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Tawjihi Science, Technology & Engineering Field study materials
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Tawjihi Science, Technology & Engineering Field practice questions

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Question 1Mathematics

If f(x) = x³ − 3x + 2, what is the value of the derivative f′(x) at x = 2?

  1. 9 Correct answer.
  2. 3
  3. 6
  4. 12

Explanation

Differentiating term by term gives f′(x) = 3x² − 3. Substituting x = 2 yields 3(4) − 3 = 9, so the slope of the curve at that point is 9.

Question 2Mathematics

The matrix A = [[2, 1], [4, 3]] has determinant equal to which value?

  1. 10
  2. −2
  3. 2 Correct answer.
  4. 6

Explanation

For a 2×2 matrix [[a, b], [c, d]] the determinant is ad − bc. Here ad = 2·3 = 6 and bc = 1·4 = 4, so the determinant is 6 − 4 = 2.

Question 3Mathematics

What is the sum of the first 20 terms of the arithmetic sequence whose first term is 3 and whose common difference is 4?

  1. 840 Correct answer.
  2. 860
  3. 820
  4. 800

Explanation

The sum of an arithmetic series is Sₙ = (n/2)(2a + (n−1)d). With n = 20, a = 3, d = 4 we get S₂₀ = 10(6 + 76) = 10 × 82 = 820.

Question 4Mathematics

Which expression is equal to sin²θ + cos²θ for every real angle θ?

  1. 2 sinθ cosθ
  2. 1 Correct answer.
  3. 0
  4. tan²θ

Explanation

The Pythagorean identity states that sin²θ + cos²θ = 1 for all θ. It follows directly from the coordinates of a point on the unit circle, where x² + y² = 1.

Question 5Mathematics

The complex number z = 3 + 4i is given. What is the modulus |z|?

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

Explanation

The modulus of a + bi is √(a² + b²). Here √(9 + 16) = √25 = 5, which is the distance from the origin to the point (3, 4) in the complex plane.

Question 6Mathematics

A fair six-sided die is rolled once. What is the probability that the outcome is a prime number?

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

Explanation

The primes among 1 to 6 are 2, 3 and 5, giving three favourable outcomes out of six equally likely ones. The probability is therefore 3/6 = 1/2.

Question 7Mathematics

The function f(x) = x² − 4x + 7 is a quadratic. What are the coordinates of its vertex?

  1. (−2, 19)
  2. (4, 7)
  3. (2, 3) Correct answer.
  4. (2, 7)

Explanation

The vertex x-coordinate of ax² + bx + c is −b/(2a) = 4/2 = 2. Substituting gives f(2) = 4 − 8 + 7 = 3, so the vertex is (2, 3).

Question 8Mathematics

What is the value of the definite integral ∫₀¹ (3x² + 2x) dx?

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

Explanation

An antiderivative is x³ + x². Evaluating from 0 to 1 gives (1 + 1) − (0 + 0) = 2, which is the signed area under the curve on that interval.

Question 9Mathematics

Solve the equation log₂(x + 3) = 4 for x.

  1. 13 Correct answer.
  2. 19
  3. 1
  4. 16

Explanation

Rewriting the logarithm in exponential form gives x + 3 = 2⁴ = 16. Subtracting 3 gives x = 13, and this value keeps the argument positive.

Question 10Mathematics

A line passes through the points A(1, 2) and B(4, 8). What is the equation of the line perpendicular to AB passing through A?

  1. y − 2 = ½(x − 1)
  2. y − 2 = −½(x − 1) Correct answer.
  3. y − 2 = 2(x − 1)
  4. y − 2 = −2(x − 1)

Explanation

The slope of AB is (8 − 2)/(4 − 1) = 2, so a perpendicular line has slope −½. Using point-slope form through A(1, 2) gives y − 2 = −½(x − 1).

Question 11Mathematics

What is the coefficient of x² in the expansion of (2x + 3)⁴?

  1. 96
  2. 432
  3. 216 Correct answer.
  4. 864

Explanation

The general term is C(4, k)(2x)^k·3^(4−k). For x² we take k = 2, giving C(4,2)·2²·3² = 6 × 4 × 9 = 216.

