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ICSE Class 10 ML Aggarwal Page 2
ML Aggarwal

The bisectors of angles A and B of a scalene triangle ABC meet at O. (i) What is the point O called? (ii) OR and OQ is drawn a perpendicular to AB and CA respectively. What is the relation between OR and OQ? (iii) What is the relation between ∠ACO and ∠BCO?

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Solution: (i) The point O where the angle bisectors meet is called the incenter of the triangle. (ii) The perpendicular drawn from point O to AB and CA are equal. i.e., OR and OQ. (iii) ∠ACO = ∠BCO....

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(i) Conduct a triangle ABC with BC = 6.4 cm, CA = 5.8 cm and ∠ ABC = 60°. Draw its incircle. Measure and record the radius of the incircle. (ii) Construct a ∆ABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm. Construct the incircle of the triangle. Measure and record the radius of the incircle. (2014)

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Solution: Steps to construct: Step 1: Draw a line segment BC = 6.4cm. Step 2: Construct an angle of 60o at point B. Step 3: With C as center and radius CA = 5.8cm, draw an arc cutting BD at A. Step...

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Using a ruler and compasses only: (i) Construe a triangle ABC with the following data: Base AB = 6 cm, AC = 5.2 cm and ∠CAB = 60°. (ii) In the same diagram, draw a circle which passes through the points A, B and C. and mark its centre O.

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Solution: Steps to construct: Step 1: Draw a line segment AB = 6cm. Step 2: At point A, draw a ray making an angle of 60o. Step 3: With B as the center and radius 5.2cm, draw an arc which intersects...

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Use a ruler and compass only in this question. (i) Draw a circle, centre O and radius 4 cm. (ii) Mark a point P such that OP = 7 cm. Construct the two tangents to the circle from P. Measure and record the length of one of the tangents.

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Solution: Steps to construct: Step 1: Draw a circle with center O and radius 4cm and mark that point as A. Step 2: Take a point P such that OP = 7cm. Step 3: Bisect OB at M. Step 4: With center M...

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(a) In the figure (i) given below, triangle ABC is equilateral. Find ∠BDC and ∠BEC. (b) In the figure (ii) given below, AB is a diameter of a circle with center O. OD is perpendicular to AB and C is a point on the arc DB. Find ∠BAD and ∠ACD

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Solution: (a) triangle ABC is an equilateral triangle Each angle = 60o ∠A = 60o But ∠A = ∠D (Angles in the same segment) ∠D = 600 Now ABEC is a cyclic quadrilateral, ∠A = ∠E = 180o 60o + ∠E = 180o...

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(a) In the figure (i) given below, O is the center of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, find the radius of the circle. (b) In the figure (ii) given below, from an external point P, tangents PA and PB are drawn to a circle. CE is a tangent to the circle at D. If AP = 15 cm, find the perimeter of the triangle PEC.

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Solution: (i) Join OB ∠OBA = 90° (Radius through the point of contact is perpendicular to the tangent) OB2 = OA2 – AB2 r2 = (r + 7.5)2 – 152 r2 = r2 + 56.25 + 15r – 225 15r = 168.75 r = 11.25 Hence,...

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(a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB. (b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values of x, y and z.

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Solution: (a) Construction: Join BC, and AC then ABCD is a cyclic quadrilateral. Now in ∆DCF Ext. ∠2 = x + z and in ∆CBE Ext. ∠1 = x + y Adding (i) and (ii) x + y + x + z = ∠1 + ∠2 2 x + y + z =...

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(a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC. (b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find: (i)∠BAD (ii) DBCD.

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Solution: (a) ADFE is a cyclic quadrilateral Ext. ∠FEB = ∠ADF ⇒ ∠ADF = 80° ABCD is a parallelogram ∠B = ∠D = ∠ADF = 80° or ∠ABC = 80° (b)In trapezium ABCD, AD || BC (i) ∠B + ∠A = 180° ⇒ 70° + ∠A =...

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(a)In the figure (i) given below, O is the centre of the circle and ∠PBA = 42°. Calculate the value of ∠PQB (b) In the figure (ii) given below, AB is a diameter of the circle whose centre is O. Given that ∠ECD = ∠EDC = 32°, calculate (i) ∠CEF (ii) ∠COF.

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Solution: In ∆APB, ∠APB = 90° (Angle in a semi-circle) But ∠A + ∠APB + ∠ABP = 180° (Angles of a triangle) ∠A + 90° + 42°= 180° ∠A + 132° = 180° ⇒ ∠A = 180° – 132° = 48° But ∠A = ∠PQB (Angles in the...

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Find a and b if \[\left[ \begin{align} & a-b\,\,\,\,\,b-4 \\ & b+4\,\,\,\,\,a-2 \\ \end{align} \right]\left[ \begin{align} & 2\,\,\,\,0 \\ & 0\,\,\,\,\,2 \\ \end{align} \right]=\left[ \begin{align} & -2\,\,\,\,-2 \\ & 14\,\,\,\,\,\,\,\,0 \\ \end{align} \right]\]

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On comparing the corresponding terms, we have \[\begin{array}{*{35}{l}} 2a\text{ }\text{ }4\text{ }=\text{ }0  \\ 2a\text{ }=\text{ }4  \\ a\text{ }=\text{ }4/2  \\ a\text{ }=\text{ }2  \\...

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If \[\left[ \begin{align} & -1\,\,\,\,0 \\ & 0\,\,\,\,\,\,\,1 \\ \end{align} \right]\left[ \begin{align} & a\,\,\,\,b \\ & c\,\,\,\,\,d \\ \end{align} \right]=\left[ \begin{align} & 1\,\,\,\,0 \\ & 0\,\,\,\,-1 \\ \end{align} \right]\] find a, b, c and d.

