In the below figure, ∠AOC and ∠BOC form a linear pair. If a - 2b = 30°, find a and b?
Question: In the below figure, AOC and BOC form a linear pair. If a - 2b = 30, find a and b? Solution: Given that, AOC and BOCform a linear pair If a - b = 30 AOC = a, BOC = b Therefore,a + b= 180... (1) Given a - 2b = 30 ... (2) By subtracting (1) and (2) a + b - a + 2b = 180 - 30 3b = 150 b = 150/3 b = 50 Since a - 2b = 30 a - 2(50) = 30 a = 30 + 100 a = 130 Hence, the values of a and b are 130 and 50 respectively....
Read More →Write the set
Question: Write the set $\left\{\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}, \frac{7}{50}\right\}$ in the set-builder form. Solution: The set-builder form of the set $\left\{\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}, \frac{7}{50}\right\}$ is $\left\{\frac{n}{n^{2}+1}: n \in N, n \leq 7\right\}$....
Read More →Write the set of all positive integers whose cube is odd.
Question: Write the set of all positive integers whose cube is odd. Solution: The set of all positive integers whose cube is odd is $\{2 n+1: n \in Z, n \geq 0\}$....
Read More →In the given below figure rays OA, OB, OC, OP and OE have the common end point O.
Question: In the given below figure rays OA, OB, OC, OP and OE have the common end point O. Show that AOB + BOC + COD + DOE + EOA = 360 Solution: Given that OA, OB, OD and OE have the common end point O. A ray opposite to OA is drawn SinceAOB, BOFare linear pairs, AOB + BOF = 180 AOB + BOC + COF = 180 .... (1) Also, AOE and EOFare linear pairs AOE + EOF = 180 AOE + DOF + DOE = 180.... (2) By adding (1) and (2) equations we get AOB + BOC + COF + AOE + DOF + DOE = 180 AOB + BOC + COD + DOE + EOA =...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{5}{x+1}-\frac{2}{y-1}=\frac{1}{2}$ $\frac{10}{x+1}+\frac{2}{y-1}=\frac{5}{2}$ where $x \neq-1$ and $y \neq 1$ Solution: The given equations are: $\frac{5}{x+1}-\frac{2}{y-1}=\frac{1}{2}$ $\frac{10}{x+1}+\frac{2}{y-1}=\frac{5}{2}$ Let $\frac{1}{x+1}=u$ and $\frac{1}{y-1}=v$ then equations are $5 u-2 v=\frac{1}{2} \ldots(i)$ $10 u+2 v=\frac{5}{2} \ldots(i i)$ Add both equations, we get $5 u-2 v=\frac{1}{2}$ Put the value of $u$ in equation...
Read More →Write the set of all vowels in the English alphabet which precede q.
Question: Write the set of all vowels in the English alphabet which precedeq. Solution: The set of vowels in the English alphabet that precedeqis {a, e, i, o}....
Read More →Prove
Question: $\frac{x^{2}}{1-x^{6}}$ Solution: Let $x^{3}=t$ $\therefore 3 x^{2} d x=d t$ $\Rightarrow \int \frac{x^{2}}{1-x^{6}} d x=\frac{1}{3} \int \frac{d t}{1-t^{2}}$ $=\frac{1}{3}\left[\frac{1}{2} \log \left|\frac{1+t}{1-t}\right|\right]+\mathrm{C}$ $=\frac{1}{6} \log \left|\frac{1+x^{3}}{1-x^{3}}\right|+\mathrm{C}$...
Read More →Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:
Question: Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form: (i) $\quad\{A, P, L, E\}$ (i) $\quad x: x+5=5, x \in Z$ (ii) $\{5,-5\}$ (ii) $\{x: x$ is a prime natural number and a divisor of 10$\}$ (iii) $\{0\}$ (iii) $\{x: x$ is a letter of the word "RAJASTHAN" $\}$ (iv) $\{1,2,5,10,$, (iv) $\{x: x$ is a natural number and divisor of 10$\}$ (v) $\{A, H, J, R, S, T, N\}$ (v) $\quad x: x^{2}-25=0$ (vi) $\{2,5\}$ (vi) $\{x: x$ is ...
Read More →In the given below figure, find x. Further find
Question: In the given below figure, find x. Further find COD, AOD, BOC Solution: Since AOD and BODform a line pair, AOD + BOD = 180 AOD + BOC + COD = 180 Given that, AOD = (x + 10), COD = x, BOC = (x + 20) (x + 10) + x + (x + 20) = 180 3x + 30 = 180 3x = 180 - 30 3x = 150/3 x = 50 Therefore,AOD = (x + 10) = 50 + 10 = 60 COD = x = 50 COD = (x + 20) = 50 + 20 = 70 AOD = 60COD = 50BOC = 70...
