If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP,
Question: IfSbe the sum,Pthe product andRbe the sum of the reciprocals ofnterms of a GP, thenP2is equal to (a)S/R (b)R/S (c) (R/S)n (d) (S/R)n Solution: (d) $\left(\frac{S}{R}\right)^{n}$ Sum of $n$ terms of the G.P., $\mathrm{S}=\frac{a\left(r^{n}-1\right)}{(r-1)}$ Product of $n$ terms of the G.P., $\mathrm{P}=a^{n} r\left[\frac{n(n-1)}{2}\right]$ Sum of the reciprocals of $n$ terms of the G.P., $\mathrm{R}=\frac{\left[\frac{1}{r^{n}}-1\right]}{a\left(\frac{1}{r}-1\right)}=\frac{\left(r^{n}-1\r...
Read More →If a, b, c are in G.P. and a
Question: Ifa,b,care in G.P. anda1/x=b1/y=c1/z, thenxyzare in (a) AP (b) GP (c) HP (d) none of these Solution: (a) AP $a, b$ and $c$ are in G.P. $\therefore b^{2}=a c$ Taking log on both the sides : $2 \log \mathrm{b}=\log \mathrm{a}+\log \mathrm{c}$ ....(i) Now, $a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}$ Taking log on both the side $s:$ $\frac{\log \mathrm{a}}{\mathrm{x}}=\frac{\log \mathrm{b}}{\mathrm{y}}=\frac{\log \mathrm{c}}{\mathrm{z}} \quad \ldots \ldots$ (ii) Now, comparing (i) an...
Read More →Let A and B be square matrices of the same order. Does
Question: Let $A$ and $B$ be square matrices of the same order. Does $(A+B)^{2}=A^{2}+2 A B+B^{2}$ hold? If not, why? Solution: LHS $=(A+B)^{2}$ $=(A+B)(A+B)$ $=A(A+B)+B(A+B)$ $=A^{2}+A B+B A+B^{2}$ We know that a matrix does not have commutative property. So,ABBAThus, $(A+B)^{2} \neq A^{2}+2 A B+B^{2}$...
Read More →Prove that the points (3, 0), (4, 5), (−1, 4) and (−2 −1),
Question: Prove that the points (3, 0), (4, 5), (1, 4) and (2 1), taken in order, form a rhombus. Also, find its area. Solution: The distancedbetween two pointsandis given by the formula $d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$ In a rhombus all the sides are equal in length. And the area A of a rhombus is given as $A=\frac{1}{2}$ (Product of both diagonals) Here the four points areA(3,0), B(4,5),C(1,4) andD(2,1). First let us check if all the four sides are equal. $A ...
Read More →Give examples of matrices
Question: Give examples of matrices (i)AandBsuch thatABBA(ii)AandBsuch thatAB=ObutA 0,B 0.(iii)AandBsuch thatAB=ObutBAO.(iv)A,BandCsuch thatAB=ACbutBC,A 0. Solution: (i) Let $A=\left[\begin{array}{cc}1 -2 \\ 3 2\end{array}\right]$ and $B=\left[\begin{array}{cc}2 3 \\ -1 2\end{array}\right]$ $A B=\left[\begin{array}{cc}1 -2 \\ 3 2\end{array}\right]\left[\begin{array}{cc}2 3 \\ -1 2\end{array}\right]$ $\Rightarrow A B=\left[\begin{array}{ll}2+2 3-4 \\ 6-2 9+4\end{array}\right]$ $\Rightarrow A B=\l...
Read More →In the given figure, BD = DC and ∠CBD = 30°, find m(∠BAC).
Question: In the given figure,BD=DCandCBD= 30, find m(BAC). Solution: BD = DC⇒BCD =CBD= 30In ΔBCD,we have:BCD+CBD+CDB= 180 (Angle sum property of a triangle)⇒ 30+ 30+CDB= 180⇒CDB= (180 60) = 120The opposite angles of a cyclic quadrilateral are supplementary.Thus,CDB+BAC= 180⇒ 120+BAC= 180⇒BAC= (180 120) = 60BAC= 60...