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Given, On comparing the corresponding elements, we have \[\begin{array}{*{35}{l}} -a\text{ }=\text{ }1\Rightarrow a\text{ }=\text{ }-1  \\ -b\text{ }=\text{ }0\Rightarrow b\text{ }=\text{ }0  \\...

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If A=\[\left[ \begin{align} & 3\,\,\,\,2 \\ & 0\,\,\,\,\,5 \\ \end{align} \right]\] and B=\[\left[ \begin{align} & 1\,\,\,\,0 \\ & 1\,\,\,\,2 \\ \end{align} \right]\] , find the each of the following and state it they are equal: (i) (A + B) (A – B) (ii) \[{{\mathbf{A}}^{\mathbf{2}}}~\text{ }{{\mathbf{B}}^{\mathbf{2}}}\]

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Given, Hence, its clearly seen that \[\left( A\text{ }+\text{ }B \right)\text{ }\left( A\text{ }\text{ }B \right)\text{ }\ne \text{ }{{A}^{2}}~\text{ }{{B}^{2}}\].

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(i) Find the matrix B if A=\[\left[ \begin{align} & 4\,\,\,\,\,1 \\ & 2\,\,\,\,\,3 \\ \end{align} \right]\] and \[{{A}^{2}}=A+2B\] (ii) If A= \[\left[ \begin{align} & 1\,\,\,\,\,2 \\ & -3\,\,\,4 \\ \end{align} \right]\], B= \[\left[ \begin{align} & 0\,\,\,\,\,\,1 \\ & -2\,\,\,5 \\ \end{align} \right]\] and C= \[\left[ \begin{align} & -2\,\,\,\,\,\,0 \\ & -1\,\,\,\,\,\,1 \\ \end{align} \right]\] find \[A(4B-3C)\]

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  Comparing the corresponding elements, we have \[\begin{array}{*{35}{l}} 4\text{ }+\text{ }2a\text{ }=\text{ }18  \\ 2a\text{ }=\text{ }18\text{ }\text{ }4\text{ }=\text{ }14  \\ a\text{...

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Find a, b, c and d if \[3\left[ \begin{align} & a\,\,\,\,\,\,b \\ & c\,\,\,\,\,\,\,d \\ \end{align} \right]=\left[ \begin{align} & 4\,\,\,\,\,\,\,\,\,\,a+b \\ & c+d\,\,\,\,\,\,\,3 \\ \end{align} \right]+\left[ \begin{align} & a\,\,\,\,\,\,\,6 \\ & -1\,\,\,\,\,2d \\ \end{align} \right]\]

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Given \[3\left[ \begin{align} & a\,\,\,\,\,\,b \\ & c\,\,\,\,\,\,\,d \\ \end{align} \right]=\left[ \begin{align} & 4\,\,\,\,\,\,\,\,\,\,a+b \\ & c+d\,\,\,\,\,\,\,3 \\ \end{align}...

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Find the values of a and b if \[\left[ \begin{align} & a+3\,\,\,\,\,\,\,{{b}^{2}}+2 \\ & \,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-6 \\ \end{align} \right]=\left[ \begin{align} & 2a+1\,\,\,\,\,\,\,\,\,\,\,\,3b \\ & \,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,{{b}^{2}}-5b \\ \end{align} \right]\]

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Given \[\left[ \begin{align} & a+3\,\,\,\,\,\,\,{{b}^{2}}+2 \\ & \,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-6 \\ \end{align} \right]=\left[ \begin{align} & 2a+1\,\,\,\,\,\,\,\,\,\,\,\,3b \\...

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Find the values of x and y if \[\left[ \begin{align} & x+y\,\,\,\,\,\,y \\ & 2x\,\,\,\,\,\,\,\,x-y \\ \end{align} \right]\left[ \begin{align} & 2 \\ & -1 \\ \end{align} \right]=\left[ \begin{align} & 3 \\ & 2 \\ \end{align} \right]\]

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Given, On comparing the corresponding elements, we have \[2x\text{ }+\text{ }y\text{ }=\text{ }3\]… (i) \[3x\text{ }+\text{ }y\text{ }=\text{ }2\]… (ii) Subtracting, we get \[-x\text{ }=\text{...

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.(i) Find x and y if \[\left[ \begin{align} & -3\,\,\,\,\,\,2 \\ & 0\,\,\,\,\,\,\,\,-5 \\ \end{align} \right]\left[ \begin{align} & x \\ & 2 \\ \end{align} \right]=\left[ \begin{align} & -5 \\ & y \\ \end{align} \right]\] (ii) Find x and y if \[\left[ \begin{align} & 2x\,\,\,\,\,\,x \\ & y\,\,\,\,\,\,\,\,3y \\ \end{align} \right]\left[ \begin{align} & 3 \\ & 2 \\ \end{align} \right]=\left[ \begin{align} & 16 \\ & 9 \\ \end{align} \right]\]

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Comparing the corresponding elements, \[\begin{array}{*{35}{l}} \text{ }3x\text{ }+\text{ }4\text{ }=\text{ }-5  \\ -3x\text{ }=\text{ }-5\text{ }\text{ }4\text{ }=\text{ }-9  \\ x\text{ }=\text{...

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IF A=\[\left[ \begin{align} & 2\,\,\,\,a \\ & -3\,\,\,5 \\ \end{align} \right]\] and B=\[\left[ \begin{align} & -2\,\,\,\,3 \\ & 7\,\,\,\,\,\,\,b \\ \end{align} \right]\], C=\[\left[ \begin{align} & c\,\,\,\,\,\,\,9 \\ & -1\,\,\,\,-11 \\ \end{align} \right]\] and \[\mathbf{5A}\text{ }+\text{ }\mathbf{2B}\text{ }=\text{ }\mathbf{C}\], find the values of a, b and c.

