Question 1

If x + y = 12 and xy = 32, then find the value of x² + y².

Accepted Answer

Step 1: We use the algebraic identity (x + y)² = x² + 2xy + y²
Step 2: Rearranging: x² + y² = (x + y)² - 2xy
Step 3: Substitute the given values: x² + y² = (12)² - 2(32)
Step 4: Calculate: x² + y² = 144 - 64 = 80
Therefore, x² + y² = 80

Question 2

If x + 1/x = 5, then find the value of x⁴ + 1/x⁴.

Accepted Answer

Step 1: Use (x + 1/x)² = x² + 2 + 1/x² → 25 = x² + 2 + 1/x² → x² + 1/x² = 23. Step 2: Use (x² + 1/x²)² = x⁴ + 2 + 1/x⁴ → 529 = x⁴ + 2 + 1/x⁴ → x⁴ + 1/x⁴ = 527. Therefore, answer is 527.

Question 3

Expand: (2a - 3b)(2a + 3b)

Accepted Answer

Step 1: Recognize this as the difference of squares pattern (x - y)(x + y) = x² - y². Step 2: Here, x = 2a and y = 3b. Step 3: Apply the identity: (2a)² - (3b)² = 4a² - 9b².

Question 4

If x - 1/x = 4, find the value of x² + 1/x².

Accepted Answer

Step 1: Square both sides of x - 1/x = 4. Step 2: (x - 1/x)² = 16. Step 3: Expand using (a - b)² = a² - 2ab + b²: x² - 2(x)(1/x) + 1/x² = 16. Step 4: Simplify: x² - 2 + 1/x² = 16. Step 5: Therefore, x² + 1/x² = 16 + 2 = 18.

Question 5

If (a + b)² = 81 and ab = 18, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Substitute (a + b)² = 81 and ab = 18 into the identity: 81 = a² + 2(18) + b². Step 3: Simplify: 81 = a² + 36 + b². Step 4: Therefore, a² + b² = 81 - 36 = 45.

Question 6

Simplify: (x + 5)² - (x - 5)²

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². Step 2: Expand (x + 5)² = x² + 10x + 25. Step 3: Expand (x - 5)² = x² - 10x + 25. Step 4: Subtract: (x² + 10x + 25) - (x² - 10x + 25) = x² + 10x + 25 - x² + 10x - 25 = 20x.

Question 7

Simplify: (x + 5)² - (x - 5)² using algebraic identities.

Accepted Answer

Step 1: Use the identity (a + b)² - (a - b)² = 4ab. Step 2: Here, a = x and b = 5. Step 3: Apply the formula: (x + 5)² - (x - 5)² = 4 × x × 5 = 20x.

Question 8

If a + b + c = 12 and ab + bc + ca = 47, find the value of a² + b² + c².

Accepted Answer

Step 1: Use the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 2: Substitute the known values: (12)² = a² + b² + c² + 2(47). Step 3: Simplify: 144 = a² + b² + c² + 94. Step 4: Therefore, a² + b² + c² = 144 - 94 = 50.

Question 9

Simplify: (p + q)³ - (p - q)³

Accepted Answer

Step 1: Use the identity (a + b)³ = a³ + 3a²b + 3ab² + b³ and (a - b)³ = a³ - 3a²b + 3ab² - b³. Step 2: Expand (p + q)³ = p³ + 3p²q + 3pq² + q³. Step 3: Expand (p - q)³ = p³ - 3p²q + 3pq² - q³. Step 4: Subtract: (p³ + 3p²q + 3pq² + q³) - (p³ - 3p²q + 3pq² - q³) = 6p²q + 2q³ = 2q(3p² + q²).

Question 10

If a - b = 7 and ab = 12, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a - b)² = a² - 2ab + b². Step 2: Rearrange to get a² + b² = (a - b)² + 2ab. Step 3: Substitute the given values: a² + b² = 7² + 2(12) = 49 + 24 = 73.

Question 11

Simplify: (2x + 3y)² - (2x - 3y)² and find the coefficient of xy.

Accepted Answer

Step 1: Use the identity (a + b)² - (a - b)² = 4ab. Step 2: Here, a = 2x and b = 3y. Step 3: Apply the formula: (2x + 3y)² - (2x - 3y)² = 4 × (2x) × (3y) = 4 × 6xy = 24xy. Step 4: The coefficient of xy is 24.

Question 12

If x - y = 4 and xy = 5, then find the value of x³ - y³.

Accepted Answer

Step 1: Use the identity x³ - y³ = (x - y)(x² + xy + y²). Step 2: We know x - y = 4 and xy = 5. Step 3: Find x² + y² using (x - y)² = x² - 2xy + y². Step 4: (4)² = x² - 2(5) + y², so 16 = x² + y² - 10, thus x² + y² = 26. Step 5: Calculate x² + xy + y² = 26 + 5 = 31. Step 6: x³ - y³ = (x - y)(x² + xy + y²) = 4 × 31 = 124.

