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MATHS 3 Marks Question

Algebraic Expressions · Rajasthan Board (RBSE) STD 6 (Rajasthan - English Medium). 20 questions.

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Question 13 Marks
Simplify: $2a - 3b - [3a - 2b - {a - c - (a - 2b)}]$
Answer
Removing the innermost grouping symbol $( )$ first, then $\{ \}$ and then $[ ],$
we have: $2a - 3b - [3a - 2b - {a - c - (a - 2b)}] $
$= 2a - 3b - [3a - 2b - {a - c - a + 2b}] $
$= 2a - 3b - [3a - 2b - {- c + 2b}] $
$= 2a - 3b - [3a - 2b + c - 2b] $
$= 2a - 3b - 3a + 2b - c + 2b $
$= 2a - 3a - 3b + 2b + 2b - c $
$= -a + b - c$
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Question 23 Marks
Simplify: $xy - yz - zx - {yx - (3y - xz) - (xy - zy)}$
Answer
Removing the innermost grouping symbol $( )$ first, then $\{ \}$ and then $[ ]$,
we have: $xy - [yz - zx - {yx - (3y - xz) - (xy - zy)}] $
$= xy - [yz - zx - {yx - 3y + xz - xy + zy}] $
$= xv - [yz - zx - {-3y + xz + zy}] $
$= xy - [yz - zx + 3y - xz - zy] $
$= xy - [-2xz + 3y] $
$= xy + 2xz - 3y$
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Question 33 Marks
If $A=7 x^2+5 x y-9 y^2, B=-4 x^2+x y+5 y^2$ and $C=4 y^2-3 x^2-6 x y$ then show that $A + B + C = 0.$
Answer
L.H.S. $=\left(7 x^2+5 x y-9 y^2\right)+\left(-4 x^2+x y+5 y^2\right)+\left(4 y^2-3 x^2-6 x y\right)$
$=7 x^2+5 x y-9 y^2-4 x^2+x y+5 y^2+4 y^2-3 x^2-6 x y $
$ =7 x^2-4 x^2-3 x^2+5 x y+x y-6 x y-9 y^2+5 y^2+4 y^2$
$=7 x^2-7 x^2+6 x y-6 x y-9 y^2+9 y^2$
$= 0 = R.H.S$. (Proved)
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Question 43 Marks
Simplify: $\text{2a}-[4\text{b}-\{\text{4a}-(3\text{b}-\overline{2\text{a}+2\text{b}})\}]$
Answer
Removing the innermost grouping symbol ‘—’ first, then $( )$, then $\{ \}$ and then $[ ]$,
we have: $\text{2a}-[4\text{b}-\{\text{4a}-(3\text{b}-\overline{2\text{a}+2\text{b}})\}]$
$= 2a - [4b - {4a - (3b - 2a - 2b)}] $
$= 2a - [4b - {4a - (b - 2a)}] $
$= 2a - [4b - {4a - b + 2a}] $
$= 2a - [4b - {6a - b}] $
$= 2a - [4b - 6a + b] $
$= 2a - [5b - 6a] $
$= 2a - 5b + 6a $
$= 8a - 5b.$
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Question 53 Marks
Let $P=a^2-b^2+2 a b, Q=a^2+4 b^2-6 a b, R=b^2+6, S=a^2-4 a b$ and $T=-2 a^2+b^2-a b+a$. Find $P + Q + R + S - T.$
Answer
$ =a^2-b^2+2 a b+a^2+4 b^2-6 a b+b^2+6+a^2-4 a b-\left(-2 a^2+b^2-a b+a\right) $
$ =a^2-b^2+2 a b+a^2+4 b^2-6 a b+b^2+6+a^2-4 a b+2 a^2-b^2+a b-a $
$ =5 a^2+3 b^2-7 a b+6-a $
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Question 63 Marks
Subtract the sum of $5x - 4y + 6z$ and $-8x + y - 2z$ from the sum of $12x - y + 3z$ and $-3x + 5y - 8z.$
Answer
$= (12x - y + 3z - 3x + 5y - 8z) - (5x - 4y + 6z - 8x + y - 2z)$
$= 12x - y + 3z - 3x + 5y - 8z - 5x + 4y - 6z + 8x - y + 2z$
$= 12x + 7y - 9z$
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Question 73 Marks
What must be added to $5 x^3-2 x^2+6 x+7$ to make the sum $x^3+3 x^2-x+1$?
