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Lebanese Baccalaureate - General Sciences (GS) study materials
Lebanese Baccalaureate - General Sciences (GS) study resources

Lebanese Baccalaureate - General Sciences (GS) practice questions

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Question 1Mathematics

What is the value of the derivative of f(x) = 3x² − 5x + 2 at x = 2?

  1. 2
  2. 12
  3. 7 Correct answer.
  4. 17

Explanation

The derivative is f'(x) = 6x − 5. Substituting x = 2 gives f'(2) = 12 − 5 = 7, the instantaneous rate of change at that point.

Question 2Mathematics

The sequence defined by uₙ = 3 + 2n for n ≥ 0 is arithmetic. What is its common difference?

  1. n
  2. 5
  3. 2 Correct answer.
  4. 3

Explanation

In an arithmetic sequence the difference uₙ₊₁ − uₙ is constant. Here (3 + 2(n+1)) − (3 + 2n) = 2, so the common difference is 2.

Question 3Mathematics

What is the limit as x tends to 0 of (sin 3x)/x?

  1. 0
  2. 1
  3. 1/3
  4. 3 Correct answer.

Explanation

Using the standard limit sin(ax)/x → a as x → 0, or rewriting as 3·(sin 3x)/(3x), the expression tends to 3·1 = 3.

Question 4Mathematics

In the complex plane, what is the modulus of z = 3 − 4i?

  1. 1
  2. 25
  3. 5 Correct answer.
  4. 7

Explanation

The modulus is |z| = √(a² + b²) = √(9 + 16) = √25 = 5. It represents the distance from the origin to the point (3, −4).

Question 5Mathematics

What is the value of the integral from 0 to 1 of 2x dx?

  1. 0
  2. 1 Correct answer.
  3. 1/2
  4. 2

Explanation

An antiderivative of 2x is x². Evaluating x² between 0 and 1 gives 1² − 0² = 1, the area under the line y = 2x on that interval.

Question 6Mathematics

If log₂(x) = 5, what is x?

  1. 25
  2. 32 Correct answer.
  3. 7
  4. 10

Explanation

By definition of logarithm, log₂(x) = 5 means 2⁵ = x, so x = 32. The logarithm is the exponent to which the base must be raised.

Question 7Mathematics

A fair six-sided die is rolled once. What is the probability of obtaining an even number?

  1. 1/3
  2. 1/2 Correct answer.
  3. 1/6
  4. 2/3

Explanation

The even outcomes are 2, 4, and 6, giving 3 favorable cases out of 6 equally likely outcomes. The probability is 3/6 = 1/2.

Question 8Mathematics

What is the sum of the first 10 positive integers, 1 + 2 + … + 10?

  1. 55 Correct answer.
  2. 100
  3. 50
  4. 45

Explanation

The sum of the first n integers is n(n+1)/2. For n = 10 this gives 10·11/2 = 55, the standard arithmetic-series formula.

Question 9Mathematics

The function f(x) = x² − 4x + 7 has a minimum. At which value of x does it occur?

  1. 4
  2. −2
  3. 7
  4. 2 Correct answer.

Explanation

For a parabola ax² + bx + c with a > 0, the vertex is at x = −b/(2a) = 4/2 = 2. Since a > 0 this vertex is the minimum.

Question 10Mathematics

What is the derivative of f(x) = x·eˣ?

  1. eˣ(x − 1)
  2. eˣ
  3. eˣ(x + 1) Correct answer.
  4. x·eˣ

Explanation

Apply the product rule: (uv)' = u'v + uv'. With u = x and v = eˣ, this gives eˣ + x·eˣ = eˣ(x + 1).

Question 11Mathematics

The complex number z = 1 + i is given. What is z²?

  1. 2i Correct answer.
  2. −2i
  3. 1 + 2i
  4. 2

Explanation

Expand (1 + i)² = 1 + 2i + i². Since i² = −1, this becomes 1 + 2i − 1 = 2i.

Question 12Mathematics

What is the limit as x tends to infinity of (2x² + 3)/(x² − 1)?

  1. 0
  2. ∞
  3. 2 Correct answer.
  4. 3

Explanation

For a rational function, the limit at infinity equals the ratio of leading coefficients when numerator and denominator have equal degree. Both are degree 2, so the limit is 2/1 = 2.

Question 13Mathematics

A geometric sequence has first term u₀ = 3 and common ratio q = 2. What is the sum u₀ + u₁ + … + u₅ of its first 6 terms?

  1. 96
  2. 63
  3. 189 Correct answer.
  4. 192

Explanation

The sum of n terms of a geometric sequence is u₀(1 − qⁿ)/(1 − q). Here 3·(1 − 64)/(1 − 2) = 3·(−63)/(−1) = 189.

