Question 1

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: x² + 1/x² = 5² - 2 = 25 - 2 = 23.

Question 2

Expand (2x + 3y)² using the algebraic identity.

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Here, a = 2x and b = 3y. Step 3: Apply: (2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y².

Question 3

Simplify: (p + q)² + (p - q)² using algebraic identities.

Accepted Answer

Step 1: Expand (p + q)² = p² + 2pq + q². Step 2: Expand (p - q)² = p² - 2pq + q². Step 3: Add them: (p + q)² + (p - q)² = p² + 2pq + q² + p² - 2pq + q² = 2p² + 2q² = 2(p² + q²).

Question 4

If (a + b)² = 49 and ab = 12, find the value of a² + b².

Accepted Answer

Step 1: Expand (a + b)² using the identity (a + b)² = a² + 2ab + b². Step 2: Substitute the given values: 49 = a² + 2(12) + b². Step 3: Simplify: 49 = a² + 24 + b². Step 4: Rearrange: a² + b² = 49 - 24 = 25.

Question 5

If a - b = 7 and ab = 18, 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 = 7² + 2(18) = 49 + 36 = 85.

Question 6

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 7

If a² + b² = 34 and ab = 15, find (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 the given values: (a + b)² = 34 + 2(15) = 34 + 30 = 64.

Question 8

If (a + b + c) = 12 and (ab + bc + ca) = 47, find 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 the given values: a² + b² + c² = 12² - 2(47) = 144 - 94 = 50.

Question 9

Simplify: (2x + 3y)² - (2x - 3y)²

Accepted Answer

Step 1: Use the identity a² - b² = (a + b)(a - b). Here a = (2x + 3y) and b = (2x - 3y). Step 2: (a + b) = (2x + 3y) + (2x - 3y) = 4x. Step 3: (a - b) = (2x + 3y) - (2x - 3y) = 6y. Step 4: (2x + 3y)² - (2x - 3y)² = 4x × 6y = 24xy.

Question 10

If (a + b)² = 81 and (a - b)² = 25, 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 = 25. Step 2: Subtract the second equation from the first: (a² + 2ab + b²) - (a² - 2ab + b²) = 81 - 25, which gives 4ab = 56. Step 3: Therefore, ab = 14.

Question 11

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

Question 12

If p - q = 7 and p² - q² = 147, find the value of p + q.

Accepted Answer

Step 1: Use the identity p² - q² = (p + q)(p - q). Step 2: Substitute the known values: 147 = (p + q)(7). Step 3: Solve for p + q: p + q = 147 ÷ 7 = 21.

Question 13

If m + n = 12 and m² + n² = 90, find the value of (m - n)².

Accepted Answer

Step 1: Use the identity (m + n)² = m² + 2mn + n². Step 2: Substitute: 12² = 90 + 2mn, so 144 = 90 + 2mn, giving 2mn = 54 and mn = 27. Step 3: Use the identity (m - n)² = m² - 2mn + n² = (m² + n²) - 2mn. Step 4: (m - n)² = 90 - 54 = 36.

Question 14

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 15

If a² + b² + c² = 50 and ab + bc + ca = 25, find the value of (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 16

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

Accepted Answer

Step 1: Use the identity A² - B² = (A + B)(A - B), where 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: (2x + 3y)² - (2x - 3y)² = 4x × 6y = 24xy. Step 4: Therefore, k = 24.

SSC MTS Algebraic Identities

SSC MTS Algebraic Identities — Past Exam Questions

Algebraic Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Algebraic Identities