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On comparing the corresponding terms, we get \[\begin{array}{*{35}{l}} 5a\text{ }+\text{ }6\text{ }=\text{ }9  \\ 5a\text{ }=\text{ }9\text{ }\text{ }6  \\ 5a\text{ }=\text{ }3  \\ a\text{ }=\text{...

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If \[\left[ \begin{align} & a\,\,\,\,\,3 \\ & 4\,\,\,\,\,\,2 \\ \end{align} \right]+\left[ \begin{align} & 2\,\,\,\,\,b \\ & 1\,\,\,\,\,-2 \\ \end{align} \right]-\left[ \begin{align} & 1\,\,\,\,\,1 \\ & 1\,\,\,\,\,-2 \\ \end{align} \right]=\left[ \begin{align} & 5\,\,\,\,\,\,0 \\ & 7\,\,\,\,\,\,3 \\ \end{align} \right]\] Find the value of a, b and c.

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Next, on comparing the corresponding terms, we have \[\begin{array}{*{35}{l}} a\text{ }+\text{ }1\text{ }=\text{ }5\Rightarrow a\text{ }=\text{ }4  \\ b\text{ }+\text{ }2\text{ }=\text{...

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If \[\left[ \begin{align} & 5\,\,\,\,\,\,\,\,2 \\ & -1\,\,\,\,\,y+1 \\ \end{align} \right]-2\left[ \begin{align} & 1\,\,\,\,\,\,2x-1 \\ & 3\,\,\,\,\,\,\,\,\,\,\,-2 \\ \end{align} \right]=\left[ \begin{align} & 3\,\,\,\,\,-8 \\ & -7\,\,\,\,\,\,2 \\ \end{align} \right]\] Find the values of x and y

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Now, comparing the corresponding terms, we get \[\begin{array}{*{35}{l}} 4\text{ }\text{ }4x\text{ }=\text{ }-8  \\ 4\text{ }+\text{ }8\text{ }=\text{ }4x  \\ 12\text{ }=\text{ }4x  \\ x\text{...

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IF \[2\left[ \begin{align} & 3\,\,\,\,4 \\ & 5\,\,\,\,\,x \\ \end{align} \right]+\left[ \begin{align} & 1\,\,\,\,y \\ & 0\,\,\,\,1 \\ \end{align} \right]=\left[ \begin{align} & z\,\,\,\,0 \\ & 10\,\,\,5 \\ \end{align} \right]\] Find the values of x and y

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On comparing the corresponding terms, we have \[\begin{array}{*{35}{l}} 2x\text{ }+\text{ }1\text{ }=\text{ }5  \\ 2x\text{ }=\text{ }5\text{ }-1\text{ }=\text{ }4  \\ x\text{ }=\text{ }4/2\text{...

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If \[2\left[ \begin{align} & 3\,\,\,\,4 \\ & 5\,\,\,\,\,x \\ \end{align} \right]+\left[ \begin{align} & 1\,\,\,\,y \\ & 0\,\,\,\,1 \\ \end{align} \right]=\left[ \begin{align} & 7\,\,\,\,0 \\ & 10\,\,\,5 \\ \end{align} \right]\] Find the values of x and y

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On comparing the corresponding elements, we have \[\begin{array}{*{35}{l}} 8\text{ }+\text{ }y\text{ }=\text{ }0  \\ Then,\text{ }y\text{ }=\text{ }-8  \\ And,\text{ }2x\text{ }+\text{ }1\text{...

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Construct a \[\mathbf{2}\text{ }\times \text{ }\mathbf{2}\] matrix whose elements aij are given by (i) \[{{\mathbf{a}}_{\mathbf{ij}}}~=\text{ }\mathbf{2i}\text{ }\text{ }\mathbf{j}\] (ii) \[{{\mathbf{a}}_{\mathbf{ij}}}~=\mathbf{i}.\mathbf{j}\]

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(i) Given \[{{\mathbf{a}}_{\mathbf{ij}}}~=\text{ }\mathbf{2i}\text{ }\text{ }\mathbf{j}\] Therefore matrix of order \[\mathbf{2}\text{ }\times \text{ }\mathbf{2}\]is \[\left[ \begin{align} &...

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Classify the following matrices: (v) \[\left[ \begin{align} & 2\,\,\,\,\,\,\,\,7\,\,\,\,\,\,\,8 \\ & -1\,\,\sqrt{2}\,\,\,\,\,\,0 \\ \end{align} \right]\] (vi) \[\left[ \begin{align} & 0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,0\, \\ & 0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,0 \\ \end{align} \right]\]

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It is a matrix of order \[2\text{ }\times \text{ }3\] (vi) \[\left[ \begin{align} & 0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,0\, \\ & 0\,\,\,\,\,\,\,\,0\,\,\,\,\,\,0 \\ \end{align} \right]\] Solution:...

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When a polynomial f(x) is divided by \[(x-1)\], the remainder is 5 and when it is, divided by \[(x-2)\], the remainder is \[7\]. Find the remainder when it is divided by \[\left( \mathbf{x}\text{ }\text{ }\mathbf{1} \right)\text{ }\left( \mathbf{x}\text{ }\text{ }\mathbf{2} \right).\]

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From the question it is given that, Polynomial f(x) is divided by \[(x-1)\], Remainder = \[5\] Let us assume \[x-1=0\] x = \[1\] \[f\left( 1 \right)\text{ }=\text{ }5\] and the divided be \[(x-2)\],...