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Question: $\frac{3 x}{1+2 x^{4}}$ Solution: Let $\sqrt{2} x^{2}=t$ $\therefore 2 \sqrt{2} x d x=d t$ $\Rightarrow \int \frac{3 x}{1+2 x^{4}} d x=\frac{3}{2 \sqrt{2}} \int \frac{d t}{1+t^{2}}$ $=\frac{3}{2 \sqrt{2}}\left[\tan ^{-1} t\right]+\mathrm{C}$ $=\frac{3}{2 \sqrt{2}} \tan ^{-1}\left(\sqrt{2} x^{2}\right)+\mathrm{C}$...
Read More →In the below figure, write all pairs of adjacent angles and all the linear pairs.
Question: In the below figure, write all pairs of adjacent angles and all the linear pairs. Solution: Adjacent angles are: (i) AOC, COB (ii) AOD BOD (i) AOD, COD (i) BOC, COD Linear pairs: AOD, BOD, AOC, BOC...
Read More →Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:
Question: Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form: (i) $\quad\{A, P, L, E\}$ (i) $\quad x: x+5=5, x \in Z$ (ii) $\{5,-5\}$ (ii) $\{x: x$ is a prime natural number and a divisor of 10$\}$ (iii) $\{0\}$ (iii) $\{x: x$ is a letter of the word "RAJASTHAN" $\}$ (iv) $\{1,2,5,10,$, (iv) $\{x: x$ is a natural number and divisor of 10$\}$ (v) $\{A, H, J, R, S, T, N\}$ (v) $\quad x: x^{2}-25=0$ (vi) $\{2,5\}$ (vi) $\{x: x$ is ...
Read More →In the below Fig. OA and OB are opposite rays:
Question: In the below Fig. OA and OB are opposite rays: (i) If x = 25, what is the value of y? (ii) If y = 35, what is the value of x? Solution: (i) Given that, x = 25 SinceAOC and BOCform a linear pair AOC + BOC = 180 Given thatAOC = 2y + 5andBOC = 3x AOC + BOC = 180 (2y + 5) + 3x = 180 (2y +5) + 3(25) = 180 2y + 5 + 75 = 180 2y + 80 = 180 2y = 180 - 80= 100 y = 100/2 = 50 y = 50 (ii) Given that, y = 35 AOC + BOC = 180 (2y + 5) + 3x = 180 (2(35) + 5) + 3x = 180 (70 + 5) + 3x = 180 3x = 180 - 7...
Read More →Prove
Question: $\frac{1}{\sqrt{9-25 x^{2}}}$ Solution: Let $5 x=t$ $\therefore 5 d x=d t$ $\Rightarrow \int \frac{1}{\sqrt{9-25 x^{2}}} d x=\frac{1}{5} \int \frac{1}{9-t^{2}} d t$ $=\frac{1}{5} \int \frac{1}{\sqrt{3^{2}-t^{2}}} d t$ $=\frac{1}{5} \sin ^{-1}\left(\frac{t}{3}\right)+\mathrm{C}$ $=\frac{1}{5} \sin ^{-1}\left(\frac{5 x}{3}\right)+\mathrm{C}$...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{1}{2(x+2 y)}+\frac{5}{3(3 x-2 y)}=\frac{-3}{2}$ $\frac{5}{4(x+2 y)}-\frac{3}{5(3 x-2 y)}=\frac{61}{60}$ Solution: The given equations are: $\frac{1}{2(x+2 y)}+\frac{5}{3(3 x-2 y)}=-\frac{3}{2}$ $\frac{5}{4(x+2 y)}-\frac{3}{5(3 x-2 y)}=\frac{61}{60}$ Let $\frac{1}{x+2 y}=u$ and $\frac{1}{3 x-2 y}=v$ then equations are $\frac{1}{2} u+\frac{5}{3} v=-\frac{3}{2} \ldots$$\ldots(i)$ $\frac{5}{4} u-\frac{3}{5} v=\frac{61}{60}$$. .(i i)$ Multipl...
Read More →Prove
Question: $\frac{1}{\sqrt{(2-x)^{2}+1}}$ Solution: Let $2-x=t$ $\Rightarrow-d x=d t$ $\Rightarrow \int \frac{1}{\sqrt{(2-x)^{2}+1}} d x=-\int \frac{1}{\sqrt{t^{2}+1}} d t$ $=-\log \left|t+\sqrt{t^{2}+1}\right|+\mathrm{C}$ $\left[\int \frac{1}{\sqrt{x^{2}+a^{2}}} d t=\log \left|x+\sqrt{x^{2}+a^{2}}\right|\right]$ $=-\log \left|2-x+\sqrt{(2-x)^{2}+1}\right|+\mathrm{C}$ $=\log \left|\frac{1}{(2-x)+\sqrt{x^{2}-4 x+5}}\right|+C$...
