Question 1

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: 81 = a² + 2(18) + b². Step 3: 81 = a² + 36 + b². Step 4: a² + b² = 81 - 36 = 45.

Question 2

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. Step 4: a - b = (p + q + r) - (p + q - r) = 2r. Step 5: (p + q + r)² - (p + q - r)² = (2p + 2q)(2r) = 4r(p + q).

Question 3

Expand: (2x + 3y)(2x - 3y)

Accepted Answer

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

Question 4

If x + 1/x = 5, find the value of x² + 1/x².

Accepted Answer

Step 1: Use the identity (x + 1/x)² = x² + 2 + 1/x². Step 2: Rearrange: x² + 1/x² = (x + 1/x)² - 2. Step 3: Substitute x + 1/x = 5: x² + 1/x² = 5² - 2 = 25 - 2 = 23.

Question 5

Simplify: (x + 5)² - (x - 5)²

Accepted Answer

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

Question 6

If x + y = 7 and xy = 12, find the value of (x − y)².

Accepted Answer

Step 1: Use the identity (x − y)² = (x + y)² − 4xy. Step 2: Substitute x + y = 7 and xy = 12: (x − y)² = (7)² − 4(12) = 49 − 48 = 1.

Question 7

Simplify: (2a + 3b)² − (2a − 3b)².

Accepted Answer

Step 1: Use the identity A² − B² = (A + B)(A − B). Step 2: Let A = (2a + 3b) and B = (2a − 3b). Step 3: A + B = (2a + 3b) + (2a − 3b) = 4a. Step 4: A − B = (2a + 3b) − (2a − 3b) = 6b. Step 5: Therefore, (2a + 3b)² − (2a − 3b)² = (4a)(6b) = 24ab.

Question 8

If a + b + c = 9 and ab + bc + ca = 26, 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: Rearrange: a² + b² + c² = (a + b + c)² − 2(ab + bc + ca). Step 3: Substitute a + b + c = 9 and ab + bc + ca = 26: a² + b² + c² = (9)² − 2(26) = 81 − 52 = 29.

Question 9

If x + 1/x = 5, then find the value of x² + 1/x².

Accepted Answer

Step 1: Use the identity (x + 1/x)² = x² + 2 + 1/x². Step 2: Substitute x + 1/x = 5: (5)² = x² + 2 + 1/x². Step 3: 25 = x² + 2 + 1/x². Step 4: x² + 1/x² = 25 - 2 = 23.

Question 10

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 11

If a − b = 3 and ab = 10, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a − b)² = a² − 2ab + b². Step 2: Rearrange: a² + b² = (a − b)² + 2ab. Step 3: Substitute a − b = 3 and ab = 10: a² + b² = (3)² + 2(10) = 9 + 20 = 29.

Question 12

If (x + y)³ = 216 and (x - y)³ = 64, find the value of x.

Accepted Answer

Step 1: Take cube roots: x + y = 6 and x - y = 4. Step 2: Add both equations: (x + y) + (x - y) = 6 + 4, so 2x = 10. Step 3: Therefore, x = 5.

Question 13

If a + b = 12 and a² - b² = 60, find the value of (a - b).

Accepted Answer

Step 1: Use the identity a² - b² = (a + b)(a - b). Step 2: Substitute: 60 = 12(a - b). Step 3: Solve for (a - b): a - b = 60/12 = 5.

Question 14

If (p + q)² - (p - q)² = 96, find the value of pq.

Accepted Answer

Step 1: Expand both squares: (p + q)² = p² + 2pq + q² and (p - q)² = p² - 2pq + q². Step 2: Subtract: (p² + 2pq + q²) - (p² - 2pq + q²) = 4pq. Step 3: Therefore, 4pq = 96, so pq = 24.

Question 15

If (a + b)² = 81 and (a - b)² = 49, find the value of ab.

Accepted Answer

Step 1: Expand both identities: (a + b)² = a² + 2ab + b² = 81 and (a - b)² = a² - 2ab + b² = 49. Step 2: Subtract the second from the first: (a² + 2ab + b²) - (a² - 2ab + b²) = 81 - 49, which gives 4ab = 32. Step 3: Therefore, ab = 8.

Question 16

If x + 1/x = 5, then 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: x³ + 1/x³ = 5³ - 3(5) = 125 - 15 = 110.

Question 17

If a² + b² + c² = 50 and ab + bc + ca = 25, then find (a + b + c)².

Accepted Answer

Step 1: Use the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 2: Substitute the given values: (a + b + c)² = 50 + 2(25) = 50 + 50 = 100. Step 3: Therefore, (a + b + c)² = 100.

SSC GD Constable Algebraic Identities

SSC GD Constable Algebraic Identities — Past Exam Questions

Algebraic Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Algebraic Identities