Question 1

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

Accepted Answer

Step 1: We use the algebraic identity (x + 1/x)² = x² + 2(x)(1/x) + 1/x² = x² + 2 + 1/x². Step 2: Substitute x + 1/x = 5 into the identity: (5)² = x² + 2 + 1/x². Step 3: 25 = x² + 2 + 1/x². Step 4: x² + 1/x² = 25 - 2 = 23. Therefore, the answer is 23.

Question 2

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 3

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 values: (12)² = x² + 2(35) + y². Step 3: 144 = x² + y² + 70. Step 4: x² + y² = 144 - 70 = 74.

Question 4

If a - b = 5 and ab = 24, 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 values: a² + b² = (5)² + 2(24) = 25 + 48 = 73.

Question 5

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

Accepted Answer

Step 1: Recognize this as the difference of squares pattern (p + q)(p - q) = p² - q². Step 2: Here p = 2x and q = 3y. Step 3: Apply: (2x)² - (3y)² = 4x² - 9y².

Question 6

Simplify: (m + n)³ - (m - n)³

Accepted Answer

Step 1: Use the identity a³ - b³ = (a - b)(a² + ab + b²). Step 2: Let a = (m + n) and b = (m - n). Step 3: a - b = (m + n) - (m - n) = 2n. Step 4: a² = (m + n)² = m² + 2mn + n². Step 5: b² = (m - n)² = m² - 2mn + n². Step 6: ab = (m + n)(m - n) = m² - n². Step 7: a² + ab + b² = (m² + 2mn + n²) + (m² - n²) + (m² - 2mn + n²) = 3m² + n². Step 8: Result = 2n(3m² + n²) = 6m²n + 2n³.

Question 7

If a - b = 5 and ab = 24, then find the value of a³ - b³.

Accepted Answer

Step 1: Use the identity a³ - b³ = (a - b)(a² + ab + b²). Step 2: Find a² + b² using (a - b)² = a² - 2ab + b², so a² + b² = (a - b)² + 2ab = 25 + 48 = 73. Step 3: Calculate a² + ab + b² = 73 + 24 = 97. Step 4: Therefore, a³ - b³ = 5 × 97 = 485.

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), or expand directly. Step 2: (a + b)³ = a³ + 3a²b + 3ab² + b³. Step 3: (a - b)³ = a³ - 3a²b + 3ab² - b³. Step 4: Subtract: (a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) = 6a²b + 2b³ = 2b(3a² + b²).

Question 9

If p + q + r = 15 and pq + qr + rp = 72, then find the value of p² + q² + r².

Accepted Answer

Step 1: Use the identity (p + q + r)² = p² + q² + r² + 2(pq + qr + rp). Step 2: Rearrange to get p² + q² + r² = (p + q + r)² - 2(pq + qr + rp). Step 3: Substitute values: p² + q² + r² = (15)² - 2(72) = 225 - 144 = 81.

Question 10

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

Accepted Answer

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

Question 11

If x + y = 10 and xy = 21, find the value of x² + y².

Accepted Answer

Step 1: Use the identity x² + y² = (x + y)² − 2xy. Step 2: Substitute the given values: x² + y² = (10)² − 2(21) = 100 − 42 = 58. Therefore, the answer is 58.

Question 12

If p + q + r = 15 and pq + qr + rp = 72, find the value of p² + q² + r².

Accepted Answer

Step 1: Use the identity (p + q + r)² = p² + q² + r² + 2(pq + qr + rp). Step 2: Rearrange: p² + q² + r² = (p + q + r)² - 2(pq + qr + rp). Step 3: Substitute: p² + q² + r² = (15)² - 2(72) = 225 - 144 = 81.

Question 13

Simplify: (a + b)³ - (a - b)³.

Accepted Answer

Step 1: Use the identity (p)³ - (q)³ = (p - q)(p² + pq + q²). Step 2: Let p = (a + b) and q = (a - b). Then p - q = (a + b) - (a - b) = 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 = 2b(3a² + b²).