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If a polynomial f(x)= \[{{\mathbf{x}}^{\mathbf{4}}}-\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}\text{ }\mathbf{ax}\text{ }+\text{ }\mathbf{b}\] leaves reminder \[5\] and \[19\] when divided by (x – 1) and (x + 1) respectively, Find the values of a and b. Hence determined the reminder when f(x) is divided by (x-2).

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From the question it is given that, f(x) = \[{{\mathbf{x}}^{\mathbf{4}}}-\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}\text{ }\mathbf{ax}\text{ }+\text{...

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If \[(2x+1)\] is a factor of both the expressions \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{p}\] and \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{q}\], find the value of p and q. Hence find the other factors of both the polynomials.

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Let us assume \[2x\text{ }+\text{ }1\text{ }=\text{ }0\] Then, \[2x\text{ }=\text{ }-1\] \[x\text{ }=\text{ }-{\scriptscriptstyle 1\!/\!{ }_2}\] Given, p(x) =...

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If \[{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{px}\text{ }+\text{ }\mathbf{q}\] has a factor \[(x+2)\] and leaves a remainder \[9\], when divided by \[(x+1)\], find the values of p and q. With these values of p and q, factorize the given polynomial completely.

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From the question it is given that, \[(x+2)\] is a factor of the expression \[{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{px}\text{ }+\text{...

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Use factor theorem to factorize the following polynomials completely: (i) \[\mathbf{4}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{9x}\text{ }\text{ }\mathbf{9}~\] (ii) \[{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{19x}\text{ }\text{ }\mathbf{30}\]

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Let us assume x = \[-1\], Given, f(x) = \[\mathbf{4}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{9x}\text{ }\text{ }\mathbf{9}~\] Now, substitute the...

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When \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{p}\] is divided by \[(x-2)\], the remainder is \[3\]. Find the value of p. Also factorize the polynomial \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{p}\text{ }\text{ }\mathbf{3}\].

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Let us assume \[x\text{ }\text{ }2\text{ }=\text{ }0\] Then, x = \[2\] Given, f(x) = \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{p}\] Now, substitute the...

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Given \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{bx}\text{ }+\text{ }\mathbf{2}\text{ }\mathbf{and}\text{ }\mathbf{g}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{b}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{ax}\text{ }+\text{ }\mathbf{1}\]. If \[x-2\] is a factor of f(x) but leaves the remainder \[-15\] when it divides g(x), find the values of a and b. With these values of a and b, factorize the expression. \[\mathbf{f}\left( \mathbf{x} \right)\text{ }+\text{ }\mathbf{g}\left( \mathbf{x} \right)\text{ }+\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{7x}\]

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From the question it is given that, \[\mathbf{f}\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{bx}\text{ }+\text{ }\mathbf{2}\text{...

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If \[\mathbf{a}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{bx}\text{ }\text{ }\mathbf{3}\] has a factor \[(2x+3)\] and leaves remainder \[-3\] when divided by \[(x+2)\], find the values of a and b. With these values of a and b, factorize the given expression.

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Let us assume, \[\begin{array}{*{35}{l}} ~2x\text{ }+\text{ }3\text{ }=\text{ }0  \\ Then,\text{ }2x\text{ }=\text{ }-3  \\ x\text{ }=\text{ }-3/2  \\ \end{array}\] Given, f(x) =...

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\[(x-2)\] is a factor of the expression \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{bx}\text{ }+\text{ }\mathbf{6}\]. When this expression is divided by \[(x-3)\], it leaves the remainder \[3\]. Find the values of a and b.

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From the question it is given that, \[(x-2)\] is a factor of the expression \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{bx}\text{ }+\text{...

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If \[\left( \mathbf{x}\text{ }+\text{ }\mathbf{2} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{x}\text{ }\text{ }\mathbf{3} \right)\] are factors of \[{{x}^{3}}~+\text{ }ax\text{ }+\text{ }b\], find the values of a and b. With these values of a and b, factorize the given expression.

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Let us assume \[x\text{ }+\text{ }2\text{ }=\text{ }0\] Then, x = \[-2\] Given, f(x) = \[{{x}^{3}}~+\text{ }ax\text{ }+\text{ }b\] Now, substitute the value of x in f(x), \[\begin{array}{*{35}{l}}...

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(i) Find the value of the constants a and b, if \[\left( \mathbf{x}\text{ }\text{ }\mathbf{2} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{x}\text{ }+\text{ }\mathbf{3} \right)\] are both factors of the expression \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{bx}\text{ }\text{ }\mathbf{12}\] (ii) If \[\left( \mathbf{x}\text{ }+\text{ }\mathbf{2} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{x}\text{ }+\text{ }\mathbf{3} \right)\] are factors of \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{ax}\text{ }+\text{ }\mathbf{b}\] , Find the values of a and b.

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Let us assume \[x\text{ }\text{ }2\text{ }=\text{ }0\] Then, x = \[2\] Given, f(x) = \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{a}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{bx}\text{ }\text{...

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If \[(x-2)\] is a factor of \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{px}\text{ }\text{ }\mathbf{2}\], then (i) find the value of p. (ii) with this value of p, factorize the above expression completely.

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Let us assume \[x\text{ }-2\text{ }=\text{ }0\] Then, \[x=2\] Given, f(x) = \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{px}\text{ }\text{...

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Use the remainder theorem to factorize the following expression. (iii) \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{9x}\text{ }\text{ }\mathbf{10}\] (iv) \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{10}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{37x}\text{ }+\text{ }\mathbf{26}\]

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Given, f(x) = \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{9x}\text{ }\text{ }\mathbf{10}\] Let us assume, x = \[-1\]...

