What value of y would make AOB a line in the below figure,
Question: What value of y would make AOB a line in the below figure, If AOB = 4y and BOC = (6y + 30)? Solution: Since,AOC and BOCare linear pairs AOC + BOC = 180 6y + 30 + 4y = 180 10y + 30 = 180 10y = 180 - 30 10y = 150 y = 150/10 y = 15 Hence value of y that will make AOB a line is 15...
Read More →In Fig: a is greater than b by one third of a right angle. Find the value of a and b?
Question: In Fig: a is greater than b by one third of a right angle. Find the value of a and b? Solution: Since a and b are linear a + b = 180 a = 180 b ... (1) From given data, a is greater than b by one third of a right angle a = b + 90/3 a = b + 30 a b = 30 ... (2) Equating (1) and (2) 180 b = b + 30 180 30 = 2b b = 150 / 2 b = 75 From (1) a = 180 b a = 180 75 a = 105 Hence the values of a and b are 105 and 75 respectively....
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{2}{3 x+2 y}+\frac{3}{3 x-2 y}=\frac{17}{5}$ $\frac{5}{3 x+2 y}+\frac{1}{3 x-2 y}=2$ Solution: The given equations are: $\frac{2}{3 x+2 y}+\frac{3}{3 x-2 y}=\frac{17}{5}$ $\frac{5}{3 x+2 y}+\frac{1}{3 x-2 y}=2$ Let $\frac{1}{3 x+2 y}=u$ and $\frac{1}{3 x-2 y}=v$ then equations are $2 u+3 v=\frac{17}{5}$ $\ldots(i)$ $5 u+v=2 \ldots(i i)$ Multiply equation (ii) by 3 and subtract (ii) from (i), we get $2 u+3 v=\frac{17}{5}$ Put the value of ...
Read More →If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
Question: IfA= {1, 2, 3},B= {4},C= {5}, then verify that:(i)A (BC) = (AB) (AC) (ii)A (BC) = (AB) (AC) (iii)A (BC) = (AB) (AC) Solution: Given: A= {1, 2, 3},B= {4} andC= {5} (i)A (BC) = (AB) (AC) We have: (BC) = {4, 5} LHS:A (BC) = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} Now, (AB) = {(1, 4), (2, 4), (3, 4)} And, (AC) = {(1, 5), (2, 5), (3, 5)} RHS: (AB) (AC) = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)} LHS = RHS (ii)A (BC) = (AB) (AC) We have: $(B \cap C)=\phi$ $\mathrm{LHS}: A \ti...
Read More →Prove
Question: $\frac{1}{\sqrt{7-6 x-x^{2}}}$ Solution: $7-6 x-x^{2}$ can be written as $7-\left(x^{2}+6 x+9-9\right)$ Therefore, $7-\left(x^{2}+6 x+9-9\right)$ $=16-\left(x^{2}+6 x+9\right)$ $=16-(x+3)^{2}$ $=(4)^{2}-(x+3)^{2}$ $\therefore \int \frac{1}{\sqrt{7-6 x-x^{2}}} d x=\int \frac{1}{\sqrt{(4)^{2}-(x+3)^{2}}} d x$ Let $x+3=t$ $\Rightarrow d x=d t$ $\Rightarrow \int \frac{1}{\sqrt{(4)^{2}-(x+3)^{2}}} d x=\int \frac{1}{\sqrt{(4)^{2}-(t)^{2}}} d t$ $=\sin ^{-1}\left(\frac{t}{4}\right)+C$ $=\sin ...
Read More →In the given figure, Given ∠POR = 3x and ∠QOR = 2x + 10, Find the value of x for which POQ will be a line?
Question: In the given figure, Given POR = 3x and QOR = 2x + 10, Find the value of x for which POQ will be a line? Solution: For the case that POR is a line POR and QOR are linear parts POR + QOR = 180 Also, given that, POR = 3x and QOR = 2x + 10 2x + 10 + 3x = 180 5x + 10 = 180 5x = 180 10 5x = 170 x = 34 Hence the value of x is 34...
Read More →In the below figure, ACB is a line such that ∠DCA = 5x and ∠DCB = 4x. Find the value of x?
Question: In the below figure, ACB is a line such that DCA = 5x and DCB = 4x. Find the value of x? Solution: Here,ACD + BCD = 180 [Since they are linear pairs] DCA = 5x and DCB = 4x 5x + 4x = 180 9x = 180 x = 20 Hence, the value of x is 20...