Answer
$\left(x^3+3 x^2-x+1\right)-\left(5 x^3-2 x^2+6 x+7\right)$
$ =x^3+3 x^2-x+1-5 x^3+2 x^2-6 x-7$
$ =-4 x^3+5 x^2-7 x-6$
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Question 83 Marks
Simplify:
$3 - [x - {2y - (5x + y - 3) + 2x^2} - (x^2 - 3y)]$
Answer
Removing the innermost grouping symbol ( ) first, then { } and then [ ], we have:
$3 - [x - {2y - (5x + y - 3) + 2x^2} - (x^2 - 3y)]$
$= 3 - [x - {2y - 5x - y + 3 + 2x^2} - x^2+ 3y]$
$= 3 - [x - {y - 5x + 3 + 2x^2} - x^2+ 3y]$
$= 3 - [x - y + 5x - 3 - 2x^2 - x^2+ 3y]$
$= 3 - [6x + 2y - 3 - 3x^2]$
$= 3 - 6x - 2y + 3 + 3x^2$
$= 6 - 6x - 2y + 3x^2$
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Question 93 Marks
Simplify:
$(a^2+ b^2+ 2ab) - (a^2+ b^2- 2ab)$
Answer
We have:
$(a^2+ b^2+ 2ab) - (a^2+ b^2- 2ab)$
$= a^2+ b^2+ 2ab - a^2- b^2+ 2ab $
$= a^2- a^2+ b^2- b^2+ 2ab + 2ab $
$= 0 + 0 + (2 + 2)ab $
$= 4ab$
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Question 103 Marks
Simplify: $-x + [5y - {x - (5y - 2x)}]$
Answer
We have: Removing the innermost grouping symbol $( )$ first, then $\{ \}$ and then $[ ],$
we have: $-x + [5y - {x - (5y - 2x)}] $
$= -x + [5y - {x - 5y + 2x}] $
$= -x + [5y - {3x - 5y}] $
$= -x + [5y - 3x + 5y] $
$= -x + [10y - 3x] $
$= -x + 10y - 3x $
$= -x - 3x + 10y $
$= -4x + 10y$
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Question 113 Marks
Simplify: $-a - [a + {a + b - 2a - (a - 2b)} - b]$
Answer
Removing the innermost grouping symbol $()$ first, then $\{ \}$ and ten $[ ]$, we have:
$-a - [a + {a + b - 2a - (a - 2b)} - b] $
$= -a - [a + {a + b - 2a - a + 2b} - b] $
$= -a - [a + {- 2a + 3b} - b] $
$= - a - [a - 2a + 3b - b] $
$= -a - a + 2a - 3b + b $
$= -2a + 2a - 2b $
$= -2b$
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Question 123 Marks
Simplify:
$ 5a - [a^2- {2a(1 - a + 4a^2) - 3a(a^2- 5a - 3)}] - 8a$
Answer
Removing the innermost grouping symbol ( ) first, then { } and then [ ], we have:
$ 5 a-\left[a^2-\left\{2 a\left(1-a+4 a^2\right)-3 a\left(a^2-5 a-3\right)\right\}\right]-8 a $
$ =5 a-\left[a^2-\left\{2 a-2 a^2+8 a^3-3 a^3+15 a^2+9 a\right\}\right]-8 a $
$ =5 a-\left[a^2-\left\{2 a+9 a-2 a^2+15 a^2+8 a^3-3 a^3\right\}\right]-8 a $
$ =5 a-\left[a^2-\left\{11 a+13 a^2+5 a^3\right\}\right]-8 a $
$ =5 a-\left[a^2-11 a-13 a^2-5 a^3\right]-8 a $
$ =5 a-a^2+11 a+13 a^2+5 a^3-8 a $
$ =5 a+11 a-8 a-a^2+13 a^2+5 a^3 $
$ =8 a+12 a^2+5 a^3 $
$ =5 a 3+12 a^2+8 a $
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Question 133 Marks
Simplify:
$12x - [3x^3+ 5x^2- {7x^2- (4 - 3x - x^3) + 6x^3} - 3x]$
Answer
Removing the innermost grouping symbol () first, then { } and then [ ], we have
$12 x-\left[3 x^3+5 x^2-\left\{7 x^2-\left(4-3 x-x^3\right)+6 x^3\right\}-3 x\right] $
$ =12 x-\left[3 x^3-5 x^2-\left\{7 x^2-4+3 x+x^3+6 x^3\right\}-3 x\right] $
$ =12 x-\left[3 x^3+5 x^2-\left\{7 x^2-4+3 x+7 x^3\right\}-3 x\right] $
$ =12 x-\left[3 x^3+5 x^2-7 x^2+4-3 x-7 x^3-3 x\right] $
$ =12 x-\left[3 x^3-7 x^3+5 x^2-7 x^2+4-3 x-3 x\right] $
$ =12 x-\left[-4 x^3+2 x^2+4-6 x\right] $
$ =12 x+4 x^3+2 x^2-4+6 x $
$ =12 x+6 x+4 x^3+2 x^2-4 $
$ =18 x+4 x^3+2 x^2-4 $
$ =4 x^3+2 x^2+18 x-4 $
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Question 143 Marks
From the sum of $3x^2- 5x + 2$ and $-5x^2- 8x + 6$, subtract $4x^2- 9x + 7$.
