Determine the domain and range of the following relations:
Question: Determine the domain and range of the following relations: (i) $R=\{(a, b): a \in N, a5, b=4\}$ (ii) $S=\{(a, b): b=|a-1|, a \in Z$ and $|a| \leq 3\}$ Solution: (i)R= {(a,b) :a N,a 5,b= 4} We have: a= 1, 2, 3, 4 b= 4 R= {(1, 4), (2, 4), (3, 4), (4, 4)} Domain (R) = {1, 2, 3, 4} Range (R) = {4} (ii) $S=\{(a, b): b=|a-1|, a \in Z$ and $|a| \leq 3\}$ Now, $a=-3,-2,-1,0,1,2,3$ $b=|-3-1|=4$ $b=|-2-1|=3$ $b=|-1-1|=2$ $b=|0-1|=1$ $b=|1-1|=0$ $b=|2-1|=1$ $b=|3-1|=2$ Thus, we have: b= 4, 3, 2, ...
Read More →In the below fig, I, m and n are parallel lines intersected by transversal
Question: In the below fig, I, m and n are parallel lines intersected by transversal p at X. Y and Z respectively. Find 1, 2 and 3 Solution: From the given figure: 3 + mYZ = 180 [Linear pair] ⟹3 = 180 120 ⟹3 = 60 Now line I parallel to m 1 = 3 [Corresponding angles] 1 = 60 Also m parallel to n ⟹2 = 120 [Alternative interior angle] Hence,1 = 3 = 60 2 = 120...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{10}{x+y}+\frac{2}{x-y}=4$ $\frac{15}{x+y}-\frac{9}{x-y}=-2$ Solution: The given equations are: $\frac{10}{x+y}+\frac{2}{x-y}=4$ $\frac{15}{x+y}-\frac{9}{x-y}=-2$ Let $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$ then equations are $10 u+2 v=4 \ldots(i)$ $15 u-9 v=-2 \ldots(i i)$ Multiply equation $(i)$ by 9 and equation $(i i)$ by 2 and add both equations, we get Put the value of $u$ in equation $(i)$, we get $10 \times \frac{32}{120}+2 v=4$ $...
Read More →In below fig. AB CD and ∠1 and ∠2 are in the ratio 3: 2. Determine all angles from 1 to 8.
Question: In below fig. AB CD and1 and 2are in the ratio 3: 2. Determine all angles from 1 to 8. Solution: Let1 = 3x and2 = 2x 1 and 2are linear pair of angle Now,1 and 2 ⟹ 3x + 2x = 180 ⟹ 5x = 180 ⟹ x = 180/5 ⟹ x = 36 1 = 3x = 108,2 = 2x = 72 We know, Vertically opposite angles are equal 1 = 3 = 108 2 = 4 = 72 6 = 7 = 108 5 = 8 = 72 We also know, corresponding angles are equal 1 = 5 = 108 2 = 6 = 72...
Read More →Determine the domain and range of the relation R defined by
Question: Determine the domain and range of the relation R defined by(i) R = [(x,x+ 5):x (0, 1, 2, 3, 4, 5)] (ii) R = {(x,x3) :xis a prime number less than 10} Solution: (i) R = {(x,x+ 5):x (0, 1, 2, 3, 4, 5)} We have: R = {(0, 0 + 5), (1, 1 + 5), (2, 2 + 5), (3, 3 + 5), (4, 4 + 5), (5, 5 + 5)} Or, R = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} Domain (R) = {0, 1, 2, 3, 4, 5} Range (R) = {5, 6, 7, 8, 9, 10} (ii) R = {(x,x3) :xis a prime number less than 10} We have: x= 2, 3, 5, 7 x3= 8, 2...
Read More →Let A = [1, 2] and B = [3, 4].
Question: Let A = [1, 2] and B = [3, 4]. Find the total number of relation from A into B. Solution: We have: A = {1, 2} and B = {3, 4} Now, $n(A \times B)=n(A) \times n(B)=2 \times 2=4$ There are 2nrelations from A to B, wherenis the number of elements in their Cartesian product. Number of relations from A to B is 24= 16....
