أسئلة تدريبية في الرياضيات – الصف الثاني عشر – المنهج اللبناني

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سجّل على edurise.me واشترك (10 دولارات شهرياً) لتتدرّب على كل فصول الرياضيات للصف الثاني عشر، وتقدّم الامتحانات المؤقّتة وتتابع تقدّمك.

Question 1

Calculate lim(x→−∞) (2x⁴ + 7x³).

Accepted Answer

+∞. A polynomial behaves at −∞ like its highest-degree term: lim(x→−∞) 2x⁴ = +∞.

Question 2

The asymptotes of the curve of f(x) = (2x − 4)/(x − 4) are:

Accepted Answer

x = 4 and y = 2. x − 4 = 0 ⟺ x = 4 (vertical asymptote); at ±∞, f(x) → 2x/x = 2 (horizontal asymptote).

Question 3

The point A(5 ; −1) belongs to the curve of f. Which point belongs to the curve of f⁻¹?

Accepted Answer

(−1 ; 5). C_f and C_{f⁻¹} are symmetric about y = x: exchange the coordinates, (5 ; −1) → (−1 ; 5).

Question 4

Let f(x) = 2x and g(x) = x² + 3. Calculate (g∘f)(−3).

Accepted Answer

39. (g∘f)(−3) = g(f(−3)) = g(−6) = (−6)² + 3 = 39.

Question 5

What is the period of f(x) = cos(2x − π/6)?

Accepted Answer

π. The period of cos(ax + b) is 2π/|a| with |a| = 2: T = π. The constant b does not change the period.

Question 6

What is the period of f(x) = sin(2x + π/6)?

Accepted Answer

π. The period of sin(ax + b) is 2π/|a| with |a| = 2: T = π. The constant b does not change the period.

Question 7

‖u→‖ = 2, ‖v→‖ = 4 and the angle of u→ and v→ is π/6. Then ‖u→ ∧ v→‖ =

Accepted Answer

4. ‖u→ ∧ v→‖ = ‖u→‖ × ‖v→‖ × sin θ = 2 × 4 × sin(π/6) = 4.

Question 8

u→(−2 ; 4 ; 0) and v→(1 ; −4 ; 0). The area of the parallelogram built on u→ and v→ is:

Accepted Answer

4. u→ ∧ v→ = (0 ; 0 ; −2 × (−4) − 4 × 1) = (0 ; 0 ; 4); the area is ‖u→ ∧ v→‖ = 4.

Question 9

Which point belongs to the plane P: −3x − y + z − 6 = 0?

Accepted Answer

(−1 ; 1 ; 4). Replace x, y, z by the coordinates: −3(−1) − (1) + (4) − 6 = 0 ✓.

Question 10

Which point belongs to the plane P: x + 2z − 5 = 0?

Accepted Answer

(−1 ; 3 ; 3). Replace x, y, z by the coordinates: (−1) + 2(3) − 5 = 0 ✓.

Question 11

Calculate i^40.

Accepted Answer

1. 40 = 4 × 10 + 0 and i⁴ = 1, so i^40 = i^0 = 1.

Question 12

Write (5 − i)(5 − 4i) in algebraic form.

Accepted Answer

21 − 25i. (5 − i)(5 − 4i) = 25 − 20i − 5i + 4i² = 21 − 25i (i² = −1).

Question 13

If ∫₀² f(x) dx = −1 and ∫₂⁵ f(x) dx = −4, then ∫₀⁵ f(x) dx =

Accepted Answer

−5. Chasles: ∫₀⁵ f = ∫₀² f + ∫₂⁵ f = −1 + (−4) = −5.

Question 14

If ∫₁⁴ f(x) dx = 1, then ∫₄¹ f(x) dx − ∫₁¹ f(x) dx =

Accepted Answer

−1. ∫₄¹ f = −∫₁⁴ f = −1 and ∫₁¹ f = 0, so the result is −1.

Question 15

A solution prepared in a Beirut laboratory has [H₃O⁺] = 10⁻⁵ mol/L. Its pH = −log[H₃O⁺] is:

Accepted Answer

5. pH = −log 10⁻⁵ = −(−5) = 5.

أسئلة الرياضيات الصف الثاني عشر مع الحل

15 سؤالاً تدريبياً في الرياضيات – الصف الثاني عشر

1. Calculate lim(x→−∞) (2x⁴ + 7x³).

2. The asymptotes of the curve of f(x) = (2x − 4)/(x − 4) are:

3. The point A(5 ; −1) belongs to the curve of f. Which point belongs to the curve of f⁻¹?

4. Let f(x) = 2x and g(x) = x² + 3. Calculate (g∘f)(−3).

5. What is the period of f(x) = cos(2x − π/6)?

6. What is the period of f(x) = sin(2x + π/6)?

7. ‖u→‖ = 2, ‖v→‖ = 4 and the angle of u→ and v→ is π/6. Then ‖u→ ∧ v→‖ =

8. u→(−2 ; 4 ; 0) and v→(1 ; −4 ; 0). The area of the parallelogram built on u→ and v→ is:

9. Which point belongs to the plane P: −3x − y + z − 6 = 0?

10. Which point belongs to the plane P: x + 2z − 5 = 0?

11. Calculate i^40.

12. Write (5 − i)(5 − 4i) in algebraic form.

13. If ∫₀² f(x) dx = −1 and ∫₂⁵ f(x) dx = −4, then ∫₀⁵ f(x) dx =

14. If ∫₁⁴ f(x) dx = 1, then ∫₄¹ f(x) dx − ∫₁¹ f(x) dx =

15. A solution prepared in a Beirut laboratory has [H₃O⁺] = 10⁻⁵ mol/L. Its pH = −log[H₃O⁺] is:

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