Question 1

What is the degree of the polynomial q(x) = 7x⁴ − 3x² + 2x + 9?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial q(x) = 7x⁴ − 3x² + 2x + 9. Step 2: The highest power is x⁴, so the degree is 4. Therefore, the answer is 4.

Question 2

Solve: 2x + 3x = 35

Accepted Answer

Step 1: Combine like terms on the left side: 2x + 3x = 5x. Step 2: Now the equation is 5x = 35. Step 3: Divide both sides by 5: x = 7. Therefore, x = 7.

Question 3

If p + q + r = 15 and pq + qr + rp = 72, find the value of p² + q² + r².

Accepted Answer

Step 1: Use the identity (p + q + r)² = p² + q² + r² + 2(pq + qr + rp). Step 2: Rearrange: p² + q² + r² = (p + q + r)² - 2(pq + qr + rp). Step 3: Substitute: p² + q² + r² = (15)² - 2(72) = 225 - 144 = 81.

Question 4

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

Accepted Answer

Step 1: We use the algebraic identity (x + 1/x)² = x² + 2(x)(1/x) + 1/x² = x² + 2 + 1/x². Step 2: Substitute x + 1/x = 5 into the identity: (5)² = x² + 2 + 1/x². Step 3: 25 = x² + 2 + 1/x². Step 4: x² + 1/x² = 25 - 2 = 23. Therefore, the answer is 23.

Question 5

A number when multiplied by 6 and then reduced by 8 equals 40. Find the number.

Accepted Answer

Step 1: Let the number be x. According to the problem: 6x - 8 = 40. Step 2: Add 8 to both sides: 6x = 40 + 8 = 48. Step 3: Divide both sides by 6: x = 48 ÷ 6 = 8. Therefore, the number is 8.

Question 6

If 6x - 4 = 2x + 12, find x.

Accepted Answer

Step 1: Start with 6x - 4 = 2x + 12. Step 2: Subtract 2x from both sides: 4x - 4 = 12. Step 3: Add 4 to both sides: 4x = 16. Step 4: Divide by 4: x = 4. Therefore, x = 4.

Question 7

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 8

Simplify: (a + b)² - (a - b)²

Accepted Answer

Step 1: Expand (a + b)² = a² + 2ab + b². Step 2: Expand (a - b)² = a² - 2ab + b². Step 3: Subtract: (a² + 2ab + b²) - (a² - 2ab + b²) = a² + 2ab + b² - a² + 2ab - b² = 4ab.

Question 9

The sum of two numbers is 48 and their difference is 14. What is the larger number?

Accepted Answer

Step 1: Let the two numbers be x and y where x > y. So x + y = 48 and x - y = 14. Step 2: Add both equations: 2x = 62, so x = 31. Step 3: y = 48 - 31 = 17. The larger number is 31. Therefore, the answer is 31.

Question 10

If a - b = 5 and ab = 24, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a - b)² = a² - 2ab + b². Step 2: Rearrange: a² + b² = (a - b)² + 2ab. Step 3: Substitute values: a² + b² = (5)² + 2(24) = 25 + 48 = 73.

Question 11

Simplify: (m + n)³ - (m - n)³

Accepted Answer

Step 1: Use the identity a³ - b³ = (a - b)(a² + ab + b²). Step 2: Let a = (m + n) and b = (m - n). Step 3: a - b = (m + n) - (m - n) = 2n. Step 4: a² = (m + n)² = m² + 2mn + n². Step 5: b² = (m - n)² = m² - 2mn + n². Step 6: ab = (m + n)(m - n) = m² - n². Step 7: a² + ab + b² = (m² + 2mn + n²) + (m² - n²) + (m² - 2mn + n²) = 3m² + n². Step 8: Result = 2n(3m² + n²) = 6m²n + 2n³.

Question 12

Simplify: (a + b)³ - (a - b)³

Accepted Answer

Step 1: Use the identity (x)³ - (y)³ = (x - y)(x² + xy + y²), or expand directly. Step 2: (a + b)³ = a³ + 3a²b + 3ab² + b³. Step 3: (a - b)³ = a³ - 3a²b + 3ab² - b³. Step 4: Subtract: (a³ + 3a²b + 3ab² + b³) - (a³ - 3a²b + 3ab² - b³) = 6a²b + 2b³ = 2b(3a² + b²).

Question 13

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

Accepted Answer

Step 1: Start with 3(y - 4) = 18. Step 2: Divide both sides by 3: y - 4 = 18 ÷ 3 = 6. Step 3: Add 4 to both sides: y = 6 + 4 = 10. Therefore, y = 10.

Question 14

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, x = 8.

Question 15

If x + y = 12 and xy = 35, then find the value of x² + y².

Accepted Answer

Step 1: Use the identity (x + y)² = x² + 2xy + y². Step 2: Rearrange to get x² + y² = (x + y)² - 2xy. Step 3: Substitute values: x² + y² = (12)² - 2(35) = 144 - 70 = 74.

Question 16

If x + y = 10 and xy = 21, find the value of x² + y².

Accepted Answer

Step 1: Use the identity x² + y² = (x + y)² − 2xy. Step 2: Substitute the given values: x² + y² = (10)² − 2(21) = 100 − 42 = 58. Therefore, the answer is 58.

Question 17

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

Accepted Answer

Step 1: Identify the highest power of x in the polynomial. Step 2: The terms are 4x⁵, −3x³, 2x, and −7. Step 3: The highest power is 5. Therefore, the degree is 5.

Question 18

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, y = 7.

Question 19

If the polynomial x² − 5x + m has roots α and β such that α + β = 5 and αβ = 6, find the value of m.

Accepted Answer

Step 1: Use Vieta's formulas for a quadratic x² + bx + c with roots α and β: α + β = −b and αβ = c. Step 2: For x² − 5x + m, we have −b = −(−5) = 5 and c = m. Step 3: Given αβ = 6, we have m = 6. Therefore, m = 6.

Question 20

If p(x) = 2x³ + 5x² - 3x + 4, then find the value of p(2).

Accepted Answer

Step 1: We need to substitute x = 2 into the polynomial p(x) = 2x³ + 5x² - 3x + 4.
Step 2: p(2) = 2(2)³ + 5(2)² - 3(2) + 4
Step 3: p(2) = 2(8) + 5(4) - 6 + 4
Step 4: p(2) = 16 + 20 - 6 + 4
Step 5: p(2) = 34
Therefore, p(2) = 34.

SSC CGL Algebra Questions

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