Question 12Mathematics

If A = [[1, 2], [3, 4]], what is the inverse matrix A⁻¹?

  1. [[4, −2], [−3, 1]]
  2. [[4, 2], [3, 1]]
  3. [[−2, 1], [1.5, −0.5]] Correct answer.
  4. [[−1, 0.5], [1.5, −0.5]]

Explanation

The determinant is 1·4 − 2·3 = −2. The inverse is (1/−2)[[4, −2], [−3, 1]], which simplifies to [[−2, 1], [1.5, −0.5]].

Question 13Mathematics

For the geometric sequence 2, 6, 18, 54, …, what is the sum of the first 6 terms?

  1. 728 Correct answer.
  2. 486
  3. 242
  4. 729

Explanation

The first term is 2 and the ratio is 3. The sum formula Sₙ = a(rⁿ − 1)/(r − 1) gives S₆ = 2(3⁶ − 1)/(3 − 1) = 2(729 − 1)/2 = 728.

Question 14Mathematics

The function f(x) = x³ − 3x is being analysed for monotonicity. On which interval is f strictly increasing?

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

Explanation

The derivative f′(x) = 3x² − 3 is positive when x² > 1, i.e. on (−∞, −1) and (1, ∞). On those intervals the function is strictly increasing, while it decreases between −1 and 1.

Question 15Mathematics

What is the general solution of the differential equation dy/dx = 2xy?

  1. y = Ce^(−x²)
  2. y = Ce^(x²) Correct answer.
  3. y = Cx²
  4. y = Ce^(2x)

Explanation

Separating variables gives dy/y = 2x dx. Integrating both sides yields ln|y| = x² + C, so y = Ce^(x²) after exponentiating.

Question 16Mathematics

A vector u = (3, −4) in the plane. What is the unit vector in the direction of u?

  1. (3/25, −4/25)
  2. (−3/5, 4/5)
  3. (3/5, −4/5) Correct answer.
  4. (3/7, −4/7)

Explanation

The magnitude of u is √(9 + 16) = 5. Dividing each component by 5 gives the unit vector (3/5, −4/5), which has length 1 and the same direction.

Question 17Mathematics

Using the binomial distribution, a biased coin lands heads with probability 0.6. If it is tossed 5 times, what is the probability of exactly 3 heads?

  1. 0.216
  2. 0.6000
  3. 0.3456 Correct answer.
  4. 0.2592

Explanation

The binomial probability is C(5,3)(0.6)³(0.4)² = 10 × 0.216 × 0.16 = 0.3456. This accounts for the number of ways to arrange three heads among five tosses.

Question 18Mathematics

What is the equation of the circle whose centre is (2, −3) and which passes through the point (6, 0)?

  1. (x − 2)² + (y + 3)² = 25 Correct answer.
  2. (x − 6)² + (y)² = 25
  3. (x − 2)² + (y + 3)² = 5
  4. (x + 2)² + (y − 3)² = 25

Explanation

The radius squared is (6 − 2)² + (0 + 3)² = 16 + 9 = 25. With centre (h, k) = (2, −3), the standard form is (x − 2)² + (y + 3)² = 25.

Question 19Mathematics

The limit lim (x→0) (sin 3x)/x equals which value?

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

Explanation

Using the standard limit lim (x→0) sin(kx)/x = k, the value here is 3. Equivalently, sin 3x ≈ 3x for small x, so the ratio approaches 3.

Question 20Mathematics

In triangle ABC, angle A = 30°, side a = 5 cm and side b = 8 cm. How many distinct triangles satisfy these conditions?

  1. None
  2. One
  3. Infinitely many
  4. Two Correct answer.

Explanation

The height from C is b·sin A = 8·sin 30° = 4 cm. Since 4 < a = 5 < b = 8, the ambiguous case of the sine rule gives two possible triangles.

Question 21Mathematics

Find the value of the real number k for which the vectors u = (k, 2, 1) and v = (1, k, −3) are perpendicular.

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

Explanation

Perpendicularity requires u·v = 0. Computing k·1 + 2·k + 1·(−3) = 3k − 3, setting it to zero gives k = 1.

