Question 1

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

Accepted Answer

Step 1: We need to find p(2) by substituting x = 2 into the polynomial p(x) = 2x² + 5x - 3.
Step 2: p(2) = 2(2)² + 5(2) - 3
Step 3: p(2) = 2(4) + 10 - 3
Step 4: p(2) = 8 + 10 - 3
Step 5: p(2) = 15
Therefore, the answer is 15.

Question 2

The sum of two numbers is 48. If one number is twice the other, what is the smaller number?

Accepted Answer

Step 1: Let the smaller number be x. Then the larger number is 2x. Step 2: Form the equation: x + 2x = 48 → 3x = 48. Step 3: x = 48 / 3 = 16. Therefore, the smaller number is 16.

Question 3

If x + y = 12 and xy = 32, then find the value of x² + y².

Accepted Answer

Step 1: We use the algebraic identity (x + y)² = x² + 2xy + y²
Step 2: Rearranging: x² + y² = (x + y)² - 2xy
Step 3: Substitute the given values: x² + y² = (12)² - 2(32)
Step 4: Calculate: x² + y² = 144 - 64 = 80
Therefore, x² + y² = 80

Question 4

Ramesh is currently 4 years older than Priya. If the sum of their ages is 36 years, then in how many years will Ramesh's age be twice Priya's current age?

Accepted Answer

Step 1: Let Priya's current age = x years. Then Ramesh's current age = (x + 4) years. Step 2: Sum of their ages is 36. So, x + (x + 4) = 36. This gives 2x + 4 = 36, therefore 2x = 32, so x = 16. Step 3: Priya's current age = 16 years, Ramesh's current age = 20 years. Step 4: We need to find after how many years (let's call it y) Ramesh's age will be twice Priya's current age. So, 20 + y = 2 × 16. Step 5: 20 + y = 32, therefore y = 12 years.

Question 5

The sum of the digits of a two-digit number is 13. If 27 is subtracted from the number, the digits get reversed. What is the original number?

Accepted Answer

Step 1: Let the tens digit be x and units digit be y. Then x + y = 13 and 10x + y − 27 = 10y + x. Step 2: From the second equation: 9x − 9y = 27 → x − y = 3. Step 3: Solve: x + y = 13 and x − y = 3 → 2x = 16 → x = 8, y = 5. Step 4: Original number = 85. Verify: 8 + 5 = 13 ✓; 85 − 27 = 58 (reversed) ✓. Therefore, answer is 85.

Question 6

If α and β are roots of x² − 5x + 6 = 0, and γ and δ are roots of x² − 7x + 12 = 0, find the value of (α + β)(γ + δ) − αβ − γδ.

Accepted Answer

Step 1: For x² − 5x + 6 = 0, by Vieta's formulas: α + β = 5 and αβ = 6. Step 2: For x² − 7x + 12 = 0, by Vieta's formulas: γ + δ = 7 and γδ = 12. Step 3: Substitute into the expression: (5)(7) − 6 − 12 = 35 − 6 − 12 = 17.

Question 7

If x + 1/x = 5, then find the value of x⁴ + 1/x⁴.

Accepted Answer

Step 1: Use (x + 1/x)² = x² + 2 + 1/x² → 25 = x² + 2 + 1/x² → x² + 1/x² = 23. Step 2: Use (x² + 1/x²)² = x⁴ + 2 + 1/x⁴ → 529 = x⁴ + 2 + 1/x⁴ → x⁴ + 1/x⁴ = 527. Therefore, answer is 527.

Question 8

If α and β are the roots of the equation 2x² − 7x + 3 = 0, find the value of α³ + β³.

Accepted Answer

Step 1: By Vieta's formulas, α + β = 7/2 and αβ = 3/2. Step 2: α² + β² = (α + β)² − 2αβ = (49/4) − 3 = 37/4. Step 3: α³ + β³ = (α + β)(α² − αβ + β²) = (α + β)[(α² + β²) − αβ] = (7/2)[(37/4) − (3/2)] = (7/2)(37/4 − 6/4) = (7/2)(31/4) = 217/8. Therefore, α³ + β³ = 217/8.

Question 9

A polynomial p(x) leaves remainder 3 when divided by (x − 2) and remainder 5 when divided by (x − 3). Find the remainder when p(x) is divided by (x − 2)(x − 3).

Accepted Answer

Step 1: By the Remainder Theorem, p(2) = 3 and p(3) = 5. Step 2: Since the divisor (x − 2)(x − 3) is quadratic, the remainder R(x) must be linear: R(x) = ax + b. Step 3: Write p(x) = (x − 2)(x − 3)·q(x) + ax + b. Step 4: Substitute x = 2: p(2) = 0 + 2a + b = 3, so 2a + b = 3. Step 5: Substitute x = 3: p(3) = 0 + 3a + b = 5, so 3a + b = 5. Step 6: Subtract: a = 2, then b = 3 − 4 = −1. Step 7: Remainder = 2x − 1.

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