A)\}-\cos \{B+(C-A)\}]+$$\frac{1}{2}[\cos \{C-(A-B)\}-\cos \{C+(A-B)\}]$ $=\frac{1}{2}[\cos (A-B+C)-\...
Read More →Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
Question: Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$ Solution: We have, $\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{b}=\hat{i}-2 \hat{j}+\hat{k}$ Then, $\vec{c}=\vec{a}+\vec{b}=(2+1) \hat{i}+(3-2) \hat{j}+(-1+1) \hat{k}=3 \hat{i}+\hat{j}$ $\therefore|\vec{c}|=\sqrt{3^{2}+1^{2}}=\sqrt{9+1}=\sqrt{10}$ $\therefore \hat{c}=\frac{\vec{c}}{|\vec{c}|}=\frac{(3 \hat{i}+\hat{j})}{\sqrt{...
Read More →The following table gives the route length (in thousand kilometres) of the Indian Railways in some of the years:
Question: The following table gives the route length (in thousand kilometres) of the Indian Railways in some of the 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 route lengths in thousand km respectively. We have to draw 5 bars of different lengths given in the table. At first we mark 5 points ...
Read More →The production of saleable steel in some of the steel plants of our country during 1999 is given below:
Question: The production of saleable steel in some of the steel plants of our country during 1999 is given below: Construct a bar graph to represent the above data on a graph paper by using the scale 1 big divisions = 20 thousand tonnes. 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 plants and the production in thousand tonnes respectively. We have to draw 4 bars of di...
Read More →Find the value of
Question: Find the value of $x$ for which $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector. Solution: $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector if $|x(\hat{i}+\hat{j}+\hat{k})|=1$ Now, $|x(\hat{i}+\hat{j}+\hat{k})|=1$ $\Rightarrow \sqrt{x^{2}+x^{2}+x^{2}}=1$ $\Rightarrow \sqrt{3 x^{2}}=1$ $\Rightarrow \sqrt{3} x=1$ $\Rightarrow x=\pm \frac{1}{\sqrt{3}}$ Hence, the required value of $x$ is $\pm \frac{1}{\sqrt{3}}$....
Read More →if the
Question: If $\vec{a}=\vec{b}+\vec{c}$, then is it true that $|\vec{a}|=|\vec{b}|+|\vec{c}| ?$ Justify your answer. Solution: In $\triangle \mathrm{ABC}$, let $\overrightarrow{\mathrm{CB}}=\vec{a}, \overrightarrow{\mathrm{CA}}=\vec{b}$, and $\overrightarrow{\mathrm{AB}}=\vec{c}$ (as shown in the following figure). Now, by the triangle law of vector addition, we have $\vec{a}=\vec{b}+\vec{c}$. It is clearly known that $|\vec{a}|,|\vec{b}|$, and $|\vec{c}|$ represent the sides of $\triangle \mathr...
Read More →The following data gives the number (in thousands) of applicants registered with an Employment Exchange during 1995 – 2000:
Question: The following data gives the number (in thousands) of applicants registered with an Employment Exchange during 1995 2000: 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 number of applicants registered in thousands respectively. We have to draw 6 bars of different lengths given in the table. At fi...
Read More →Prove that:
Question: Prove that: (i) $\cos 10^{\circ} \cos 30^{\circ} \cos 50^{\circ} \cos 70^{\circ}=\frac{3}{16}$ (ii) $\cos 40^{\circ} \cos 80^{\circ} \cos 160^{\circ}=-\frac{1}{8}$ (iii) $\sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ}=\frac{\sqrt{3}}{8}$ (iv) $\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}=\frac{1}{8}$ (v) $\tan 20^{\circ} \tan 40^{\circ} \tan 60^{\circ} \tan 80^{\circ}=3$ (vi) $\tan 20^{\circ} \tan 30^{\circ} \tan 40^{\circ} \tan 80^{\circ}=1$ (vii) $\sin 10^{\circ} \sin 50^{\circ} ...
Read More →If sin θ=1213, find the value of sin2 θ−cos2 θ2 sin θ cos θ×1tan2 θ.
