Question 1

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 2

If m + n = 15 and m² + n² = 117, find the value of mn.

Accepted Answer

Step 1: Use the identity (m + n)² = m² + 2mn + n². Step 2: Substitute known values: (15)² = 117 + 2mn. Step 3: 225 = 117 + 2mn. Step 4: 2mn = 225 - 117 = 108. Step 5: mn = 54.

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

Simplify: (x + 4)² - (x + 2)²

Accepted Answer

Step 1: Use the difference of squares identity a² - b² = (a + b)(a - b). Step 2: Let a = (x + 4) and b = (x + 2). Step 3: [(x + 4) + (x + 2)][(x + 4) - (x + 2)] = (2x + 6)(2) = 4x + 12.

Question 5

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 6

If p - q = 8 and pq = 20, 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 values: p² + q² = (8)² + 2(20) = 64 + 40 = 104.

Question 7

If x² + 1/x² = 7, then find the value of x + 1/x (assuming x > 0).

Accepted Answer

Step 1: Use the identity (x + 1/x)² = x² + 2 + 1/x². Step 2: Rearrange: (x + 1/x)² = (x² + 1/x²) + 2 = 7 + 2 = 9. Step 3: Take the positive square root (since x > 0): x + 1/x = 3.

Question 8

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

Question 9

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 the given values: 12² = 50 + 2(ab + bc + ca). Step 3: 144 = 50 + 2(ab + bc + ca). Step 4: 2(ab + bc + ca) = 94, so ab + bc + ca = 47.

Question 10

Simplify: (p + q)³ - (p - q)³ and find its value when p = 2 and q = 1.

Accepted Answer

Step 1: Use the identity A³ - B³ = (A - B)(A² + AB + B²). Here A = (p + q) and B = (p - q). Step 2: (A - B) = (p + q) - (p - q) = 2q. Step 3: A² = (p + q)² = p² + 2pq + q², B² = (p - q)² = p² - 2pq + q², AB = (p + q)(p - q) = p² - q². Step 4: A² + AB + B² = (p² + 2pq + q²) + (p² - q²) + (p² - 2pq + q²) = 3p² + q². Step 5: (p + q)³ - (p - q)³ = 2q(3p² + q²). Step 6: Substitute p = 2, q = 1: 2(1)(3·4 + 1) = 2(13) = 26.

Question 11

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

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 and (A - B) = (2x + 3y) - (2x - 3y) = 6y. Step 3: (2x + 3y)² - (2x - 3y)² = 4x × 6y = 24xy. Step 4: Therefore, k = 24.

Question 12

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 13

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. Step 3: Therefore (a + b + c)² = 100.

Question 14

If p³ + q³ = 152 and p + q = 8, find the value of pq.

Accepted Answer

Step 1: Use the identity p³ + q³ = (p + q)³ - 3pq(p + q). Step 2: Substitute known values: 152 = 8³ - 3pq(8). Step 3: 152 = 512 - 24pq. Step 4: 24pq = 512 - 152 = 360. Step 5: pq = 360/24 = 15.

Question 15

If (m + n)² = 144 and (m - n)² = 64, find the value of m² + n².

Accepted Answer

Step 1: Expand both identities: (m + n)² = m² + 2mn + n² = 144 and (m - n)² = m² - 2mn + n² = 64. Step 2: Add both equations: (m² + 2mn + n²) + (m² - 2mn + n²) = 144 + 64 = 208. Step 3: 2m² + 2n² = 208. Step 4: m² + n² = 104.

Question 16

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

Accepted Answer

Step 1: Square the given equation: (x + 1/x)² = 25, which expands to 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 17

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), 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: Therefore (2x + 3y)² - (2x - 3y)² = 4x · 6y = 24xy. Step 4: Comparing with k·xy, we get k = 24.

SBI Clerk Algebraic Identities

SBI Clerk Algebraic Identities — Past Exam Questions

Algebraic Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Algebraic Identities