Read More →Find the equation of the perpendicular bisector of the
Question: Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3,5). Solution: TO FIND: The equation of perpendicular bisector of line segment joining points (7, 1) and (3, 5) Let P(x,y) be any point on the perpendicular bisector of AB. Then, PA=PB $\Rightarrow \sqrt{(x-7)^{2}+(y-1)^{2}}=\sqrt{(x-3)^{2}+(y-5)^{2}}$ $\Rightarrow(x-7)^{2}+(y-1)^{2}=(x-3)^{2}+(y-5)^{2}$ $\Rightarrow \mathrm{x}^{2}-14 x+49+y^{2}-2 y+1=\mathrm{x}^{2}-6 x+9+y^{2}-10 y+25$ $\Ri...
Read More →In the given figure, ABCD is a cyclic quadrilateral in which AE is drawn parallel to CD,
Question: In the given figure,ABCDis a cyclic quadrilateral in whichAEis drawn parallel toCD, andBAis produced. IfABC= 92 and FAE= 20, find BCD. Solution: Given:ABCDis a cyclic quadrilateral.ThenABC +ADC= 180⇒ 92+ADC= 180⇒ADC= (180 92) = 88Again,AEparallel toCD.Thus,EAD =ADC= 88 (Alternate angles)We know that the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.BCD =DAF⇒BCD =EAD +EAF= 88+ 20= 108Hence,BCD= 108...
Read More →Name the quadrilateral formed, if any,
Question: Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:(i) A(1,2) B(1, 0), C (1, 2), D(3, 0)(ii) A(3, 5) B(3, 1), C (0, 3), D(1, 4)(iii) A(4, 5) B(7, 6), C (4, 3), D(1, 2) Solution: (i) A (1,2) , B(1,0), C(1,2), D(3,0) Let A, B, C and D be the four vertices of the quadrilateral ABCD. We know the distance between two points Pand Qis given by distance formula: $\mathrm{PQ}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$ Hence ...
Read More →In the given figure, O is the centre of the circle and arc ABC subtends an angle of 130° at the centre.
Question: In the given figure,Ois the centre of the circle and arcABCsubtends an angle of 130 at the centre. IfABis extended toP, find PBC. Solution: Reflex AOC+ AOC =360∘⇒Reflex AOC +130∘+x =360∘⇒Reflex AOC =360∘130∘⇒Reflex AOC =230∘We know that the angle subtended by an arc of a circle at the centre is double the angle subtended by it on the remaining part of the circle.Here, arcACsubtends reflex AOCat the centre and ABCatBon the circle. AOC= 2ABC $\Rightarrow \angle A B C=\frac{230^{\circ}}{2...
Read More →The first three of four given numbers are in G.P. and their last three are in A.P.
Question: The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is (a) 2 (b) 4 (c) 6 (d) 8 Solution: (d) 8 The first and the last numbers are equal. Let the four given numbers be $p, q, r$ and $p$. The first three of four given numbers are in G.P. $\therefore \mathrm{q}^{2}=\mathrm{p} \cdot \mathrm{r}$ ...(i) And, the last three numbers are in A.P. with common difference $6 .$ We h...
Read More →A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist.
Question: A matrixXhasa+brows anda+ 2 columns while the matrixYhasb+ 1 rows anda+ 3 columns. Both matricesXYandYXexist. Findaandb. Can you sayXYandYXare of the same type? Are they equal. Solution: Here, $[X]_{(a+b) \times(a+2)}$ $[Y]_{(b+1) \times(a+3)}$ Since $X Y$ exists, the number of columns in $X$ is equal to the number of rows in $Y$. $\Rightarrow a+2=b+1 \quad \ldots(1)$ Similarly, $\sin c_{e} Y X$ exists, the number of columns in $Y$ is equal to the number of rows in $X$. $\Rightarrow a+...
Read More →In the given figure, POQ is a diameter and PQRS is a cyclic quadrilateral.
Question: In the given figure,POQis a diameter andPQRSis a cyclic quadrilateral. IfPSR= 150, find RPQ. Solution: In cyclic quadrilateralPQRS,we have:PSR+PQR= 180⇒ 150+ PQR= 180⇒ PQR= (180 150) = 30 PQR= 30...(i)Also, PRQ= 90(Angle in a semicircle) ...(ii)Now, in ΔPRQ, we have:PQR+PRQ+RPQ= 180⇒ 30+ 90+ RPQ= 180 [From(i) and (ii)]⇒ RPQ= 180 120= 60 RPQ= 60...