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15. Use the remainder theorem to factorize the following expression. (i) \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{13x}\text{ }+\text{ }\mathbf{6}\] (ii) \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{19x}\text{ }+\text{ }\mathbf{6}\]

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Let us assume x = \[2\], Then, f(x) = \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{13x}\text{ }+\text{ }\mathbf{6}\] Now, substitute the value of x in...

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Use factor theorem to factorize the following polynomials completely. (i) \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{6}\] (ii) \[{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{13x}\text{ }\text{ }\mathbf{12}\]

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Let us assume \[x=-1\], Given, f(x) = \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{6}\] Now, substitute the value of x in...

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Show that \[2x+7\] is a factor of \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{5}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{11x}\text{ }\text{ }\mathbf{14}\]. Hence factorize the given expression completely, using the factor theorem.

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Let us assume \[2x+7=0\] Then, \[\begin{array}{*{35}{l}} 2x\text{ }=\text{ }-7  \\ X\text{ }=\text{ }-7/2  \\ \end{array}\] Given, f(x) = \[2{{x}^{3}}~+\text{ }5{{x}^{2}}~\text{ }11x\text{ }\text{...

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Show that \[(x-2)\] is a factor of \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{x}\text{ }\text{ }\mathbf{10}\] . Hence factories \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{x}\text{ }\text{ }\mathbf{10}\]

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Let us assume \[x\text{ }\text{ }2\text{ }=\text{ }0\] Then, x = \[2\] Given, f(x) = \[\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{x}\text{ }\text{ }\mathbf{10}\] Now, substitute the value...

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Without actual division, prove that \[{{\mathbf{x}}^{\mathbf{4}}}~+\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2x}\text{ }+\text{ }\mathbf{3}\] is exactly divisible by \[{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2x}\text{ }\text{ }\mathbf{3}\].

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Consider \[{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2x}\text{ }\text{ }\mathbf{3}\] By factor method, \[{{x}^{2}}~+\text{ }3x\text{ }\text{ }x\text{ }\text{ }3\] \[\begin{array}{*{35}{l}}...

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By factor theorem, show that \[\left( \mathbf{x}\text{ }+\text{ }\mathbf{3} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{2x}\text{ }\text{ }\mathbf{1} \right)\] are factors of \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{3}\].

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Let us assume, \[x\text{ }+\text{ }3\text{ }=\text{ }0\] Then, \[x\text{ }=\text{ }\text{ }3\] Given, f(x) = \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{5x}\text{ }\text{ }\mathbf{3}\]...

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Using remainder theorem, find the remainders obtained when \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\left( \mathbf{kx}\text{ }+\text{ }\mathbf{8} \right)\mathbf{x}\text{ }+\text{ }\mathbf{k}\] Is divided by \[\mathbf{x}\text{ }+\text{ }\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{2}\]. Hence, find k if the sum of two remainders is \[1\].

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Let us assume p(x) = \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\left( \mathbf{kx}\text{ }+\text{ }\mathbf{8} \right)\mathbf{x}\text{ }+\text{ }\mathbf{k}\] From the question it is given that, the sum...

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(iii) The polynomials \[\mathbf{a}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{3}\] and \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{a}\] when divided by \[\mathbf{x}\text{ }\text{ }\mathbf{4}\] leave the remainder r1 and r2 respectively. If , then find the value of a.

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Let us assume p(x) =  \[\mathbf{a}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{3}\] and q(x) = \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{...

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(i) When divided by \[x-3\] the polynomials \[{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{p}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{6}\] and \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\left( \mathbf{p}\text{ }+\text{ }\mathbf{3} \right)\text{ }\mathbf{x}\text{ }\text{ }\mathbf{6}\] leave the same remainder. Find the value of ‘p’. (ii) Find ‘a’ if the two polynomials \[\mathbf{a}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{9}\] and \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{4x}\text{ }+\text{ }\mathbf{a}\], leaves the same remainder when divided by \[\mathbf{x}\text{ }+\text{ }\mathbf{3}\].

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From the question it is given that, by dividing  \[{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{p}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{x}\text{ }+\text{ }\mathbf{6}\]and...

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(i) What number must be divided be subtracted from \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\] so that the resulting polynomial leaves the remainder \[2\], when divided by \[\mathbf{2x}\text{ }+\text{ }\mathbf{1}\] ? (ii) What number must be added to \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{7}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2x}\] so that the resulting polynomial leaves the remainder \[-2\] when divided by \[\mathbf{2x}\text{ }\text{ }\mathbf{3}\] ?

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let us assume ‘p’ be subtracted from \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\] So, dividing  \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\] by \[\mathbf{2x}\text{...

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Using remainder theorem, find the value of ‘a’ if the division of \[{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{5}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{ax}\text{ }+\text{ }\mathbf{6}\text{ }\mathbf{by}\text{ }\left( \mathbf{x}\text{ }\text{ }\mathbf{1} \right)\] leaves the remainder \[2a\].

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Let us assume \[x\text{ }-1\text{ }=\text{ }0\] Then, x = \[1\] Given, f(x) = \[{{x}^{3}}~+\text{ }5{{x}^{2}}~\text{ }ax\text{ }+\text{ }6\] Now, substitute the value of x in f(x),...

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Using remainder theorem, find the value of k if on dividing \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{kx}\text{ }+\text{ }\mathbf{5}\text{ }\mathbf{by}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{2}\] leaves a remainder \[7\].

