ABCD is a square, X and Y are points on sides AD and BC respectively such that
Question: ABCD is a square, X and Y are points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and BAY = ABX. Solution: Given that ABCD is a square, X and Y are points on sides AD and BC respectively such that AY = BX. To prove BY = AX andBAY =ABX Join B and X, A and Y. Since, ABCD is a square DAB =CBA = 90 XAB =YAB = 90 .... (i) Now, consider triangle XAB and YBA We have XAB =YBA = 90. [From (i)] BX = AY [given] And AB = BA [Common side] So, by RHS congruence criterion, we...
Read More →In figure, AD ⊥ CD and CB ⊥ CD.
Question: In figure, AD CD and CB CD. If AQ = BP and DP = CQ, prove that DAQ = CBP. Solution: Given in the figure, ADCD and CBCD. And AQ = BP and DP = CQ, To prove thatDAQ =CBP Given that DP = QC Add PQ on both sides DP + PQ = PQ + QC DQ = PC ... (i) Now, consider triangle DAQ and CBP, We have ADQ =BCP = 90 [given] AQ = BP [given] And DQ = PC [From (i)] So, by RHS congruence criterion, we have ΔDAQ ΔCBP Now, DAQ =CBP [Corresponding parts of congruent triangles are equal] Hence proved...
Read More →If perpendiculars from any point within an angle on its arms are congruent.
Question: If perpendiculars from any point within an angle on its arms are congruent. Prove that it lies on the bisector of that angle. Solution: Given that, if perpendicular from any point within, an angle on its arms is congruent, prove that it lies on the bisector of that angle. Now, Let us consider an angle ABC and let BP be one of the arm within the angle Draw perpendicular PN and PM on the arms BC and BA such that they meet BC and BA in N and M respectively. Now, inΔBPM andΔBPN We haveBMP ...
Read More →Find the value of k for which each of the following system of equations have infinitely many solutions :
Question: Find the value ofkfor which each of the following system of equations have infinitely many solutions : $8 x+5 y=9$ $k x+10 y=18$ Solution: GIVEN: $8 x+5 y=9$ $k x+10 y=18$ To find: To determine for what value ofkthe system of equation has 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 infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Here, $\frac{8}{k}=\frac{5}{10}=\frac{9}{18}$ $\f...
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Question: $x \tan ^{-1} x$ Solution: Let $I=\int x \tan ^{-1} x d x$ Taking $\tan ^{-1} x$ as first function and $x$ as second function and integrating by parts, we obtain $I=\tan ^{-1} x \int x d x-\int\left\{\left(\frac{d}{d x} \tan ^{-1} x\right) \int x d x\right\} d x$ $=\tan ^{-1} x\left(\frac{x^{2}}{2}\right)-\int \frac{1}{1+x^{2}} \cdot \frac{x^{2}}{2} d x$ $=\frac{x^{2} \tan ^{-1} x}{2}-\frac{1}{2} \int \frac{x^{2}}{1+x^{2}} d x$ $=\frac{x^{2} \tan ^{-1} x}{2}-\frac{1}{2} \int\left(\frac...
Read More →ABC is a triangle in which BE and CF are, respectively,
Question: ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles Solution: Given that ABC is a triangle in which BE and CF are perpendicular to the sides AC and AS respectively such that BE = CF. To prove, ΔABC is isosceles Now, considerΔBCF andΔCBE, We have BFC = CEB = 90 [Given] BC = CB [Common side] And CF = BE [Given] So, by RHS congruence criterion, we have ΔBFC ΔCEB Now, FBC =EBC [Incongruent triangles cor...
Read More →ABC is a triangle and D is the mid-point of BC.
Question: ABC is a triangle and D is the mid-point of BC. The perpendiculars from B to AB and AC are equal. Prove that the triangle is isosceles. Solution: Given that, in two right triangles one side and acute angle of one are equal to the corresponding side and angle of the other We have to prove that the triangles are congruent Let us consider two right triangles such that B =E = 90 ... (i) AB = DE ... (ii) C =F ... (iii) Now observe the two triangles ABC and DEF C =F [From (iii)] B =E [From (...