Read More →Fill in Inc blanks so as to make the following statements true:
Question: Fill in Inc blanks so as to make the following statements true: (i) If one angle of a linear pair is acute then its other angle will be______ (ii) A ray stands on a line, then the sum of the two adjacent angles so formed is ______ (iii) If the sum of two adjacent angles is 180, then the ______ arms of the two angles are opposite rays. Solution: (i) Obtuse angle (ii) 180 degrees (iii) Uncommon...
Read More →Which of the following statements are true (T) and which are false (F)?
Question: Which of the following statements are true (T) and which are false (F)? (i) Angles forming a linear pair are supplementary. (ii) If two adjacent angles are equal and then each angle measures 90 (iii) Angles forming a linear pair can both acute angles. (iv) If angles forming a linear pair are equal, then each of the angles have a measure of 90 Solution: (i) True (ii) False (iii) False (iv) true...
Read More →Let A = (3, 5) and B = (7, 11).
Question: Let A = (3, 5) and B = (7, 11). Let R = {(a,b) :a A,b B,abis odd}. Show that R is an empty relation from A into B. Solution: Given: A = (3, 5) and B = (7, 11) Also, R = {(a,b) :a A,b B,abis odd} aare the elements of A andbare the elements of B. $\therefore a-b=3-7,3-11,5-7,5-11$ $\Rightarrow a-b=-4,-8,-2,-6$ Here, $a-b$ is always an even number. So, R is an empty relation from A to B. Hence proved....
Read More →In below figure, lines AB and CD intersect at O.
Question: In below figure, lines AB and CD intersect at O. If AOC + BOE = 70 and BOD = 40, find BOE and reflex COE? Solution: Given that AOC + BOE = 70 and BOD = 40 To find BOE Here, BOD and AOC an vertically opposite angles BOD = AOC = 40 Given, AOC + BOE = 70, ⟹ 40 + BOF = 70 ⟹ BOF = 70 - 40 ⟹ BOE = 30 AOC and BOC are lines pair of angles ⟹ AOC + COF + BOE = 180 ⟹ COE = 180 - 30 - 40 ⟹ COE = 110 Hence, Reflex COE = 360 - 110 - 250....
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{5}{x-1}+\frac{1}{y-2}=2$ $\frac{6}{x-1}-\frac{3}{y-2}=1$ Solution: The given equations are: $\frac{5}{x-1}+\frac{1}{y-2}=2$ $\frac{6}{x-1}-\frac{3}{y-2}=1$ Let $\frac{1}{x-1}=u$ and $\frac{1}{y-2}=v$ then equations are $5 u+v=2 \ldots(i)$ $6 u-3 v=1 \ldots(i i)$ Multiply equation $(i)$ by 3 and add both equations, we get Put the value of $u$ in equation $(i)$, we get $5 \times \frac{1}{3}+v=2$ $\Rightarrow v=\frac{1}{3}$ Then $\frac{1}{x...
Read More →Let R be a relation in N defined by (x, y)
Question: LetRbe a relation in N defined by (x,y) R ⇔x+ 2y=8. Express R and R1as sets of ordered pairs. Solution: LetRbe a relation in N defined by (x,y) R ⇔x+ 2y= 8. We have: x= 8-2y Fory= 3, 2, 1, we have: x= 2, 4, 6 R= {(2, 3), (4, 2), (6, 1)} And, R1= {(3, 2), (2, 4), (1, 6)}...
Read More →AB, CD and EF are three concurrent lines passing through the point O such that OF bisects BOD. If BOF = 35. Find BOC and AOD.
Question: AB, CD and EF are three concurrent lines passing through the point O such that OF bisects BOD. If BOF = 35. Find BOC and AOD. Solution: Given OF bisects BOD BOF = 35 Angles BOC and AOD are unknown BOD = 2 BOF = 70 [since BOD is bisected] BOD = AOC = 70[BOD and AOC are vertically opposite angles] Now, BOC + AOC = 180 BOC +70 = 180 BOC =110 AOD = BOC = 110 (Vertically opposite angles]...