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Let us assume, \[x\text{ }\text{ }2\text{ }=\text{ }0\] Then, x = \[2\] Given, \[2{{x}^{3}}~+\text{ }3{{x}^{2}}~\text{ }kx\text{ }+\text{ }5\] Now, substitute the value of x in f(x),...

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Find the remainder (without division) on dividing f(x) by (\[2x+1\]) where, (i) f(x) = \[4{{x}^{2}}~+\text{ }5x\text{ }+\text{ }3\] (ii) f(x) = \[\mathbf{3}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{7}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{4x}\text{ }+\text{ }\mathbf{11}\]

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Let us assume \[~2x\text{ }+\text{ }1\text{ }=\text{ }0\] Then, \[2x\text{ }=\text{ }-1\] \[X\text{ }=\text{ }-{\scriptscriptstyle 1\!/\!{ }_2}\] Given, f(x) = \[4{{x}^{2}}~+\text{ }5x\text{...

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Using the remainder theorem, find the remainder on dividing f(x) by (x + \[3\]) where (i) f(x) = \[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{1}\] (ii) f(x) = \[\mathbf{3}{{\mathbf{x}}^{\mathbf{3}}}~+\text{ }\mathbf{7}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{1}\]

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Let us assume \[x\text{ }+\text{ }3\text{ }=\text{ }0\] Then, x = \[-3\] Given, f(x) =\[\mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }+\text{ }\mathbf{1}\] Now, substitute the...

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Find the remainder (without division) on dividing f(x) by (x – \[2\]) where (i) f(x) = \[\mathbf{5}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{7x}\text{ }+\text{ }\mathbf{4}\] (ii) f(x) = \[\mathbf{2}{{\mathbf{x}}^{\mathbf{3}}}~\text{ }\mathbf{7}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{3}\]

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Let us assume \[x\text{ }\text{ }2\text{ }=\text{ }0\] Then, x = \[2\] Given, f(x) = \[\mathbf{5}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{7x}\text{ }+\text{ }\mathbf{4}\] Now, substitute the...

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If \[(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}):\text{ }(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{11}:\text{ }\mathbf{9}\], find the value of \[\frac{3{{x}^{4}}+25{{y}^{4}}}{3{{x}^{4}}-25{{y}^{4}}}\]

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It is given that \[(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~+\text{ }\mathbf{2}{{\mathbf{y}}^{\mathbf{2}}}):\text{ }(\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{...

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If \[\left( \mathbf{a}\text{ }+\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right),\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{c} \right)\text{ }\mathbf{and}\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right)\] are in continued proportion, prove that b is the mean proportional between a and c.

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It is given that \[\left( \mathbf{a}\text{ }+\text{ }\mathbf{2b}\text{ }+\text{ }\mathbf{c} \right),\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{c} \right)\text{ }\mathbf{and}\text{ }\left(...

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In an examination, the number of those who passed and the number of those who failed were in the ratio of \[3:1\]. Had \[8\] more appeared, and \[6\] less passed, the ratio of passed to failures would have been \[2:1\]. Find the number of candidates who appeared.

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Consider the number of passed = \[3x\] Number of failed = x So the total candidates appeared = \[3x\text{ }+\text{ }x\text{ }=\text{ }4x\] In the second case Number of candidates appeared =...

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Find the compound ratio of \[{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{2}}}:\text{ }{{\left( \mathbf{a}\text{ }\text{ }\mathbf{b} \right)}^{\mathbf{2}}},\text{ }({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}}):\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{a}}^{\mathbf{4}}}~\text{ }{{\mathbf{b}}^{\mathbf{4}}}):\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b} \right)}^{\mathbf{4}}}\]

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\[\begin{array}{*{35}{l}} {{\left( a\text{ }+\text{ }b \right)}^{2}}:\text{ }{{\left( a\text{ }\text{ }b \right)}^{2}}  \\ ({{a}^{2}}~\text{ }{{b}^{2}}):\text{ }({{a}^{2}}~+\text{ }{{b}^{2}})  \\...

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Using properties of properties, find x from the following equations: (iii) \[\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}=\frac{a}{b}\] (iv) \[\frac{\sqrt{12x+1}+\sqrt{2x-3}}{\sqrt{12x+1}-\sqrt{2x-3}}=\frac{3}{2}\]

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By cross multiplication \[\begin{array}{*{35}{l}} 50x\text{ }\text{ }75\text{ }=\text{ }12x\text{ }+\text{ }1  \\ 50x\text{ }\text{ }12x\text{ }=\text{ }1\text{ }+\text{ }75  \\ \end{array}\] So we...

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If \[(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~\text{ }\mathbf{13}{{\mathbf{d}}^{\mathbf{2}}})\text{ }=\text{ }(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{d}}^{\mathbf{2}}})\], prove that a: b :: c: d.

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It is given that \[(\mathbf{11}{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }\mathbf{13}{{\mathbf{b}}^{\mathbf{2}}})\text{ }(\mathbf{11}{{\mathbf{c}}^{\mathbf{2}}}~\text{...

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If \[~\left( \mathbf{4a}\text{ }+\text{ }\mathbf{5b} \right)\text{ }\left( \mathbf{4c}\text{ }\text{ }\mathbf{5d} \right)\text{ }=\text{ }\left( \mathbf{4a}\text{ }\text{ }\mathbf{5d} \right)\text{ }\left( \mathbf{4c}\text{ }+\text{ }\mathbf{5d} \right)\], prove that a, b, c, d are in proportion.

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It is given that \[~\left( \mathbf{4a}\text{ }+\text{ }\mathbf{5b} \right)\text{ }\left( \mathbf{4c}\text{ }\text{ }\mathbf{5d} \right)\text{ }=\text{ }\left( \mathbf{4a}\text{ }\text{ }\mathbf{5d}...

