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Cambodia Bac II Science study materials
Cambodia Bac II Science study resources

Cambodia Bac II Science practice questions

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Question 1Mathematics

The roots of the quadratic equation x² − 5x + 6 = 0 are:

  1. x = −2 and x = −3
  2. x = 2 and x = 3 Correct answer.
  3. x = −1 and x = −6
  4. x = 1 and x = 6

Explanation

Factor the quadratic as (x − 2)(x − 3) = 0. The zero-product property then gives x − 2 = 0 or x − 3 = 0, so the roots are x = 2 and x = 3. Checking with Vieta, the roots sum to 5 and multiply to 6, as required.

Question 2Mathematics

The exact value of sin 30° is:

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

Explanation

In a 30°–60°–90° right triangle, the side opposite the 30° angle is half the hypotenuse, so sin 30° = 1/2. This value also follows from the unit-circle coordinates at 30°.

Question 3Mathematics

Given f(x) = 3x² − 4x + 7, the derivative f′(x) is:

  1. 6x + 7
  2. 3x − 4
  3. 6x² − 4x
  4. 6x − 4 Correct answer.

Explanation

Differentiate term by term using the power rule: d/dx(3x²) = 6x and d/dx(−4x) = −4. The derivative of the constant 7 is zero, so f′(x) = 6x − 4.

Question 4Mathematics

In an arithmetic sequence the first term is 5 and the common difference is 3. The 10th term is:

  1. 35
  2. 29
  3. 32 Correct answer.
  4. 26

Explanation

For an arithmetic sequence, a_n = a₁ + (n − 1)d. With a₁ = 5, d = 3 and n = 10, a₁₀ = 5 + 9 × 3 = 32.

Question 5Mathematics

The value of lim_{x→2} (x² − 4)/(x − 2) is:

  1. 0
  2. 2
  3. 4 Correct answer.
  4. The limit does not exist

Explanation

Factor the numerator as (x − 2)(x + 2), then cancel the common factor x − 2 for x ≠ 2. The expression becomes x + 2 near x = 2, so the limit is 2 + 2 = 4.

Question 6Mathematics

A bag contains 3 red balls and 5 blue balls, all identical except colour. One ball is drawn at random. The probability that it is red is:

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

Explanation

There are 3 + 5 = 8 equally likely balls in the bag, and 3 of them are red. Therefore P(red) = 3/8.

Question 7Mathematics

The circle x² + y² − 4x + 6y − 12 = 0 has centre and radius:

  1. (2, −3), radius 5 Correct answer.
  2. (2, −3), radius 25
  3. (−2, 3), radius 5
  4. (−2, 3), radius 25

Explanation

Group and complete the square: (x² − 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9 = 25. This gives (x − 2)² + (y + 3)² = 25, so the centre is (2, −3) and the radius is √25 = 5.

Question 8Mathematics

If log₃(x − 1) = 2, then x equals:

  1. 9
  2. 5
  3. 7
  4. 10 Correct answer.

Explanation

Convert the logarithm to exponential form: x − 1 = 3² = 9. Then add 1 to both sides to obtain x = 10.

Question 9Mathematics

The value of ∫₀¹ (3x² + 2x) dx is:

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

Explanation

An antiderivative of 3x² + 2x is x³ + x², since d/dx(x³ + x²) = 3x² + 2x. Evaluating from 0 to 1 gives (1 + 1) − (0 + 0) = 2.

Question 10Mathematics

The solutions of 2sin²x − sin x − 1 = 0 in the interval 0 ≤ x ≤ 2π are:

  1. x = π/2, π/6, 7π/6
  2. x = π/6, 5π/6, π/2
  3. x = π/2, 7π/6, 11π/6 Correct answer.
  4. x = 7π/6, 11π/6 only

Explanation

Factor the equation as (2 sin x + 1)(sin x − 1) = 0, so sin x = −1/2 or sin x = 1. In 0 ≤ x ≤ 2π, sin x = 1 at x = π/2, and sin x = −1/2 at x = 7π/6 and x = 11π/6.

Question 11Mathematics

The modulus of the complex number z = 3 − 4i is:

  1. 5 Correct answer.
  2. √7
  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 12Mathematics

The determinant of the matrix [[2, 3], [1, 4]] is:

  1. 11
  2. 8
  3. 10
  4. 5 Correct answer.

Explanation

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

Question 13Mathematics

The distance between the points A(1, 2) and B(4, 6) is:

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

Explanation

Distance = √((x₂−x₁)² + (y₂−y₁)²). Here it is √((4−1)² + (6−2)²) = √(9+16) = √25 = 5.

