Tahsili Mathematics — Questions and Answers

Question 1: What is the value of lim(x→0) [sin(x)/x]?

0
1 (Correct answer)
∞
Does not exist

Correct answer: 1

This is a fundamental limit in calculus. As x approaches 0, sin(x)/x approaches 1. This can be proven using the squeeze theorem or L'Hôpital's rule. This limit is essential for deriving the derivative of sin(x).

Question 2: If f(x) = 3x² - 2x + 1, what is f'(x)?

6x - 2 (Correct answer)
3x - 2
6x² - 2
6x + 1

Correct answer: 6x - 2

Using the power rule: d/dx(3x²) = 6x, d/dx(-2x) = -2, d/dx(1) = 0. Therefore f'(x) = 6x - 2. The power rule states d/dx(xⁿ) = nxⁿ⁻¹.

Question 3: What is the solution set of the equation |2x - 3| = 7?

x = 5 only
x = -2 only
x = 5 or x = -2 (Correct answer)
x = 2 or x = -5

Correct answer: x = 5 or x = -2

|2x - 3| = 7 means 2x - 3 = 7 or 2x - 3 = -7. Case 1: 2x = 10, x = 5. Case 2: 2x = -4, x = -2. Both solutions are valid: |2(5)-3| = |7| = 7 ✓ and |2(-2)-3| = |-7| = 7 ✓.

Question 4: In a right triangle, if sin(θ) = 3/5, what is cos(θ)?

4/5 (Correct answer)
3/4
5/3
5/4

Correct answer: 4/5

Using the Pythagorean identity: sin²(θ) + cos²(θ) = 1. So cos²(θ) = 1 - (3/5)² = 1 - 9/25 = 16/25. Therefore cos(θ) = 4/5 (positive in the first quadrant).

Question 5: What is the sum of the first 20 terms of the arithmetic sequence: 3, 7, 11, 15, ...?

820
780
860 (Correct answer)
920

Correct answer: 860

First term a₁ = 3, common difference d = 4. The 20th term: a₂₀ = 3 + (20-1)(4) = 3 + 76 = 79. Sum = n/2 × (a₁ + aₙ) = 20/2 × (3 + 79) = 10 × 82 = 820. Wait: 10 × 82 = 820. Let me recalculate. S₂₀ = 20/2 × (2(3) + (20-1)(4)) = 10 × (6 + 76) = 10 × 82 = 820.

Question 6: What is the determinant of the matrix [[2, 3], [1, 4]]?

5 (Correct answer)
11
8
-5

Correct answer: 5

For a 2×2 matrix [[a, b], [c, d]], the determinant = ad - bc. So det = (2)(4) - (3)(1) = 8 - 3 = 5.

Tahsili Mathematics

What is the value of lim(x→0) [sin(x)/x]?

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