Edudel, Directorate of Education Govt. of NCT of Delhi Class 8 Mental Maths Question Bank Chapter 9 Algebraic Expressions and Identities Questions Solution. In this chapter, there is total 45 math problems. We have given each questions solution step by step.
algebraic Expressions and identities class 8 mental math(1) Simplify: (2x2 – 3xy2) 2
Answer:
Here we use identity.
(a – b) 2 = a2 – 2ab + b2
a = 2, b = 3xy2
(2x2 – 3xy2) 2 = (2x2) x (-2 x 2x2 x 3xy2) + (3xy2) 2
(2x2 – 3xy2) 2 = 4x4 – 12x3y2 + 9x2y4
(2) Simplify: (25x2 – 16y2) ÷ (5x + 4y)
Answer:
(25x2 – 16y2) = we write this as, a2 – b2
a = 5x b 4y
We know,
(a + b)(a – b) = a2 – b2
(5x + 4y) (5x – 4y) = (25x2 – 16y2)
(25x2 – 16y2) ÷ (5x + 4y) = (5x + 4y) (5x – 4y) ÷ (5x + 4y)
(25x2 – 16y2) ÷ (5x + 4y) = (5x – 4y)
(3) Simplify: (3a + 2b) (3a -2b) – 5b2
Answer:
We know,
(a + b)(a – b) = a2 – b2
We put, a = 3a, b = 2b
(3a + 2b) (3a – 2b) = 3a2 – 2b2
(3a + 2b) (3a – 2b) – 5b2 = (3a) 2 – (2b) 2 – 5b2
(3a + 2b) (3a – 2b) – 5b2 = 9a2 – 9b2
(4) Simplify: (4x – 3y) 2 + 24xy
Answer:
Here (4x – 3y) 2 + 24xy
We use (a – b) 2 = a2 – 2ab + b2
a =4, b = 3y
= (4x) 2 x (-2 x 4x x 3y) + (3y) 2
(4x – 3y) 2 = 16x2 – 24xy + 9y2
16x2 – 24xy + 9y2 + 24xy
(4x – 3y) 2 + 24xy = 16x2 + 9y2
(5) Simplify: 9x2 + 24xy + 16y2 / 3x + 4y
Answer:
We know,
(a + b) 2 = a2 + 2ab + b2
9x2 +24xy+16y2 = (3x + 4y) 2
9x2 +24xy+16y2 / 3x+4y = (3x + 4y) 2 / 3x+4y
9x2 +24xy+16y2 / 3x+4y = 3x+4y
(6) Simplify: (x2 + 5x +6) ÷ (x + 2)
Answer:
Here we us,
(x + a)(x + b) = x2 + (a + b) x + ab
a = 2 and b =3
(x2 + 5x + 6) we put this as (x + 2)(x +3)
(x + 2) (x + 3) ÷ (x + 2)
(x2 + 5x + 6) ÷ (x + 2) = (x + 3)
(7) Simplify: (6. 25p2 – 2. 25q2) ÷ (2. 5p + 1. 5q)
Answer:
We know,
(a + b)(a – b) = a2 –b2
We put, a = 6. 25p, b = 2. 25q
(6. 25p2 – 2. 25q2) = (6. 25p + 2. 25q) (6. 25p – 2. 25q)
(6. 25p2 – 2. 25q2) ÷ (2. 5p + 1. 5p) = (6. 25p + 2. 25q) (6. 25p – 2. 25q) ÷ (2. 5p + 1. 5q)
= 2. 5p – 1. 5q
(8) Simplify: (7x + 4y) 2 – 49x2 – 16y2
Answer:
a = 7x b = 4y
7x2 + 2x 7x x 4y + 4y2
(7x + 4y2) = 49x2 + 56xy + 16y2
(7x + 4y2) – 49x2 – 16y2 = 49x2 + 56xy + 16y2 – 49x2 – 16y2
(7x + 4y2) – 49x2 – 16y2 = 56xy
(9) Simplify: (x2 + 10x + 25) ÷ (x + 5)
Answer:
(x2 + 10x + 25) Here we use (a + b)2 = a2 + 2ab + b2
a = x b = 5
(x2 + 10x + 25) = (x + 5)2
(x2 + 10x + 25) ÷ (x + 5) = (x + 5)2 ÷ (x + 5)
(x2 + 10x + 25) ÷ (x + 5) = x + 5
(10) Simplify: (a + b) (a2 + b2 – ab)
Answer:
We Know, Standard identify,
A3 + b3 = (a + b) (a2 + b2 – ab)
(a + b) (a2 + b2 – ab) = a3 + b3
