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Chapter 12

Fundamental Concepts of Algebra — Exercise 12(C)

Class - 6 Concise Mathematics Selina



Exercise 12(C)

Question 1(i)

Find the sum of:

3a+4b+7c,−5a+3b−6c3a + 4b + 7c, -5a + 3b - 6c and 4a−2b−4c4a - 2b - 4c

Answer

Solving,

3a+4b+7c−5a+3b−6c+4a−2b−4c2a+5b−3c\begin{array}{rrrr} &3a &+ 4b &+ 7c \\ &-5a &+ 3b &- 6c \\ + &4a &- 2b &- 4c \\ \hline &2a &+ 5b &- 3c \end{array}

Hence, the required sum = 2a+5b−3c\bm{2a + 5b - 3c}.

Question 1(ii)

Find the sum of:

2x2+xy−y2,−x2+2xy+3y22x^2 + xy - y^2, -x^2 + 2xy + 3y^2 and 3x2−10xy+4y23x^2 - 10xy + 4y^2

Answer

Solving,

2x2+xy−y2−x2+2xy+3y2+3x2−10xy+4y24x2−7xy+6y2\begin{array}{rrrr} &2x^2 &+ xy &- y^2 \\ &-x^2 &+ 2xy &+ 3y^2 \\ + &3x^2 &- 10xy &+ 4y^2 \\ \hline &4x^2 &- 7xy &+ 6y^2 \end{array}

Hence, the required sum = 4x2−7xy+6y2\bm{4x^2 - 7xy + 6y^2}.

Question 1(iii)

Find the sum of:

x2−x+1,−5x2+2x−2x^2 - x + 1, -5x^2 + 2x - 2 and 3x2−3x+13x^2 - 3x + 1

Answer

Solving,

x2−x+1−5x2+2x−2+3x2−3x+1−x2−2x+0\begin{array}{rrrr} &x^2 &- x &+ 1 \\ &-5x^2 &+ 2x &- 2 \\ + &3x^2 &- 3x &+ 1 \\ \hline &-x^2 &- 2x &+ 0 \end{array}

Hence, the required sum = −x2−2x\bm{-x^2 - 2x}.

Question 1(iv)

Find the sum of:

a2−ab+bc,2ab+bc−2a2a^2 - ab + bc, 2ab + bc - 2a^2 and −3bc+3a2+ab-3bc + 3a^2 + ab

Answer

Solving,

a2−ab+bc−2a2+2ab+bc+3a2+ab−3bc2a2+2ab−bc\begin{array}{rrrr} &a^2 &- ab &+ bc \\ &-2a^2 &+ 2ab &+ bc \\ + &3a^2 &+ ab &- 3bc \\ \hline &2a^2 &+ 2ab &- bc \end{array}

Hence, the required sum = 2a2+2ab−bc\bm{2a^2 + 2ab - bc}.

Question 1(v)

Find the sum of:

4x2+7−3x,4x−x2+84x^2 + 7 - 3x, 4x - x^2 + 8 and −10+5x−2x2-10 + 5x - 2x^2

Answer

Solving,

4x2−3x+7−x2+4x+8+−2x2+5x−10x2+6x+5\begin{array}{rrrr} &4x^2 &- 3x &+ 7 \\ &-x^2 &+ 4x &+ 8 \\ + &-2x^2 &+ 5x &- 10 \\ \hline &x^2 &+ 6x &+ 5 \end{array}

Hence, the required sum = x2+6x+5\bm{x^2 + 6x + 5}.

Question 1(vi)

Find the sum of:

3x+4xy−y2,xy−4x+2y23x + 4xy - y^2, xy - 4x + 2y^2 and 3y2−xy+6x3y^2 - xy + 6x

Answer

Solving,

3x+4xy−y2−4x+xy+2y2+6x−xy+3y25x+4xy+4y2\begin{array}{rrrr} &3x &+ 4xy &- y^2 \\ &-4x &+ xy &+ 2y^2 \\ + &6x &- xy &+ 3y^2 \\ \hline &5x &+ 4xy &+ 4y^2 \end{array}

Hence, the required sum = 5x+4xy+4y2\bm{5x + 4xy + 4y^2}.

