The value of
Question: The value of $\frac{\sin 70^{\circ}+\cos 40^{\circ}}{\cos 70^{\circ}+\sin 40^{\circ}}$ is_____________________ Solution: $\frac{\sin 70^{\circ}+\cos 40^{\circ}}{\cos 70^{\circ}+\sin 40^{\circ}}$ $=\frac{\sin 70^{\circ}+\cos \left(90^{\circ}-50^{\circ}\right)}{\cos \left(90^{\circ}-20^{\circ}\right)+\sin 40^{\circ}}$$\left[\because \cos \left(90^{\circ}-\theta\right)-\sin \theta\right]$ $=\frac{\sin 70^{\circ}+\sin 50^{\circ}}{\sin 20^{\circ}+\sin 40^{\circ}}$ $=\frac{2 \sin \left(\frac...
Read More →Define cumulative frequency distribution.
Question: Define cumulative frequency distribution. Solution: Cumulative frequency distribution: A table which displays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution or cumulative frequency distribution table....
Read More →If either vector
Question: If either vector $\vec{a}=\overrightarrow{0}$ or $\vec{b}=\overrightarrow{0}$, then $\vec{a} \cdot \vec{b}=0$. But the converse need not be true. Justify your answer with an example. Solution: Consider $\vec{a}=2 \hat{i}+4 \hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}+3 \hat{j}-6 \hat{k}$. Then, $\vec{a} \cdot \vec{b}=2.3+4.3+3(-6)=6+12-18=0$ We now observe that: $|\vec{a}|=\sqrt{2^{2}+4^{2}+3^{2}}=\sqrt{29}$ $\therefore \vec{a} \neq \overrightarrow{0}$ $|\vec{b}|=\sqrt{3^{2}+3^{2}+(-6)^{2...
Read More →Thirty children were asked about the number of hours they watched TV programes in the previous week.
Question: Thirty children were asked about the number of hours they watched TV programes in the previous week. The results were found as follows: 1, 6, 2, 3, 5, 12, 5, 8, 4, 8 10, 3, 4, 12, 2, 8, 15, 1, 17, 6 3, 2, 8, 5, 9, 6, 8, 7, 14, 2. (i) Make a frequency distribution table for this data, taking class width 5 and one of the class intervals as 5 - 10. (ii) How many children watched television for 15 or more hours a week. Solution: (i)Class intervals will be 0 - 5, 5 - 10, 10 - 15 The grouped...
Read More →The value of sin
Question: The value of $\sin \frac{\pi}{18}+\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}+\sin \frac{5 \pi}{18}$ is given by (a) $\sin \frac{7 \pi}{18}+\sin \frac{4 \pi}{9}$ (b) 1 (c) $\cos \frac{\pi}{6}+\cos \frac{3 \pi}{7}$ (d) $\cos \frac{\pi}{9}+\sin \frac{\pi}{9}$ Solution: $\sin \frac{\pi}{18}+\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}+\sin \frac{5 \pi}{18}$ $=\sin \frac{\pi}{18}+\sin \frac{5 \pi}{18}+\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}$ $=2 \sin \left(\frac{\frac{5 \pi}{18}+\frac{\pi}{18}}{2}\...
Read More →The value of sin
Question: The value of $\sin \frac{\pi}{18}+\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}+\sin \frac{5 \pi}{18}$ is given by (a) $\sin \frac{7 \pi}{18}+\sin \frac{4 \pi}{9}$ (b) 1 (c) $\cos \frac{\pi}{6}+\cos \frac{3 \pi}{7}$ (d) $\cos \frac{\pi}{9}+\sin \frac{\pi}{9}$ Solution: $\sin \frac{\pi}{18}+\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}+\sin \frac{5 \pi}{18}$ $=\sin \frac{\pi}{18}+\sin \frac{5 \pi}{18}+\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}$ $=2 \sin \left(\frac{\frac{5 \pi}{18}+\frac{\pi}{18}}{2}\...
