Write the class -size in each of the following:
Question: Write the class -size in each of the following: (1) 0 - 4, 5 - 9, 10 14 (2) 10 - 19, 20 - 29, 30 - 39 (3) 100 - 120, 120 - 140, 160 - 180 (4) 0 - 0.25, 0.25 - 00.50, 0.50 - 0.75 (5) 5 - 5.01, 5.01 - 5.02, 5.02 - 5.03. Solution: (1)0 - 4, 5 - 9, 10 - 14 True class limits are 0.5 - 4.5, 4.5 - 9.5, 9.5 - 14.5 Therefore class size = 14.5 - 9.5 = 5 (2)10 - 19, 20 - 29, 30 - 39 True class limits 19.5 - 19.5, 19.5 - 29.5, 29.5 - 29.5 Class size = 39.5 - 29.5 = 10 (3)100 - 120, 120 - 140, 160 ...
Read More →If sin 2 θ + sin 2 ϕ
Question: If $\sin 2 \theta+\sin 2 \phi=\frac{1}{2}$ and $\cos 2 \theta+\cos 2 \phi=\frac{3}{2}$, then $\cos ^{2}(\theta-\phi)=$ (a) $\frac{3}{8}$ (b) $\frac{5}{8}$ (c) $\frac{3}{4}$ (d) $\frac{5}{4}$ Solution: (b) $\frac{5}{8}$ Given: $\sin 2 \theta+\sin 2 \phi=\frac{1}{2}$ ...(i) and $\cos 2 \theta+\cos 2 \phi=\frac{3}{2}$ ....(ii) Squaring and adding (i) and (ii), we get: $(\sin 2 \theta+\sin 2 \phi)^{2}+(\cos 2 \theta+\cos 2 \phi)^{2}=\frac{1}{4}+\frac{9}{4}$ $\Rightarrow\left[2 \sin \left(\...
Read More →If sin 2 θ + sin 2 ϕ
Question: If $\sin 2 \theta+\sin 2 \phi=\frac{1}{2}$ and $\cos 2 \theta+\cos 2 \phi=\frac{3}{2}$, then $\cos ^{2}(\theta-\phi)=$ (a) $\frac{3}{8}$ (b) $\frac{5}{8}$ (c) $\frac{3}{4}$ (d) $\frac{5}{4}$ Solution: (b) $\frac{5}{8}$ Given: $\sin 2 \theta+\sin 2 \phi=\frac{1}{2}$ ...(i) and $\cos 2 \theta+\cos 2 \phi=\frac{3}{2}$ ....(ii) Squaring and adding (i) and (ii), we get: $(\sin 2 \theta+\sin 2 \phi)^{2}+(\cos 2 \theta+\cos 2 \phi)^{2}=\frac{1}{4}+\frac{9}{4}$ $\Rightarrow\left[2 \sin \left(\...
Read More →The monthly pocket money of six friends is given below:
Question: The monthly pocket money of six friends is given below: Rs 45, Rs 30, Rs 40, Rs 25, Rs 45. (i) What is the lowest pocket money? (ii) What is the highest pocket money? (iii) What is the range? (iv) Arrange the amounts of pocket money in ascending order Solution: The monthly pocket money of six friends is given below: Rs 45,Rs 30, Rs 40, Rs 25, Rs 45. (1) Highest pocket money = Rs 50 (2) Lowest pocket money = Rs 25 (3) Range = 50 25 = 25 (4) The amounts of pocket money in an ascending or...
Read More →Given 15 cot A = 8, find sin A and sec A.
Question: Given 15 cotA= 8, find sinAand secA. Solution: Given: $15 \cot A=8$ To Find: $\sin A, \sec A$ Since $15 \cot A=8$ By taking 15 on R.H.S We get, $\cot A=\frac{8}{15} \ldots \ldots .(1)$ By definition, $\cot A=\frac{1}{\tan A}$ Hence, $\cot A=\frac{1}{\frac{\text { Perpendicular side opposite to } \angle \mathrm{A}}{\text { Base side adjacent to } \angle \mathrm{A}}}$ $\cot A=\frac{\text { Base side adjacent to } \angle \mathrm{A}}{\text { Perpendicular side opposite to } \angle \mathrm{...
Read More →The ages of ten students of a group are given below.