Answer
$ =\left(3 x^2-5 x+2\right)+\left(-5 x^2-8 x+6\right)-\left(4 x^2-9 x+7\right) $
$ =3 x^2-5 x+2-5 x^2-8 x+6-4 x^2+9 x-7 $
$ =3 x^2-5 x^2-4 x^2-5 x-8 x+9 x+2+6-7 $
$ =-6 x^2-4 x+1 $
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Question 153 Marks
Simplify: $a - [2b - {3a - (2b - 3c)}]$
Answer
We have:a $- [2b - {3a - (2b - 3c)}]$
$= a - [2b - {3a - 2b + 3c}]$
[Removing grouping symbol $( )]$
$= a - [2b - 3a + 2b - 3c]$
(Removing grouping symbol $\{\})$
$= a - [4b - 3a - 3c]$
$= a - 4b + 3a + 3c$
(Removing grouping symbol $[ ])$
$= 4a - 4b + 3c$
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Question 163 Marks
Simplify:
$-2\left(x^2-y^2+x y\right)-3\left(x^2+y^2-x y\right)$
Answer
We have:
$ -2\left(x^2-y^2+x y\right)-3\left(x^2+y^2-x y\right) $
$ =-2 x^2+2 y^2-2 x y-3 x^2-3 y^2+3 x y$
$ =-2 x^2-3 x^2+2 y^2-3 y^2-2 x y+3 x y$
$ =(-2-3) x^2+(2-3) y^2+(-2+3) x y$
$ =-5 x^2-y^2+x y$
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Question 173 Marks
Simplify:
$-4 x^2+\left\{\left(2 x^2-3\right)-\left(4-3 x^2\right)\right\}$
Answer
We have:
$-4 x^2+\left\{\left(2 x^2-3\right)-\left(4-3 x^2\right)\right\}$
$=-4 x^2+\left\{2 x^2-3-4+3 x^2\right\}$
[Removing grouping symbol]
$=-4 x^2+\left(5 x^2-7\right)$
$=-4 x^2+5 x^2-7$
(Removing grouping symbol {})
$=x^2-7$
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Question 183 Marks
Simplify: $- 3(a + b) + 4(2a - 3b) - (2a - b)$
Answer
We have: $-3(a + b) + 4(2a - 3b) - (2a - b) $
$= -3a - 3b + 8a - 12b - 2a + b $
$= -3a + 8a - 2a - 3b - 12b + b $
$= ( -3 + 8 - 2)a + ( -3 - 12 + 1)b $
$= 3a - 14b$
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Question 193 Marks
Simplify: $86 - [15x - 7(6x - 9) - 2{10x - 5(2 - 3x)}]$
Answer
Removing the innermost grouping symbol $( )$ first, then $\{ \}$ and then $[ ],$
we have: $86 - [15x - 7 (6x - 9) - 2{10x - 5(2 - 3x)}] $
$= 86 - [15x - 42x + 63 - 2{10x - 10 + 15x} $
$= 86 - [15x - 42x + 63 - 2{25x - 10}] $
$= 86 - [15x - 42x + 63 - 50x + 20] $
$= 86 - 15x + 42x - 63 + 50x - 20 $
$= (86 - 63 - 20) - 15x + 42x + 50x $
$= (86 - 83) + (-15 + 42 + 50)x $
$= 3 + 77x$
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Question 203 Marks
Simplify: $5x - [4y - {7x - (3z - 2y) + 4z - 3(x + 3y - 2z)}]$
Answer
Removing the innermost grouping symbol $( )$ first, then $\{\}$ and then $[ ]$, we have:
$5x - [4y - {7x - (3z - 2y) + 4z - 3(x + 3y - 2z)}] $
$= 5x - [4y - {7x - 3z + 2y + 4z - 3x - 9y + 6z}] $
$= 5x - [4y - {4x + 7z - 7y}] $
$= 5x - [4y - 4x - 7z + 7y] $
$= 5x - [11y - 4x - 7z] $
$= 5x - 11y + 4x + 7z $
$= 9x - 11y + 7z$
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