Question 14Mathematics

In the expansion of (2 + x)⁵, the coefficient of x³ is:

  1. 20
  2. 40 Correct answer.
  3. 10
  4. 80

Explanation

The term containing x³ is C(5,3)·2²·x³. Since C(5,3) = 10 and 2² = 4, the coefficient is 10×4 = 40.

Question 15Mathematics

The vectors u = (2, k) and v = (3, −4) are perpendicular. The value of k is:

  1. −6
  2. 6
  3. 1.5 Correct answer.
  4. −1.5

Explanation

Perpendicular vectors have dot product zero: u·v = 2×3 + k×(−4) = 6 − 4k. Setting 6 − 4k = 0 gives k = 1.5.

Question 16Mathematics

For a geometric sequence with first term 2 and common ratio 3, the sum of the first 5 terms is:

  1. 162
  2. 242 Correct answer.
  3. 240
  4. 80

Explanation

The sum of the first n terms is S_n = a(rⁿ − 1)/(r − 1). Here S₅ = 2(3⁵ − 1)/(3 − 1) = 2(243 − 1)/2 = 242.

Question 17Mathematics

The value of ∫₁² (2x + 1) dx is:

  1. 3
  2. 5
  3. 4 Correct answer.
  4. 6

Explanation

An antiderivative of 2x + 1 is x² + x. Evaluating from 1 to 2 gives (4 + 2) − (1 + 1) = 6 − 2 = 4.

Question 18Mathematics

If sin θ = 3/5 and θ lies in quadrant II, then cos θ equals:

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

Explanation

Using sin²θ + cos²θ = 1, cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5. In quadrant II cosine is negative, hence cos θ = −4/5.

Question 19Mathematics

Using the definition of the derivative, the value of f′(4) for f(x) = √x is:

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

Explanation

f′(x) = lim_{h→0} (√(x+h) − √x)/h. Rationalising gives 1/(√(x+h)+√x), which tends to 1/(2√x). At x = 4, f′(4) = 1/(2×2) = 1/4.

Question 20Mathematics

How many four-digit numbers with all digits distinct can be formed using the digits 1, 2, 3, 4, 5?

  1. 625
  2. 24
  3. 120 Correct answer.
  4. 100

Explanation

The first digit can be any of 5 digits, the second any of the remaining 4, the third any of 3, and the fourth any of 2. Thus the count is 5×4×3×2 = 120, which equals ⁵P₄.

Question 21Mathematics

The solution set of the inequality 5 − 2x ≥ 1 is

  1. x = 2
  2. x ≥ 2
  3. x < 2
  4. x ≤ 2 Correct answer.

Explanation

Subtract 5 from both sides to obtain −2x ≥ −4. Dividing both sides by −2 reverses the inequality sign, so x ≤ 2. Checking x = 2 gives 5 − 4 = 1, so equality is included.

Question 22Mathematics

The domain of the real-valued function f(x) = √(2x − 6) is

  1. x ≠ 3
  2. x > 3
  3. x ≥ 3 Correct answer.
  4. x ≤ 3

Explanation

For the square root to be a real number, the radicand must be non-negative: 2x − 6 ≥ 0. Solving gives x ≥ 3, and x = 3 is included because √0 is defined.

Question 23Mathematics

Events A and B are independent with P(A) = 1/3 and P(B) = 1/4. The value of P(A ∩ B) is

  1. 7/12
  2. 1/7
  3. 1/12 Correct answer.
  4. 1/6

Explanation

For independent events, the multiplication rule gives P(A ∩ B) = P(A)P(B). Therefore P(A ∩ B) = (1/3)(1/4) = 1/12.

Question 24Mathematics

If f(x) = (3x − 1)/2, then f⁻¹(x) equals

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

Explanation

Write y = (3x − 1)/2, then interchange x and y to get x = (3y − 1)/2. Solving for y gives 2x = 3y − 1, hence y = (2x + 1)/3.

Question 25Mathematics

If α and β are the roots of 2x² − 5x + 1 = 0, then 1/α + 1/β equals

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

Explanation

By Vieta's formulas, α + β = 5/2 and αβ = 1/2. Then 1/α + 1/β = (α + β)/(αβ) = (5/2)/(1/2) = 5.

Question 26Mathematics

The slope of a line perpendicular to the line 3x − 2y + 5 = 0 is

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

Explanation

Rearranging 3x − 2y + 5 = 0 gives y = (3/2)x + 5/2, so the original slope is 3/2. A perpendicular line has slope equal to the negative reciprocal, which is −2/3.

