Let A and B be two sets.
Question: LetAandBbe two sets. Show that the setsABandBAhave elements in commoniffthe setsAandBhave an elements in common. Solution: Case (i): Let: A =(a, b, c) B= (e, f) Now, we have: AB ={(a, e}), (a, f),(b,e), (b, f), (c, e), (c, f)} BA = {(e, a), (e, b), (e, c), (f, a), (f, b), (f, c)} Thus, they have no elements in common. Case (ii): Let: A =(a, b, c) B =(a, f) Thus, we have: AB ={(a, a}), (a,f), (b, a), (b, f), (c,a), (c, f)} BA= {(a, a), (a, b), (a, c), (f, a), (f, b), (f, c)} Here,ABandB...
Read More →Show that
Question: $x^{2} e^{x}$ Solution: Let $I=\int x^{2} e^{x} d x$ Taking $x^{2}$ as first function and $e^{x}$ as second function and integrating by parts, we obtain $\begin{aligned} I =x^{2} \int e^{x} d x-\int\left\{\left(\frac{d}{d x} x^{2}\right) \int e^{x} d x\right\} d x \\ =x^{2} e^{x}-\int 2 x \cdot e^{x} d x \\ =x^{2} e^{x}-2 \int x \cdot e^{x} d x \end{aligned}$ Again integrating by parts, we obtain $=x^{2} e^{x}-2\left[x \cdot \int e^{x} d x-\int\left\{\left(\frac{d}{d x} x\right) \cdot ...
Read More →If A and B are two set having 3 elements in common.
Question: IfAandBare two set having 3 elements in common. Ifn(A) = 5,n(B) = 4, findn(AB) andn[(AB) (BA)]. Solution: Given: n(A)= 5 andn(B)= 4 Thus, we have: n(AB)= 5(4)= 20 AandBare two sets having 3 elements in common. Now, Let: A=(a, a, a, b, c)andB=(a, a, a, d) Thus, we have: (A B)= {(a, a), (a, a), (a, a), (a, d), (a, a), (a, a), (a, a), (a, d), (a, a), (a, a), (a, a), (a, d), (b, a), (b, a), (b, a), (b, d), (c, a), (c, a), (c, a), (c, d)}(B A)= {(a, a), (a, a), (a, a), (a, b), (a, c), (a, a...
Read More →ABC is a triangle in which ∠B = 2∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD.
Question: ABC is a triangle in which B = 2C. D is a point on BC such that AD bisects BAC and AB = CD. Prove that [BAC = 72. Solution: Given that in ABC,B = 2C and D is a point on BC such that AD bisectorsBAC and AB = CD. We have to prove that BAC = 72 Now, draw the angular bisector ofABC, which meets AC in P. Join PD Let C =ACB = y B =ABC = 2C = 2y and also LetBAD =DAC BAC = 2x [AD is the bisector ofBAC] Now, inΔBPC, CBP = y [BP is the bisector ofABC] PCB = y CBP =PCB = y [PC = BP] Consider,ΔABP...
Read More →In each of the following systems of equations determine whether the system has a unique solution,
Question: In each of the following systems of equations determine whether the system has a unique solution,no solution or infinitely many solutions. In case there is a unique solution, find it $3 x-5 y=20$ $6 x-10 y=40$ Solution: GIVEN: $3 x-5 y=20$ $6 x-10 y=40$ To find: To determine whether the system has a unique solution, no solution or infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For unique solution $\frac{a_{1}}{a_{2}} \neq ...
Read More →If A = {1, 2, 3} and B = {2, 4},
Question: IfA= {1, 2, 3} andB= {2, 4}, what areAB,BA,AA,BBand (AB) (BA)? Solution: Given : A= {1, 2, 3} andB= {2, 4} Now, AB ={(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)} BA ={(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)} AA ={(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} BB ={(2, 2), (2, 4), (4, 2), (4, 4)} We observe: (AB) (BA) = {(2, 2)}...
Read More →Show that
Question: $x \sin 3 x$ Solution: Let $I=\int x \sin 3 x d x$ Takingxas first function and sin 3xas second function and integrating by parts, we obtain $I=x \int \sin 3 x d x-\int\left\{\left(\frac{d}{d x} x\right) \int \sin 3 x d x\right\}$ $=x\left(\frac{-\cos 3 x}{3}\right)-\int 1 \cdot\left(\frac{-\cos 3 x}{3}\right) d x$ $=\frac{-x \cos 3 x}{3}+\frac{1}{3} \int \cos 3 x d x$ $=\frac{-x \cos 3 x}{3}+\frac{1}{9} \sin 3 x+C$...
Read More →Angles ΔA, B, C of a triangle ABC are equal to each other. Prove that ABC is equilateral.