Question 2(i)

Add the following expressions:

−17x2−2xy+23y2,−9y2+15x2+7xy-17x^2 - 2xy + 23y^2, -9y^2 + 15x^2 + 7xy and 13x2+3y2−4xy13x^2 + 3y^2 - 4xy

Answer

Solving,

−17x2−2xy+23y215x2+7xy−9y2+13x2−4xy+3y211x2+xy+17y2\begin{array}{rrrr} &-17x^2 &- 2xy &+ 23y^2 \\ &15x^2 &+ 7xy &- 9y^2 \\ + &13x^2 &- 4xy &+ 3y^2 \\ \hline &11x^2 &+ xy &+ 17y^2 \end{array}

Hence, the required sum = 11x2+xy+17y2\bm{11x^2 + xy + 17y^2}.

Question 2(ii)

Add the following expressions:

−x2−3xy+3y2+8,3x2−5y2−3+4xy-x^2 - 3xy + 3y^2 + 8, 3x^2 - 5y^2 - 3 + 4xy and −6xy+2x2−2+y2-6xy + 2x^2 - 2 + y^2

Answer

Solving,

−x2−3xy+3y2+83x2+4xy−5y2−3+2x2−6xy+y2−24x2−5xy−y2+3\begin{array}{rrrrr} &-x^2 &- 3xy &+ 3y^2 &+ 8 \\ &3x^2 &+ 4xy &- 5y^2 &- 3 \\ + &2x^2 &- 6xy &+ y^2 &- 2 \\ \hline &4x^2 &- 5xy &- y^2 &+ 3 \end{array}

Hence, the required sum = 4x2−5xy−y2+3\bm{4x^2 - 5xy - y^2 + 3}.

Question 2(iii)

Add the following expressions:

a3−2b3+a,b3−2a3+ba^3 - 2b^3 + a, b^3 - 2a^3 + b and −2b+2b3−5a+4a3-2b + 2b^3 - 5a + 4a^3

Answer

Solving,

a3−2b3+a−2a3+b3+b+4a3+2b3−5a−2b3a3+b3−4a−b\begin{array}{rrrrr} &a^3 &- 2b^3 &+ a & \\ &-2a^3 &+ b^3 & &+ b \\ + &4a^3 &+ 2b^3 &- 5a &- 2b \\ \hline &3a^3 &+ b^3 &- 4a &- b \end{array}

Hence, the required sum = 3a3+b3−4a−b\bm{3a^3 + b^3 - 4a - b}.

Question 3(i)

Evaluate:

3a−(a+2b)3a - (a + 2b)

Answer

Solving,

+3a+a+2b−−+2a−2b\begin{array}{rrr} &+3a & \\ &+a &+ 2b \\ &- &- \\ \hline &+2a &- 2b \end{array}

Hence, the value of the given expression = 2a−2b\bm{2a - 2b}.

Question 3(ii)

Evaluate:

(5x−3y)−(x+y)(5x - 3y) - (x + y)

Answer

Solving,

+5x−3y+x+y−−+4x−4y\begin{array}{rrr} &+5x &- 3y \\ &+x &+ y \\ &- &- \\ \hline &+4x &- 4y \end{array}

Hence, the value of the given expression = 4x−4y\bm{4x - 4y}.

Question 3(iii)

Evaluate:

(8a+15b)−(3b−7a)(8a + 15b) - (3b - 7a)

Answer

Solving,

+8a+15b−7a+3b+−+15a+12b\begin{array}{rrr} &+8a &+ 15b \\ &-7a &+ 3b \\ &+ &- \\ \hline &+15a &+ 12b \end{array}

Hence, the value of the given expression = 15a+12b\bm{15a + 12b}.

Question 3(iv)

Evaluate:

(8x+7y)−(4y−3x)(8x + 7y) - (4y - 3x)

Answer

Solving,

+8x+7y−3x+4y+−+11x+3y\begin{array}{rrr} &+8x &+ 7y \\ &-3x &+ 4y \\ &+ &- \\ \hline &+11x &+ 3y \end{array}

Hence, the value of the given expression = 11x+3y\bm{11x + 3y}.

Question 3(v)

Evaluate:

7−(4a−5)7 - (4a - 5)

Answer

Solving,

⇒7−(4a−5)⇒7−4a+5⇒12−4a.\Rightarrow 7 - (4a - 5) \\[1em] \Rightarrow 7 - 4a + 5 \\[1em] \Rightarrow 12 - 4a.

Hence, the value of the given expression = 12−4a\bm{12 - 4a}.

Question 4(i)

Subtract:

5a−3b+2c5a - 3b + 2c from a−4b−2ca - 4b - 2c

Answer

Solving,

+a−4b−2c+5a−3b+2c−+−−4a−b−4c\begin{array}{rrrr} &+a &- 4b &- 2c \\ &+5a &- 3b &+ 2c \\ &- &+ &- \\ \hline &-4a &- b &- 4c \end{array}

Hence, the required difference = −4a−b−4c\bm{-4a - b - 4c}.