Read More →Write the following relation as the sets of ordered pairs:
Question: Write the following relation as the sets of ordered pairs:(i) A relation R from the set [2, 3, 4, 5, 6] to the set [1, 2, 3] defined byx= 2y. (ii) A relation R on the set [1, 2, 3, 4, 5, 6, 7] defined by (x,y) R ⇔xis relatively prime toy. (iii) A relation R on the set [0, 1, 2, ....., 10] defined by 2x+ 3y= 12. (iv) A relation R from a set A = [5, 6, 7, 8] to the set B = [10, 12, 15, 16,18] defined by (x,y) R ⇔xdividesy. Solution: (i) A relation R from the set [2, 3, 4, 5, 6] to the se...
Read More →In the below fig. lines AB. CD and EF intersect at O. Find the measures of ∠AOC, ∠COF, ∠DOE and ∠BOF.
Question: In the below fig. lines AB. CD and EF intersect at O. Find the measures of AOC, COF, DOE and BOF. Solution: AOE and EOB are linear pair of angles AOE + EOB = 180 AOE + DOE + BOD = 180 ⟹ DOE = 180 - 40 - 35 = 105 Vertically opposite side angles are equal DOE = COF =105 Now, AOE + AOF = 180 [Linear pair] AOE + AOC + COF = 180 ⟹ 40 + AOC +105 = 180 ⟹ AOC = 180 - 145 ⟹ AOC = 35 Also, BOF = AOE = 40 (Vertically opposite angles are equal)...
Read More →In the below fig. rays AB and CD intersect at O.
Question: In the below fig. rays AB and CD intersect at O. (I) Determine y when x = 60 (ii) Determine x when y = 40 Solution: (i) Given x = 60 AOC, BOC are linear pair of angles AOC + BOC = 180 ⟹ 2x + y = 180 ⟹ 2(60) + y= 180 [since x = 60] ⟹ y = 60 (ii) Given y = 40 AOC and BOC are linear pair of angles AOC + BOC = 180 ⟹ 2x + y = 180 ⟹ 2x + 40 = 180 ⟹2x =180 - 140 ⟹ 2x = 140 ⟹ x = 70...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{2}{x}+\frac{3}{y}=13$ $\frac{5}{x}-\frac{4}{y}=-2$ Solution: The given equations are: $\frac{2}{x}+\frac{3}{y}=13$..(i) $\frac{5}{x}-\frac{4}{y}=-2 \ldots(i i)$ Multiply equation (i) by 4 and equation (ii) by 3 and add both equations we get Put the value of $x$ in equation $(i)$, we get $\frac{2}{\frac{1}{2}}+\frac{3}{y}=13$ $\Rightarrow \frac{3}{y}=9$ $\Rightarrow y=\frac{1}{3}$ Hence the value of $x=\frac{1}{2}$ and $y=\frac{1}{3}$....
Read More →Find the inverse relation
Question: Find the inverse relation R1in each of the following cases:(i) R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)} (ii) R = {(x,y), :x,y N,x+ 2y= 8} (iii) R is a relation from {11, 12, 13} to (8, 10, 12] defined byy=x 3. Solution: (i) R = {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)} R1= {(2, 1), (3, 1), (3, 2), (2, 3), (6, 5)}(ii) R = {(x,y) :x,y N,x+ 2y= 8} On solvingx+ 2y= 8, we get: x= 8-2y On puttingy= 1, we getx= 6. On puttingy= 2, we getx= 4. On puttingy= 3, we getx= 2. R = {(6, 1), (4, 2),...
Read More →If one of the four angles formed by two intersecting lines is a right angle.