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If a: b :: c: d, prove that (iii) \[\left( \mathbf{2a}\text{ }+\text{ }\mathbf{3b} \right)\text{ }\left( \mathbf{2c}\text{ }\text{ }\mathbf{3d} \right)\text{ }=\text{ }\left( \mathbf{2a}\text{ }\text{ }\mathbf{3b} \right)\text{ }\left( \mathbf{2c}\text{ }+\text{ }\mathbf{3d} \right)\] (iv) (la + mb): (lc + mb) :: (la – mb): (lc – mb)

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(iii) We know that If a: b :: c: d we get a/b = c/d By multiplying \[2/3\] \[2a/3b\text{ }=\text{ }2c/3d\] By applying componendo and dividendo \[\left( 2a\text{ }+\text{ }3b \right)/\text{ }\left(...

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If a, b, c, d are in continued proportion, prove that: (V) \[{{\left( \frac{a-b}{c}+\frac{a-c}{b} \right)}^{2}}-{{\left( \frac{d-b}{c}+\frac{d-c}{b} \right)}^{2}}={{(a-d)}^{2}}\left( \frac{1}{{{c}^{2}}}-\frac{1}{{{b}^{2}}} \right)\]

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It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

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If a, b, c, d are in continued proportion, prove that: (iii) \[\left( \mathbf{a}\text{ }+\text{ }\mathbf{d} \right)\text{ }\left( \mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }\text{ }\left( \mathbf{a}\text{ }+\text{ }\mathbf{c} \right)\text{ }\left( \mathbf{b}\text{ }+\text{ }\mathbf{d} \right)\text{ }=\text{ }{{\left( \mathbf{b}\text{ }\text{ }\mathbf{c} \right)}^{\mathbf{2}}}\] (iv) a: d = triplicate ratio of (a – b): (b – c)

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It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

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If a, b, c, d are in continued proportion, prove that: (i) \[\frac{{{a}^{3}}+{{b}^{3}}+{{c}^{3}}}{{{b}^{3}}+{{c}^{3}}+{{d}^{3}}}=\frac{a}{d}\] (ii) \[({{\mathbf{a}}^{\mathbf{2}}}~\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }({{\mathbf{c}}^{\mathbf{2}}}~\text{ }{{\mathbf{d}}^{\mathbf{2}}})\text{ }=\text{ }{{({{\mathbf{b}}^{\mathbf{2}}}~\text{ }{{\mathbf{c}}^{\mathbf{2}}})}^{\mathbf{2}}}\]

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It is given that a, b, c, d are in continued proportion Here we get a/b = b/c = c/d = k \[c\text{ }=\text{ }dk,\text{ }b\text{ }=\text{ }ck\text{ }=\text{ }dk\text{ }.\text{ }k\text{ }=\text{...

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If a, b, c are in continued proportion, prove that: (v) \[\mathbf{abc}\text{ }{{\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)}^{\mathbf{3}}}~=\text{ }{{\left( \mathbf{ab}\text{ }+\text{ }\mathbf{bc}\text{ }+\text{ }\mathbf{ca} \right)}^{\mathbf{3}}}\] (vi) \[\left( \mathbf{a}\text{ }+\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }\left( \mathbf{a}\text{ }\text{ }\mathbf{b}\text{ }+\text{ }\mathbf{c} \right)\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}\]

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It is given that a, b, c are in continued proportion So we get a/b = b/c = k (v) LHS = \[abc\text{ }{{\left( a\text{ }+\text{ }b\text{ }+\text{ }c \right)}^{3}}\] We can write it as \[=\text{...

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If a, b, c are in continued proportion, prove that: (iii) \[\mathbf{a}:\text{ }\mathbf{c}\text{ }=\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}}):\text{ }({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}})\] (iv) \[~{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}~({{\mathbf{a}}^{-\mathbf{4}}}~+\text{ }{{\mathbf{b}}^{-\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{-\mathbf{4}}})\text{ }=\text{ }{{\mathbf{b}}^{-\mathbf{2}}}~({{\mathbf{a}}^{\mathbf{4}}}~+\text{ }{{\mathbf{b}}^{\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{\mathbf{4}}})\]

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It is given that a, b, c are in continued proportion So we get a/b = b/c = k   (iii) \[~a:\text{ }c\text{ }=\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}):\text{ }({{b}^{2}}~+\text{ }{{c}^{2}})\] We...

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If x, y, z are in continued proportion, prove that:\[{{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y} \right)}^{\mathbf{2}}}/\text{ }{{\left( \mathbf{y}\text{ }+\text{ }\mathbf{z} \right)}^{\mathbf{2}}}~=\text{ }\mathbf{x}/\mathbf{z}\]

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It is given that x, y, z are in continued proportion Consider x/y = y/z = k So we get y = kz \[x\text{ }=\text{ }yk\text{ }=\text{ }kz\text{ }\times \text{ }k\text{ }=\text{ }{{k}^{2}}z\] Therefore,...

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If a, b, c and d are in proportion, prove that: (vii) \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}+{{d}^{2}}}=\frac{ab+ad-bc}{bc+cd-ad}\] (viii) \[abcd\left[ \frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}+\frac{1}{{{c}^{2}}}+\frac{1}{{{d}^{2}}} \right]={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}\]

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It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk Therefore, LHS = RHS. So we get = d2 (1 + k2) + b2 (1 + k2) = (1 + k2) (b2 + d2) RHS = a2 + b2 + c2 + d2 We can...

