Question 1

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 2

If r(x) = 3x² - 12, find r(-2).

Accepted Answer

Step 1: Substitute x = -2 into r(x) = 3x² - 12. Step 2: r(-2) = 3(-2)² - 12 = 3(4) - 12 = 12 - 12 = 0.

Question 3

Solve for y: 4y + 8 = 2y + 20

Accepted Answer

Step 1: Start with 4y + 8 = 2y + 20. Step 2: Subtract 2y from both sides: 4y - 2y + 8 = 20, which gives 2y + 8 = 20. Step 3: Subtract 8 from both sides: 2y = 20 - 8 = 12. Step 4: Divide both sides by 2: y = 12 ÷ 2 = 6. Therefore, y = 6.

Question 4

Solve: 7x + 5 = 3x + 25

Accepted Answer

Step 1: Start with 7x + 5 = 3x + 25. Step 2: Subtract 3x from both sides: 7x - 3x + 5 = 25, which gives 4x + 5 = 25. Step 3: Subtract 5 from both sides: 4x = 25 - 5 = 20. Step 4: Divide both sides by 4: x = 20 ÷ 4 = 5. Therefore, x = 5.

Question 5

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 6

Simplify: (x + 3)(x - 2) and identify the coefficient of x in the resulting polynomial.

Accepted Answer

Step 1: Expand (x + 3)(x - 2) using FOIL. Step 2: = x² - 2x + 3x - 6 = x² + x - 6. Step 3: The coefficient of x is 1.

Question 7

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 8

If 2x - 5 = x + 3, what is the value of x?

Accepted Answer

Step 1: Start with 2x - 5 = x + 3. Step 2: Subtract x from both sides: 2x - x - 5 = 3, which gives x - 5 = 3. Step 3: Add 5 to both sides: x = 3 + 5 = 8. Therefore, x = 8.

Question 9

What is the degree of the polynomial 4x⁵ + 3x³ + 2x + 7?

Accepted Answer

Step 1: The degree of a polynomial is the highest power of the variable x. Step 2: In the polynomial 4x⁵ + 3x³ + 2x + 7, the highest power is 5. Step 3: Therefore, the degree is 5.

Question 10

If 6x - 3 = 2x + 13, find x.

Accepted Answer

Step 1: Start with 6x - 3 = 2x + 13. Step 2: Subtract 2x from both sides: 6x - 2x - 3 = 13, which gives 4x - 3 = 13. Step 3: Add 3 to both sides: 4x = 13 + 3 = 16. Step 4: Divide both sides by 4: x = 16 ÷ 4 = 4. Therefore, x = 4.

Question 11

Simplify: (p + q + r)² - (p + q - r)²

Accepted Answer

Step 1: Use the identity a² - b² = (a + b)(a - b). Step 2: Let a = (p + q + r) and b = (p + q - r). Step 3: a + b = (p + q + r) + (p + q - r) = 2p + 2q. Step 4: a - b = (p + q + r) - (p + q - r) = 2r. Step 5: (p + q + r)² - (p + q - r)² = (2p + 2q)(2r) = 4r(p + q).

Question 12

If one root of the equation x² - 6x + k = 0 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: -8 + k = 0, so k = 8. Therefore, the answer is 8.

Question 13

The quadratic equation x² + 4x + 4 = 0 has roots. Which statement is true?

Accepted Answer

Step 1: Factor the equation: x² + 4x + 4 = (x + 2)² = 0. Step 2: This gives x + 2 = 0, so x = -2 (a repeated root). Step 3: The equation has equal roots (both roots are -2). Therefore, the answer is that it has equal roots or discriminant = 0.

Question 14

Simplify: (x + 5)² - (x - 5)²

Accepted Answer

Step 1: Use the identity (a + b)² - (a - b)² = 4ab. Step 2: Here a = x and b = 5. Step 3: (x + 5)² - (x - 5)² = 4 × x × 5 = 20x.

Question 15

If 5x + 12 = 37, find the value of x.

Accepted Answer

Step 1: Start with 5x + 12 = 37. Step 2: Subtract 12 from both sides: 5x = 37 - 12 = 25. Step 3: Divide both sides by 5: x = 25 ÷ 5 = 5. Therefore, x = 5.

Question 16

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: 81 = a² + 2(18) + b². Step 3: 81 = a² + 36 + b². Step 4: a² + b² = 81 - 36 = 45.

Question 17

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 18

Solve: 3(x - 4) = 15

Accepted Answer

Step 1: Expand the left side: 3x - 12 = 15. Step 2: Add 12 to both sides: 3x = 15 + 12 = 27. Step 3: Divide both sides by 3: x = 27 ÷ 3 = 9. Therefore, x = 9.

Question 19

If q(x) = x² - 6x + 8, find the value of q(4).

Accepted Answer

Step 1: Substitute x = 4 into q(x) = x² - 6x + 8. Step 2: q(4) = (4)² - 6(4) + 8 = 16 - 24 + 8 = 0.

Question 20

What is the sum of the coefficients of the polynomial s(x) = 2x³ + 5x² - 3x + 4?

Accepted Answer

Step 1: The sum of coefficients of any polynomial is found by substituting x = 1. Step 2: s(1) = 2(1)³ + 5(1)² - 3(1) + 4 = 2 + 5 - 3 + 4 = 8.

SSC GD Constable Algebra Questions

SSC GD Constable Algebra — Past Exam Questions

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