Question: If one of the four angles formed by two intersecting lines is a right angle. Then show that each of the four angles is a right angle. Solution: Given, AB and CD are two lines intersecting at O, such that BOC = 90, AOC = 90 AOD = 90 and BOD = 90 Proof: Given that B0C = 90 Vertically opposite angles are equal BOC = AOD = 90 A0C, BOC are a Linear pair of angles AOC + BOC = 180 [Linear pair] AOC + 90 = 180 AOC = 90 Vertically opposite angles Therefore, AOC = BOD = 90 Hence, AOC = BOC = BOD...
Read More →Prove
Question: $\frac{x+2}{\sqrt{x^{2}-1}}$ Solution: Let $x+2=A \frac{d}{d x}\left(x^{2}-1\right)+B$ $\Rightarrow x+2=A(2 x)+B$ Equating the coefficients ofxand constant term on both sides, we obtain $2 A=1 \Rightarrow A=\frac{1}{2}$ $B=2$ From (1), we obtain $(x+2)=\frac{1}{2}(2 x)+2$ Then, $\int \frac{x+2}{\sqrt{x^{2}-1}} d x=\int \frac{\frac{1}{2}(2 x)+2}{\sqrt{x^{2}-1}} d x$ $=\frac{1}{2} \int \frac{2 x}{\sqrt{x^{2}-1}} d x+\int \frac{2}{\sqrt{x^{2}-1}} d x$ ...(1) In $\frac{1}{2} \int \frac{2 x...
Read More →If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus
Question: If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle. Solution: Let AB and CD intersect at a point O Also, let us draw the bisector OP of AOC Therefore, AOP = POC ... (i) Also, let us extend OP to Q. We need to show that, OQ bisects BOD Let us assume that OQ bisects BOD, now we shall prove that POQ is a line. We know that, AOC and DOB are vertically opposite angles. Therefore, thes...
Read More →Solve the following systems of equations:
Question: Solve the following systems of equations: $\frac{4}{x}+5 y=7$ $\frac{3}{x}+4 y=5$ Solution: The given equations are: $\frac{4}{x}+5 y=7 \ldots(i)$ $\frac{3}{x}+4 y=5$ ...(ii) Multiply equation $(i)$ by 4 and equation $(i i)$ by 5 and subtract (ii) from (i) we get $\frac{16}{x}+20 y=28$ Put the value of $x$ in equation $(i)$, we get $\frac{4}{\frac{1}{3}}+5 y=7$ $\Rightarrow 5 y=-5$ $\Rightarrow y=-1$ Hence the value of $x=\frac{1}{3}$ and $y=-1$...
Read More →Let A be the set of first five natural numbers and let R be a relation on A defined as follows:
Question: Let A be the set of first five natural numbers and let R be a relation on A defined as follows: (x,y) R ⇔xy Express R and R1as sets of ordered pairs. Determine also (i) the domain of R1 (ii) the range of R. Solution: Given: A is the set of the first five natural numbers. A = {1, 2, 3, 4, 5} The relation is defined as: (x,y) R ⇔xy Now, R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5), (4, 4), (4, 5), (5, 5)} R-1= {(1, 1), (2, 1), (3, 1)...
Read More →Prove that bisectors of a pair of vertically opposite angles are in the same straight line.
Question: Prove that bisectors of a pair of vertically opposite angles are in the same straight line. Solution: Given, Lines A0B and COD intersect at point O, such that AOC - BOO Also OE is the bisector of ADC and OF is the bisector of BOD To prove: EOF is a straight line, vertically opposite angles are equal AOD = BOC = 5x ... (1) Also, AOC + BOD 2 AOD = 2 DOF... (2) We know, Sum of the angles around a point is 360 2AOD + 2AOE + 2DOF = 360 AOD +AOE + DOF = 180 From this we can conclude that EOF...
Read More →In the below fig. find the value of x?
Question: In the below fig. find the value of x? Solution: Vertically opposite angles are equal AOE = BOF = 5x Linear pair COA + AOE + EOD = 180 3x + 5x + 2x = 180 10x = 180 x = 180/10 x = 18 Hence, the value of x = 18...
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