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Question: $x \sin ^{-1} x$ Solution: Let $I=\int x \sin ^{-1} x d x$ Taking $\sin ^{-1} x$ as first function and $x$ as second function and integrating by parts, we obtain $I=\sin ^{-1} x \int x d x-\int\left\{\left(\frac{d}{d x} \sin ^{-1} x\right) \int x d x\right\} d x$ $=\sin ^{-1} x\left(\frac{x^{2}}{2}\right)-\int \frac{1}{\sqrt{1-x^{2}}} \cdot \frac{x^{2}}{2} d x$ $=\frac{x^{2} \sin ^{-1} x}{2}+\frac{1}{2} \int \frac{-x^{2}}{\sqrt{1-x^{2}}} d x$ $=\frac{x^{2} \sin ^{-1} x}{2}+\frac{1}{2} ...
Read More →If A = {1, 2, 4} and B = {1, 2, 3},
Question: If A = {1, 2, 4} and B = {1, 2, 3}, represent following sets graphically:(i) A B (ii) B A (iii) A A (iv) B B Solution: Given: A = {1, 2, 4} and B = {1, 2, 3}(i) A B = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}(ii) B A = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 2), (3, 4)}(iii) A A = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (4, 1), (4, 2), (4, 4)}(iv) B B = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}...
Read More →Find the value of k for which each of the following system of equations have infinitely many solutions :
Question: Find the value ofkfor which each of the following system of equations have infinitely many solutions : $k x-2 y+6=0$ $4 x-3 y+9=0$ Solution: GIVEN: $k x-2 y+6=0$ $4 x-3 y+9=0$ To find: To determine for what value ofkthe system of equation has 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 infinitely many solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$ Here, $\frac{k}{4}=\frac{-2}{-3}=\frac{6}{9}$...
Read More →If A = {1, 2},
Question: IfA = {1, 2}, from the setAAA. Solution: Given: A= {1, 2} Now, AA ={(1, 1), (1, 2), (2, 1), (2, 2)} AAA ={(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}...
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Question: $x^{2} \log x$ Solution: Let $I=\int x^{2} \log x d x$ Taking $\log x$ as first function and $x^{2}$ as second function and integrating by parts, we obtain $\begin{aligned} I =\log x \int x^{2} d x-\int\left\{\left(\frac{d}{d x} \log x\right) \int x^{2} d x\right\} d x \\ =\log x\left(\frac{x^{3}}{3}\right)-\int \frac{1}{x} \cdot \frac{x^{3}}{3} d x \\ =\frac{x^{3} \log x}{3}-\int \frac{x^{2}}{3} d x \\ =\frac{x^{3} \log x}{3}-\frac{x^{3}}{9}+\mathrm{C} \end{aligned}$...
Read More →In a Δ PQR. IF PQ = QR and L, M and N are the mid-points of the sides PQ,
Question: In a Δ PQR. IF PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN. Solution: Given that inΔPQR, PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively We have to prove LN = MN. Join L and M, M and N, N and L We have PL = LQ, QM = MR and RN = NP [Since, L, M and N are mid-points of Pp. QR and RP respectively] And also PQ = QR PL = LQ = QM = MR =PQ/2=QR/2... (i) Using mid-point theorem, We have MN∥PQ and MN =PQ...
Read More →State whether each of the following statements are true or false.
Question: State whether each of the following statements are true or false. If the statements is false, re-write the given statements correctly:(i) If P = {m,n} and Q = {n,m}, then P Q = {(m,n), (n,m)} (ii) If A and B are non-empty sets, then A B is a non-empty set of ordered pairs (x,y) such thatx B andy A. (iii) If A = {1, 2}, B = {3, 4}, then A (B ϕ) = ϕ. Solution: (i) False Correct statement: If P = {m,n} and Q = {n,m}, then P Q = {(m,n), (m, m),(n, n), (n, m)}.(ii) False Correct statement: ...
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Question: $x \log 2 x$ Solution: Let $I=\int x \log 2 x d x$ Taking log 2xas first function andxas second function and integrating by parts, we obtain $I=\log 2 x \int x d x-\int\left\{\left(\frac{d}{d x} 2 \log x\right) \int x d x\right\} d x$ $=\log 2 x \cdot \frac{x^{2}}{2}-\int \frac{2}{2 x} \cdot \frac{x^{2}}{2} d x$ $=\frac{x^{2} \log 2 x}{2}-\int \frac{x}{2} d x$ $=\frac{x^{2} \log 2 x}{2}-\frac{x^{2}}{4}+\mathrm{C}$...
Read More →Find the value of k for which the following system of equations has a unique solution:
Question: Find the value ofkfor which the following system of equations has a unique solution: $4 x+k y+8=0$ $2 x+2 y+2=0$ Solution: GIVEN: $4 x+k y+8=0$ $2 x+2 y+2=0$ To find: To determine to value ofkfor which the system has a unique solution. 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 $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ Here, $\frac{4}{2} \neq \frac{k}{2}$ $k \neq \frac{4 \times 2}{2}$ $k \neq 4$ Hence for $k \neq 4$ the...