Read More →Three coins were tossed 30 times. Each time the number of heads occurring was noted down as follows:
Question: Three coins were tossed 30 times. Each time the number of heads occurring was noted down as follows: 0, 1, 2, 2, 1, 2, 3, 1, 3, 0 1, 3, 1, 1, 2, 2, 0, 1, 2, 1 3, 0, 1, 1, 2, 3, 2, 2, 0 Prepare a frequency distribution table for the data given above. Solution: By observing the data given above, the following frequency table can be constructed:...
Read More →If tan θ=ab, find the value of cos θ+sin θ cos θ−sin θ.
Question: If $\tan \theta=\frac{a}{b}$, find the value of $\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}$. Solution: Given: $\tan \theta=\frac{a}{b} \ldots \ldots$(1) Now, we know that $\tan \theta=\frac{\sin \theta}{\cos \theta}$ Therefore equation (1) becomes as follows $\frac{\sin \theta}{\cos \theta}=\frac{a}{b}$ Now, by applying invertendo We get, $\frac{\cos \theta}{\sin \theta}=\frac{b}{a}$ Now, by applying Compenendo-dividendo We get, $\frac{\cos \theta+\sin \theta}{\cos \theta...
Read More →The blood groups of 30 students of class VIII are recorded as follows:
Question: The blood groups of 30 students of class VIII are recorded as follows: A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O Represent this data in the form of a frequency distribution table .Find out which is the most common and which is the most rarest blood group among these students. Solution: Here 9 students have blood group A, 6 as B, 3 as AB and 12 as O So the table representing the data is as follows: As 12 students have their blood group O...
Read More →if the
Question: If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, find the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a} .$ Solution: It is given that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$. $\therefore \vec{a} \cdot(\vec{a}+\vec{b}+\vec{c})=\vec{a} \cdot \overrightarrow{0}$ $\Rightarrow \vec{a} \cdot \vec{a}+\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}=\vec{a} \cdot \overrightarrow{0}$ [Distributivity of sca...
Read More →If sin x + sin y =
Question: If $\sin x+\sin y=\sqrt{3}(\cos y-\cos x)$, then $\sin 3 x+\sin 3 y=$ (a) 2 sin 3x (b) 0 (c) 1 (d) none of these Solution: We have, $\sin x+\sin y=\sqrt{3}(\cos y-\cos x)$ $\Rightarrow 2 \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)=2 \sqrt{3} \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)$ $\Rightarrow 2 \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)-2 \sqrt{3} \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)=0$ $\Righ...
Read More →If sin x + sin y =
Question: If $\sin x+\sin y=\sqrt{3}(\cos y-\cos x)$, then $\sin 3 x+\sin 3 y=$ (a) 2 sin 3x (b) 0 (c) 1 (d) none of these Solution: We have, $\sin x+\sin y=\sqrt{3}(\cos y-\cos x)$ $\Rightarrow 2 \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)=2 \sqrt{3} \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)$ $\Rightarrow 2 \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)-2 \sqrt{3} \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)=0$ $\Righ...
Read More →The daily minimum temperatures in degree Celsius recorded in a certain arctic region are as follows:
Question: The daily minimum temperatures in degree Celsius recorded in a certain arctic region are as follows: -12.5, -10.8, -18.6, -8.4-10.8, -4.2, -4.8, -6.7, -13.2, -11.8, -2.3, -1.2, -2.6, 0, 2.4, 0, 3.2, 2.7, 3.4, 0, -2.4, -2.4, 0, 3.2, 2.7, 3.4, 0, -2.4, -5.8, -8.9, -14.6, -12.3, -11.5, -7.8, -2.9. Represent them as frequency distribution table taking -19.9 to -15 as the first class interval. Solution: Since first class interval is -19.9 to -15, frequency distribution with lower limit incl...