(11) Simplify: (m2 + mn + n2) (m – n)
Answer:
We Know, standard identify
A3 – b3 = (a – b) (a2 + ab + b2)
Here, a = m b = n
(m2 + mn + n2) (m – n) = m3 – n3
(12) If a = 3, b = 4, then find the value of (a + b)2 + ab
Answer:
Given, a = 3, b = 4 we put this value in (a + b)2 + ab
(a + b)2 + ab = (3 + 4)2 + (3 × 4)
(a + b)2 + ab = (9 + 24 + 16) + 12
(a + b)2 + ab = 49 + 12
(a + b)2 + ab = 61
(13) If m = -5, n = 7, a = 3, b = 3, then find the value of (a – b)3 + (m + n)2
Answer:
Here, We use
(a – b)3 = a3 – b3 – 3ab (a – b)
Here a = 3 and b = 3
(a – b)3 = 33 – 33 – 3 × 3 × 3 (3 – 3)
(a – b)3 = 0
(m + n)2 = m2 + 2mn + n2
Here m = -5, n = 7
(m + n)2 = -52 + 2x -5 × 7 + 72
(m + n)2 = 25 – 70 + 49
(m + n)2 = 4
(a – b)3 + (m + n)2 = 0 + 4
(a – b)3 + (m + n)2 = 4
(14) If P = 7, q = -5 then find the value of (q + p)2 – qp
Answer:
= (q + p)2 – qp
Here we use,
= (q + p)2 = q2 + 2qp + p2
Here, p = 7, q = -5
= (q + p)2 = -52 + 2 × -5 × 7 + 72
= (q + p)2 = 25 – 70 + 49
= (q + p)2 = 4
= (q + p)2 – qp = 4 – (7 × – 5)
= (q + p)2 – qp = 39
(15) If x = 10, y = 17 then find the value of (x – y)2 + (x + y)
Answer:
= (x – y)2 + (x + y)
Here we use,
= (x – y)2 = 2 – 2xy + y2
= (x – y)2 + (x + y) = x2 – 2xy + y2 (x + y)
= X = 10, y = 17
= 102 – 2 × 10 × 17 + 172 (10 + 17)
= 100 – 340 + 289 + 27
= (x – y)2 + (x + y) = 76
(16) If p = 1.5, q = 0.5 then find the value of (p + q)2 – (2)3
Answer:
= (p + q)2 – (2)3
Here we use,
= (p + q)2 = p2 + 2pq + q2
= (p + q)2 – (2)3 = p2 + 2pq + q2 – 8
= P = 1.5, q = 0.5
= p2 + 2pq + q2 – 8 = 1.52 + 2 × 1.5 × 0.5 + 0.52 – 8
= 2.25 + 1.5 + 0.28 – 8
= 4 – 8
= (p + q)2 – (2)3 = – 4
(17) If a = 10, b = 5, C= 2, then find the value of (a + b + c)3
Answer:
= (a + b + c)3
= (10 – 5 + 2)3
= (7)3
(a + b + c)3 = 343
(18) If x = 100 √25 + (1° × 0) then find the value of x2
Answer:
= X = 100/ √25 + (1° × 0)
We have to find value of x2
= X2 = (100/√25 + (1° × 0)2
= X2 = 1002/25
= X2 = 400
(19) If a = 1012 – 1002 then find the value of a – 1
Answer:
A = 1012 – 1002
We have to find a – 1
We know,
= (a + b) 9a – b) = a2 – b2
We Put, a = 101, b = 100
= (101 + 100) (101 – 100) = 1012 – 1002
= 201 × 1 = 1012 – 1002
= 1012 – 1002 = 201
The value of a – 1 = 201 – 1 = 200
(20) If m = 1002 – 982 then find the value of m + 4
Answer:
M = 1002 – 982
(a + b) (a – b) = a2 – b2
We Put, a = 100, b = 98
(100 + 98) (100 – 98) = 190 × 2
1002 – 982 = 396
The Value of m + 4 = 400
(21) If a = 100, b = 98 then find the value of (a – b)2 + (a + b)
Answer:
A = 100, b = 98