Question: Angles ΔA, B, C of a triangle ABC are equal to each other. Prove that ABC is equilateral. Solution: Given that angles A, B, C of a triangle ABC equal to each other. We have to prove that ΔABC is equilateral We have,A =B =C Now, A =B BC = AC [opposite sides to equal angles are equal] AndB =C AC = AB From the above we get AB = BC = AC ΔABC is equilateral....
Read More →Let A = {1, 2, 3} and B = {3, 4}.
Question: LetA= {1, 2, 3} andB= {3, 4}. FindABand show it graphically. Solution: Given: A= {1, 2, 3} andB= {3, 4} Now, AB ={(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)} To representABgraphically, follow the given steps: (a) Draw two mutually perpendicular linesone horizontal and one vertical. (b) On the horizontal line, represent the elements of set A; and on the vertical line, represent the elements of set B. (c) Draw vertical dotted lines through points representing elements of set A on the...
Read More →Prove that each angle of an equilateral triangle is 60°.
Question: Prove that each angle of an equilateral triangle is 60. Solution: Given to prove each angle of an equilateral triangle is 60. Let us consider an equilateral triangle ABC. Such that AB = BC = CA Now, AB = BC A =C ... (i) [Opposite angles to equal sides are equal] And BC = AC B =A ... (ii) From (i) and (ii), we get A =B =C... (iii) We know that Sum of angles in a triangle = 180 A +B +C = 180 A +A +A = 180 3A = 180 A = 60 A =B =C = 60 Hence, each angle of an equilateral triangle is 60....
Read More →Show that
Question: $x \sin x$ Solution: Let $I=\int x \sin x d x$ Taking xas first function and sinxas second function and integrating by parts, we obtain $\begin{aligned} I =x \int \sin x d x-\int\left\{\left(\frac{d}{d x} x\right) \int \sin x d x\right\} d x \\ =x(-\cos x)-\int 1 \cdot(-\cos x) d x \\ =-x \cos x+\sin x+\mathrm{C} \end{aligned}$...
Read More →If A = {1, 2} and B = {1, 3},
Question: IfA= {1, 2} andB= {1, 3}, findABandBA. Solution: Given: A= {1, 2} andB= {1, 3} Now, AB ={(1, 1), (1, 3), (2, 1), (2, 3)} BA ={(1, 1), (1, 2), (3, 1), (3, 2)}...
Read More →In each of the following systems of equations determine whether the system has a unique solution,
Question: In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it $2 x+y=5$ $4 x+2 y=10$ Solution: GIVEN: $2 x+y=5$ $4 x+2 y=10$ To find: To determine whether the system has a unique solution, no solution or infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For unique solution $\frac{a_{1}}{a_{2}} \neq \frac{b...
Read More →If a
Question: Ifa [2, 4, 6, 9] andb [4, 6, 18, 27], then form the set of all ordered pairs (a,b) such that a dividesbandab. Solution: Given: a [2, 4, 6, 9] andb [4, 6, 18, 27] Here, 2 divides 4, 6 and 18 and 2 is less than all of them. 6 divides 18 and 6 and 6 is less than 18. 9 divides 18 and 27 and 9 is less than 18 and 27. Now, Set of all ordered pairs (a,b) such that a dividesbandab= {(2, 4), (2, 6), (2, 18), (6, 18), (9, 18), (9, 27)}...
Read More →P is a point on the bisector of an angle ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is isosceles.
Question: P is a point on the bisector of an angle ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is isosceles. Solution: Given that P is a point on the bisector of an angle ABC, and PQ∥AB. We have to prove thatΔBPQ is isosceles. Since, BP is bisector ofABC =ABP =PBC.... (i) Now, PQ∥AB BPQ =ABP ... (ii) [alternative angles] From (i) and (ii), we get BPQ =PBC (or)BPQ =PBQ Now, In BPQ, BPQ =PBQ ΔBPQ is an isosceles triangle. Hence proved...
Read More →If a ∈ [−1, 2, 3, 4, 5] and b ∈ [0, 3, 6],
Question: Ifa [1, 2, 3, 4, 5] andb [0, 3, 6], write the set of all ordered pairs (a,b) such thata+b= 5. Solution: Given: a [1, 2, 3, 4, 5] andb [0, 3, 6] We know: 1 + 6 = 5, 2 + 3 = 5 and 5 + 0 = 5 Thus, possible ordered pairs (a,b) are {(1, 6), (2, 3), (5, 0)} such thata+b= 5....
Read More →In each of the following systems of equations determine whether the system has a unique solution,
Question: In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it: $x-3 y=3$ $3 x-9 y=2$ Solution: GIVEN: $x-3 y=3$ $3 x-9 y=2$ To find: To determine whether the system has a unique solution, no solution or infinitely many solutions We know that the system of equations $a_{1} x+b_{1} y=c_{1}$ $a_{2} x+b_{2} y=c_{2}$ For unique solution $\frac{a_{1}}{a_{2}} \neq \frac{b_...