Question 14Mathematics

What is the value of the integral from 0 to π of sin x dx?

  1. 2 Correct answer.
  2. 0
  3. 1
  4. π

Explanation

An antiderivative of sin x is −cos x. Evaluating from 0 to π gives (−cos π) − (−cos 0) = 1 + 1 = 2.

Question 15Mathematics

The equation x² + y² − 4x + 6y − 3 = 0 represents a circle. What is its center?

  1. (−2, 3)
  2. (−4, 6)
  3. (4, −6)
  4. (2, −3) Correct answer.

Explanation

Completing the square gives (x − 2)² + (y + 3)² = 16. The center is (2, −3) and the radius is 4.

Question 16Mathematics

The equation z³ = 1 has three complex solutions. Which set lists them correctly?

  1. 0, 1, −1
  2. 1, e^(2iπ/3), e^(4iπ/3) Correct answer.
  3. 1, i, −i
  4. 1, −1, i

Explanation

The cube roots of unity are e^(2ikπ/3) for k = 0, 1, 2, giving 1, e^(2iπ/3), and e^(4iπ/3). They lie on the unit circle at angles 0, 120°, and 240°.

Question 17Mathematics

Consider f(x) = ln(x) for x > 0. What is the equation of the tangent line to the curve at x = 1?

  1. y = 1
  2. y = x − 1 Correct answer.
  3. y = x
  4. y = x + 1

Explanation

The derivative of ln x is 1/x, so the slope at x = 1 is 1. The point is (1, 0), giving y − 0 = 1·(x − 1), i.e. y = x − 1.

Question 18Mathematics

A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

  1. 2/15 Correct answer.
  2. 1/5
  3. 4/10
  4. 8/25

Explanation

The probability the first is red is 4/10, and given that, the second is 3/9. Multiplying gives (4/10)·(3/9) = 12/90 = 2/15.

Question 19Mathematics

What is the limit as x tends to 1 of (x² − 1)/(x − 1)?

  1. undefined
  2. 0
  3. 2 Correct answer.
  4. 1

Explanation

Factor the numerator: x² − 1 = (x − 1)(x + 1). Cancelling (x − 1) for x ≠ 1 leaves x + 1, whose limit as x → 1 is 2.

Question 20Mathematics

Let f be defined on ℝ by f(x) = x³ − 3x. On which interval is f strictly increasing?

  1. ]−∞, 0[
  2. ]0, +∞[
  3. ]−∞, −1[ ∪ ]1, +∞[ Correct answer.
  4. ]−1, 1[

Explanation

The derivative f'(x) = 3x² − 3 = 3(x − 1)(x + 1) is positive when x < −1 or x > 1. On those intervals f is strictly increasing.

Question 21Mathematics

Evaluate the integral from 0 to 1 of x·eˣ dx.

  1. 1 Correct answer.
  2. e − 1
  3. 2e − 1
  4. e

Explanation

Integrating by parts with u = x, dv = eˣ dx gives [x·eˣ − eˣ] from 0 to 1. This equals (e − e) − (0 − 1) = 1.

Question 22Mathematics

A geometric sequence has u₁ = 1 and common ratio q with |q| < 1. Its infinite sum is 4. What is q?

  1. 1/4
  2. 3/4 Correct answer.
  3. 1/2
  4. 4/3

Explanation

The infinite sum is u₁/(1 − q) = 1/(1 − q). Setting this equal to 4 gives 1 − q = 1/4, so q = 3/4, which satisfies |q| < 1.

Question 23Mathematics

In the complex plane, the set of points z satisfying |z − 1| = |z + 1| is which geometric object?

  1. The unit circle
  2. A circle centered at 1
  3. The imaginary axis Correct answer.
  4. The real axis

Explanation

Writing z = x + iy, the condition becomes (x − 1)² + y² = (x + 1)² + y². Expanding and simplifying gives −2x = 2x, so x = 0, the imaginary axis.

Question 24Mathematics

Let f(x) = x³ − 3x + 2. How many distinct real roots does the equation f(x) = 0 have?

  1. 1
  2. 3
  3. 0
  4. 2 Correct answer.

Explanation

Since f(1) = 0, x − 1 is a factor: f(x) = (x − 1)²(x + 2). The roots are x = 1 (double) and x = −2, giving two distinct real roots.

Question 25Mathematics

A function f is continuous on [0, 2] with f(0) = −1 and f(2) = 3. Which statement is guaranteed by the intermediate value theorem?

  1. There exists c in ]0, 2[ with f(c) = 1 Correct answer.
  2. f attains its maximum at x = 2
  3. f is increasing on [0, 2]
  4. f is differentiable on ]0, 2[

Explanation

The intermediate value theorem states that a continuous function on an interval takes every value between f(0) and f(2). Since 1 lies between −1 and 3, some c in ]0, 2[ satisfies f(c) = 1.

