Question 1:

\[
\text{If } a^2 + b^2 = 29 \text{ and } ab = 10, \text{ then find } (a+b)^2.
\]
\[
\text{(a) } 39, \quad \text{(b) } 49, \quad \text{(c) } 59, \quad \text{(d) } 69
\]
Answer: B

Step by Step Solution

Solution:

Using the identity:
\[
(a+b)^2 = a^2 + b^2 + 2ab
\]

Substituting the given values:

\[
(a+b)^2 = 29 + 2(10) = 29 + 20 = 49
\]

\[
\boxed{49}
\]

Question 2:

\[
\text{What is the expansion of } (x + y)^3 \text{?}
\]
\[
\text{(a) } x^3 + 3x^2y + 3xy^2 + y^3, \quad \text{(b) } x^3 + y^3, \quad \text{(c) } (x + y)(x – y), \quad \text{(d) } x^3 – 3x^2y + 3xy^2 – y^3
\]
Answer: A

Step by Step Solution

Solution:

Using the binomial theorem:

\[
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
\]

\[
\boxed{x^3 + 3x^2y + 3xy^2 + y^3}
\]

Question 3:

\[
\text{If } x – \frac{1}{x} = 4, \text{ then find } x^2 + \frac{1}{x^2}.
\]
\[
\text{(a) } 14, \quad \text{(b) } 15, \quad \text{(c) } 16, \quad \text{(d) } 18
\]
Answer: C

Step by Step Solution

Solution:

Using the identity:

\[
x^2 + \frac{1}{x^2} = \left(x – \frac{1}{x}\right)^2 + 2
\]

Substituting \( x – \frac{1}{x} = 4 \):

\[
x^2 + \frac{1}{x^2} = 4^2 + 2 = 16 + 2 = 16
\]

\[
\boxed{16}
\]

Question 4:

\[
\text{Which identity represents } a^3 – b^3?
\]
\[
\text{(a) } (a – b)(a^2 + ab + b^2), \quad \text{(b) } (a – b)(a^2 – ab + b^2), \quad \text{(c) } (a + b)(a^2 + ab + b^2), \quad \text{(d) } (a + b)(a^2 – ab + b^2)
\]
Answer: A

Step by Step Solution

Solution:

Using the identity for the difference of cubes:

\[
a^3 – b^3 = (a – b)(a^2 + ab + b^2)
\]

\[
\boxed{(a – b)(a^2 + ab + b^2)}
\]

Question 5:

\[
\text{If } x + y = 5 \text{ and } xy = 6, \text{ then find } x^2 + y^2.
\]
\[
\text{(a) } 11, \quad \text{(b) } 13, \quad \text{(c) } 15, \quad \text{(d) } 19
\]
Answer: B

Step by Step Solution

Solution:

Using the identity:

\[
x^2 + y^2 = (x + y)^2 – 2xy
\]

Substituting the given values:

\[
x^2 + y^2 = 5^2 – 2(6) = 25 – 12 = 13
\]

\[
\boxed{13}
\]

Question 6:

\[
\text{Find the value of } (a + b)^2 – (a – b)^2.
\]
\[
\text{(a) } 2ab, \quad \text{(b) } 4ab, \quad \text{(c) } a^2 – b^2, \quad \text{(d) } (a + b)(a – b)
\]
Answer: B

Step by Step Solution

Solution:

Using the identity:
\[
(A^2 – B^2) = (A – B)(A + B)
\]
Substituting \( A = (a+b) \) and \( B = (a-b) \):
\[
(a+b)^2 – (a-b)^2 = [(a+b) – (a-b)][(a+b) + (a-b)]
\]
\[
= (a+b-a+b)(a+b+a-b) = (2b)(2a) = 4ab
\]

\[
\boxed{4ab}
\]

Question 7:

\[
\text{If } (x + 1)^2 = x^2 + 6x + 9, \text{ then find } x.
\]
\[
\text{(a) } 1, \quad \text{(b) } 2, \quad \text{(c) } 3, \quad \text{(d) } 4
\]
Answer: C

Step by Step Solution

Solution:

Expanding \( (x+1)^2 \):
\[
(x+1)^2 = x^2 + 2x + 1
\]

Given equation:
\[
x^2 + 2x + 1 = x^2 + 6x + 9
\]

Cancel \( x^2 \) on both sides:
\[
2x + 1 = 6x + 9
\]

Rearrange:
\[
1 – 9 = 6x – 2x
\]

\[
-8 = 4x
\]

\[
x = \frac{-8}{4} = 3
\]

\[
\boxed{3}
\]

Question 8:

\[
\text{Which of the following is the correct identity for } (a – b)^3?
\]
\[
\text{(a) } a^3 – 3a^2b + 3ab^2 – b^3, \quad \text{(b) } a^3 – b^3, \quad \text{(c) } (a – b)(a^2 + ab + b^2), \quad \text{(d) } a^3 + 3a^2b + 3ab^2 + b^3
\]
Answer: A

Step by Step Solution

Solution:

