Prove that the points (3, 0), (6, 4) and (−1, 3)
Question: Prove that the points (3, 0), (6, 4) and (1, 3) are vertices of a right-angled isosceles triangle. 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 an isosceles triangle there are two sides which are equal in length. Here the three points areA(3,0), B(6,4) andC(1,3). Let us check the length of the three sides of the triangle. $A B=\sqr...
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Question: If $A=\left[\begin{array}{lll}1 2 2 \\ 2 1 2 \\ 2 2 1\end{array}\right]$, then prove that $A^{2}-4 A-5 /=0$ Solution: Given : $A=\left[\begin{array}{lll}1 2 2 \\ 2 1 2 \\ 2 2 1\end{array}\right]$ Now. $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{lll}1 2 2 \\ 2 1 2 \\ 2 2 1\end{array}\right]\left[\begin{array}{lll}1 2 2 \\ 2 1 2 \\ 2 2 1\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{lll}1+4+4 2+2+4 2+4+2 \\ 2+2+4 4+1+4 4+2+2 \\ 2+4+2 4+2+2 4+4+1\end{array}\right]$ $\Ri...
Read More →Prove that two different circles cannot intersect each other at more than two points.
Question: Prove that two different circles cannot intersect each other at more than two points. Solution: Given: Two distinct circlesTo prove:Two distinct circles cannot intersect each other in more than two points.Proof:Suppose that two distinct circles intersect each other in more than two points. These points are non-collinear points.Three non-collinear points determine one and only one circle. There should be only one circle.This contradicts the given, which shows that our assumption is wron...
Read More →Prove that the points A(1, 7), B (4, 2),
Question: Prove that the pointsA(1, 7),B(4, 2),C(1, 1)D(4, 4) are the vertices of a square. Solution: The distance $d$ between two points $\left(x_{1}, y_{\mathrm{I}}\right)$ 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 equal in length. Also, the diagonals are equal in length in a square. Here the four points areA(1,7), B(4,2),C(1,1) andD(4,4). First let us check if all the four sides ar...
Read More →If a, b, c, d are in G.P., prove that:
Question: If a, b, c, d are in G.P., prove that: (i) $\frac{a b-c d}{b^{2}-c^{2}}=\frac{a+c}{b}$ (ii) (a+b+c+d)2= (a+b)2+ 2 (b+c)2+ (c+d)2 (iii) (b+ c) (b +d) = (c+a) (c+d) Solution: a, b, c and d are in G.P. $\therefore b^{2}=a c$ ...(i) $b c=a d$ $c^{2}=b d$ (i) $\mathrm{LHS}=\frac{a b-c d}{b^{2}-c^{2}}$ $=\frac{a b-c d}{a c-b d} \quad[\operatorname{Using}(1)]$ $=\frac{(a b-c d) b}{(a c-b d) b}$ $=\frac{a b^{2}-b c d}{(a c-b d) b}$ $=\frac{a(a c)-c\left(c^{2}\right)}{(a c-b d) b} \quad[\operat...
Read More →Prove that the diameter of a circle perpendicular to one of the two parallel chords of a circle is perpendicular to the other and bisects it.
Question: Prove that the diameter of a circle perpendicular to one of the two parallel chords of a circle is perpendicular to the other and bisects it. Solution: Given:ABandCDare two parallel chords of a circle with centreO.POQis a diameter which is perpendicular toAB.To prove:PFCDandCF = FDProof:AB || CDandPOQis a diameter.PEB= 90 (Given)PFD = PEB (∵AB || CD,Corresponding angles)Thus,PFCDOFCDWe know that the perpendicular from the centre to a chord bisects the chord.i.e.,CF = FDHence,POQis perp...
Read More →If a, b, c, d are in G.P., prove that:
Question: If a, b, c, d are in G.P., prove that: (i) $\frac{a b-c d}{b^{2}-c^{2}}=\frac{a+c}{b}$ (ii) (a+b+c+d)2= (a+b)2+ 2 (b+c)2+ (c+d)2 (iii) (b+ c) (b +d) = (c+a) (c+d) Solution: a, b, c and d are in G.P. $\therefore b^{2}=a c$ ...(i) $b c=a d$ $c^{2}=b d$ (i) $\mathrm{LHS}=\frac{a b-c d}{b^{2}-c^{2}}$ $=\frac{a b-c d}{a c-b d} \quad[\operatorname{Using}(1)]$ $=\frac{(a b-c d) b}{(a c-b d) b}$ $=\frac{a b^{2}-b c d}{(a c-b d) b}$ $=\frac{a(a c)-c\left(c^{2}\right)}{(a c-b d) b} \quad[\operat...
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Question: If $A=\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]$, then show that $\mathrm{A}$ is a root of the polynomial $f(x)=x^{3}-6 x^{2}+7 x+2$ Solution: Given : $f(x)=x^{3}-6 x^{2}+7 x+2$ $f(A)=A^{3}-6 A^{2}+7 A+2 I_{3}$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{lll}1+0+4 0+0+0 2+0+6 \\ 0+0+2 0+4+0 0+2+3 \\ 2+...