Question 26Mathematics

What is the value of the expression (3 + 2i)(1 − i)?

  1. 5 − i Correct answer.
  2. 1 + i
  3. 5 + i
  4. 1 − i

Explanation

Expand the product: (3)(1) + (3)(−i) + (2i)(1) + (2i)(−i) = 3 − 3i + 2i − 2i². Since i² = −1, this becomes 3 − i + 2 = 5 − i.

Question 27Mathematics

What is the modulus of the complex number z = 3 − 4i?

  1. 1
  2. 5 Correct answer.
  3. 25
  4. 7

Explanation

The modulus of a + bi is √(a² + b²). Here a = 3 and b = −4, so |z| = √(9 + 16) = √25 = 5.

Question 28Mathematics

A function f is defined on ℝ by f(x) = 2x + 1. What is the value of f⁻¹(5)?

  1. 11
  2. 1/2
  3. 2 Correct answer.
  4. 3

Explanation

To find f⁻¹(5), solve 2x + 1 = 5. Subtracting 1 gives 2x = 4, so x = 2.

Question 29Mathematics

What is the value of cos(π/3)?

  1. 1/2 Correct answer.
  2. √3/2
  3. √2/2
  4. 1

Explanation

π/3 radians equals 60°. The cosine of 60° is 1/2, a standard exact value from the unit circle.

Question 30Mathematics

What is the value of sin(π/2)?

  1. 1 Correct answer.
  2. 1/2
  3. −1
  4. 0

Explanation

π/2 radians equals 90°. The sine of 90° is 1, the maximum value of the sine function.

Question 31Mathematics

What is the derivative of f(x) = cos(2x)?

  1. −sin(2x)
  2. 2 sin(2x)
  3. −2 sin(2x) Correct answer.
  4. −2 cos(2x)

Explanation

By the chain rule, the derivative of cos(u) is −sin(u)·u′. With u = 2x, u′ = 2, so f′(x) = −2 sin(2x).

Question 32Mathematics

What is the derivative of f(x) = x³?

  1. x²
  2. 3x² Correct answer.
  3. 3x
  4. x⁴/4

Explanation

The power rule states that the derivative of xⁿ is n·xⁿ⁻¹. For n = 3, this gives 3x².

Question 33Mathematics

What is the value of the integral from 0 to 1 of x² dx?

  1. 1
  2. 1/2
  3. 1/4
  4. 1/3 Correct answer.

Explanation

An antiderivative of x² is x³/3. Evaluating from 0 to 1 gives (1³/3) − (0³/3) = 1/3.

Question 34Mathematics

What is the limit as x tends to 0 of (sin x)/x?

  1. Does not exist
  2. ∞
  3. 1 Correct answer.
  4. 0

Explanation

This is a standard limit: lim_{x→0} (sin x)/x = 1. It follows from the fact that sin x ≈ x for small x.

Question 35Mathematics

What is the general solution of the differential equation y′ = 3y?

  1. y = 3x + C
  2. y = C e^{3x} Correct answer.
  3. y = C e^{x/3}
  4. y = C e^{−3x}

Explanation

The equation y′ = ky has general solution y = C e^{kx}. Here k = 3, so y = C e^{3x}.

Question 36Mathematics

The sequence (uₙ) is defined by u₀ = 1 and uₙ₊₁ = 2uₙ + 1. What is u₃?

  1. 9
  2. 15 Correct answer.
  3. 7
  4. 31

Explanation

Compute successively: u₁ = 2(1) + 1 = 3; u₂ = 2(3) + 1 = 7; u₃ = 2(7) + 1 = 15.

Question 37Mathematics

An arithmetic sequence has first term u₁ = 2 and common difference 3. What is the sum of its first 10 terms?

  1. 155 Correct answer.
  2. 145
  3. 150
  4. 160

Explanation

The sum of the first n terms of an arithmetic sequence is Sₙ = n/2 · (2u₁ + (n−1)d). Here S₁₀ = 10/2 · (4 + 27) = 5 · 31 = 155.

Question 38Mathematics

A fair coin is tossed three times. What is the probability of obtaining exactly two heads?

  1. 1/2
  2. 1/8
  3. 5/8
  4. 3/8 Correct answer.

Explanation

There are 2³ = 8 equally likely outcomes. Exactly two heads occur in C(3,2) = 3 outcomes (HHT, HTH, THH), so the probability is 3/8.

Question 39Mathematics

A committee of 2 students is to be chosen from a group of 5. How many different committees are possible?