We have to find (a – b) 2 + (a + b)
We Know,
= (a – b)2 = a2 – 2ab + b2
= (a – b)2 + (a + b) = a2 – 2ab + b + (a + b)
= 1002 – 2 × 100 × 98 + 982 + (100 + 98)
= (a – b)2 + (a + b) = 201
(22) Simplify: (√25 – √16) (√16 – √9) + √36 – √16
Answer:
= √25 = 5
= √9 = 3
= √16 = 4
= √36 = 6
= (√25 – √16) (√16 – √9) + √36 – √16 = (5 – 4) × (4 – 3) + 6 – 4
= (5 – 4) × (4 – 3) + 6 – 4 = 1 × 1 + 2
= (√25 – √16) (√16 – √9) + √36 – √16 = 3
(23) Simplify: (2a b)2 – (a + 2b)2 + 3b2 + ab
Answer:
Here We use,
= (2a – b)2 = (a – b)2 = a2 – 2ab + b2
a = 2a and b = b
= (2a – b)2 = 4a2 – 4ab + b2
= (a + 2b)2 = (a + b)2 = a2 + 2ab + b2
= a= a and b = 2b
= (a + 2b)2 = a2 + 4ab + 4b2
= (2a – b)2 – (a + 2b)2 + 3b2 + b = 4a2 – 4ab + b2 – a2 + 4ab + 4b2 + 3b2 + ab
= 4a2 – 4ab + b2 –a2 + 4ab + 4b2 +3b2 + ab
On Solving,
= (2a – b)2 – (a + 2b)2 + 3b2 + ab = 3a2 – 7ab
(24) Simplify: (a + b)2 + (a – b)2 – ab
Answer:
Here we Use,
= (a – b)2 = a2 – 2ab + b2
= (a + b)2 = a2 + 2ab + b2
= (a + b)2 + (a – b)2 – ab = a2 + 2ab + b2 + a2 – 2ab + b2 – ab
= (a + b)2 + (a – b)2 – ab = 2a2 + 2b2 – ab
(25) Simplify: (a – b) (a + b) + (a + b) (a – b) + a2 – b2
Answer:
We Know,
= (a – b) (a + b) = a2 – b2
= (a – b) (a + b) + (a + b) (a – b) + a2 – b2 + a2 – b2 + a2 – b2
= a2 – b2 + a2 – b2 + a2 – b2 = 3a2 – 3b2
(26) If (5)2 – (4.9)2 = x, find the value of (0.01 × x)
Answer:
Here We use,
= (a – b) (a + b) = a2 – b2
= a = 5 and b = 4.9
(5 – 4.9) × (5 + 4.9) = (5)2 – (4.9)2
X = 0. × 9.9
= 0.99
The Value of (0.01 × x) = (0.01 × 0.1 × 9.9)
= (0.01 × x)
= 0.0099
(28) Simplify: (x + y)2 – (x – y)2 +x – y
Answer:
Here we use,
= (x + y)2 = x2 + 2xy + y2
= (x – y)2 =x2 – 2xy + y2
= (x + y)2 – (x – y)32 + x – y = x2 +2xy + y2 – x2 – 2xy +y2 + x – y
= x2 + 2xy + y2 – x2 – 2xy + y2 + x – y = 4xy + x – y
= (x + y)2 – (x – y)2 + x – y = 4xy + x – y
(29) What should be Subtracted from (a + b)2 to make it (a – b)2?
Answer:
We Know,
= (a – b)2 = a2 – 2ab + b2
= (a + b)2 = a2 + 2ab + b2
4ab subtracted from (a +b)2 to make it (a – b)2
(30) What should be added in 25x2 + 16y2 to make it (5x + 4y)2?
Answer:
= (a +b)2 = a2 + 2ab + b2
= (5x + 4y)2 = 5x2 + 2 × 5x × 4y + 4y2
= (5x + 4y)2 = 25x2 + 16y2 + 40xy
= 40xy added in 25x2 + 16y2 to make it (5x + 4y)2
(31) 121m2 – 100n2 should be divided by which expression to get 11m + 10n?
Answer:
= 121m2 – 100n2/x = 11m + 10n
By cross multiplication,
= 121m2 – 100n2/11m + 10n = x
= (a – b) (a + b) = a2 – b2
= 121m2 – 100n2 = (11m + 10n) (11m – 10n)
On Solving,
X = 11m – 10n.