Read More →In figure, It is given that AB = CD and AD = BC. Prove that ΔADC ≅ ΔCBA.
Question: In figure, It is given that AB = CD and AD = BC. Prove that ΔADC ΔCBA. Solution: Given that in the figure AB = CD and AD = BC. We have to proveΔADC ΔCBA Now, ConsiderΔADC andΔCBA. We have AB = CD [Given] BC = AD [Given] And AC = AC [Common side] So, by SSS congruence criterion, we have ΔADC ΔCBA Hence proved...
Read More →If A = {−1, 1},
Question: IfA= {1, 1}, findAAA. Solution: Given: A= {1, 1} Thus, we have: AA ={(1, 1), (1, 1), (1, 1), (1, 1)} And, AAA ={(1, 1, 1), (1, 1, 1), (1, 1, 1), (1, 1, 1), (1, 1, 1), (1, 1, 1), (1, 1, 1), (1, 1, 1)}...
Read More →Let A = {1, 2, 3, 4}
Question: LetA= {1, 2, 3, 4} and R = {(a,b) :aA,bA,adividesb}. WriteRexplicitly. Solution: Given: A= {1, 2, 3, 4} R= {(a,b) :aA,bA,adividesb} We know: 1 divides 1, 2, 3 and 4. 2 divides 2 and 4. 3 divides 3. 4 divides 4. R= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}...
Read More →Find the value of k for which the following system of equations has a unique solution:
Question: Find the value ofkfor which the following system of equations has a unique solution: $k x+2 y=5$ $3 x+y=1$ Solution: GIVEN: $k x+2 y=5$ $3 x+y=1$ To find: To determine to value ofkfor which the system has a unique solution. 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 $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ Here, $\frac{k}{3} \neq \frac{2}{1}$ $k \neq 6$ Hence for $k \neq 6$ the system of equation has unique solution....
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Question: $x \log x$ Solution: Let $I=\int x \log x d x$ Taking logxas first function andxas second function and integrating by parts, we obtain $I=\log x \int x d x-\int\left\{\left(\frac{d}{d x} \log x\right) \int x d x\right\} d x$ $=\log x \cdot \frac{x^{2}}{2}-\int \frac{1}{x} \cdot \frac{x^{2}}{2} d x$ $=\frac{x^{2} \log x}{2}-\int \frac{x}{2} d x$ $=\frac{x^{2} \log x}{2}-\frac{x^{2}}{4}+\mathrm{C}$...
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Question: $x \log x$ Solution: Let $I=\int x \log x d x$ Taking logxas first function andxas second function and integrating by parts, we obtain $I=\log x \int x d x-\int\left\{\left(\frac{d}{d x} \log x\right) \int x d x\right\} d x$ $=\log x \cdot \frac{x^{2}}{2}-\int \frac{1}{x} \cdot \frac{x^{2}}{2} d x$ $=\frac{x^{2} \log x}{2}-\int \frac{x}{2} d x$ $=\frac{x^{2} \log x}{2}-\frac{x^{2}}{4}+\mathrm{C}$...
Read More →Let A and B be two sets such that n(A) = 3 and n(B) = 2.
Question: LetAandBbe two sets such thatn(A) = 3 andn(B) = 2. If (x, 1), (y, 2), (z, 1) are inAB, findAandB, wherex,y,zare distinct elements. Solution: A is the set of all first entries in ordered pairs inA BandBis the set of all second entries in ordered pairs inAB. Also, n(A) = 3 andn(B) = 2 A ={x,y,z} andB ={1, 2}...
Read More →ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
Question: ABC is a right angled triangle in which A = 90 and AB = AC. Find B and C. Solution: Given that ABC is a right angled triangle such thatA = 90 and AB = AC Since, AB = AC ΔABC is also isosceles. Therefore, we can say thatΔABC is right angled isosceles triangle. C =B andA = 90 ... (i) Now, we have sum of angled in a triangle = 180 A +B +C = 180 90 +B +B = 180 [From (i)] 2B = 180 - 90 B = 45 Therefore, B = C = 45...
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 $x-2 y=8$ $5 x-10 y=10$ Solution: GIVEN: $\begin{aligned} x-2 y =8 \\ 5 x-10 y =10 \end{aligned}$ 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...
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