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 \( x^3 + y^3 \): \[ x^3 + y^3 = (10)(58 – 21) = (10)(37) = 490 \] \[ \boxed{490} \]

More MCQs on Algebra (Collected from Past Papers)

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