Question 1

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 2

If (a + b)² = 81 and ab = 18, 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: 81 = a² + 2(18) + b². Step 3: Simplify: 81 = a² + 36 + b². Step 4: Therefore, a² + b² = 81 - 36 = 45.

Question 3

If a - b = 7 and ab = 12, find the value of (a - b)² + 4ab.

Accepted Answer

Step 1: We know (a - b)² = 49 (since a - b = 7). Step 2: We are given ab = 12. Step 3: Calculate (a - b)² + 4ab = 49 + 4(12) = 49 + 48 = 97.

Question 4

Expand (2x + 3y)² and find the coefficient of xy.

Accepted Answer

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

Question 5

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

Accepted Answer

Step 1: Square both sides of x + 1/x = 5. Step 2: (x + 1/x)² = 25. Step 3: Expand using (a + b)² = a² + 2ab + b²: x² + 2(x)(1/x) + 1/x² = 25. Step 4: Simplify: x² + 2 + 1/x² = 25. Step 5: Therefore, x² + 1/x² = 25 - 2 = 23.

Question 6

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 7

If (m + n)² = 100 and (m - n)² = 36, find the value of m² + n².

Accepted Answer

Step 1: Expand both identities: (m + n)² = m² + 2mn + n² = 100 and (m - n)² = m² - 2mn + n² = 36. Step 2: Add both equations: 2m² + 2n² = 136, so m² + n² = 68.

Question 8

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

Question 9

If x + 1/x = 5, 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 10

If a² + b² = 34 and ab = 15, find the value of (a + b)² - (a - b)².

Accepted Answer

Step 1: Expand (a + b)² - (a - b)² = (a² + 2ab + b²) - (a² - 2ab + b²) = 4ab. Step 2: Substitute ab = 15: (a + b)² - (a - b)² = 4 × 15 = 60.

Question 11

If p + q = 12 and p² + q² = 90, find the value of p³ + q³.

Accepted Answer

Step 1: Find pq using (p + q)² = p² + 2pq + q². So 144 = 90 + 2pq, giving pq = 27. Step 2: Use the identity p³ + q³ = (p + q)³ - 3pq(p + q). Step 3: p³ + q³ = 12³ - 3(27)(12) = 1728 - 972 = 756.

Question 12

If x - 1/x = 4, 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 = 4: 16 = x² - 2 + 1/x². Step 3: Therefore, x² + 1/x² = 18.

Question 13

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). Let 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: Therefore (2x + 3y)² − (2x − 3y)² = 4x · 6y = 24xy. Step 5: So k = 24.

Question 14

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: Simplify: 2m² + 2n² = 208. Step 4: m² + n² = 104.

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: (a + b)² − (a − b)² = 81 − 49 = 32. Step 3: Simplify left side: (a² + 2ab + b²) − (a² − 2ab + b²) = 4ab = 32. Step 4: Therefore ab = 8.

Question 16

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 17

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 = (8)³ − 3pq(8). Step 3: 152 = 512 − 24pq. Step 4: 24pq = 512 − 152 = 360. Step 5: pq = 360/24 = 15.

SBI PO Algebraic Identities

SBI PO Algebraic Identities — Past Exam Questions

Algebraic Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Algebraic Identities