Read More →The measure of an angle is twice the measure of its supplementary angle
Question: The measure of an angle is twice the measure of its supplementary angle. Find the measure of the angles? Solution: Let the angle in case be 'x' The supplementary of a angle x is (180 - x) Applying given data: x = 2 (180 - x) x = 360 - 2x 3x = 360 x = 360/3 x = 120 Therefore the value of the angle in case is 120...
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Question: $\frac{1}{\sqrt{1+4 x^{2}}}$ Solution: Let $2 x=t$ $\therefore 2 d x=d t$ $\Rightarrow \int \frac{1}{\sqrt{1+4 x^{2}}} d x=\frac{1}{2} \int \frac{d t}{\sqrt{1+t^{2}}}$ $=\frac{1}{2}\left[\log \left|t+\sqrt{t^{2}+1}\right|\right]+\mathrm{C}$ $\left[\int \frac{1}{\sqrt{x^{2}+a^{2}}} d t=\log \left|x+\sqrt{x^{2}+a^{2}}\right|\right]$ $=\frac{1}{2} \log \left|2 x+\sqrt{4 x^{2}+1}\right|+\mathrm{C}$...
Read More →An angle is 14° more than its complementary angle. What is its measure?
Question: An angle is 14 more than its complementary angle. What is its measure? Solution: Let the angle in case be 'x' Complementary angle of 'x' is (90 - x) From given data, x - (90 - x) = 14 x - 90 + x = 14 2x = 90 + 14 2x = 104 x = 104/2 x = 52 Hence the angle in case is found to be 52...
Read More →If the supplement of an angle is two third of itself. Determine the angle and its supplement?
Question: If the supplement of an angle is two third of itself. Determine the angle and its supplement? Solution: Supplementary of an angle =2/3 angle Let the angle in case be 'x', Supplementary of angle x will be (180 - x) It is given that 180 - x =2/3 x (180 - x) 3 = 2x 540 - 3x = 2x 5x = 540 x = 540/5 x = 108 Hence, supplementary angle = 180 - 108 = 72 Therefore, angles in case are 108 and supplementary angle is 72...
Read More →If the supplement of an angle is 3 times its complement, find its angle?
Question: If the supplement of an angle is 3 times its complement, find its angle? Solution: Let the angle in case be 'x' Given that, Supplement of an angle = 3 times its complementary angle Supplementary angle = 180 - x Complementary angle = 90 - x Applying given data, 180 - x = 3 (90 - x) 3x - x = 270 - 180 2x = 90 x = 90/2 x = 45 Therefore, the angle in case is 45...
Read More →If an angle differs from its complement by (10)°, find the angle ?
Question: If an angle differs from its complement by (10), find the angle ? Solution: Let the measured angle be 'x' say Given that, The angles measured will differ by(20) x - (90 - x) = 10 x - 90 + x = 10 2x = 90 + 10 2x = 100 x = 100/2 x = 50 Therefore the measure of the angle is (50)...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{3}{x+y}+\frac{2}{x-y}=2$ $\frac{9}{x+y}-\frac{4}{x-y}=1$ Solution: The given equations are: $\frac{3}{x+y}+\frac{2}{x-y}=2$ $\frac{9}{x+y}-\frac{4}{x-y}=1$ Let $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$ then equations are $3 u+2 v=2 \ldots(i)$ $9 u-4 v=1 \ldots(i i)$ Multiply equation $(i)$ by 2 and add both equations, we get Put the value of $u$ in equation $(i)$, we get $3 \times \frac{1}{3}+2 v=2$ $\Rightarrow 2 v=1$ $\Rightarrow v=\frac...
Read More →If the compliment of an angle is equal to the supplement of Thrice of itself, find the measure of the angle?
Question: If the compliment of an angle is equal to the supplement of Thrice of itself, find the measure of the angle? Solution: Let the angle measured be 'x' say. Its complementary angle is (90 - x) and Its supplementary angle is (180 - 3x) Given that, Supplementary of 4 times the angle = (180 - 3x) According to the given information; (90 - x) = (180 - 3x) 3x - x = 180 - 90 2x = 90 x = 90/2 x = 45 Therefore, the measured angle x = (45)...
Read More →If the angles (2x − 10)° and (x − 5)° are complementary, find x?
Question: If the angles (2x 10) and (x 5) are complementary, find x? Solution: Given that(2x 10) and(x 5)are complementary Since angles are complementary, their sum will be 90 (2x - 10) + (x - 5) = 90 3x -15 = 90 3x = 90 + 15 3x = 105 x = 105/3 x = 35 Hence, the value of x = (35)...
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