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If a, b, c and d are in proportion, prove that: (iii)\[({{\mathbf{a}}^{\mathbf{4}}}~+\text{ }{{\mathbf{c}}^{\mathbf{4}}}):\text{ }({{\mathbf{b}}^{\mathbf{4}}}~+\text{ }{{\mathbf{d}}^{\mathbf{4}}})\text{ }=\text{ }{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}:\text{ }{{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{d}}^{\mathbf{2}}}\] (iv) \[\frac{{{a}^{2}}+ab}{{{c}^{2}}+cd}=\frac{{{b}^{2}}-2ab}{{{d}^{2}}-2cd}\]

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It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk (iii) \[({{a}^{4}}~+\text{ }{{c}^{4}}):\text{ }({{b}^{4}}~+\text{ }{{d}^{4}})\text{ }=\text{...

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If a, b, c and d are in proportion, prove that: (i) \[\left( \mathbf{5a}\text{ }+\text{ }\mathbf{7v} \right)\text{ }\left( \mathbf{2c}\text{ }\text{ }\mathbf{3d} \right)\text{ }=\text{ }\left( \mathbf{5c}\text{ }+\text{ }\mathbf{7d} \right)\text{ }\left( \mathbf{2a}\text{ }\text{ }\mathbf{3b} \right)\] (ii) (ma + nb): b = (mc + nd): d

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It is given that a, b, c, d are in proportion Consider a/b = c/d = k a = b, c = dk (i) LHS = \[\left( 5a\text{ }+\text{ }7b \right)\text{ }\left( 2c\text{ }\text{ }3d \right)\] Substituting the...

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If a/b = c/d = e/f prove that: \[\frac{{{a}^{2}}}{{{b}^{2}}}+\frac{{{c}^{2}}}{{{d}^{2}}}+\frac{{{e}^{2}}}{{{f}^{2}}}=\frac{ac}{bd}+\frac{ce}{df}+\frac{ae}{df}\] (iv) \[bdf{{\left[ \frac{a+b}{b}+\frac{c+d}{d}+\frac{c+f}{f} \right]}^{3}}=27(a+b)(c+d)(e+f)\]

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Consider a/b = c/d = e/f = k So we get a = bk, c = dk, e = fk Therefore, LHS = RHS. So we get \[=\text{ }bdf\text{ }{{\left( k\text{ }+\text{ }1\text{ }+\text{ }k\text{ }+\text{ }1\text{ }+\text{...

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If a/b = c/d = e/f prove that: (i) \[({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{d}}^{\mathbf{2}}}~+\text{ }{{\mathbf{f}}^{\mathbf{2}}})\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}~+\text{ }{{\mathbf{e}}^{\mathbf{2}}})\text{ }=\text{ }{{\left( \mathbf{ab}\text{ }+\text{ }\mathbf{cd}\text{ }+\text{ }\mathbf{ef} \right)}^{\mathbf{2}}}\] (ii) \[\frac{{{({{a}^{3}}+{{c}^{3}})}^{2}}}{{{({{b}^{3}}+{{d}^{3}})}^{2}}}=\frac{{{e}^{6}}}{{{f}^{6}}}\]

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Consider a/b = c/d = e/f = k So we get a = bk, c = dk, e = fk (i) LHS = \[({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~+\text{ }{{e}^{2}})\] We can write it...

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16. If x/a = y/b = z/c, prove that (i) \[\frac{{{x}^{3}}}{{{a}^{2}}}+\frac{{{y}^{3}}}{{{b}^{2}}}+\frac{{{z}^{3}}}{{{c}^{2}}}=\frac{{{(x+y+z)}^{3}}}{{{(a+b+c)}^{2}}}\] (ii)\[{{\left[ \frac{{{a}^{2}}{{x}^{2}}+{{b}^{2}}{{y}^{2}}+{{c}^{2}}{{z}^{2}}}{{{a}^{3}}x+{{b}^{3}}y+{{c}^{3}}z} \right]}^{3}}=\frac{xyz}{abc}\]

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It is given that x/a = y/b = z/c We can write it as x = ak, y = bk and z = ck Therefore, LHS = RHS. Therefore, LHS = RHS.

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If y is mean proportional between x and z, prove that \[\mathbf{xyz}\text{ }{{\left( \mathbf{x}\text{ }+\text{ }\mathbf{y}\text{ }+\text{ }\mathbf{z} \right)}^{\mathbf{3}}}~=\text{ }{{\left( \mathbf{xy}\text{ }+\text{ }\mathbf{yz}\text{ }+\text{ }\mathbf{zx} \right)}^{\mathbf{3}}}\]

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It is given that y is mean proportional between x and z We can write it as \[{{y}^{2}}~=\text{ }xz\]…… (1) Consider LHS = \[xyz\text{ }{{\left( x\text{ }+\text{ }y\text{ }+\text{ }z \right)}^{3}}\]...

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If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between \[({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{b}}^{\mathbf{2}}})\text{ }\mathbf{and}\text{ }({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}})\]

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It is given that b is the mean proportional between a and c \[{{b}^{2}}~=\text{ }ac\]…. (1) Here (ab + bc) is the mean proportional between \[({{\mathbf{a}}^{\mathbf{2}}}~+\text{...

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If \[\mathbf{k}\text{ }+\text{ }\mathbf{3},\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2},\text{ }\mathbf{3k}\text{ }\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{2k}\text{ }\text{ }\mathbf{3}\] are in proportion, find k.

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It is given that \[\mathbf{k}\text{ }+\text{ }\mathbf{3},\text{ }\mathbf{k}\text{ }+\text{ }\mathbf{2},\text{ }\mathbf{3k}\text{ }\text{ }\mathbf{7}\text{ }\mathbf{and}\text{ }\mathbf{2k}\text{...

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