Question 1

If x + y = 12 and xy = 35, find the value of x² + y².

Accepted Answer

Step 1: Use the identity (x + y)² = x² + 2xy + y². Step 2: Substitute: (12)² = x² + 2(35) + y². Step 3: 144 = x² + 70 + y². Step 4: x² + y² = 144 - 70 = 74.

Question 2

If p - q = 5 and pq = 24, find the value of p² + q².

Accepted Answer

Step 1: Use the identity (p - q)² = p² - 2pq + q². Step 2: Rearrange: p² + q² = (p - q)² + 2pq. Step 3: Substitute: p² + q² = (5)² + 2(24) = 25 + 48 = 73.

Question 3

Expand: (2m + 3n)²

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Here a = 2m and b = 3n. Step 3: (2m)² + 2(2m)(3n) + (3n)² = 4m² + 12mn + 9n².

Question 4

If a + b + c = 15 and ab + bc + ca = 72, 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: a² + b² + c² = (15)² - 2(72) = 225 - 144 = 81.

Question 5

Simplify: (x + 4)(x - 4) - (x + 2)²

Accepted Answer

Step 1: Use (x + 4)(x - 4) = x² - 16 (difference of squares). Step 2: Use (x + 2)² = x² + 4x + 4. Step 3: Subtract: (x² - 16) - (x² + 4x + 4) = x² - 16 - x² - 4x - 4 = -4x - 20.

Question 6

Simplify: (a + b)² - (a - b)²

Accepted Answer

Step 1: Expand (a + b)² = a² + 2ab + b². Step 2: Expand (a - b)² = a² - 2ab + b². Step 3: Subtract: (a² + 2ab + b²) - (a² - 2ab + b²) = a² + 2ab + b² - a² + 2ab - b² = 4ab.

Question 7

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

Accepted Answer

Step 1: We 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 8

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). Then x - y = 2b. Step 3: Alternatively, expand directly: (a+b)³ = a³ + 3a²b + 3ab² + b³ and (a-b)³ = a³ - 3a²b + 3ab² - b³. Step 4: Subtract: (a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) = 6a²b + 2b³. Step 5: Factor: 2b(3a² + b²).

Question 9

If a - b = 3 and ab = 4, find the value of a³ - b³.

Accepted Answer

Step 1: Use the identity a³ - b³ = (a - b)(a² + ab + b²). Step 2: We know a - b = 3 and ab = 4. Step 3: Find a² + b²: (a - b)² = a² - 2ab + b², so 9 = a² + b² - 8, thus a² + b² = 17. Step 4: Therefore a² + ab + b² = 17 + 4 = 21. Step 5: a³ - b³ = 3 × 21 = 63.

Question 10

If x² + y² = 34 and xy = 15, find the value of (x + y)².

Accepted Answer

Step 1: Use the identity (x + y)² = x² + 2xy + y². Step 2: Rearrange: (x + y)² = (x² + y²) + 2xy. Step 3: Substitute x² + y² = 34 and xy = 15: (x + y)² = 34 + 2(15) = 34 + 30 = 64.

Question 11

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: Therefore (A + B)(A - B) = (2p + 2q)(2r) = 4r(p + q).

Question 12

If m + n = 8 and m² + n² = 40, find the value of m³ + n³.

Accepted Answer

Step 1: Find mn using (m + n)² = m² + 2mn + n². Step 2: 64 = 40 + 2mn, so 2mn = 24, thus mn = 12. Step 3: Use the identity m³ + n³ = (m + n)(m² - mn + n²). Step 4: m² - mn + n² = (m² + n²) - mn = 40 - 12 = 28. Step 5: m³ + n³ = 8 × 28 = 224.

Question 13

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

Accepted Answer

Step 1: From x + 1/x = 5, square both sides: (x + 1/x)² = 25, which gives x² + 2 + 1/x² = 25, so 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 14

If a + b = 7 and a² + b² = 29, then find the value of a⁴ + b⁴.

Accepted Answer

Step 1: From a + b = 7, square both sides: a² + 2ab + b² = 49, so 29 + 2ab = 49, giving ab = 10. Step 2: Square a² + b² = 29: (a² + b²)² = a⁴ + 2a²b² + b⁴ = 841. Step 3: Calculate a²b² = (ab)² = 100. Step 4: Therefore, a⁴ + b⁴ = 841 - 2(100) = 841 - 200 = 641.

Question 15

If p + q + r = 12 and p² + q² + r² = 60, then find the value of (p + q + r)² - 3(pq + qr + rp).

Accepted Answer

Step 1: Use the identity (p + q + r)² = p² + q² + r² + 2(pq + qr + rp). Step 2: Substitute known values: 144 = 60 + 2(pq + qr + rp), so pq + qr + rp = 42. Step 3: Calculate (p + q + r)² - 3(pq + qr + rp) = 144 - 3(42) = 144 - 126 = 18.

Question 16

If (x + y)² = 81 and (x - y)² = 49, then find the value of xy.

Accepted Answer

Step 1: Expand both identities: (x + y)² = x² + 2xy + y² = 81 and (x - y)² = x² - 2xy + y² = 49. Step 2: Subtract the second from the first: (x² + 2xy + y²) - (x² - 2xy + y²) = 81 - 49, which gives 4xy = 32. Step 3: Therefore, xy = 8.

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.

Question 18

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] = 26 × 24 = 624.

SSC CPO Algebraic Identities

SSC CPO Algebraic Identities — Past Exam Questions

Algebraic Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Algebraic Identities