Question 13

Simplify: (a + b)³ - (a - b)³.

Accepted Answer

Step 1: Use the identity (x)³ - (y)³ = (x - y)(x² + xy + y²). Step 2: Let x = (a + b) and y = (a - b). Step 3: x - y = (a + b) - (a - b) = 2b. Step 4: Expand (a + b)³ = a³ + 3a²b + 3ab² + b³ and (a - b)³ = a³ - 3a²b + 3ab² - b³. Step 5: Subtract: (a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) = 6a²b + 2b³ = 2b(3a² + b²).

Question 14

If a² + b² = 34 and ab = 15, find the value of (a + b)².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Rearrange: (a + b)² = (a² + b²) + 2ab. Step 3: Substitute a² + b² = 34 and ab = 15: (a + b)² = 34 + 2(15) = 34 + 30 = 64.

Question 15

If x - y = 7 and xy = 18, find the value of x² + y².

Accepted Answer

Step 1: Use the identity (x - y)² = x² - 2xy + y². Step 2: Rearrange: x² + y² = (x - y)² + 2xy. Step 3: Substitute x - y = 7 and xy = 18: x² + y² = (7)² + 2(18) = 49 + 36 = 85.

Question 16

Simplify: (p + q + r)² - (p + q - r)².

Accepted Answer

Step 1: Use the identity a² - b² = (a + b)(a - b). Step 2: Let a = (p + q + r) and b = (p + q - r). Step 3: a + b = (p + q + r) + (p + q - r) = 2p + 2q = 2(p + q). Step 4: a - b = (p + q + r) - (p + q - r) = 2r. Step 5: (a + b)(a - b) = 2(p + q) × 2r = 4r(p + q).

Question 17

If a² + b² = 13 and ab = 6, then find the value of (a + b)².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Rearrange: (a + b)² = (a² + b²) + 2ab. Step 3: Substitute a² + b² = 13 and ab = 6: (a + b)² = 13 + 2(6) = 13 + 12 = 25.

Question 18

If x + 1/x = 5, find the value of x³ + 1/x³.

Accepted Answer

Step 1: Find x² + 1/x² using the identity (x + 1/x)² = x² + 2 + 1/x². So 25 = x² + 2 + 1/x², giving x² + 1/x² = 23. Step 2: Use the identity x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x). Step 3: Substitute: x³ + 1/x³ = 5³ - 3(5) = 125 - 15 = 110.

Question 19

If (2x + 3y)² - (2x - 3y)² = k·xy, find the value of k.

Accepted Answer

Step 1: Use the difference of squares identity A² - B² = (A + B)(A - B). Let A = (2x + 3y) and B = (2x - 3y). Step 2: (A + B) = (2x + 3y) + (2x - 3y) = 4x and (A - B) = (2x + 3y) - (2x - 3y) = 6y. Step 3: Therefore, (2x + 3y)² - (2x - 3y)² = 4x · 6y = 24xy. Step 4: Comparing with k·xy, we get k = 24.

Question 20

If a + b + c = 12 and a² + b² + c² = 60, find the value of ab + bc + ca.

Accepted Answer

Step 1: Use the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 2: Substitute the known values: 12² = 60 + 2(ab + bc + ca). Step 3: 144 = 60 + 2(ab + bc + ca). Step 4: 2(ab + bc + ca) = 84, so ab + bc + ca = 42.

Question 21

If x² + y² = 13 and xy = 6, then find the value of (x + y)⁴ − (x − y)⁴.

Accepted Answer

Step 1: Calculate (x + y)² = x² + 2xy + y² = 13 + 12 = 25, so x + y = ±5. Step 2: Calculate (x − y)² = x² − 2xy + y² = 13 − 12 = 1, so x − y = ±1. Step 3: Use the identity a⁴ − b⁴ = (a² − b²)(a² + b²). Step 4: (x+y)⁴ − (x−y)⁴ = [(x+y)² − (x−y)²][(x+y)² + (x−y)²] = (25 − 1)(25 + 1) = 24 × 26 = 624.

Question 22

If a + b + c = 12 and a² + b² + c² = 60, then find the value of ab + bc + ca.

Accepted Answer

Step 1: Use the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 2: Substitute: 144 = 60 + 2(ab + bc + ca). Step 3: Solve: 2(ab + bc + ca) = 84, so ab + bc + ca = 42.

SSC CHSL Algebraic Identities

SSC CHSL Algebraic Identities — Past Exam Questions

Algebraic Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Algebraic Identities — Practice Questions

60-Second Revision — Algebraic Identities