Question 4(ii)

Subtract:

4x−6y+3z4x - 6y + 3z from 12x+7y−21z12x + 7y - 21z

Answer

Solving,

+12x+7y−21z+4x−6y+3z−+−+8x+13y−24z\begin{array}{rrrr} &+12x &+ 7y &- 21z \\ &+4x &- 6y &+ 3z \\ &- &+ &- \\ \hline &+8x &+ 13y &- 24z \end{array}

Hence, the required difference = 8x+13y−24z\bm{8x + 13y - 24z}.

Question 4(iii)

Subtract:

5−a−4b+4c5 - a - 4b + 4c from 5a−7b+2c5a - 7b + 2c

Answer

Solving,

+5a−7b+2c−a−4b+4c+5++−−+6a−3b−2c−5\begin{array}{rrrrr} &+5a &- 7b &+ 2c & \\ &-a &- 4b &+ 4c &+ 5 \\ &+ &+ &- &- \\ \hline &+6a &- 3b &- 2c &- 5 \end{array}

Hence, the required difference = 6a−3b−2c−5\bm{6a - 3b - 2c - 5}.

Question 4(iv)

Subtract:

−8x−12y+17z-8x - 12y + 17z from x−y−zx - y - z

Answer

Solving,

+x−y−z−8x−12y+17z++−+9x+11y−18z\begin{array}{rrrr} &+x &- y &- z \\ &-8x &- 12y &+ 17z \\ &+ &+ &- \\ \hline &+9x &+ 11y &- 18z \end{array}

Hence, the required difference = 9x+11y−18z\bm{9x + 11y - 18z}.

Question 4(v)

Subtract:

2ab+cd−ac−2bd2ab + cd - ac - 2bd from ab−2cd+2ac+bdab - 2cd + 2ac + bd

Answer

Solving,

+ab−2cd+2ac+bd+2ab+cd−ac−2bd−−++−ab−3cd+3ac+3bd\begin{array}{rrrr} &+ab &- 2cd &+ 2ac &+ bd \\ &+2ab &+ cd &- ac &- 2bd \\ &- &- &+ &+ \\ \hline &-ab &- 3cd &+ 3ac &+ 3bd \end{array}

Hence, the required difference = −ab−3cd+3ac+3bd\bm{-ab - 3cd + 3ac + 3bd}.

Question 5(i)

Take −ab+bc−ca-ab + bc - ca from bc−ca+abbc - ca + ab.

Answer

Solving,

+ab+bc−ca−ab+bc−ca+−++2ab\begin{array}{rrrr} &+ab &+ bc &- ca \\ &-ab &+ bc &- ca \\ &+ &- &+ \\ \hline &+2ab & & \end{array}

Hence, the required difference = 2ab\bm{2ab}.

Question 5(ii)

Take 5x+6y−3z5x + 6y - 3z from 3x+5y−4z3x + 5y - 4z.

Answer

Solving,

+3x+5y−4z+5x+6y−3z−−+−2x−y−z\begin{array}{rrrr} &+3x &+ 5y &- 4z \\ &+5x &+ 6y &- 3z \\ &- &- &+ \\ \hline &-2x &- y &- z \end{array}

Hence, the required difference = −2x−y−z\bm{-2x - y - z}.

Question 5(iii)

Take −32p+q−r\dfrac{-3}{2}p + q - r from 12p−13q−32r\dfrac{1}{2}p - \dfrac{1}{3}q - \dfrac{3}{2}r.

Answer

Solving,

⇒(12p−13q−32r)−(−32p+q−r)⇒12p−13q−32r+32p−q+r⇒12p+32p−13q−q−32r+r⇒(12+32)p+(−13−1)q+(−32+1)r⇒1+32p+−1−33q+−3+22r⇒42p−43q−12r⇒2p−43q−12r.\Rightarrow \left(\dfrac{1}{2}p - \dfrac{1}{3}q - \dfrac{3}{2}r\right) - \left(-\dfrac{3}{2}p + q - r\right) \\[1em] \Rightarrow \dfrac{1}{2}p - \dfrac{1}{3}q - \dfrac{3}{2}r + \dfrac{3}{2}p - q + r \\[1em] \Rightarrow \dfrac{1}{2}p + \dfrac{3}{2}p - \dfrac{1}{3}q - q - \dfrac{3}{2}r + r \\[1em] \Rightarrow \left(\dfrac{1}{2} + \dfrac{3}{2}\right)p + \left(-\dfrac{1}{3} - 1\right)q + \left(-\dfrac{3}{2} + 1\right)r \\[1em] \Rightarrow \dfrac{1 + 3}{2}p + \dfrac{-1 - 3}{3}q + \dfrac{-3 + 2}{2}r \\[1em] \Rightarrow \dfrac{4}{2}p - \dfrac{4}{3}q - \dfrac{1}{2}r \\[1em] \Rightarrow 2p - \dfrac{4}{3}q - \dfrac{1}{2}r.