Question 27Mathematics

The constant term in the expansion of (x + 2/x)⁶ is

  1. 120
  2. 240
  3. 64
  4. 160 Correct answer.

Explanation

The general term is C(6,r)x^(6−r)(2/x)^r = C(6,r)2^r x^(6−2r). The term is constant when 6 − 2r = 0, so r = 3. Hence the constant term is C(6,3)2³ = 20 × 8 = 160.

Question 28Mathematics

The area of the region bounded by y = x², the x-axis, x = 0 and x = 2 is

  1. 16/3
  2. 2/3
  3. 8/3 Correct answer.
  4. 4

Explanation

The area is given by the definite integral ∫₀² x² dx because x² is non-negative on this interval. Evaluating gives [x³/3]₀² = 8/3 − 0 = 8/3.

Question 29Mathematics

The maximum value of f(x) = x³ − 3x on the interval −2 ≤ x ≤ 2 is

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

Explanation

Differentiating gives f′(x) = 3x² − 3, so the critical points are x = −1 and x = 1. Evaluating f at the critical points and endpoints gives f(−2) = −2, f(−1) = 2, f(1) = −2 and f(2) = 2, so the maximum value is 2.

Question 30Mathematics

If z = (1 + i)/(1 − i), then z²⁰²⁶ equals

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

Explanation

Multiply numerator and denominator by 1 + i: z = (1 + i)²/(1² − i²) = (2i)/2 = i. Since i⁴ = 1 and 2026 = 4 × 506 + 2, we get z²⁰²⁶ = i²⁰²⁶ = i² = −1.

Question 31Mathematics

The argument of the complex number z = −1 + i√3, measured in radians in the interval [0, 2π), is

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

Explanation

The point (−1, √3) lies in quadrant II, so the argument is π minus the reference angle. Since tan α = √3/1 = √3, the reference angle is π/3. Therefore arg z = π − π/3 = 2π/3.

Question 32Mathematics

In an arithmetic sequence, the first term is 5 and the common difference is 3. The 10th term is

  1. 29
  2. 35
  3. 27
  4. 32 Correct answer.

Explanation

For an arithmetic sequence, a_n = a_1 + (n − 1)d. Substituting a_1 = 5, d = 3 and n = 10 gives 5 + 9(3) = 32.

Question 33Mathematics

Events A and B are mutually exclusive. If P(A) = 0.4 and P(B) = 0.3, then P(A ∪ B) is

  1. 1.00
  2. 0.70 Correct answer.
  3. 0.12
  4. 0.58

Explanation

Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. Therefore P(A ∪ B) = P(A) + P(B) = 0.4 + 0.3 = 0.70.

Question 34Mathematics

The solution set of x² − 5x + 6 < 0 is

  1. −3 < x < −2
  2. x < 2 or x > 3
  3. 2 < x < 3 Correct answer.
  4. x ≤ 2 or x ≥ 3

Explanation

Factor x² − 5x + 6 = (x − 2)(x − 3). The upward-opening parabola is negative strictly between its roots, so the solution is 2 < x < 3.

Question 35Mathematics

When the polynomial P(x) = x³ − 2x² + 3x − 4 is divided by x − 2, the remainder is

  1. 6
  2. 0
  3. 2 Correct answer.
  4. −2

Explanation

By the remainder theorem, the remainder on division by x − 2 is P(2). P(2) = 2³ − 2·2² + 3·2 − 4 = 8 − 8 + 6 − 4 = 2.

Question 36Mathematics

If cos θ = −5/13 and θ lies in quadrant III, then tan θ equals

  1. 12/5 Correct answer.
  2. −5/12
  3. 5/12
  4. −12/5

Explanation

In quadrant III both sine and cosine are negative. From sin²θ = 1 − cos²θ = 1 − 25/169 = 144/169, sin θ = −12/13. Hence tan θ = sin θ / cos θ = (−12/13)/(−5/13) = 12/5.

Question 37Mathematics

The equation of the tangent line to the curve y = x³ − 2x at the point where x = 1 is

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

Explanation

For y = x³ − 2x, y′ = 3x² − 2, so at x = 1 the slope is 1. The point is (1, −1). Point-slope form gives y + 1 = 1(x − 1), or y = x − 2.