Question 14

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. Therefore, the answer is 23.

Question 15

If a² + b² = 13 and ab = 6, 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 16

If x - y = 4 and xy = 5, 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: (x - y)² = x² - 2xy + y², so 16 = x² + y² - 10, thus x² + y² = 26. Step 4: Calculate x² + xy + y² = 26 + 5 = 31. Step 5: Therefore, x³ - y³ = 4 × 31 = 124.

Question 17

Simplify: (2a + 3b)² - (2a - 3b)².

Accepted Answer

Step 1: Use the identity p² - q² = (p + q)(p - q). Step 2: Let p = (2a + 3b) and q = (2a - 3b). Step 3: p + q = (2a + 3b) + (2a - 3b) = 4a. Step 4: p - q = (2a + 3b) - (2a - 3b) = 6b. Step 5: Therefore (2a + 3b)² - (2a - 3b)² = 4a × 6b = 24ab.

Question 18

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: We know x² + y² = 34 and xy = 15. Step 3: Substitute: (x + y)² = 34 + 2(15) = 34 + 30 = 64.

Question 19

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: Calculate a² + ab + b² = 17 + 4 = 21. Step 5: Therefore a³ - b³ = 3 × 21 = 63.

Question 20

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: Substitute a + b + c = 9 and ab + bc + ca = 26. Step 3: (9)² = a² + b² + c² + 2(26). Step 4: 81 = a² + b² + c² + 52. Step 5: a² + b² + c² = 81 - 52 = 29.

Question 21

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

Accepted Answer

Step 1: Use the identity (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x). Step 2: Substitute x + 1/x = 5 into the identity: 5³ = x³ + 1/x³ + 3(5). Step 3: 125 = x³ + 1/x³ + 15. Step 4: x³ + 1/x³ = 125 - 15 = 110.

Question 22

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

Accepted Answer

Step 1: From (x + y)³ = 216, we get x + y = 6. From (x − y)³ = 64, we get x − y = 4. Step 2: Adding: 2x = 10, so x = 5. Subtracting: 2y = 2, so y = 1. Step 3: Therefore, xy = 5 × 1 = 5.

Question 23

If (a + b + c)² = 81 and a² + b² + c² = 35, then find the value of (a + b − c)² + (b + c − a)² + (c + a − b)².

Accepted Answer

Step 1: From (a + b + c)² = 81, we have a + b + c = ±9. Using (a + b + c)² = a² + b² + c² + 2(ab + bc + ca): 81 = 35 + 2(ab + bc + ca) → ab + bc + ca = 23. Step 2: Let S₁ = (a + b − c)² + (b + c − a)² + (c + a − b)². Expand each term: (a + b − c)² = a² + b² + c² + 2ab − 2ac − 2bc. Similarly for the other two. Step 3: Sum: S₁ = 3(a² + b² + c²) − 2(ab + bc + ca) = 3(35) − 2(23) = 105 − 46 = 59.

Question 24

If a + b + c = 12 and a² + b² + c² = 56, 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: (12)² = 56 + 2(ab + bc + ca), so 144 = 56 + 2(ab + bc + ca). Step 3: 2(ab + bc + ca) = 88, therefore ab + bc + ca = 44.

Question 25

If (a + b + c)² = 81 and a² + b² + c² = 35, then find the value of (a + b + c)² − 3(ab + bc + ca).

Accepted Answer

Step 1: From (a + b + c)² = 81, we have a + b + c = ±9. Step 2: Expand (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), so 81 = 35 + 2(ab + bc + ca), giving ab + bc + ca = 23. Step 3: Compute (a + b + c)² − 3(ab + bc + ca) = 81 − 3(23) = 81 − 69 = 12.

Question 26

If x² + y² = 13 and xy = 6, then find the value of (x + y)² − (x − y)².

