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Maths 3 Mark Question

Algebraic Expressions · Maharashtra Board STD 6 (MSBSHSE - 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 = 7x^2 + 5xy - 9y^2$, $B = -4x^2 + xy + 5y^2$ and $C = 4y^2 - 3x^2 - 6xy$ then show that $A + B + C = 0. $
Answer
$L.H.S. = A + B + C= (7x^2 + 5xy - 9y^2) + (-4x^2 + xy + 5y^2) + (4y^2 - 3x^2 - 6xy)$
$= 7x^2 + 5xy - 9y^2- 4x^2 + xy + 5y^2+ 4y^2 - 3x^2 - 6xy$
$= 7x^2 - 4x^2 - 3x^2 + 5xy + xy - 6xy - 9y^2+ 5y^2+ 4y^2$
$= 7x^2 - 7x^2 + 6xy - 6xy - 9y^2+ 9y^2$
$= 0 = R.H.S. (Proved)$
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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 $= a^2 - b^2 + 2ab$, Q $= a^2 + 4b^2 - 6ab$, R $= b^2 + 6, S = a^2 - 4ab$ and T $= -2a^2 + b^2 - ab + a.$
Find $ P + Q + R + S - T.$
Answer
$= a^2 - b^2 + 2ab + a^2 + 4b^2 - 6ab + b^2 + 6 + a^2 - 4ab - (-2a^2 + b^2 - ab + a)$
$= a^2 - b^2 + 2ab + a^2 + 4b^2 - 6ab + b^2 + 6 + a^2 - 4ab + 2a^2 - b^2 + ab - a$
$= 5a^2 + 3b^2 - 7ab + 6 - a$
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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$ 5x^3 - 2x^2 + 6x + 7$ to make the sum $x^3 + 3x^2 - x + 1$?
Answer
$= (x^3 + 3x^2 - x + 1) - (5x^3 - 2x^2 + 6x + 7)$
$= x^3 + 3x^2 - x + 1 - 5x^3 + 2x^2 - 6x - 7$
$= -4x^3 + 5x^2 - 7x - 6$
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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:
$5a - [a^2 - {2a (1 - a + 4a^2) - 3a (a^2 - 5a - 3)}] - 8a$
$= 5a - [a^2 - {2a - 2a^2 + 8a^3 - 3a^3 + 15a^2 + 9a}] - 8a$
$= 5a - [a^2 - {2a + 9a - 2a^2 + 15a^2 + 8a^3 - 3a^3}] - 8a$
$= 5a - [a^2 - {11a + 13a^2 + 5a^3}] - 8a$
$= 5a - [a^2 - 11a - 13a^2 - 5a^3] - 8a$
$= 5a - a^2 + 11a + 13a^2 + 5a^3 - 8a$
$= 5a + 11a - 8a - a^2 + 13a^2 + 5a^3$
$= 8a + 12a^2 + 5a^3$
$= 5a3 + 12a^2 + 8a$
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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:
$12x - [3x^3 + 5x^2 - {7x^2 - (4 - 3x - x^3) + 6x^3} - 3x]$
$= 12x - [3x^3 - 5x^2 - {7x^2 - 4 + 3x + x^3 + 6x^3} - 3x]$
$= 12x - [3x^3 + 5x^2 - {7x^2 - 4 + 3x + 7x^3} - 3x]$
$= 12x - [3x^3 + 5x^2 - 7x^2 + 4 - 3x - 7x^3 - 3x]$
$= 12x - [3x^3 - 7x^3 + 5x^2 - 7x^2 + 4 - 3x - 3x]$
$= 12x - [ - 4x^3 + 2x^2 + 4 - 6x]$
$= 12x + 4x^3 + 2x^2 - 4 + 6x$
$= 12x + 6x + 4x^3 + 2x^2 - 4$
$= 18x + 4x^3 + 2x^2 - 4$
$= 4x^3 + 2x^2 + 18x - 4$
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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
$= (3x^2 - 5x + 2) + (-5x^2 - 8x + 6) - (4x^2 - 9x + 7)$
$= 3x^2 - 5x + 2 - 5x^2 - 8x + 6 - 4x^2 + 9x - 7$
$= 3x^2 - 5x^2 - 4x^2 - 5x - 8x + 9x + 2 + 6 - 7$
$= -6x^2 - 4x + 1$
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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:
$-2(x^2 - y^2 + xy) - 3(x^2 + y^2 - xy)$
Answer
We have:
$-2(x^2 - y^2+ xy) - 3(x^2 + y^2 - xy)$
$= -2x^2 + 2y^2 - 2xy - 3x^2 - 3y^2 + 3xy$
$= -2x^2 - 3x^2 + 2y^2 - 3y^2 - 2xy + 3xy$
$= (-2 - 3)x^2 + (2 - 3) y^2 + (- 2 + 3)xy$
$= -5x^2 - y^2 + xy$
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Question 173 Marks
Simplify:
$-4x^2 + {(2x^2 - 3) - (4 - 3x^2)}$
Answer
We have:
$-4x^2 + {(2x^2 - 3) - (4 - 3x^2)}$
$= -4x^2 + {2x^2 - 3 - 4 + 3x^2}$
[Removing grouping symbol]
$= -4x^2 + (5x^2 - 7)$
$= -4x^2 + 5x^2 - 7$
(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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