Question 22Mathematics

The polynomial P(x) = x³ + ax² + bx + 6 has a factor (x − 1) and leaves remainder 12 when divided by (x − 2). What is the value of a + b?

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

Explanation

From P(1) = 0 we get 1 + a + b + 6 = 0, so a + b = −7. The remainder condition P(2) = 12 gives 8 + 4a + 2b + 6 = 12, i.e. 2a + b = −1, which is consistent with a + b = −7 and a = 6, b = −13.

Question 23Mathematics

Evaluate the integral ∫ x·e^(2x) dx using integration by parts.

  1. x·e^(2x) − e^(2x) + C
  2. (x/2)e^(2x) + (1/4)e^(2x) + C
  3. (x/2)e^(2x) − (1/4)e^(2x) + C Correct answer.
  4. (x²/2)e^(2x) + C

Explanation

Take u = x and dv = e^(2x) dx, so du = dx and v = e^(2x)/2. Then ∫x·e^(2x)dx = (x/2)e^(2x) − ∫(1/2)e^(2x)dx = (x/2)e^(2x) − (1/4)e^(2x) + C.

Question 24Mathematics

A geometric series has first term a = 4 and common ratio r = −1/2. What is the sum to infinity?

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

Explanation

Since |r| < 1, the sum to infinity is a/(1 − r) = 4/(1 + 1/2) = 4/(3/2) = 8/3. The alternating signs converge to this positive value.

Question 25Mathematics

A function f satisfies f′(x) = 3x² − 6x and f(0) = 5. What is the value of f(3)?

  1. −4
  2. 23
  3. 5 Correct answer.
  4. 14

Explanation

Integrating gives f(x) = x³ − 3x² + C. Using f(0) = 5 gives C = 5, so f(3) = 27 − 27 + 5 = 5.

Question 26Mathematics

The complex number z = 3 − 4i is given. What is the modulus |z|?

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

Explanation

The modulus of a complex number a + bi equals the square root of a² + b². Here a = 3 and b = −4, so |z| equals the square root of 9 + 16, which is the square root of 25, namely 5. The modulus measures distance from the origin in the Argand plane.

Question 27Mathematics

A student evaluates the permutation ⁵P₃ and the combination ⁵C₃. What is the difference ⁵P₃ − ⁵C₃?

  1. 10
  2. 60
  3. 54 Correct answer.
  4. 50

Explanation

⁵P₃ equals 5 × 4 × 3 = 60, and ⁵C₃ equals 60 divided by 3! = 60/6 = 10. Their difference is 60 − 10 = 54. The gap arises because permutations count orderings while combinations do not.

Question 28Mathematics

For the arithmetic sequence whose nth term is aₙ = 7 + 4(n − 1), what is the sum of the first 20 terms?

  1. 700
  2. 760
  3. 740
  4. 820 Correct answer.

Explanation

The first term is 7 and the twentieth term is 7 + 4(19) = 83. The sum of an arithmetic sequence is n/2 times (first + last), giving 20/2 × (7 + 83) = 10 × 90 = 820.

Question 29Mathematics

The mean of five observations is 12. A sixth observation equal to 24 is added. What is the mean of the six observations?

  1. 15
  2. 14 Correct answer.
  3. 16
  4. 18

Explanation

The original total is 5 × 12 = 60. Adding 24 gives a new total of 84 over 6 observations, so the new mean is 84 ÷ 6 = 14. The mean rises because the added value exceeds the original mean.

Question 30Mathematics

The matrix M = [[2, 1], [1, 2]] has eigenvalues λ. What is the sum of the two eigenvalues of M?

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

Explanation

The sum of the eigenvalues of a square matrix equals its trace, the sum of the diagonal entries. Here the trace is 2 + 2 = 4. The characteristic equation λ² − 4λ + 3 = 0 confirms eigenvalues 1 and 3, which sum to 4.

Question 31Mathematics

A curve is defined implicitly by x² + y² = 25. What is the slope of the tangent to the curve at the point (3, 4)?

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

Explanation

Differentiating implicitly gives 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y. Substituting x = 3 and y = 4 gives −3/4. This matches the tangent being perpendicular to the radius of slope 4/3.