Read More →Prove that the points (7, 10),
Question: Prove that the points (7, 10), (2, 5) and (3, 4) are the vertices of an isosceles right triangle. [CBSE 2013] Solution: Let the given points be A(7, 10), B(2, 5) and C(3, 4).Using distance formula, we have $\mathrm{AB}=\sqrt{(-2-7)^{2}+(5-10)^{2}}=\sqrt{(-9)^{2}+(-5)^{2}}=\sqrt{81+25}=\sqrt{106}$ units $\mathrm{BC}=\sqrt{[3-(-2)]^{2}+(-4-5)^{2}}=\sqrt{5^{2}+(-9)^{2}}=\sqrt{25+81}=\sqrt{106}$ units $\mathrm{CA}=\sqrt{(3-7)^{2}+(-4-10)^{2}}=\sqrt{(-4)^{2}+(-14)^{2}}=\sqrt{16+196}=\sqrt{2...
Read More →In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect at P such that
Question: In the given figure,ABCDis a cyclic quadrilateral whose diagonals intersect atPsuch thatDBC= 60 and BAC= 40. Find (i) BCD, (ii) CAD. Solution: (i) BDC = BAC = 40 (Angles in the same segment)InΔBCD, we have:BCD + DBC + BDC = 180 (Angle sum property of a triangle)⇒ BCD + 60 + 40 = 180⇒ BCD = (180 - 100) = 80(ii) CAD = CBD(Angles in the same segment)= 60...
Read More →If a point A(0, 2) is equidistant from the points
Question: If a point A(0, 2) is equidistant from the points B(3,p) and C(p, 5), then find the value ofp. [CBSE 2012, 2013] Solution: It is given that A(0, 2) is equidistant from the points B(3,p) and C(p, 5). AB = AC $\Rightarrow \sqrt{(3-0)^{2}+(p-2)^{2}}=\sqrt{(p-0)^{2}+(5-2)^{2}}$(Distance formula) Squaring on both sides, we get $9+p^{2}-4 p+4=p^{2}+9$ $\Rightarrow-4 p+4=0$ $\Rightarrow p=1$ Thus, the value ofpis 1....
Read More →If a, b, c are in A.P. and x, y, z are in G.P.,
Question: Ifa,b,care in A.P. andx,y,zare in G.P., then the value ofxbcycazabis (a) 0 (b) 1 (c)xyz (d)xaybzc Solution: (b) 1 $a, b$ and $c$ are in A.P. $\therefore 2 b=a+c$ .....(1) And, $x, y$ and $z$ are in G.P. $\therefore y^{2}=x z$ Now, $x^{b-c} y^{c-a} z^{a-b}$ $=x^{b+a-2 b} y^{2 b-a-a} z^{a-b} \quad[$ From $(\mathrm{i})]$ $=x^{a-b} y^{2(b-a)} z^{a-b}$ $=(x z)^{a-b}(x z)^{b-a} \quad\left[\right.$ From (ii),$\left.y^{2}=x z\right]$ $=(\mathrm{xz})^{0}$ $=1$...
Read More →If A is a square matrix, using mathematical induction prove that
Question: If $A$ is a square matrix, using mathematical induction prove that $\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$. Solution: Let the given statement $P(n)$, be given as $P(n):\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$. We observe that $P(1):\left(A^{T}\right)^{1}=A^{T}=\left(A^{1}\right)^{\top}$ Thus, $P(n)$ is true for $n=1$. Assume that $P(n)$ is true for $n=k \in N$. i.e., $P(k):\left(A^{T}\right)^{k}=\left(A^{k}\right)^{T}$ To prove that $P(...
Read More →If A is a square matrix, using mathematical induction prove that
Question: If $A$ is a square matrix, using mathematical induction prove that $\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$. Solution: Let the given statement $P(n)$, be given as $P(n):\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$. We observe that $P(1):\left(A^{T}\right)^{1}=A^{T}=\left(A^{1}\right)^{\top}$ Thus, $P(n)$ is true for $n=1$. Assume that $P(n)$ is true for $n=k \in N$. i.e., $P(k):\left(A^{T}\right)^{k}=\left(A^{k}\right)^{T}$ To prove that $P(...