Read More →In the given figure, O is the centre of a circle in which chords AB and CD intersect at P such that PO bisects ∠BPD.
Question: In the given figure,Ois the centre of a circle in which chordsABandCDintersect atPsuch thatPObisects BPD. Prove thatAB=CD. Solution: Given:Ois the centre of a circle in which chordsABandCDintersect atPsuch thatPObisects BPD.To prove:AB = CDConstruction:DrawOEABandOFCDProof:In ΔOEPand ΔOFP,we have:OEP = OFP (90 each)OP = OP (Common)OPE= OPF (∵ OP bisects BPD )Thus, ΔOEP ΔOFP (AAS criterion)⇒OE = OFThus, chordsABandCDare equidistant from the centreO.⇒AB = CD(∵ Chords equidistant from the...
Read More →Show that the points A (1, −2), B (3, 6), C (5, 10)
Question: Show that the pointsA(1, 2),B(3, 6),C(5, 10) andD(3, 2) are the vertices of a parallelogram. 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 parallelogram the opposite sides are equal in length. Here the four points areA(1,2), B(3,6),C(5,10) andD(3,2). Let us check the length of the opposite sides of the quadrilateral that is formed...
Read More →In the adjoining figure, OD is perpendicular to the chord AB of a circle with centre O.
Question: In the adjoining figure,ODis perpendicular to the chordABof a circle with centreO. IfBCis a diameter, show thatAC||CDandAC= 2 OD. Solution: Given:BCis a diameter of a circle with centreOandODAB.To prove:ACparallel toODandAC = 2 ODConstruction:JoinAC.Proof:We know that the perpendicular from the centre of a circle to a chord bisects the chord.Here,ODABDis the mid point ofAB.i.e.,AD = BDAlso,Ois the mid point ofBC.i.e.,OC = OB Now, in ΔABC, we have: Dis the mid point ofABandOis the mid p...
Read More →If a, b, c are in G.P., prove that:
Question: Ifa,b,care in G.P., prove that: (i) $a\left(b^{2}+c^{2}\right)=c\left(a^{2}+b^{2}\right)$ (ii) $a^{2} b^{2} c^{2}\left(\frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}\right)=a^{3}+b^{3}+c^{3}$ (iii) $\frac{(a+b+c)^{2}}{a^{2}+b^{2}+c^{2}}=\frac{a+b+c}{a-b+c}$ (iv) $\frac{1}{a^{2}-b^{2}}+\frac{1}{b^{2}}=\frac{1}{b^{2}-c^{2}}$ (v) (a+ 2b+ 2c) (a 2b+ 2c) = a2+ 4c2. Solution: a, b and c are in G.P. $\therefore b^{2}=a c$ (i) $\mathrm{LHS}=a\left(b^{2}+c^{2}\right)$ $=a b^{2}+a c^{2}$ $=a(a ...
Read More →Show that the points (−4, −1), (−2, −4) (4, 0) and
Question: Show that the points (4, 1), (2, 4) (4, 0) and (2, 3) are the vertices points of a rectangle. 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 rectangle, the opposite sides are equal in length. The diagonals of a rectangle are also equal in length. Here the four points areA(4,1), B(2,4),C(4,0) andD(2,3). First let us check the length...
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Question: If $f(x)=x^{3}+4 x^{2}-x$, find $f(A)$, where $A=\left[\begin{array}{rrr}0 1 2 \\ 2 -3 0 \\ 1 -1 0\end{array}\right]$ Solution: Given: $f(x)=x^{3}+4 x^{2}-x$ $f(A)=A^{3}+4 A^{2}-A$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left|\begin{array}{ccc}0 1 2 \\ 2 -3 0 \\ 1 -1 0\end{array}\right|\left[\begin{array}{ccc}0 1 2 \\ 2 -3 0 \\ 1 -1 0\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{lll}0+2+2 0-3-2 0+0+0 \\ 0-6+0 2+9-0 4-0+0 \\ 0-2+0 1+3-0 2-0+0\end{array}\right]$ $\Rightarrow A^...
Read More →In the given figure, a circle with centre O is given in which a diameter AB bisects the chord CD at a point E such that CE = ED = 8 cm and EB = 4 cm.
Question: In the given figure, a circle with centreOis given in which a diameterABbisects the chordCDat a pointEsuch thatCE=ED= 8 cm andEB= 4 cm. Find the radius of the circle. Solution: ABis the diameter of the circle with centreO,which bisects the chordCDat pointE.Given:CE = ED =8 cm andEB= 4 cmJoinOC. LetOC = OB=rcm (Radii of a circle)ThenOE= (r 4) cmNow, in right angled ΔOEC, we have: $O C^{2}=O E^{2}+E C^{2}$ (Pythagoras theorem) $\Rightarrow r^{2}=(r-4)^{2}+8^{2}$ $\Rightarrow r^{2}=r^{2}-...
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Question: If $f(x)=x^{3}+4 x^{2}-x$, find $f(A)$, where $A=\left[\begin{array}{rrr}0 1 2 \\ 2 -3 0 \\ 1 -1 0\end{array}\right]$ Solution: Given : $f(x)=x^{3}+4 x^{2}-x$...