Read More →Prove
Question: $\frac{1}{9 x^{2}+6 x+5}$ Solution: $\int \frac{1}{9 x^{2}+6 x+5} d x=\int \frac{1}{(3 x+1)^{2}+2^{2}} d x$ Let $(3 x+1)=t$ $\therefore 3 d x=d t$ $\Rightarrow \int \frac{1}{(3 x+1)^{2}+2^{2}} d x=\frac{1}{3} \int \frac{1}{t^{2}+2^{2}} d t$ $=\frac{1}{3 \times 2} \tan ^{-1} \frac{t}{2}+C$ $=\frac{1}{6} \tan ^{-1}\left(\frac{3 x+1}{2}\right)+C$...
Read More →If A = {2, 3},
Question: IfA= {2, 3},B= {4, 5},C={5, 6}, findA (BC), A (BC), (AB) (AC). Solution: Given:A= {2, 3},B= {4, 5} andC={5, 6} Also, (BC) = {4, 5, 6} Thus, we have: A (BC) = {(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3,6)} And, (BC) = {5} Thus, we have: A (BC) = {(2, 5), (3, 5)} Now, (AB) = {(2, 4), (2, 5), (3, 4), (3, 5)} (AC) = {(2, 5), (2, 6), (3, 5), (3, 6)} (AB) (AC) = {(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}...
Read More →In the below figure, POS is a line, Find x?
Question: In the below figure, POS is a line, Find x? Solution: SincePOQ and QOSare linear pairs POQ + QOS = 180 POQ + QOR + SOR = 180 60 + 4x + 40 = 180 4x = 180 - 100 4x = 80 x = 20 Hence, Value of x = 20...
Read More →Given A =
Question: Given A = {1, 2, 3}, B = {3, 4}, C ={4, 5, 6}, find (A B) (B C). Solution: Given: A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6} Now, (A B) = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)} (B C) = {(3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)} (A B) (B C) = {(3, 4)}...
Read More →Prove
Question: $\frac{1}{\sqrt{x^{2}+2 x+2}}$ Solution: $\int \frac{1}{\sqrt{x^{2}+2 x+2}} d x=\int \frac{1}{\sqrt{(x+1)^{2}+(1)^{2}}} d x$ $\operatorname{Let} x+1=t$ $\therefore d x=d t$ $\Rightarrow \int \frac{1}{\sqrt{x^{2}+2 x+2}} d x=\int \frac{1}{\sqrt{t^{2}+1}} d t$ $=\log \left|t+\sqrt{t^{2}+1}\right|+\mathrm{C}$ $=\log \left|(x+1)+\sqrt{(x+1)^{2}+1}\right|+\mathrm{C}$ $=\log \left|(x+1)+\sqrt{x^{2}+2 x+2}\right|+\mathrm{C}$...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $2(3 u-\mathrm{v})=5 u \mathrm{~V}$ $2(u+3 \mathrm{~V})=5 u \mathrm{~V}$ Solution: The given equations are: $2(3 u-v)=5 u v$ $6 u-2 v=5 u v$$\ldots(i)$ $2(u+3 v)=5 u v$ $2 u+6 v=5 u v$$\ldots(i i)$ Multiply equation ( $i$ ) by 3 and add both equations, we get Put the value of $v$ in equation $(i)$, we get $6 u-2 \times 1=5 u \times 1$ $\Rightarrow u=2$ Hence the value of $u=2$ and $v=1$....
Read More →In the below figure, AOC is a line, find x.
Question: In the below figure, AOC is a line, find x. Solution: SinceAOB and BOCare linear pairs, AOB + BOC = 180 70 + 2x = 180 2x = 180 - 70 2x = 110 x = 110/2 x = 55 Hence, the value of x is 55...
Read More →In the below figure, find value of x?
Question: In the below figure, find value of x? Solution: Since the sum of all the angles round a point is equal to360 3x + 3x + 150 + x = 360 7x = 360 - 150 7x = 210 x = 210/7 x = 30 Value of x is 30...
Read More →Prove
Question: $\frac{\sec ^{2} x}{\sqrt{\tan ^{2} x+4}}$ Solution: Let $\tan x=t$ $\therefore \sec ^{2} x d x=d t$ $\Rightarrow \int \frac{\sec ^{2} x}{\sqrt{\tan ^{2} x+4}} d x=\int \frac{d t}{\sqrt{t^{2}+2^{2}}}$ $=\log \left|t+\sqrt{t^{2}+4}\right|+C$ $=\log \left|\tan x+\sqrt{\tan ^{2} x+4}\right|+\mathrm{C}$...
Read More →Which of the following sets are equal?