Hence, the required difference = 2p−43q−12r\bm{2p - \dfrac{4}{3}q - \dfrac{1}{2}r}.

Question 5(iv)

Take 1−a+a21 - a + a^2 from a2+a+1a^2 + a + 1.

Answer

Solving,

+a2+a+1+a2−a+1−+−+2a\begin{array}{rrrr} &+a^2 &+ a &+ 1 \\ &+a^2 &- a &+ 1 \\ &- &+ &- \\ \hline & &+ 2a & \end{array}

Hence, the required difference = 2a\bm{2a}.

Question 6

From the sum of x+y−2zx + y - 2z and 2x−y+z2x - y + z, subtract x+y+zx + y + z.

Answer

Sum of x+y−2zx + y - 2z and 2x−y+z2x - y + z,

x+y−2z+2x−y+z3x0−z\begin{array}{rrrr} &x &+ y &- 2z \\ + &2x &- y &+ z \\ \hline &3x &0 &- z \end{array}

Now, subtracting x+y+zx + y + z from this sum,

+3x0−z+x+y+z−−−+2x−y−2z\begin{array}{rrrr} &+3x &0 &- z \\ &+x &+ y &+ z \\ &- &- &- \\ \hline &+2x &- y &- 2z \end{array}

Hence, the required difference = 2x−y−2z\bm{2x - y - 2z}.

Question 7

From the sum of 3a−2b+4c3a - 2b + 4c and 3b−2c3b - 2c, subtract a−b−ca - b - c.

Answer

Sum of 3a−2b+4c3a - 2b + 4c and 3b−2c3b - 2c,

3a−2b+4c++3b−2c3a+b+2c\begin{array}{rrrr} &3a &- 2b &+ 4c \\ + & &+ 3b &- 2c \\ \hline &3a &+ b &+ 2c \end{array}

Now, subtracting a−b−ca - b - c from this sum,

+3a+b+2c+a−b−c−+++2a+2b+3c\begin{array}{rrrr} &+3a &+ b &+ 2c \\ &+a &- b &- c \\ &- &+ &+ \\ \hline &+2a &+ 2b &+ 3c \end{array}

Hence, the required difference = 2a+2b+3c\bm{2a + 2b + 3c}.

Question 8

Subtract x−2y−zx - 2y - z from the sum of 3x−y+z3x - y + z and x+y−3zx + y - 3z.

Answer

Sum of 3x−y+z3x - y + z and x+y−3zx + y - 3z,

3x−y+z+x+y−3z4x0−2z\begin{array}{rrrr} &3x &- y &+ z \\ + &x &+ y &- 3z \\ \hline &4x &0 &- 2z \end{array}

Now, subtracting x−2y−zx - 2y - z from this sum,

+4x0−2z+x−2y−z−+++3x+2y−z\begin{array}{rrrr} &+4x &0 &- 2z \\ &+x &- 2y &- z \\ &- &+ &+ \\ \hline &+3x &+ 2y &- z \end{array}

Hence, the required difference = 3x+2y−z\bm{3x + 2y - z}.

Question 9

Subtract the sum of x+yx + y and x−zx - z from the sum of x−2zx - 2z and x+y+zx + y + z.

Answer

Sum of x−2zx - 2z and x+y+zx + y + z,

x−2z+x+y+z2x+y−z\begin{array}{rrrr} &x & &- 2z \\ + &x &+ y &+ z \\ \hline &2x &+ y &- z \end{array}

Sum of x+yx + y and x−zx - z,

x+y+x−z2x+y−z\begin{array}{rrrr} &x &+ y & \\ + &x & &- z \\ \hline &2x &+ y &- z \end{array}

Now, subtracting the second sum from the first sum,

+2x+y−z+2x+y−z−−+000\begin{array}{rrrr} &+2x &+ y &- z \\ &+2x &+ y &- z \\ &- &- &+ \\ \hline &0 &0 &0 \end{array}

Hence, the required difference = 0.

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