Question 1

Simplify: (a + b + c)² - (a² + b² + c²)

Accepted Answer

Step 1: Expand (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. Step 2: Subtract (a² + b² + c²): (a² + b² + c² + 2ab + 2bc + 2ca) - (a² + b² + c²) = 2ab + 2bc + 2ca. Step 3: Factor: 2(ab + bc + ca).

Question 2

If 4(x - 3) = 20, what is the value of x?

Accepted Answer

Step 1: Start with 4(x - 3) = 20. Step 2: Divide both sides by 4: x - 3 = 5. Step 3: Add 3 to both sides: x = 8. Therefore, the answer is 8.

Question 3

Expand: (2m + 3n)(2m - 3n)

Accepted Answer

Step 1: Use the difference of squares identity (a + b)(a - b) = a² - b². Step 2: Here a = 2m and b = 3n. Step 3: Apply: (2m + 3n)(2m - 3n) = (2m)² - (3n)² = 4m² - 9n².

Question 4

If x + 1/x = 5, find x² + 1/x².

Accepted Answer

Step 1: Use the identity (x + 1/x)² = x² + 2(x)(1/x) + 1/x² = 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 5

If p - q = 7 and pq = 12, find 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 (p - q) = 7 and pq = 12: p² + q² = 7² + 2(12) = 49 + 24 = 73.

Question 6

If 3x + 7 = 22, find the value of x.

Accepted Answer

Step 1: Start with 3x + 7 = 22. Step 2: Subtract 7 from both sides: 3x = 15. Step 3: Divide both sides by 3: x = 5. Therefore, the answer is 5.

Question 7

If the discriminant of the equation 3x² - 6x + m = 0 is 0, find the value of m.

Accepted Answer

Step 1: For a quadratic ax² + bx + c = 0, discriminant Δ = b² - 4ac. Step 2: Here a = 3, b = -6, c = m. Step 3: Δ = (-6)² - 4(3)(m) = 0. Step 4: 36 - 12m = 0. Step 5: 12m = 36. Step 6: m = 3. Therefore, m = 3.

Question 8

If (x + 3) is a factor of the polynomial x² + 7x + 12, what is the other linear factor?

Accepted Answer

Step 1: Factor the polynomial x² + 7x + 12. Step 2: Find two numbers that multiply to 12 and add to 7: these are 3 and 4. Step 3: x² + 7x + 12 = (x + 3)(x + 4). Step 4: Since (x + 3) is given as one factor, the other factor is (x + 4).

Question 9

If the polynomial p(x) = x² - 5x + 6 has roots α and β, what is the value of α + β?

Accepted Answer

Step 1: For a quadratic polynomial ax² + bx + c, the sum of roots = -b/a. Step 2: In p(x) = x² - 5x + 6, we have a = 1, b = -5, c = 6. Step 3: Sum of roots = -(-5)/1 = 5.

Question 10

What is the remainder when p(x) = x³ + 2x² - 5x + 3 is divided by (x - 1)?

Accepted Answer

Step 1: Use the Remainder Theorem: when p(x) is divided by (x - a), the remainder is p(a). Step 2: Here, we divide by (x - 1), so a = 1. Step 3: Calculate p(1) = (1)³ + 2(1)² - 5(1) + 3 = 1 + 2 - 5 + 3 = 1.

Question 11

If x² - 7x + 12 = 0, find the sum of the roots of the equation.

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the sum of roots = -b/a. Step 2: Here a = 1, b = -7, c = 12. Step 3: Sum of roots = -(-7)/1 = 7. Therefore, the answer is 7.

Question 12

What is the product of the roots of the equation 2x² - 8x + 6 = 0?

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: Here a = 2, b = -8, c = 6. Step 3: Product of roots = 6/2 = 3. Therefore, the answer is 3.

Question 13

If the roots of a quadratic equation are 4 and 5, what is the equation?

Accepted Answer

Step 1: If roots are α = 4 and β = 5, the quadratic equation is (x - α)(x - β) = 0. Step 2: (x - 4)(x - 5) = 0. Step 3: Expand: x² - 5x - 4x + 20 = 0. Step 4: x² - 9x + 20 = 0. Therefore, the equation is x² - 9x + 20 = 0.

Question 14

For the equation x² + 6x + k = 0, if one root is 2, find the value of k.

Accepted Answer

Step 1: If x = 2 is a root, it must satisfy the equation. Step 2: Substitute x = 2: (2)² + 6(2) + k = 0. Step 3: 4 + 12 + k = 0. Step 4: 16 + k = 0. Step 5: k = -16. Therefore, k = -16.

Question 15

What is the degree of the polynomial 4x³ + 2x² - 7x + 5?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial. Step 2: The term 4x³ has the highest power, which is 3. Step 3: Therefore, the degree is 3.

Question 16

Simplify: (2x + 3)(2x - 3)

Accepted Answer

Step 1: Recognize this as the difference of squares pattern: (a + b)(a - b) = a² - b². Step 2: Here, a = 2x and b = 3. Step 3: (2x + 3)(2x - 3) = (2x)² - 3² = 4x² - 9.

Question 17

Solve for y: 5y - 12 = 2y + 9

Accepted Answer

Step 1: Start with 5y - 12 = 2y + 9. Step 2: Subtract 2y from both sides: 3y - 12 = 9. Step 3: Add 12 to both sides: 3y = 21. Step 4: Divide by 3: y = 7. Therefore, the answer is 7.

Question 18

Simplify: (x + 5)² - (x - 5)²

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². Step 2: Expand (x + 5)² = x² + 10x + 25. Step 3: Expand (x - 5)² = x² - 10x + 25. Step 4: Subtract: (x² + 10x + 25) - (x² - 10x + 25) = x² + 10x + 25 - x² + 10x - 25 = 20x.

Question 19

If (a + b)² = 81 and ab = 18, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Substitute (a + b)² = 81 and ab = 18 into the identity: 81 = a² + 2(18) + b². Step 3: Simplify: 81 = a² + 36 + b². Step 4: Therefore, a² + b² = 81 - 36 = 45.

Question 20

A number is such that when 8 is subtracted from it, the result is 24. Find the number.

Accepted Answer

Step 1: Let the number be x. Step 2: According to the problem, x - 8 = 24. Step 3: Add 8 to both sides: x = 32. Therefore, the number is 32.

IBPS Clerk Algebra Questions

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