Read More →Solve this$f(A)=A^{3}+4 A^{2}-A$
Question: If $f(x)=x^{3}+4 x^{2}-x$, find $f(A)$, where $A=\left[\begin{array}{rrr}0 1 2 \\ 2 -3 0 \\ 1 -1 0\end{array}\right]$ Solution: Given : $f(x)=x^{3}+4 x^{2}-x$...
Read More →The length of a line segment is of 10 units and the coordinates
Question: The length of a line segment is of 10 units and the coordinates of one end-point are (2, 3). If the abscissa of the other end is 10, find the ordinate of the other end. 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 $d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$ Here it is given that one end of a line segment has coordinates (2,3). The abscissa of the other end of the line segment i...
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Question: If $f(x)=x^{2}-2 x$, find $f(A)$, where $A=\left[\begin{array}{lll}0 1 2 \\ 4 5 0 \\ 0 2 3\end{array}\right]$ Solution: Given: $f(x)=x^{2}-2 x$ $f(A)=A^{2}-2 A$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{lll}0 1 2 \\ 4 5 0 \\ 0 2 3\end{array}\right]\left[\begin{array}{lll}0 1 2 \\ 4 5 0 \\ 0 2 3\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{ccc}0+4+0 0+5+4 0+0+6 \\ 0+20+0 4+25+0 8+0+0 \\ 0+8+0 0+10+6 0+0+9\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{arra...
Read More →The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order,
Question: The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers. Solution: Let the first term of a G.P beaand its common ratio ber. $\therefore a_{1}+a_{2}+a_{3}=56$ $\Rightarrow a+a r+a r^{2}=56$ $\Rightarrow a\left(1+r+r^{2}\right)=56$ $\Rightarrow a=\frac{56}{1+r+r^{2}}$ ....(i) Now, according to the question: $a-1, a r-7$ and $a r^{2}-21$ are in A.P. $\therefore 2(\operatorname{ar}-7)=a-1+a r^{2}-21$ $\Rightarrow ...
Read More →Find the values of x, y if the distances of the point
Question: Find the values ofx,yif the distances of the point (x,y) from (3, 0) as well as from (3, 0) are 4. 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 $d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$ It is said that (x, y) is equidistant from both (3,0) and (3,0). Letbe the distance between (x, y) and (3,0). Letbe the distance between (x, y) and (3,0). So, using the distance formula for bo...
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Question: If $A=\left[\begin{array}{rrr}1 2 0 \\ 3 -4 5 \\ 0 -1 3\end{array}\right]$, compute $A^{2}-4 A+3 / 3$. Solution: Given : $A=\left[\begin{array}{ccc}1 2 0 \\ 3 -4 5 \\ 0 -1 3\end{array}\right]$ Now, $A^{2}=A A$ $\Rightarrow A^{2}=\left[\begin{array}{ccc}1 2 0 \\ 3 -4 5 \\ 0 -1 3\end{array}\right]\left[\begin{array}{ccc}1 2 0 \\ 3 -4 5 \\ 0 -1 3\end{array}\right]$ $\Rightarrow A^{2}=\left[\begin{array}{ccc}1+6+0 2-8-0 0+10+0 \\ 3-12+0 6+16-5 0-20+15 \\ 0-3+0 0+4-3 0-5+9\end{array}\right]...
Read More →In the given figure, the diameter CD of a circle with centre O is perpendicular to chord AB.
Question: In the given figure, the diameterCDof a circle with centreOis perpendicular to chordAB. IfAB= 12 cm andCE= 3 cm, calculate the radius of the circle. Solution: CDis the diameter of the circle with centreOand is perpendicular to chordAB.JoinOA. Given:AB= 12 cm andCE= 3 cmLetOA = OC=rcm (Radii of a circle)ThenOE= (r- 3) cmSince the perpendicular from the centre of the circle to a chord bisects the chord, we have: $A E=\left(\frac{A B}{2}\right)=\left(\frac{12}{2}\right) \mathrm{cm}=6 \mat...
Read More →Two parallel chords of lengths 30 cm and 16 cm are drawn on the opposite sides of the centre of a circle of radius 17 cm.
Question: Two parallel chords of lengths 30 cm and 16 cm are drawn on the opposite sides of the centre of a circle of radius 17 cm. Find the distance between the chords. Solution: LetABandCDbe two chords of a circle such thatABis parallel toCDand they are on the opposite sides of the centre.Given:AB= 30 cm andCD= 16 cmDrawOLABandOMCD. JoinOAandOC.OA = OC= 17 cm (Radii of a circle)The perpendicular from the centre of a circle to a chord bisects the chord. $\therefore A L=\left(\frac{A B}{2}\right...
Read More →If the points (2, 1) and (1, −2) are equidistant
Question: If the points (2, 1) and (1, 2) are equidistant from the point (x,y), show thatx+ 3y= 0. 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 $d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{-}-y_{2}\right)^{2}}$ Here it is said that the points (2,1) and (1,2) are equidistant from (x, y). Letbe the distance between (2,1)and (x,y). Letbe the distance between (1, 2)and (x, y). So, using the distance formula for bo...
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