Question: The ages of ten students of a group are given below. The ages have been recorded in years and months: 8 - 6, 9 - 0, 8 - 4, 9 - 3, 7-8, 8 - 11, 8 - 7, 9 - 2, 7 - 10, 8 - 8 (i) What is the lowest age? (ii) What is the highest age? (iii) Determine the range? Solution: The ages of ten students of a group are given below 8 - 6, 9 - 0, 8 - 4, 9 - 3, 7 - 8, 8 - 11, 8 -7, 9 - 2, 7 - 10, 8 - 8. (1) Lowest age is 7 years 8 months (2) Highest age is9 years, 3months (3) Range = Highest age-lowest ...
Read More →Explain the meaning of the following terms:
Question: Explain the meaning of the following terms: (1) Variate (2) Class-integral (3) Class-size (4) Class-mark (5) Frequency (6) Class limits (7) True class limits Solution: (1) Variate Any character that can vary from one individual to another is called variate. (2) Class interval In the data each group into which raw data is considered is called a class-interval. (3) Class-size The difference between the true upper limit and lower limit is called the class size of that class size. (4) Clas...
Read More →sin 163° cos 347° + sin 73° sin 167° =
Question: sin 163 cos 347 + sin 73 sin 167 = (a) 0 (b) $\frac{1}{2}$ (c) 1(d) None of these Solution: (b) $\frac{1}{2}$ $\sin 163^{\circ} \cos 347^{\circ}+\sin 73^{\circ} \sin 167^{\circ}$ $=\sin \left(180^{\circ}-17^{\circ}\right) \cos \left(360^{\circ}-13^{\circ}\right)+\sin \left(90^{\circ}-17^{\circ}\right) \sin \left(180^{\circ}-13^{\circ}\right)$ $=\sin 17^{\circ} \cos 13^{\circ}+\cos 17^{\circ} \sin 13^{\circ}$ $=\sin \left(17^{\circ}+13^{\circ}\right)$ $[\sin (A+B)=\sin A \cos B+\sin B \...
Read More →sin 163° cos 347° + sin 73° sin 167° =
Question: sin 163 cos 347 + sin 73 sin 167 = (a) 0 (b) $\frac{1}{2}$ (c) 1(d) None of these Solution: (b) $\frac{1}{2}$ $\sin 163^{\circ} \cos 347^{\circ}+\sin 73^{\circ} \sin 167^{\circ}$ $=\sin \left(180^{\circ}-17^{\circ}\right) \cos \left(360^{\circ}-13^{\circ}\right)+\sin \left(90^{\circ}-17^{\circ}\right) \sin \left(180^{\circ}-13^{\circ}\right)$ $=\sin 17^{\circ} \cos 13^{\circ}+\cos 17^{\circ} \sin 13^{\circ}$ $=\sin \left(17^{\circ}+13^{\circ}\right)$ $[\sin (A+B)=\sin A \cos B+\sin B \...
Read More →Show that each of the given three vectors is a unit vector:
Question: Show that each of the given three vectors is a unit vector: $\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k}), \frac{1}{7}(3 \hat{i}-6 \hat{j}+2 \hat{k}), \frac{1}{7}(6 \hat{i}+2 \hat{j}-3 \hat{k})$ Also, show that they are mutually perpendicular to each other. Solution: Let $\vec{a}=\frac{1}{7}(2 \hat{i}+3 \hat{j}+6 \hat{k})=\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}+\frac{6}{7} \hat{k}$, $\vec{b}=\frac{1}{7}(3 \hat{i}-6 \hat{j}+2 \hat{k})=\frac{3}{7} \hat{i}-\frac{6}{7} \hat{j}+\frac{2}{7} \h...
Read More →Why do we group data?
Question: Why do we group data? Solution: The data obtained in original form are called raw data .Raw data does not give any useful information and is rather confusing to mind. Data is grouped so that it becomes understandable and can be interpreted. According to various characteristics groups are formed by us. After grouping the data, we are in position to make calculations of certain values which will help us in describing and analyzing the data....
Read More →cos 40° + cos 80° + cos 160° + cos 240° =
Question: cos 40 + cos 80 + cos 160 + cos 240 = (a) 0(b) 1 (c) $\frac{1}{2}$ (d) $-\frac{1}{2}$ Solution: (d) $-\frac{1}{2}$ $\cos 40^{\circ}+\cos 80^{\circ}+\cos 160^{\circ}+\cos 240^{\circ}$ $=2 \cos \left(\frac{40^{\circ}+80^{\circ}}{2}\right) \cos \left(\frac{40^{\circ}-80^{\circ}}{2}\right)+\cos 160^{\circ}-\cos \left(180^{\circ}+60^{\circ}\right)$ $\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $=2 \cos 60^{\circ} \cos \left(-20^{\cir...