Question 38Mathematics

The inverse of the matrix A = [[2, 1], [5, 3]] is

  1. [[2, −1], [−5, 3]]
  2. [[3, −1], [−5, 2]] Correct answer.
  3. [[−3, 1], [5, −2]]
  4. [[3, 1], [5, 2]]

Explanation

For A = [[a, b], [c, d]], the inverse is (1/det A)[[d, −b], [−c, a]]. Here det A = 2·3 − 1·5 = 1, so A⁻¹ = [[3, −1], [−5, 2]].

Question 39Mathematics

The value of lim_{x→∞} (√(x² + 3x) − x) is

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

Explanation

Multiply by the conjugate: (√(x² + 3x) − x)(√(x² + 3x) + x)/(√(x² + 3x) + x) = 3x/(√(x² + 3x) + x). Dividing numerator and denominator by x gives 3/(√(1 + 3/x) + 1), which tends to 3/2.

Question 40Mathematics

The value of ∫₀¹ x e^{x²} dx is

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

Explanation

Let u = x², so du = 2x dx and x dx = du/2. The limits change from x = 0, 1 to u = 0, 1, giving (1/2)∫₀¹ e^u du = (e − 1)/2.

Question 41Mathematics

For a geometric sequence with first term 3 and common ratio 2, the sum of the first five terms is

  1. 93 Correct answer.
  2. 189
  3. 48
  4. 96

Explanation

For a geometric sequence, S_n = a(r^n − 1)/(r − 1). Here a = 3, r = 2, and n = 5, so S_5 = 3(2^5 − 1)/(2 − 1) = 3 × 31 = 93.

Question 42Mathematics

The quadratic equation x² + kx + 9 = 0 has a repeated root. The possible positive value of k is

  1. 9
  2. 3
  3. 18
  4. 6 Correct answer.

Explanation

A quadratic has a repeated root exactly when its discriminant b² − 4ac is zero. Here a = 1, b = k, and c = 9, so k² − 36 = 0, giving k = ±6; the positive value is 6.

Question 43Mathematics

If log₂ x = 5, then x equals

  1. 64
  2. 10
  3. 32 Correct answer.
  4. 25

Explanation

The equation log₂ x = 5 means that 2 raised to the power 5 equals x. Therefore x = 2^5 = 32.

Question 44Mathematics

The circle x² + y² − 6x + 4y − 12 = 0 has

  1. centre (3, −2), radius 5 Correct answer.
  2. centre (−3, 2), radius 25
  3. centre (−3, 2), radius 5
  4. centre (3, −2), radius 25

Explanation

Complete the square: x² − 6x = (x − 3)² − 9 and y² + 4y = (y + 2)² − 4. Substituting gives (x − 3)² + (y + 2)² = 25, so the centre is (3, −2) and the radius is √25 = 5.

Question 45Mathematics

For vectors a = (2, 1) and b = (1, 3), the cosine of the angle between them is

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

Explanation

Use cos θ = (a · b)/(|a||b|). The dot product is 2(1) + 1(3) = 5, and |a| = √5, |b| = √10, so cos θ = 5/√50 = 1/√2 = √2/2.

Question 46Mathematics

A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. The probability that both balls are red is

  1. 5/16
  2. 5/14 Correct answer.
  3. 3/8
  4. 15/56

Explanation

The probability is (5/8) × (4/7) because after one red ball is removed, 4 red balls remain among 7 balls. This product is 20/56 = 5/14.

Question 47Mathematics

If f(x) = x² ln x, then f′(1) equals

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

Explanation

Using the product rule, f′(x) = 2x ln x + x²(1/x) = 2x ln x + x. At x = 1, ln 1 = 0, so f′(1) = 0 + 1 = 1.

Question 48Mathematics

The number of solutions of sin x = 1/2 in the interval 0 ≤ x < 2π is

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

Explanation

Sine is positive in quadrants I and II, so sin x = 1/2 has one solution in each of these quadrants. They are x = π/6 and x = 5π/6, giving exactly two solutions.

Question 49Mathematics

The value of lim_{x→0} (sin 3x)/(tan 5x) is

  1. 3/5 Correct answer.
  2. 0
  3. 1
  4. 5/3

Explanation

For small x, sin 3x ≈ 3x and tan 5x ≈ 5x. Therefore the quotient approaches (3x)/(5x) = 3/5.

Question 50Mathematics

The value of ∫₀^{π/2} sin²x dx is

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

Explanation

Use sin²x = (1 − cos 2x)/2. Then the integral is [x/2 − sin 2x/4] from 0 to π/2, which equals π/4 − 0 = π/4.