Question 32Mathematics

A triangle has side lengths 7 cm and 9 cm with an included angle of 60°. What is the length of the third side?

  1. √103 cm
  2. √127 cm
  3. √67 cm Correct answer.
  4. √79 cm

Explanation

The cosine rule gives c² = a² + b² − 2ab·cos C = 49 + 81 − 2(7)(9)(0.5) = 130 − 63 = 67. Therefore the third side is √67 cm, roughly 8.19 cm.

Question 33Mathematics

A line L has direction vector d = (2, −1, 3) and passes through the point (1, 0, 4). Which point lies on L?

  1. (5, −2, 10) Correct answer.
  2. (3, 1, 7)
  3. (2, −1, 3)
  4. (4, −1, 9)

Explanation

Points on L have the form (1 + 2t, −t, 4 + 3t). Taking t = 2 gives (5, −2, 10). Checking each coordinate against a single value of t is the reliable test.

Question 34Mathematics

The function g(x) = (x² − 9)/(x − 3) is undefined at x = 3. What is the limit of g(x) as x approaches 3?

  1. The limit does not exist
  2. 0
  3. 6 Correct answer.
  4. 3

Explanation

Factorising the numerator gives (x − 3)(x + 3), so for x ≠ 3 the function simplifies to x + 3. As x approaches 3, this expression approaches 6. The removable discontinuity at x = 3 does not prevent the limit from existing.

Question 35Mathematics

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

  1. 4/25
  2. 1/5
  3. 2/15 Correct answer.
  4. 8/45

Explanation

The probability the first ball is red is 4/10, and given that, the second is 3/9. Multiplying gives (4/10)(3/9) = 12/90 = 2/15. Drawing without replacement changes the second probability.

Question 36Mathematics

The ellipse has equation x²/36 + y²/16 = 1. What is the distance between its two foci?

  1. 4√13
  2. 2√13
  3. 2√5
  4. 4√5 Correct answer.

Explanation

For an ellipse with a² = 36 and b² = 16, we have c² = a² − b² = 20, so c = 2√5. The foci are at (±c, 0), so the distance between them is 2c = 4√5.

Question 37Mathematics

In how many distinct ways can the letters of the word BANANA be arranged?

  1. 120
  2. 20
  3. 720
  4. 60 Correct answer.

Explanation

The word has 6 letters with A repeated 3 times and N repeated 2 times. The number of distinct arrangements is 6!/(3!·2!) = 720/(6·2) = 60. Dividing by the factorial of each repeated count removes identical permutations.

Question 38Mathematics

The polynomial P(x) = x⁴ − 2x³ + 3x² − 4x + 5 is divided by (x − 1). What is the remainder?

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

Explanation

By the remainder theorem, the remainder equals P(1). Substituting gives 1 − 2 + 3 − 4 + 5 = 3. No long division is needed once the theorem is applied.

Question 39Mathematics

Let z = 1 + i. What is the value of z⁶?

  1. 8
  2. −8i Correct answer.
  3. −8
  4. 8i

Explanation

Write z in polar form: 1 + i = √2(cos 45° + i sin 45°). By De Moivre's theorem, z⁶ = (√2)⁶(cos 270° + i sin 270°) = 8(0 − i) = −8i.

Question 40Mathematics

A differentiable function f has f(0) = 1, f(1) = 4 and f(2) = 9. Applying the mean value theorem on [0, 2], which value must occur as f′(c) for some c in (0, 2)?

  1. 9/2
  2. 5
  3. 3
  4. 4 Correct answer.

Explanation

The mean value theorem guarantees some c in (0, 2) with f′(c) equal to (f(2) − f(0))/(2 − 0) = (9 − 1)/2 = 4. The value 3 is the average rate on [0, 1] and 9/2 is not the secant slope on [0, 2].

Question 41Physics

A student measures the mass of a metal block as 250 g and its volume as 50 cm³. What is the density of the metal in kg/m³?

  1. 500 kg/m³
  2. 0.2 kg/m³
  3. 5000 kg/m³ Correct answer.
  4. 5 kg/m³

Explanation

Density equals mass divided by volume. Converting 250 g to 0.25 kg and 50 cm³ to 5×10⁻⁵ m³ gives 0.25 ÷ 5×10⁻⁵ = 5000 kg/m³. The gram-to-kilogram and cm³-to-m³ conversions must both be applied.

