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

What is the degree of the polynomial r(x) = 4x⁵ - 3x³ + 2x - 7?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial r(x) = 4x⁵ - 3x³ + 2x - 7. Step 2: The highest power is 5 (from the term 4x⁵). Step 3: Therefore, the degree of the polynomial is 5.

Question 3

The product of the roots of the equation 3x² + 6x - 9 = 0 is:

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: Here, a = 3, b = 6, c = -9. Step 3: Product of roots = -9/3 = -3. Therefore, the answer is -3.

Question 4

If q(x) = x³ - 6x² + 11x - 6, find q(3).

Accepted Answer

Step 1: Substitute x = 3 into q(x) = x³ - 6x² + 11x - 6. Step 2: q(3) = (3)³ - 6(3)² + 11(3) - 6 = 27 - 6(9) + 33 - 6 = 27 - 54 + 33 - 6 = 0.

Question 5

If t(x) = 2x³ + 3x² - 5x + 4, what is the coefficient of x² in t(x)?

Accepted Answer

Step 1: Examine the polynomial t(x) = 2x³ + 3x² - 5x + 4. Step 2: The term containing x² is 3x². Step 3: The coefficient of x² is 3.

Question 6

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 7

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 8

If the roots of the quadratic equation 2x² - 8x + k = 0 are equal, find the value of k.

Accepted Answer

Step 1: For equal roots, the discriminant Δ = b² - 4ac must equal 0. Step 2: Here, a = 2, b = -8, c = k. Step 3: (-8)² - 4(2)(k) = 0. Step 4: 64 - 8k = 0. Step 5: k = 8. Therefore, the answer is 8.

Question 9

The quadratic equation whose roots are 4 and -2 is:

Accepted Answer

Step 1: If roots are α = 4 and β = -2, the quadratic equation is (x - α)(x - β) = 0. Step 2: (x - 4)(x - (-2)) = 0. Step 3: (x - 4)(x + 2) = 0. Step 4: Expand: x² + 2x - 4x - 8 = 0. Step 5: x² - 2x - 8 = 0. Therefore, the equation is x² - 2x - 8 = 0.

Question 10

If 2x + 3 = x + 9, then x equals:

Accepted Answer

Step 1: Subtract x from both sides: x + 3 = 9. Step 2: Subtract 3 from both sides: x = 6. Therefore, x = 6.

Question 11

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 12

If 5x + 12 = 37, what is the value of x?

Accepted Answer

Step 1: Subtract 12 from both sides: 5x = 37 - 12 = 25. Step 2: Divide both sides by 5: x = 25 ÷ 5 = 5. Therefore, x = 5.

Question 13

Solve for y: 3(y - 4) = 18

Accepted Answer

Step 1: Divide both sides by 3: y - 4 = 6. Step 2: Add 4 to both sides: y = 10. Therefore, y = 10.

Question 14

A number when multiplied by 4 and then 5 is subtracted from the result gives 35. Find the number.

Accepted Answer

Step 1: Let the number be x. Form the equation: 4x - 5 = 35. Step 2: Add 5 to both sides: 4x = 40. Step 3: Divide by 4: x = 10. Therefore, the number is 10.

Question 15

If 6x - 8 = 4x + 12, what is the value of x?

Accepted Answer

Step 1: Subtract 4x from both sides: 2x - 8 = 12. Step 2: Add 8 to both sides: 2x = 20. Step 3: Divide by 2: x = 10. Therefore, x = 10.

Question 16

The sum of two consecutive integers is 47. Find the larger integer.

Accepted Answer

Step 1: Let the smaller integer be x, then the larger is x + 1. Step 2: Form equation: x + (x + 1) = 47. Step 3: Simplify: 2x + 1 = 47. Step 4: Subtract 1: 2x = 46. Step 5: Divide by 2: x = 23. Step 6: The larger integer is 23 + 1 = 24. Therefore, the answer is 24.

Question 17

If s(x) = x² - 7x + 12, find the sum of the roots of s(x).

Accepted Answer

Step 1: For a quadratic polynomial ax² + bx + c, the sum of roots = -b/a. Step 2: In s(x) = x² - 7x + 12, we have a = 1 and b = -7. Step 3: Sum of roots = -(-7)/1 = 7.

Question 18

If u(x) = x² - 4x + 3, find the product of the roots of u(x).

Accepted Answer

Step 1: For a quadratic polynomial ax² + bx + c, the product of roots = c/a. Step 2: In u(x) = x² - 4x + 3, we have a = 1 and c = 3. Step 3: Product of roots = 3/1 = 3.

Question 19

If one root of the equation x² - 4x + p = 0 is 3, find the value of p and the other root.

Accepted Answer

Step 1: Since x = 3 is a root, substitute into the equation: 3² - 4(3) + p = 0. Step 2: 9 - 12 + p = 0. Step 3: p = 3. Step 4: Using Vieta's formula, sum of roots = 4. If one root is 3, the other root = 4 - 3 = 1. Therefore, p = 3 and the other root is 1.

Question 20

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.

IBPS RRB PO Algebra Questions

IBPS RRB PO Algebra — Past Exam Questions

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