Accepted Answer

Step 1: Expand (x + y)² − (x − y)² using the difference of squares: (x + y)² − (x − y)² = [(x + y) + (x − y)][(x + y) − (x − y)] = (2x)(2y) = 4xy. Step 2: Substitute xy = 6: 4 × 6 = 24.

Question 27

If a³ + b³ = 728 and a + b = 8, then find the value of ab.

Accepted Answer

Step 1: Use the identity a³ + b³ = (a + b)³ − 3ab(a + b). Step 2: Substitute: 728 = 8³ − 3ab(8), so 728 = 512 − 24ab. Step 3: 24ab = 512 − 728 = −216, which gives ab = −9. Step 4: Verify: (a + b)³ = 512, and 512 − 3(−9)(8) = 512 + 216 = 728. ✓

Question 28

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

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: Square again: (x² + 1/x²)² = 529, which gives x⁴ + 2 + 1/x⁴ = 529, so x⁴ + 1/x⁴ = 527.

Question 29

If (a + b + c)² = 81 and a² + b² + c² = 29, then find the value of (ab + bc + ca).

Accepted Answer

Step 1: Use the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 2: Substitute the given values: 81 = 29 + 2(ab + bc + ca). Step 3: Rearrange: 2(ab + bc + ca) = 81 - 29 = 52. Step 4: Therefore, ab + bc + ca = 52 ÷ 2 = 26.

Question 30

If x + 1/x = 5, then find the value of x⁵ + 1/x⁵.

Accepted Answer

Step 1: x + 1/x = 5. Step 2: x² + 1/x² = (x + 1/x)² − 2 = 25 − 2 = 23. Step 3: x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x) = 125 − 15 = 110. Step 4: x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) − (x + 1/x) = 23 × 110 − 5 = 2530 − 5 = 2525. Therefore, answer is 2525.

Question 31

If a + b + c = 12 and a² + b² + c² = 50, 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: 12² = 50 + 2(ab + bc + ca), so 144 = 50 + 2(ab + bc + ca). Step 3: 2(ab + bc + ca) = 94, therefore ab + bc + ca = 47.

Question 32

If p + q = 8 and p² + q² = 40, find the value of pq.

Accepted Answer

Step 1: Use the identity (p + q)² = p² + 2pq + q². Step 2: Rearrange: 2pq = (p + q)² - (p² + q²). Step 3: Substitute: 2pq = (8)² - 40 = 64 - 40 = 24. Step 4: pq = 24 ÷ 2 = 12.

Question 33

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 34

If a - b = 5 and ab = 24, 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 = 5 and ab = 24: a² + b² = (5)² + 2(24) = 25 + 48 = 73.

Question 35

If a + b = 12 and ab = 35, find the value of a² + b².

Accepted Answer

Step 1: Recall the algebraic identity (a + b)² = a² + 2ab + b²
Step 2: Rearrange to find a² + b² = (a + b)² - 2ab
Step 3: Substitute the given values: a² + b² = (12)² - 2(35)
Step 4: Calculate: a² + b² = 144 - 70 = 74
Therefore, the answer is 74.

Question 36

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

Accepted Answer

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

Question 37

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

Accepted Answer

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

Question 38

Expand: (2x + 3y)²

Accepted Answer

Step 1: Use the identity (p + q)² = p² + 2pq + q². Step 2: Here p = 2x and q = 3y. Step 3: (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y².

Question 39

If a - b = 5 and ab = 24, then 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 values: a² + b² = (5)² + 2(24) = 25 + 48 = 73.

Question 40

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

Accepted Answer

Step 1: Recognize this is in the form (p + q)(p - q) where p = 2x and q = 3y. Step 2: Apply the difference of squares identity: (p + q)(p - q) = p² - q². Step 3: Calculate: (2x)² - (3y)² = 4x² - 9y².