Question: Which of the following sets are equal? (i) $A=\{1,2,3\}$; (ii) $B=\left\{x \in R: x^{2}-2 x+1=0\right\}$; (iii) $C=\{1,2,2,3\}$; (iv) $D=\left\{x \in R: x^{3}-6 x^{2}+11 x-6=0\right\}$. Solution: Two setsABare equal if every element ofAis a member ofB every element ofBis a member ofA. (i) $A=\{1,2,3\}$ (ii) $B=\left\{x \in R: x^{2}-2 x+1=0\right\}$ SetBwould be {1}. (iii) $C=\{1,2,2,3\}$ It can be written as {1, 2, 3} because we do not repeat the elements while writing the elements of ...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $x+y=2 x y$ $\frac{x-y}{x y}=6$ $x \neq 0, y \neq 0$ Solution: The given equations are: $x+y=2 x y \ldots(i)$ $\frac{x-y}{x y}=6$ $x-y=6 x y$$\ldots(i i)$ Add both equations we get $x+y=2 x y$ Put the value of $y$ in equation $(i)$, we get $x+\frac{1}{4}=2 x \times \frac{1}{4}$ $\Rightarrow \frac{-x}{2}=\frac{1}{4}$ $\Rightarrow x=-\frac{1}{2}$ Hence the value of $x=-\frac{1}{2}$ and $y=\frac{1}{4}$....
Read More →How many pairs of adjacent angles, in all, can you name in the figure below?
Question: How many pairs of adjacent angles, in all, can you name in the figure below? Solution: Pairs of adjacent angles are: EOC, DOC EOD, DOB DOC, COB EOD, DOA DOC, COA BOC, BOA BOA, BOD BOA, BOE EOC, COA EOC, COB Hence, 10 pair of adjacent angles....
Read More →Prove
Question: $\frac{x^{2}}{\sqrt{x^{6}+a^{6}}}$ Solution: Let $x^{3}=t$ $\Rightarrow 3 x^{2} d x=d t$ $\therefore \int \frac{x^{2}}{\sqrt{x^{6}+a^{6}}} d x=\frac{1}{3} \int \frac{d t}{\sqrt{t^{2}+\left(a^{3}\right)^{2}}}$ $=\frac{1}{3} \log \left|t+\sqrt{t^{2}+a^{6}}\right|+\mathrm{C}$ $=\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+\mathrm{C}$...
Read More →Which of the following sets are finite and which are infinite?
Question: Which of the following sets are finite and which are infinite? (i) Set of concentric circles in a plane; (ii) Set of letters of the English Alphabets; (iii) {xN:x 5}; (iv) {x= N:x 200}; (v) {xZ:x 5}; (vi) {xR: 0 x 1}. Solution: (i) There can be infinite concentric circles in a plane. Therefore, it is an infinite set. (ii) There are 26 letters in the set of English alphabet. Therefore, it is a finite set. (iii) {xN:x 5} = {6,7,8,9,...}. There will be infinite numbers. So, it an infinite...
Read More →How many pairs of adjacent angles are formed when two lines intersect at a point?
Question: How many pairs of adjacent angles are formed when two lines intersect at a point? Solution: Four pairs of adjacent angles will be formed when two lines intersect at a point. Considering two lines AB and CD intersecting at O The 4 pairs are: (AOD, DOB), (DOB, BOC), (COA, AOD) and (BOC, COA) Hence, 4 pairs of adjacent angles are formed when two lines intersect at a point....
Read More →Prove
Question: $\frac{x-1}{\sqrt{x^{2}-1}}$ Solution: $\int \frac{x-1}{\sqrt{x^{2}-1}} d x=\int \frac{x}{\sqrt{x^{2}-1}} d x-\int \frac{1}{\sqrt{x^{2}-1}} d x$ ...(1) For $\int \frac{x}{\sqrt{x^{2}-1}} d x$, let $x^{2}-1=t \Rightarrow 2 x d x=d t$ $\therefore \int \frac{x}{\sqrt{x^{2}-1}} d x=\frac{1}{2} \int \frac{d t}{\sqrt{t}}$ $=\frac{1}{2} \int t^{-\frac{1}{2}} d t$ $=\frac{1}{2}\left[2 t^{\frac{1}{2}}\right]$ $=\sqrt{t}$ $=\sqrt{x^{2}-1}$ From (1), we obtain $\int \frac{x-1}{\sqrt{x^{2}-1}} d x...
Read More →Which of the following are examples of empty set?
Question: Which of the following are examples of empty set? (i) Set of all even natural numbers divisible by 5; (ii) Set of all even prime numbers; (iii) {x:x22 = 0 andxis rational}; (iv) {x:xis a natural number,x 8 and simultaneouslyx 12}; (v) {x:xis a point common to any two parallel lines}. Solution: (i) All natural numbers that end with 0 are even divisible by 5. Therefore, the given set is not an example of empty set. (ii) 2 is an even prime number. Therefore, the given set is not an exampl...
Read More →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 →