(32) Simplify: (2x + 3y)2 + 24xy
Answer:
= (2x + 3y)2 + 24xy
Here we use,
= (a + b)2 = a2 + 2ab + b2
= a = 2x b = 3y
= (2x + 3y)2 = 2x2 + 2 × 2 × 3y + 3y2
= (2x + 3y)2 =4 x2 + 12xy + 9y2
= (2x + 3y)2 + 24xy = 4x2 + 12xy + 9y2 + 24xy
= (2x + 3y)2 + 24xy = 4x2 + 36xy + 9y2
(33) Simplify: (3x – 4y)2 – 16y2
Answer:
= (3x – 4y)2 – 16y2
Here we use,
= (a – b)2 = a2 – 2ab + b2
= a = 3x b = 4y
= (3x – 4y)2 = 3x2 – 2 × 3 × 4y + 4y2
= (3x – 4y)2 = 9×2 – 24xy + 16y2
= (3x – 4y)2 – 16y2 = 9x2 – 24xy + 16y2 – 16y2
= (3x – 4y)2 – 16y2 = 9×2 – 24xy
(34) Simplify: (2.5m – 0.5n)2 + 2.5mn +3.5mn
Answer:
Here we use,
= (a – b)2 = a2 – 2ab + b2
= a = 2.5m b = 0.5n
= (2.5m – 0.5n)2 = 2.5m2 – 2 × 2.5m × 0.5n + 0.5n2
= (2.5m – 0.5n)2 = 6.25m2 – 2.5 mn +0.25n2
= (2.5m – 0.5n)2 + 2.5mn+ 3.5mn
= 6.25m2 – 2.5mn + 0.25n2 + 2.5mn + 3.5mn
= (2.5m – 0.5n)2 + 2.5mn + 3.5mn = 6.25m2 + 3.5mn + 0.25n2
(35) What should be subtracted from (x + y)2 to get x2 +y2?
Answer:
= (x + y)2
We Know,
= (a + b)2 = a2 + 2ab + b2
= (x + y)2 = x2 + 2xy + y2
= x2 + 2xy + y2 – Subtraction = x2 + y2
Subtraction = x2 + 2xy + y2 – (x2 + y2)
Subtraction = 2xy
2xy Subtracted (x + y)2 to get x2 + y2
(36) The sides of a rectangle are 5x units and 7y units. Find the area and Perimeter of the rectangle.
Answer:
Given that, the sides of a rectangle are 5x units and 7y units
Area of Rectangle = Length × Breadth
Area of Rectangle = 5x × 7y = 35xy
Perimeter of Rectangle = 2x (Length + Breadth)
Perimeter of Rectangle = 2 × (5x + 7y)
Perimeter of Rectangle = 10x + 14y
(37) Find the value of 61 × 5.9
Answer:
61 × 5.9
We multiply without decimal Point. After we give decimal Point.
61 × 59 = 3599
61 × 5.9 = 359.9
(38) Find the value of 302 × 298
Answer:
302 × 298
We Write,
302 = (300 + 2)
298 = (300 – 2)
302 × 298 = (300 + 2) × (300 – 2)
We Know,
(a – b) (a + b) =a2 – b2
a = 300 and b = 2
(300 + 2) × (300 – 2) = 3002 – 22
90000 – 4
302 × 298 = 89996
(39) If x = 2 + √3 and y = 2 – √3, then find the value of (x – y)2.
Answer:
Given,
X = 2 + √3 and y = 2 – √3
We have to find (x – y)2.
= (x – y)2 = x2 – 2xy + y2
= (2 + √3 – 2 – √)2 = (2 + √3)2 – (2 × 2 – √3 × 2 + √3) + (2 – √3)2
= (2 + √3 – 2 – √3)2 = 4 + 3 – 11 + 1
= (2 + √3 – 2 – √3)2 = 12
(42) If (53)2 = (48)2 + 5x, then find the value of x.
Answer:
Given,
(53)2 = (48)2 + 5x
5x = (53)2 – (48)2
We use,
(a – b) (a + b) = a2 – b2
A = 53 and b = 48
(53 – 48) × (53 + 48) = (53)2 – (48)2
= 5 × 101 = (53)2 – (48)2
= (53)2 – (48)2 = 505
5x = (53)2 – (48)2
5x = 505
X = 505/5
X = 101
(43) If a = 0. 8 and b = 0.5, then find the value of a2 + b2 + ab.
Answer:
Given,
a = 0.8 and b = 0.5
We have to find a2 + b2 + ab.
= a2 + b2 + ab = (0.8)2 + (0.5)2 + 0.8 × 0.5
= a2 + b2 + ab = 0.64 + 0.25 + 0.40
= a2 + b2 + ab = 1.29
(44) If x = 5 and b = 3.2, then find the value of x2 + y2 – xy.
Answer:
Given,
X = 5 and b = 3.2
We have to find x2 + y2 – xy
= x2 + y2 – xy = 52 + (3.2)2 – 5 (3.2)
= x2 + y2 – xy =25 + 10.24 – 16
= x2 + y2 – xy = 19.24
(45) If X = 3 + 2 √2 and y = 17 + 12 √2, then find the value of √y – √x
Answer:
Given,
X = 3 +2 √2 and y = 17 + 12√2
We have to find √y – √x
√x = √3 + 2 √2
√y = √17 + 12 √2
√y – √x = √17 + 12 √2 – √3 + 2 √2
On Solving we get,
√y – √x = 2 + √2
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