Question: If $\sin \theta=\frac{12}{13}$, find the value of $\frac{\sin ^{2} \theta-\cos ^{2} \theta}{2 \sin \theta \cos \theta} \times \frac{1}{\tan ^{2} \theta}$ Solution: Given: $\sin \theta=\frac{12}{13}$ 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 $\sin \theta=\frac{\text { Perpendicular side opposite to } \angle \theta}{\text { Hypotenuse }}$....(2) Now when we compare equation (1...
Read More →Prove that:
Question: Prove that: (i) $\cos 10^{\circ} \cos 30^{\circ} \cos 50^{\circ} \cos 70^{\circ}=\frac{3}{16}$ (ii) $\cos 40^{\circ} \cos 80^{\circ} \cos 160^{\circ}=-\frac{1}{8}$ (iii) $\sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ}=\frac{\sqrt{3}}{8}$ (iv) $\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}=\frac{1}{8}$ (v) $\tan 20^{\circ} \tan 40^{\circ} \tan 60^{\circ} \tan 80^{\circ}=3$ (vi) $\tan 20^{\circ} \tan 30^{\circ} \tan 40^{\circ} \tan 80^{\circ}=1$ (vii) $\sin 10^{\circ} \sin 50^{\circ} ...
Read More →The following bar graph shows the results of an annual examination in a secondary school.
Question: The following bar graph shows the results of an annual examination in a secondary school. Read the bar graph and choose the correct alternative in each of the following: (i) The pair of classes in which the results of boys and girls are inversely proportional are: (a) VI, VIII (b) VI, IX (c) VII, IX (d) VIII, X (ii) The class having the lowest failure rate of girls is: (a) VI (b) X (c) IX (d) VIII (iii) The class having the lowest pass rate of students is: (a) VI (b) VII (c) VIII (d) I...
Read More →A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops.
Question: A girl walks 4 km towards west, then she walks 3 km in a direction 30 east of north and stops. Determine the girls displacement from her initial point of departure. Solution: Let O and B be the initial and final positions of the girl respectively. Then, the girls position can be shown as: Now, we have: $\overrightarrow{\mathrm{OA}}=-4 \hat{i}$ $\overrightarrow{\mathrm{AB}}=\hat{i}|\overrightarrow{\mathrm{AB}}| \cos 60^{\circ}+\hat{j}|\overrightarrow{\mathrm{AB}}| \sin 60^{\circ}$ $=\ha...
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) In which year the export is minimum? (iii) In which year the import is maximum? (iv) In which year the difference of the values of export and import is maximum? Solution: (i) The bar graph represents the import and export (in 100 Crores of rupees) from 1982 83 to 1986 87. (ii) The export is minimum in the year 1982-83 at the height of the bar corresponding to export is ...
Read More →Explain the reading and interpretation of bar graphs.
Question: Explain the reading and interpretation of bar graphs. Solution: A bar graph is a diagram consisting of a sequence of vertical or horizontal bars or rectangles, each of which represents an equal interval of the values of a variable, and has the height proportional to the quantities of the phenomenon under consideration in that interval. A bar graph may also be used to illustrate discrete data, in which case each bar represents a distinct circumstance. While drawing a bar graph, we keep ...
Read More →Find the scalar components and magnitude of the vector joining the points
Question: Find the scalar components and magnitude of the vector joining the points $\mathrm{P}\left(x_{1}, y_{1}, z_{1}\right)$ and $\mathrm{Q}\left(x_{2}, y_{2}, z_{2}\right)$. Solution: The vector joining the points $\mathrm{P}\left(x_{1}, y_{1}, z_{1}\right)$ and $\mathrm{Q}\left(x_{2}, y_{2}, z_{2}\right)$ can be obtained by, $\overrightarrow{\mathrm{PQ}}=$ Position vector of $\mathrm{Q}-$ Position vector of $\mathrm{P}$ $=\left(x_{2}-x_{1}\right) \hat{i}+\left(y_{2}-y_{1}\right) \hat{j}+\l...
Read More →Read the bar graph given in Fig. and answer the following questions:
Question: Read the bar graph given in Fig. and answer the following questions: (i) What information is given by the bar graph? (ii) Which Door darshan center covers maximum area? Also, tell the covered area. (iii) What is the difference between the areas covered by the centers at Delhi and Bombay? (iv) Which Door darshan centers are in U.P. State? What are the areas covered by them? Solution: (i) The bar graph represents the area of coverage (in 1000 square km) of some Doordarshan Centers of Ind...
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