  1. 5
  2. 10 Correct answer.
  3. 20
  4. 25

Explanation

The number of ways to choose 2 from 5 without regard to order is C(5,2) = 5!/(2!·3!) = 10.

Question 40Mathematics

What is the value of the integral from 0 to 1 of e^{2x} dx?

  1. (e² − 1)/2 Correct answer.
  2. e²/2
  3. (e − 1)/2
  4. e² − 1

Explanation

An antiderivative of e^{2x} is (1/2)e^{2x}. Evaluating from 0 to 1 gives (1/2)(e² − e⁰) = (e² − 1)/2.

Question 41Mathematics

What is the derivative of f(x) = ln(x² + 1)?

  1. 2x ln(x² + 1)
  2. 1/(x² + 1)
  3. 2x/(x² + 1) Correct answer.
  4. 2x/(x² + 1)²

Explanation

By the chain rule, the derivative of ln(u) is u′/u. With u = x² + 1, u′ = 2x, so f′(x) = 2x/(x² + 1).

Question 42Mathematics

What is the equation of the tangent line to the curve y = x² at the point (1, 1)?

  1. y = x + 1
  2. y = 2x − 1 Correct answer.
  3. y = x
  4. y = 2x + 1

Explanation

The derivative y′ = 2x gives slope 2 at x = 1. Using point-slope form: y − 1 = 2(x − 1), so y = 2x − 1.

Question 43Mathematics

What is the value of the integral from 0 to 1 of 1/(x + 1) dx?

  1. 1
  2. 1/2
  3. ln 3
  4. ln 2 Correct answer.

Explanation

An antiderivative of 1/(x + 1) is ln|x + 1|. Evaluating from 0 to 1 gives ln 2 − ln 1 = ln 2.

Question 44Mathematics

In a triangle ABC, side a = 5, side b = 7, and angle C = 60°. What is the length of side c?

  1. √109
  2. √39 Correct answer.
  3. 6
  4. √74

Explanation

By the law of cosines, c² = a² + b² − 2ab cos C = 25 + 49 − 2(5)(7)(1/2) = 74 − 35 = 39, so c = √39.

Question 45Mathematics

The curve y = x³ − 3x has a local maximum at which point?

  1. (−1, 2) Correct answer.
  2. (2, 2)
  3. (0, 0)
  4. (1, −2)

Explanation

The derivative is y′ = 3x² − 3 = 3(x − 1)(x + 1), which vanishes at x = ±1. The second derivative y″ = 6x is negative at x = −1, so a local maximum occurs at (−1, 2).

Question 46Mathematics

What is the general solution of the differential equation y″ + 4y = 0?

  1. y = A cos x + B sin x
  2. y = A cos(2x) + B sin(2x) Correct answer.
  3. y = A e^{2ix} + B e^{−2ix}
  4. y = A e^{2x} + B e^{−2x}

Explanation

The characteristic equation r² + 4 = 0 has roots r = ±2i. For purely imaginary roots ±βi, the general solution is y = A cos(βx) + B sin(βx) with β = 2.

Question 47Mathematics

What is the value of the integral from 0 to π/2 of sin x cos x dx?

  1. 1
  2. 1/2 Correct answer.
  3. 0
  4. 1/4

Explanation

Since sin x cos x = (1/2) sin(2x), an antiderivative is −(1/4) cos(2x). Evaluating from 0 to π/2 gives −(1/4)(cos π − cos 0) = −(1/4)(−1 − 1) = 1/2.

Question 48Mathematics

The complex number z = 1 + i is given. What is z³?

  1. −2 − 2i
  2. 2 + 2i
  3. −2 + 2i Correct answer.
  4. 2 − 2i

Explanation

First z² = (1 + i)² = 1 + 2i + i² = 2i. Then z³ = z²·z = 2i(1 + i) = 2i + 2i² = −2 + 2i.

Question 49Mathematics

A function f is continuous on [0, 3] with f(0) = 2 and f(3) = 8. Which statement is guaranteed by the intermediate value theorem?

  1. There exists c in (0, 3) such that f(c) = 5 Correct answer.
  2. f has a maximum at some point in (0, 3)
  3. f is increasing on [0, 3]
  4. f is differentiable on (0, 3)

Explanation

The intermediate value theorem states that a continuous function on an interval takes every value between its endpoint values. Since 5 lies between 2 and 8, there is some c in (0, 3) with f(c) = 5.

Question 50Mathematics

What is the value of the limit as x tends to 0 of (1 − cos x)/x²?

  1. 0
  2. ∞
  3. 1/2 Correct answer.
  4. 1

Explanation

Using the Taylor expansion cos x = 1 − x²/2 + o(x²), we get (1 − cos x)/x² = 1/2 + o(1), so the limit is 1/2.