Read More →Two opposite vertices of a square are (−1, 2) and (3, 2).
Question: Two opposite vertices of a square are (1, 2) and (3, 2). Find the coordinates of other two vertices. Solution: The distance $d$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by the formula $d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$ In a square all the sides are of equal length. The diagonals are also equal to each other. Also in a square the diagonal is equal totimes the side of the square. Here let the two points whic...
Read More →If the first term of a G.P.
Question: If the first term of a G.P.a1,a2,a3, ... is unity such that 4a2+ 5a3is least, then the common ratio of G.P. is (a) 2/5 (b) 3/5 (c) 2/5 (d) none of these Solution: (a) $-\frac{2}{5}$ If the first term is 1 , then, the G.P. will be $1, r, r^{2}, r^{3}, \ldots$ Now, $5 r^{2}+4 r=5\left(r^{2}+\frac{4}{5} r\right)$ $=5\left(r^{2}+\frac{4}{5} r+\frac{4}{25}-\frac{4}{25}\right)$ $=5\left(r+\frac{2}{5}\right)^{2}-\frac{4}{5}$ This will be the least when $r+\frac{2}{5}=0$, i. e. $r=-\frac{2}{5}...
Read More →In the given figure, AB and CD are two chords of a circle, intersecting each other at a point E.
Question: In the given figure,ABandCDare two chords of a circle, intersecting each other at a pointE. Prove that $\angle A E C=\frac{1}{2}$ (angle subtended by arc $C X A$ at the centre $+$ angle subtended by $\operatorname{arc} D Y B$ at the centre). Solution: Join AD We know that the angle subtended by an arc of a circle at the centre is double the angle subtended by it on the remaining part of the circle.Here, arcAXCsubtends AOCat the centre and ADCatDon the circle. AOC= 2ADC $\Rightarrow \an...
Read More →Solve this
Question: If $A=\operatorname{diag}(a, b, c)$, show that $A^{n}=\operatorname{diag}\left(a^{n}, b^{n}, c^{n}\right)$ for all positive integer $n$. Solution: We shall prove the result by the principle of mathematical induction on $n$. Step 1: If $n=1$, by definition of integral power of a matrix, we have $A^{1}=\left[\begin{array}{ccc}a^{1} 0 0 \\ 0 b^{1} 0 \\ 0 0 c^{1}\end{array}\right]=\left[\begin{array}{lll}a 0 0 \\ 0 b 0 \\ 0 0 c\end{array}\right]=A$ So, the result is true for n = 1.Step 2: ...
Read More →If in an infinite G.P., first term is equal to 10 times the sum of all successive terms,
Question: If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is (a) 1/10 (b) 1/11 (c) 1/9. (d) 1/20 Solution: (b) $\frac{1}{11}$ Let the first term of the G.P. bea. Let its common ratio ber. According to the question, we have: First term = 10 [Sum of all successive terms] $a=10\left(\frac{a r}{1-r}\right)$ $\Rightarrow a-a r=10 a r$ $\Rightarrow 11 a r=a$ $\Rightarrow r=\frac{a}{11 a}=\frac{1}{11}$...
Read More →If one A.M., A and two geometric means
Question: If one A.M.,Aand two geometric meansG1andG2inserted between any two positive numbers, show that $\frac{G_{1}^{2}}{G_{2}}+\frac{G_{2}^{2}}{G_{1}}=2 A$ Solution: Let the two positive numbers be $a$ and $b$. $a, A$ and $b$ are in A.P. $\therefore 2 A=a+b$ ....(i) Also, $a, G_{1}, G_{2}$ and $b$ are in G.P. $\therefore r=\left(\frac{b}{a}\right)^{\frac{1}{3}}$ Also, $G_{1}=a r$ and $G_{2}=a r^{2}$ ...(ii) Now, LHS $=\frac{G_{1}{ }^{2}}{G_{2}}+\frac{G_{2}{ }^{2}}{G_{1}}$ $=\frac{(a r)^{2}}{...
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