Read More →What are (1) primary data (2) secondary data?
Question: What are (1) primary data (2) secondary data? Which of the two - the primary or the secondary data - is more reliable and why? Solution: The word data means information statistical data and are of two types: (1) Primary data When an investigator collects data himself with a definite plan or design in his or her mind is calledPrimary Data. (2) Secondary Data Data which are not originally collected rather obtained from published or unpublished sources are called secondary data. Secondary...
Read More →cos 40° + cos 80° + cos 160° + cos 240° =
Question: cos 40 + cos 80 + cos 160 + cos 240 = (a) 0(b) 1 (c) $\frac{1}{2}$ (d) $-\frac{1}{2}$ Solution: (d) $-\frac{1}{2}$ $\cos 40^{\circ}+\cos 80^{\circ}+\cos 160^{\circ}+\cos 240^{\circ}$ $=2 \cos \left(\frac{40^{\circ}+80^{\circ}}{2}\right) \cos \left(\frac{40^{\circ}-80^{\circ}}{2}\right)+\cos 160^{\circ}-\cos \left(180^{\circ}+60^{\circ}\right)$ $\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $=2 \cos 60^{\circ} \cos \left(-20^{\cir...
Read More →cos 40° + cos 80° + cos 160° + cos 240° =
Question: cos 40 + cos 80 + cos 160 + cos 240 = (a) 0(b) 1 (c) $\frac{1}{2}$ (d) $-\frac{1}{2}$ Solution: (d) $-\frac{1}{2}$ $\cos 40^{\circ}+\cos 80^{\circ}+\cos 160^{\circ}+\cos 240^{\circ}$ $=2 \cos \left(\frac{40^{\circ}+80^{\circ}}{2}\right) \cos \left(\frac{40^{\circ}-80^{\circ}}{2}\right)+\cos 160^{\circ}-\cos \left(180^{\circ}+60^{\circ}\right)$ $\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $=2 \cos 60^{\circ} \cos \left(-20^{\cir...
Read More →Find the projection of the vector
Question: Find the projection of the vector $\hat{i}+3 \hat{j}+7 \hat{k}$ on the vector $7 \hat{i}-\hat{j}+8 \hat{k}$. Solution: Let $\vec{a}=\hat{i}+3 \hat{j}+7 \hat{k}$ and $\hat{b}=7 \hat{i}-\hat{j}+8 \hat{k}$. Now, projection of vector $\vec{a}$ on $\vec{b}$ is given by, $\frac{1}{|\vec{b}|}(\vec{a} \cdot \vec{b})=\frac{1}{\sqrt{7^{2}+(-1)^{2}+8^{2}}}\{1(7)+3(-1)+7(8)\}=\frac{7-3+56}{\sqrt{49+1+64}}=\frac{60}{\sqrt{114}}$...
Read More →Describe some fundamental characteristics of statistics.
Question: Describe some fundamental characteristics of statistics. Solution: Fundamental characteristics of statistics (1) A single observation does not form statistics. Statistics are a sum total of observations. (2) Statistics are expressed quantitatively not qualitatively. (3) Statistics are collected with definite purpose. (4) Statistics in an experiment are comparable and can be classified into groups....
Read More →What do you understand by the word “statistics” in:
Question: What do you understand by the word statistics in: (a) Singular form (B) Plural form Solution: The word statistics is used in both its singular and plural senses. (a)In singular sense, statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data. (b)In plural sense, statistics means numerical facts or observations collected with definite purpose. For Example: Income and expenditure of persons in a particular locality, number of pers...
Read More →If m sinθ=n sin(θ+2α),
Question: If $m \sin \theta=n \sin (\theta+2 \alpha)$, prove that $\tan (\theta+\alpha) \cot \alpha=\frac{m+n}{m-n}$.[NCERT EXEMPLAR] Solution: Given: $m \sin \theta=n \sin (\theta+2 \alpha)$ $\Rightarrow \frac{m}{n}=\frac{\sin (\theta+2 \alpha)}{\sin \theta}$ Applying componendo and dividendo, we get $\frac{m+n}{m-n}=\frac{\sin (\theta+2 \alpha)+\sin \theta}{\sin (\theta+2 \alpha)-\sin \theta}$ $\Rightarrow \frac{m+n}{m-n}=\frac{2 \sin \left(\frac{\theta+2 \alpha+\theta}{2}\right) \cos \left(\f...