Read More →If the ordered pairs
Question: If the ordered pairs (x, 1) and (5,y) belong to the set {(a,b) :b= 2a 3}, find the values ofxandy. Solution: The ordered pairs (x, 1) and (5,y) belong to the set {(a,b) :b= 2a 3}. Thus, we have: x = aand 1= bsuch thatb= 2a 3. 1 = 2x 3 or, 2x= 3 1 = 2 or,x= 1 Also, 5 =aandy = bsuch thatb= 2a 3. y= 2(5) 3 or,y= 10 3 = 7 Thus, we get: x= 1 andy =7...
Read More →In a ΔABC, it is given that AB = AC and the bisectors of B and C intersect at O.
Question: In a ΔABC, it is given that AB = AC and the bisectors of B and C intersect at O. If M is a point on BO produced, prove that MOC = ABC. Solution: Given that inΔABC, AB = AC and the bisector ofB andC intersect at O. If M is a point on BO produced We have to prove MOC = ABC Since, AB = AC ABC is isosceles B =C (or) ABC =ACB Now, BO and CO are bisectors of ABC and ACB respectively $\Rightarrow \mathrm{ABO}=\angle \mathrm{OBC}=\angle \mathrm{ACO}=\angle \mathrm{OCB}=\frac{1}{2} \angle \math...
Read More →Show that
Question: $\int \frac{x d x}{(x-1)(x-2)}$ equals A. $\log \left|\frac{(x-1)^{2}}{x-2}\right|+C$ B. $\log \left|\frac{(x-2)^{2}}{x-1}\right|+\mathrm{C}$ C. $\log \left|\left(\frac{x-1}{x-2}\right)^{2}\right|+\mathrm{C}$ D. $\log |(x-1)(x-2)|+\mathrm{C}$ Solution: Let $\frac{x}{(x-1)(x-2)}=\frac{A}{(x-1)}+\frac{B}{(x-2)}$ $x=A(x-2)+B(x-1)$ ...(1) Substitutingx= 1 and 2 in (1), we obtain A= 1 andB= 2 $\therefore \frac{x}{(x-1)(x-2)}=-\frac{1}{(x-1)}+\frac{2}{(x-2)}$ $\Rightarrow \int \frac{x}{(x-1)...
Read More →(i) If
Question: (i) If $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $a$ and $b$. (ii) If $(x+1,1)=(3, y-2)$, find the values of $x$ and $y$. Solution: (i) $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ By the definition of equality of ordered pairs, we have: $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ $\Rightarrow\left(\frac{a}{3}+1\right)=\frac{5}{3}$ and $\left(b-\frac{2}...
Read More →(i) If
Question: (i) If $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $a$ and $b$. (ii) If $(x+1,1)=(3, y-2)$, find the values of $x$ and $y$. Solution: (i) $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ By the definition of equality of ordered pairs, we have: $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ $\Rightarrow\left(\frac{a}{3}+1\right)=\frac{5}{3}$ and $\left(b-\frac{2}...
Read More →(i) If $left( rac{a}{3}+1, b- rac{2}{3} ight)=left( rac{5}{3}, rac{1}{3} ight)$,
Question: (i) If $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $a$ and $b$. (ii) If $(x+1,1)=(3, y-2)$, find the values of $x$ and $y$. Solution: (i) $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ By the definition of equality of ordered pairs, we have: $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ $\Rightarrow\left(\frac{a}{3}+1\right)=\frac{5}{3}$ and $\left(b-\frac{2}...
Read More →(i) If $left( rac{a}{3}+1, b- rac{2}{3} ight)=left( rac{5}{3}, rac{1}{3} ight)$, find the values of $a$ and $b$.
Question: (i) If $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$, find the values of $a$ and $b$. (ii) If $(x+1,1)=(3, y-2)$, find the values of $x$ and $y$. Solution: (i) $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ By the definition of equality of ordered pairs, we have: $\left(\frac{a}{3}+1, b-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$ $\Rightarrow\left(\frac{a}{3}+1\right)=\frac{5}{3}$ and $\left(b-\frac{2}...
Read More →PQR is a triangle in which PQ = PR and is any point on the side PQ.
Question: PQR is a triangle in which PQ = PR and is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT. Solution: Given that PQR is a triangle such that PQ = PR ant S is any point on the side PQ and ST ∥ QR. To Prove, PS = PT Since, PQ = PR PQR is an isosceles triangle. Q =R (or)PQR =PRQ Now,PST =PQR andPTS =PRQ [Corresponding angles as ST parallel to QR] Since,PQR =PRQ PST =PTS Now, InΔPST,PST =PTS ΔPST is an isosceles triangle There...
Read More →