Read More →If 3 cot A = 4, check whether 1−tan2 A1+tan2 A=cos2 A−sin2 A or not.
Question: If $3 \cot \mathrm{A}=4$, check whether $\frac{1-\tan ^{2} A}{1+\tan ^{2} A}=\cos ^{2} A-\sin ^{2} A$ or not. Solution: Given: $3 \cot A=4$ To check whether $\frac{1-\tan ^{2} A}{1+\tan ^{2} A}=\cos ^{2} A-\sin ^{2} A$ or not $3 \cot A=4$ Dividing by 3 on both sides, We get, $\cot A=\frac{4}{3}$ By definition, $\cot A=\frac{1}{\tan A}$ Therefore, $\cot A=\frac{\frac{1}{\text { Perpendicular side opposite to } \angle \mathrm{A}}}{\text { Base side adjacent to } \angle \mathrm{A}}$ $\cot...
Read More →Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 laborers working in a factory, taking one of the class intervals as 210 – 230 (230 not included).
Question: Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 laborers working in a factory, taking one of the class intervals as 210 230 (230 not included). 220, 268, 258, 242, 210, 268, 272, 242, 311, 290, 300, 320, 319, 304, 302, 218, 306, 292, 254, 278, 210, 240, 280, 316, 306, 215, 256, 236. Solution:...
Read More →If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P.,
Question: If sin (B+CA), sin (C+AB), sin (A+BC) are in A.P., then cotA, cotBand cotCare in (a) GP (b) HP (c) AP (d) None of these Solution: (b) HP Given: sin (B+CA), sin (C+AB) and sin (A+BC) are in A.P. $\Rightarrow \sin (C+A-B)-\sin (B+C-A)=\sin (A+B-C)-\sin (C+A-B)$ $\Rightarrow 2 \sin \left(\frac{C+A-B-B-C+A}{2}\right) \cos \left(\frac{C+A-B+B+C-A}{2}\right)=2 \sin \left(\frac{A+B-C-C-A+B}{2}\right) \cos \left(\frac{A+B-C+C+A-B}{2}\right)$ $\Rightarrow \sin (A-B) \cos C=\sin (B-C) \cos A$ $\...
Read More →If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P.,
Question: If sin (B+CA), sin (C+AB), sin (A+BC) are in A.P., then cotA, cotBand cotCare in (a) GP (b) HP (c) AP (d) None of these Solution: (b) HP Given: sin (B+CA), sin (C+AB) and sin (A+BC) are in A.P. $\Rightarrow \sin (C+A-B)-\sin (B+C-A)=\sin (A+B-C)-\sin (C+A-B)$ $\Rightarrow 2 \sin \left(\frac{C+A-B-B-C+A}{2}\right) \cos \left(\frac{C+A-B+B+C-A}{2}\right)=2 \sin \left(\frac{A+B-C-C-A+B}{2}\right) \cos \left(\frac{A+B-C+C+A-B}{2}\right)$ $\Rightarrow \sin (A-B) \cos C=\sin (B-C) \cos A$ $\...
Read More →The daily maximum temperatures (in degree Celsius) recorded in a certain city during the month of November are as follows:
Question: The daily maximum temperatures (in degree Celsius) recorded in a certain city during the month of November are as follows: 25.8, 24.5, 25.6, 20.7, 21.8, 20.5, 20.6, 20.9, 22.3, 22.7, 23.1, 22.8, 22.9, 21.7, 21.3, 20.5, 20.9, 23.1, 22.4, 21.5, 22.7, 22.8, 22.0, 23.9, 24.7, 22.8, 23.8, 24.6, 23.9, 21.1. Represent the data in the form of a frequency distribution table with class size 11C. Solution:...
Read More →Heights (in cm) of 30 students of class IX are given below:
Question: Heights (in cm) of 30 students of class IX are given below: 155, 158, 154, 158, 160, 148, 149, 150, 153, 159, 161, 148, 157, 153, 157, 162, 159, 151, 154, 156 , 152 , 156 , 160, 152, 147, 155, 163, 155 , 157 , 153. Prepare a frequency distribution table with 160 - 164 as one of the class intervals. Solution:...