Read More →Find the projection of the vector
Question: Find the projection of the vector $\hat{i}-\hat{j}$ on the vector $\hat{i}+\hat{j}$. Solution: Let $\vec{a}=\hat{i}-\hat{j}$ and $\vec{b}=\hat{i}+\hat{j}$. Now, projection of vector $\vec{a}$ on $\vec{b}$ is given by, $\frac{1}{|\vec{b}|}(\vec{a} \cdot \vec{b})=\frac{1}{\sqrt{1+1}}\{1.1+(-1)(1)\}=\frac{1}{\sqrt{2}}(1-1)=0$ Hence, the projection of vector $\vec{a}$ on $\vec{b}$ is 0 ....
Read More →If $sin A
Question: If $\sin A=\frac{9}{41}$, compute $\cos A$ and $\tan A$. Solution: Given: $\sin A=\frac{9}{41}$...(1) To Find: $\cos A, \tan A$ By definition, $\sin A=\frac{\text { Perpendicular side opposite to } \angle \mathrm{A}}{\text { Hypotenuse }}$... (2) By Comparing (1) and (2) We get, Perpendicular side = 9 and Hypotenuse = 41 Now using the perpendicular side and hypotenuse we can construct $\triangle A B C$ as shown below Length of side $\mathrm{AB}$ is unknown in right angled $\triangle A ...
Read More →If m sinθ=n sin(θ+2α),
Question: If $m \sin \theta=n \sin (\theta+2 \alpha)$, prove that $\tan (\theta+\alpha) \cot \alpha=\frac{m+n}{m-n}$.[NCERT EXEMPLAR] Solution: Given: $m \sin \theta=n \sin (\theta+2 \alpha)$ $\Rightarrow \frac{m}{n}=\frac{\sin (\theta+2 \alpha)}{\sin \theta}$ Applying componendo and dividendo, we get $\frac{m+n}{m-n}=\frac{\sin (\theta+2 \alpha)+\sin \theta}{\sin (\theta+2 \alpha)-\sin \theta}$ $\Rightarrow \frac{m+n}{m-n}=\frac{2 \sin \left(\frac{\theta+2 \alpha+\theta}{2}\right) \cos \left(\f...
Read More →If x cosθ=y cos
Question: If $x \cos \theta=y \cos \left(\theta+\frac{2 \pi}{3}\right)=z \cos \left(\theta+\frac{4 \pi}{3}\right)$, prove that $x y+y z+z x=0$ [NCERT EXEMPLAR] Solution: $x \cos \theta=y \cos \left(\theta+\frac{2 \pi}{3}\right)=z \cos \left(\theta+\frac{4 \pi}{3}\right)$ $\Rightarrow \frac{\cos \theta}{\frac{1}{x}}=\frac{\cos \left(\theta+\frac{2 \pi}{3}\right)}{\frac{1}{y}}=\frac{\cos \left(\theta+\frac{4 \pi}{3}\right)}{\frac{1}{z}}$ $\Rightarrow \frac{\cos \theta}{\frac{1}{x}}=\frac{\cos \lef...
Read More →Find the angle between the vectors
Question: Find the angle between the vectors $\hat{i}-2 \hat{j}+3 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$ Solution: The given vectors are $\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}-2 \hat{j}+\hat{k}$. $|\vec{a}|=\sqrt{1^{2}+(-2)^{2}+3^{2}}=\sqrt{1+4+9}=\sqrt{14}$ $|\vec{b}|=\sqrt{3^{2}+(-2)^{2}+1^{2}}=\sqrt{9+4+1}=\sqrt{14}$ Now, $\vec{a} \cdot \vec{b}=(\hat{i}-2 \hat{j}+3 \hat{k})(3 \hat{i}-2 \hat{j}+\hat{k})$ $=1.3+(-2)(-2)+3.1$ $=3+4+3$ $=10$ Also, we know that $\vec{a} \...
Read More →In the given figure, find tan P and cot R. Is tan P = cot R?
Question: In the given figure, find tan P and cot R. Is tan P = cot R? Solution: The given figure is below: To Find: $\tan P, \cot R$ In the given right angled ΔPQR, length of side QR is unknown. Therefore, to find length of side QR we use Pythagoras Theorem Hence, by applying Pythagoras theorem in ΔPQR, We get, $P R^{2}=P Q^{2}+Q R^{2}$ Now, we substitute the length of given side PR and PQ in the above equation $13^{2}=12^{2}+Q R^{2}$ $Q R^{2}=13^{2}-12^{2}$ $Q R^{2}=169-144$ $O R^{2}=25$ $Q R=...
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