Question 42Physics

A car accelerates uniformly from rest to 20 m/s in 8 s. What distance does it travel during this interval?

  1. 80 m Correct answer.
  2. 40 m
  3. 2.5 m
  4. 160 m

Explanation

For uniform acceleration from rest, distance equals average velocity multiplied by time. The average velocity is (0 + 20)/2 = 10 m/s, so the distance is 10 × 8 = 80 m. Using s = ut + ½at² with a = 2.5 m/s² gives the same result.

Question 43Physics

Which quantity is a scalar?

  1. Acceleration
  2. Work Correct answer.
  3. Displacement
  4. Velocity

Explanation

A scalar has magnitude only, while a vector has both magnitude and direction. Work is defined as the product of force and displacement along the force, and it is fully described by a number with a unit. Displacement, velocity and acceleration all require a stated direction.

Question 44Physics

Two forces of 6 N and 8 N act at a single point at right angles to each other. What is the magnitude of their resultant?

  1. 48 N
  2. 2 N
  3. 14 N
  4. 10 N Correct answer.

Explanation

Perpendicular vectors combine by the Pythagorean theorem, so the resultant is the square root of (6² + 8²) = 36 + 64 = 100, giving 10 N. Adding the magnitudes directly would only be valid if the forces pointed the same way.

Question 45Physics

A 2 kg object moving at 3 m/s collides head-on with a stationary 4 kg object, and they stick together. What is their common velocity after the collision?

  1. 1 m/s Correct answer.
  2. 1.5 m/s
  3. 0.5 m/s
  4. 3 m/s

Explanation

Momentum is conserved in a perfectly inelastic collision. Initial momentum is 2 × 3 = 6 kg·m/s, and the combined mass is 6 kg, so the common velocity is 6 ÷ 6 = 1 m/s. Kinetic energy is not conserved here, so it cannot be used to find the answer.

Question 46Physics

A constant force of 20 N pulls a box 5 m along a horizontal floor while friction of 8 N acts on it. What is the net work done on the box?

  1. 60 J Correct answer.
  2. 40 J
  3. 140 J
  4. 100 J

Explanation

The net force along the motion is 20 − 8 = 12 N. Net work equals net force times displacement, so 12 × 5 = 60 J. The work done by friction is negative because friction opposes the displacement.

Question 47Physics

A machine lifts a 300 N load through 2 m in 4 s. What is its useful output power?

  1. 75 W
  2. 2400 W
  3. 600 W
  4. 150 W Correct answer.

Explanation

Useful work output is force times distance, 300 × 2 = 600 J. Power is work divided by time, so 600 ÷ 4 = 150 W. Power measures the rate of energy transfer, not the total energy.

Question 48Physics

A spring obeys Hooke's law with a spring constant of 200 N/m. What force is needed to stretch it by 0.15 m?

  1. 3000 N
  2. 15 N
  3. 1333 N
  4. 30 N Correct answer.

Explanation

Hooke's law states that the restoring force equals the spring constant times the extension, F = kx. Substituting k = 200 N/m and x = 0.15 m gives 30 N. The extension must be in metres for the units to be consistent.

Question 49Physics

A simple pendulum has a period of 2 s on Earth. If it is taken to a planet where the gravitational field strength is four times larger, what is its new period?

  1. 1 s Correct answer.
  2. 0.5 s
  3. 4 s
  4. 8 s

Explanation

The period of a simple pendulum depends on length and gravitational field strength as T = 2π√(L/g). Quadrupling g divides the period by the square root of four, which is two. The new period is therefore 2 ÷ 2 = 1 s.

Question 50Physics

A wave has a frequency of 50 Hz and a wavelength of 4 m. What is its speed?

  1. 12.5 m/s
  2. 200 m/s Correct answer.
  3. 0.08 m/s
  4. 54 m/s

Explanation

The wave equation relates speed, frequency and wavelength as v = fλ. Multiplying 50 Hz by 4 m gives 200 m/s. Frequency is the number of waves passing a point per second, and wavelength is the distance between successive crests.