Read More →If A, B, C are in A.P.,
Question: If $A, B, C$ are in A.P., then $\frac{\sin A-\sin C}{\cos C-\cos A}=$ (a) tanB (b) cotB (c) tan 2B (d) None of these Solution: (b) cotB SinceA,BandC are in A.P, $B-A=C-B$ or, $2 B=A+C$ $\frac{\sin A-\sin C}{\cos C-\cos A}$ $=\frac{2 \sin \left(\frac{A-C}{2}\right) \cos \left(\frac{A+C}{2}\right)}{-2 \sin \left(\frac{C+A}{2}\right) \sin \left(\frac{C-A}{2}\right)}$ $\left[\because \sin A-\sin B=2 \sin \left(\frac{A-B}{2}\right) \cos \left(\frac{A+B}{2}\right)\right.$ and $\left.\cos A-\...
Read More →If A, B, C are in A.P.,
Question: If $A, B, C$ are in A.P., then $\frac{\sin A-\sin C}{\cos C-\cos A}=$ (a) tanB (b) cotB (c) tan 2B (d) None of these Solution: (b) cotB SinceA,BandC are in A.P, $B-A=C-B$ or, $2 B=A+C$ $\frac{\sin A-\sin C}{\cos C-\cos A}$ $=\frac{2 \sin \left(\frac{A-C}{2}\right) \cos \left(\frac{A+C}{2}\right)}{-2 \sin \left(\frac{C+A}{2}\right) \sin \left(\frac{C-A}{2}\right)}$ $\left[\because \sin A-\sin B=2 \sin \left(\frac{A-B}{2}\right) \cos \left(\frac{A+B}{2}\right)\right.$ and $\left.\cos A-\...
Read More →Marks scored by 40 students of class IX in mathematics are given below:
Question: Marks scored by 40 students of class IX in mathematics are given below: 81, 55, 68, 79, 85, 43, 29, 68, 54, 73, 47, 35, 72, 64, 95, 44, 50, 77, 64, 35, 79, 52, 45, 54, 70, 83, 62, 64, 72, 92, 84, 76, 63, 43, 54, 38, 73, 68, 52, 54. Prepare a frequency distribution with class size of 10 marks. Solution:...
Read More →If cos A = m cos B,
Question: If $\cos A=m \cos B$, then $\cot \frac{A+B}{2} \cot \frac{B-A}{2}=$ (a) $\frac{m-1}{m+1}$ (b) $\frac{m+2}{m-2}$ (c) $\frac{m+1}{m-1}$ (d) None of these Solution: (c) $\frac{m+1}{m-1}$ 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{B+A}{2}\right) \sin \left(\frac{B-A}{2}\right)}=\frac{m+1}{m-...
Read More →Following data gives the number of children in 40 families:
Question: Following data gives the number of children in 40 families: 1, 2, 6, 5, 1, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 0, 4, 4, 3, 2, 2, 0, 0, 1, 2, 2, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 4, 1, 6, 2, 2. Represent it in the form of a frequency distribution. Solution:...
Read More →Write the class size and class limits in each of the following:
Question: Write the class size and class limits in each of the following: (i) 104, 114, 124, 134, 144, 154 and 164. (ii) 47, 52, 57, 62, 67, 72, 78, 82, 87, 92, 97, 102. (iii) 12.5, 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5. Solution: (i) 104, 114, 124, 134, 144, 154 and 164. Class size = 114 104 = 10 (ii) 47, 52, 57, 62, 67, 72, 78, 82, 87, 92, 97, 102. Class size = 52 47 = 5 (iii) 12.5, 17.5, 22.5, 27.5, 32.5, 37.5, 42.5, 47.5. Class size = 17.5 - 12.5 = 5...
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