Question 41

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 values: (12)² = x² + 2(35) + y². Step 3: 144 = x² + y² + 70. Step 4: x² + y² = 144 - 70 = 74.

Question 42

If a - b = 7 and ab = 18, then find the value of a² + b².

Accepted Answer

Step 1: We use the identity a² + b² = (a - b)² + 2ab. Step 2: Substitute a - b = 7 and ab = 18: a² + b² = (7)² + 2(18). Step 3: a² + b² = 49 + 36 = 85.

Question 43

If m + n = 7 and m² + n² = 29, then find the value of m³ + n³.

Accepted Answer

Step 1: Find mn using (m + n)² = m² + 2mn + n². Step 2: 7² = 29 + 2mn, so 49 = 29 + 2mn, thus mn = 10. Step 3: Use the identity m³ + n³ = (m + n)³ - 3mn(m + n). Step 4: m³ + n³ = (7)³ - 3(10)(7) = 343 - 210 = 133.

Question 44

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

Accepted Answer

Step 1: Expand (x + y)² = x² + 2xy + y² and (x - y)² = x² - 2xy + y². Step 2: (x + y)² - (x - y)² = (x² + 2xy + y²) - (x² - 2xy + y²). Step 3: = x² + 2xy + y² - x² + 2xy - y² = 4xy. Step 4: Substitute xy = 15: (x + y)² - (x - y)² = 4(15) = 60.

Question 45

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

Accepted Answer

Step 1: Use the identity (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x). Step 2: Substitute x + 1/x = 5 into the identity: 5³ = x³ + 1/x³ + 3(5). Step 3: 125 = x³ + 1/x³ + 15. Step 4: x³ + 1/x³ = 125 - 15 = 110.

Question 46

Simplify: (p + q + r)² - (p² + q² + r²)

Accepted Answer

Step 1: Expand (p + q + r)² = p² + q² + r² + 2pq + 2qr + 2rp. Step 2: Subtract (p² + q² + r²): (p² + q² + r² + 2pq + 2qr + 2rp) - (p² + q² + r²) = 2pq + 2qr + 2rp. Step 3: Factor: 2(pq + qr + rp).

Question 47

If x - y = 4 and xy = 5, then find 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 = 4 and xy = 5: x² + y² = (4)² + 2(5) = 16 + 10 = 26.

Question 48

Simplify: (a + b)³ - (a - b)³

Accepted Answer

Step 1: Use the identity (p)³ - (q)³ = (p - q)(p² + pq + q²). Step 2: Let p = (a + b) and q = (a - b). Step 3: p - q = (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 49

Simplify: (p + q)³ - (p - q)³.

Accepted Answer

Step 1: Use the identity a³ - b³ = (a - b)(a² + ab + b²). Step 2: Let a = (p + q) and b = (p - q). Step 3: a - b = (p + q) - (p - q) = 2q. Step 4: Expand (p + q)³ - (p - q)³ = [(p + q) - (p - q)][(p + q)² + (p + q)(p - q) + (p - q)²]. Step 5: Simplify: = 2q[(p² + 2pq + q²) + (p² - q²) + (p² - 2pq + q²)]. Step 6: = 2q[3p² + q²]. Step 7: = 6p²q + 2q³ = 2q(3p² + q²).

Question 50

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: We know a² + b² = 13 and ab = 6. Step 3: Substitute: (a + b)² = 13 + 2(6) = 13 + 12 = 25.

Question 51

If a + b = 8 and a² − b² = 32, then find the value of a.

Accepted Answer

Step 1: Use the identity a² − b² = (a + b)(a − b). Step 2: Substitute: 32 = 8(a − b), so a − b = 4. Step 3: Now solve the system: a + b = 8 and a − b = 4. Step 4: Add the equations: 2a = 12, so a = 6.

CDS Algebraic Identities

CDS Algebraic Identities — Past Exam Questions

Algebraic Identities — Practice Questions