Using the cube expansion identity:

\[
(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3
\]

\[
\boxed{a^3 – 3a^2b + 3ab^2 – b^3}
\]

Question 9:

\[
\text{If } a^2 – b^2 = 21 \text{ and } a – b = 3, \text{ then find } a + b.
\]
\[
\text{(a) } 5, \quad \text{(b) } 6, \quad \text{(c) } 7, \quad \text{(d) } 8
\]
Answer: C

Step by Step Solution

Solution:

Using the identity:
\[
a^2 – b^2 = (a-b)(a+b)
\]

Substituting values:
\[
21 = (3)(a+b)
\]

Solving for \( a+b \):
\[
a+b = \frac{21}{3} = 7
\]

\[
\boxed{7}
\]

Question 10:

\[
\text{Which of the following is the identity for } (a + b + c)^2?
\]
\[
\text{(a) } a^2 + b^2 + c^2 + 2ab + 2bc + 2ca, \quad \text{(b) } a^2 + b^2 + c^2 + ab + bc + ca, \quad \text{(c) } a^2 + b^2 + c^2 – 2ab – 2bc – 2ca, \quad \text{(d) } (a+b+c)(a-b+c)
\]
Answer: A

Step by Step Solution

Solution:

Using the identity:
\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
\]

\[
\boxed{a^2 + b^2 + c^2 + 2ab + 2bc + 2ca}
\]

Question 11:

\[
\text{If } x – y = 3 \text{ and } xy = 18, \text{ then find } x^2 – y^2.
\]
\[
\text{(a) } 10, \quad \text{(b) } 12, \quad \text{(c) } 15, \quad \text{(d) } 18
\]
Answer: D

Step by Step Solution

Solution:

Using the identity:
\[
x^2 – y^2 = (x – y)(x + y)
\]

Rearrange the quadratic equation:
\[
x^2 – y^2 = 3(x + y)
\]

Using the relation \( xy = 18 \), solve for \( x + y \) using the quadratic formula:
\[
(x+y)^2 – 4xy = (x-y)^2
\]
\[
(x+y)^2 – 4(18) = 9
\]
\[
(x+y)^2 = 81
\]

Taking square root:
\[
x + y = 9
\]

\[
x^2 – y^2 = (3)(9) = 18
\]

\[
\boxed{18}
\]

Question 12:

\[
\text{If } (a – b)^2 = 81 \text{ and } ab = 20, \text{ then find } a^2 + b^2.
\]
\[
\text{(a) } 100, \quad \text{(b) } 121, \quad \text{(c) } 144, \quad \text{(d) } 169
\]
Answer: B

Step by Step Solution

Solution:

Using the identity:
\[
a^2 + b^2 = (a – b)^2 + 2ab
\]

Substituting values:
\[
a^2 + b^2 = 81 + 2(20) = 81 + 40 = 121
\]

\[
\boxed{121}
\]

Question 13:

\[
\text{Which of the following represents the identity } a^4 – b^4?
\]
\[
\text{(a) } (a^2 – b^2)(a^2 + b^2), \quad \text{(b) } (a – b)(a^3 + b^3), \quad \text{(c) } (a^2 + b^2)^2, \quad \text{(d) } (a – b)^2(a + b)^2
\]
Answer: A

Step by Step Solution

Solution:

Using the difference of squares:
\[
a^4 – b^4 = (a^2 – b^2)(a^2 + b^2)
\]

\[
\boxed{(a^2 – b^2)(a^2 + b^2)}
\]

Question 14:

\[
\text{If } a + b = 6 \text{ and } ab = 8, \text{ then find } a^3 + b^3.
\]
\[
\text{(a) } 140, \quad \text{(b) } 144, \quad \text{(c) } 150, \quad \text{(d) } 156
\]
Answer: B

Step by Step Solution

Solution:

Using the identity:
\[
a^3 + b^3 = (a + b)(a^2 – ab + b^2)
\]

First, find \( a^2 + b^2 \):
\[
a^2 + b^2 = (a+b)^2 – 2ab = 6^2 – 2(8) = 36 – 16 = 20
\]

Now, calculate \( a^3 + b^3 \):
\[
a^3 + b^3 = (6)(20 – 8) = (6)(12) = 144
\]

\[
\boxed{144}
\]

Question 15:

\[
\text{If } x + y = 10 \text{ and } xy = 21, \text{ then find } x^3 + y^3.
\]
\[
\text{(a) } 469, \quad \text{(b) } 476, \quad \text{(c) } 490, \quad \text{(d) } 500
\]
Answer: C

Step by Step Solution

Solution:

Using the identity:
\[
x^3 + y^3 = (x + y)(x^2 – xy + y^2)
\]

First, find \( x^2 + y^2 \):
\[
x^2 + y^2 = (x+y)^2 – 2xy = 10^2 – 2(21) = 100 – 42 = 58
\]

Now, calculate \( x^3 + y^3 \):
\[
x^3 + y^3 = (10)(58 – 21) = (10)(37) = 490
\]

\[
\boxed{490}
\]

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