Question 1

If p - q = 7 and pq = 12, find the value of p² + q².

Accepted Answer

Step 1: Use the identity (p - q)² = p² - 2pq + q². Step 2: Rearrange to get p² + q² = (p - q)² + 2pq. Step 3: Substitute values: p² + q² = 7² + 2(12) = 49 + 24 = 73.

Question 2

Expand (2m + 3n)² and find the coefficient of mn.

Accepted Answer

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

Question 3

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: Rearrange: a² + b² = 81 - 36 = 45.

Question 4

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 5

Simplify: (3a - 2b)(3a + 2b) using the difference of squares identity.

Accepted Answer

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

Question 6

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 x + 1/x = 5: x² + 1/x² = 5² - 2 = 25 - 2 = 23.

Question 7

If a² + b² = 25 and ab = 12, find the value of (a + b)² - (a - b)².

Accepted Answer

Step 1: Recall that (a + b)² - (a - b)² = (a² + 2ab + b²) - (a² - 2ab + b²) = 4ab. Step 2: Substitute ab = 12: (a + b)² - (a - b)² = 4 × 12 = 48.

Question 8

If (2x + 3y)² - (2x - 3y)² = 144, find the value of xy.

Accepted Answer

Step 1: Use the identity A² - B² = (A + B)(A - B). Let 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: Given that 24xy = 144, so xy = 6.

Question 9

Simplify: (x + 1/x)² - (x - 1/x)² and find its value when x = 2.

Accepted Answer

Step 1: Use the identity A² - B² = (A + B)(A - B). Let A = (x + 1/x) and B = (x - 1/x). Step 2: A + B = (x + 1/x) + (x - 1/x) = 2x and A - B = (x + 1/x) - (x - 1/x) = 2/x. Step 3: Therefore, (x + 1/x)² - (x - 1/x)² = 2x × 2/x = 4. Step 4: The value is 4 regardless of x (when x ≠ 0). When x = 2, the answer is 4.

Question 10

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 equation from the first: (a² + 2ab + b²) - (a² - 2ab + b²) = 81 - 49, which gives 4ab = 32. Step 3: Therefore, ab = 8.

Question 11

Simplify: (a + b + c)² - (a + b - c)² and express in terms of c and (a + b).

Accepted Answer

Step 1: Use the identity A² - B² = (A + B)(A - B). Let A = (a + b + c) and B = (a + b - c). Step 2: A + B = (a + b + c) + (a + b - c) = 2(a + b) and A - B = (a + b + c) - (a + b - c) = 2c. Step 3: Therefore, (a + b + c)² - (a + b - c)² = 2(a + b) × 2c = 4c(a + b).

Question 12

If x + y = 10 and x² + y² = 58, find the value of x³ + y³.

Accepted Answer

Step 1: From x + y = 10, square both sides: (x + y)² = 100, so x² + 2xy + y² = 100. Step 2: Since x² + y² = 58, substitute: 58 + 2xy = 100, giving 2xy = 42, so xy = 21. Step 3: Use the identity x³ + y³ = (x + y)³ - 3xy(x + y). Step 4: x³ + y³ = 10³ - 3(21)(10) = 1000 - 630 = 370.

Question 13

If (p + q)² - (p - q)² = 48, find the value of pq.

Accepted Answer

Step 1: Expand both squares: (p + q)² = p² + 2pq + q² and (p - q)² = p² - 2pq + q². Step 2: Subtract: (p² + 2pq + q²) - (p² - 2pq + q²) = 4pq. Step 3: Therefore, 4pq = 48. Step 4: pq = 12.

Question 14

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 15

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 16

If (x + y + z)² = 144 and x² + y² + z² = 60, find the value of xy + yz + zx.

Accepted Answer

Step 1: Use the identity (x + y + z)² = x² + y² + z² + 2(xy + yz + zx). Step 2: Substitute: 144 = 60 + 2(xy + yz + zx). Step 3: 2(xy + yz + zx) = 144 - 60 = 84. Step 4: xy + yz + zx = 42.

Question 17

If a + b = 12 and a² - b² = 60, find the value of a.

Accepted Answer

Step 1: Use the identity a² - b² = (a + b)(a - b). Step 2: Substitute: 60 = 12(a - b), so a - b = 5. Step 3: Now solve the system: a + b = 12 and a - b = 5. Step 4: Adding both equations: 2a = 17, so a = 8.5.

IBPS RRB PO Algebraic Identities

IBPS RRB PO Algebraic Identities — Past Exam Questions

Algebraic Identities— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Algebraic Identities