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ECE Board Exam study materials
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ECE Board Exam practice questions

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Question 1Mathematics

What is the value of log₃ 81?

  1. 27
  2. 4 Correct answer.
  3. 9
  4. 3

Explanation

The logarithm log₃ 81 asks for the exponent x such that 3^x = 81. Since 3⁴ = 81, the value is 4.

Question 2Mathematics

Which pair of numbers are the roots of x² − 5x + 6 = 0?

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

Explanation

For a quadratic with leading coefficient 1, the roots must add to the coefficient of x with opposite sign (5) and multiply to the constant term (6). The pair 2 and 3 satisfies both conditions, since (x − 2)(x − 3) expands to x² − 5x + 6.

Question 3Mathematics

For any angle θ, the expression sin²θ + cos²θ simplifies to which value?

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

Explanation

The Pythagorean identity states that the squares of the sine and cosine of the same angle always sum to 1. This follows from the definitions of sine and cosine on a unit circle of radius 1.

Question 4Mathematics

A circle has a radius of 5 cm. What is its area?

  1. 25π cm² Correct answer.
  2. 50π cm²
  3. 10π cm²
  4. 5π cm²

Explanation

The area of a circle is πr². Substituting r = 5 gives π(5)² = 25π square centimeters.

Question 5Mathematics

If i² = −1, which of the following is equal to i³?

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

Explanation

Write i³ as i² · i and substitute i² = −1. This gives (−1)(i) = −i.

Question 6Mathematics

An arithmetic sequence begins 3, 7, 11, 15, ... What is its 10th term?

  1. 39 Correct answer.
  2. 35
  3. 31
  4. 43

Explanation

The first term is 3 and the common difference is 4. Using aₙ = a₁ + (n − 1)d with n = 10 gives 3 + 9(4) = 39.

Question 7Mathematics

What is the derivative of 3x² with respect to x?

  1. 6x Correct answer.
  2. 6x²
  3. 9x²
  4. 3x

Explanation

The power rule states that the derivative of axⁿ is anxⁿ⁻¹. Here a = 3 and n = 2, so the derivative is 3 · 2 · x¹ = 6x.

Question 8Mathematics

Evaluate lim (4x − 5) as x approaches 3.

  1. 7 Correct answer.
  2. 6
  3. 4
  4. 5

Explanation

A polynomial is continuous everywhere, so the limit equals the function value at x = 3. Substituting gives 4(3) − 5 = 7.

Question 9Mathematics

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

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

Explanation

By the binomial theorem, the term containing x³ is C(5,3)x³(2)². This gives 10 · x³ · 4 = 40x³, so the coefficient is 40.

Question 10Mathematics

How many distinct arrangements can be made from all the letters of the word LEVEL?

  1. 30 Correct answer.
  2. 20
  3. 120
  4. 60

Explanation

The word has 5 letters with L repeated twice and E repeated twice. The number of distinct arrangements is 5!/(2!·2!) = 120/4 = 30.

Question 11Mathematics

Evaluate the definite integral of (2x + 1) dx from x = 0 to x = 1.

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

Explanation

An antiderivative of 2x + 1 is x² + x. Evaluating from 0 to 1 gives (1 + 1) − (0 + 0) = 2.

Question 12Mathematics

What is the determinant of the 2×2 matrix whose first row is (3, 4) and second row is (2, 5)?

  1. 23
  2. 7 Correct answer.
  3. 15
  4. 11

Explanation

For a 2×2 matrix with entries a, b in the first row and c, d in the second, the determinant is ad − bc. Here that is (3)(5) − (4)(2) = 15 − 8 = 7.

Question 13Mathematics

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

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

Explanation

The prime outcomes on a die are 2, 3, and 5, giving three favorable outcomes out of six equally likely ones. The probability is 3/6 = 1/2.

Question 14Mathematics

If sin θ = 3/5 and θ lies in the first quadrant, what is cos θ?

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

Explanation

Apply the identity sin²θ + cos²θ = 1: cos²θ = 1 − 9/25 = 16/25. In the first quadrant cosine is positive, so cos θ = 4/5.

Question 15Mathematics

An amount P is invested at 10% per year compounded annually. What is the accumulated amount after 2 years?

  1. 1.21P Correct answer.
  2. 1.20P
  3. 1.22P
  4. 1.10P

Explanation

Compound interest follows A = P(1 + r)ᵗ. With r = 0.10 and t = 2, A = P(1.10)² = 1.21P.

Question 16Mathematics

Which equation represents the line that passes through the point (2, 3) and has slope 2?

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

Explanation

Use the point-slope form y − y₁ = m(x − x₁) with m = 2 and (x₁, y₁) = (2, 3). This gives y − 3 = 2(x − 2), which simplifies to y = 2x − 1.

Question 17Mathematics

For the vectors A = (2, −1, 3) and B = (1, 4, 2), what is the dot product A · B?

  1. 0
  2. −8
  3. 12
  4. 4 Correct answer.

Explanation

The dot product is the sum of the products of corresponding components: (2)(1) + (−1)(4) + (3)(2) = 2 − 4 + 6 = 4.

Question 18Mathematics

What is the sum of the infinite geometric series 4 + 2 + 1 + 1/2 + ...?

  1. 10
  2. 8 Correct answer.
  3. 16
  4. 6

Explanation

Here the first term is 4 and the common ratio is 1/2, so the sum is a/(1 − r) = 4/(1 − 1/2) = 8.

Question 19Mathematics

What is the sum of the roots of 2x² − 8x + 6 = 0?

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

Explanation

By Vieta's formulas, the sum of the roots of ax² + bx + c = 0 is −b/a. Here that is −(−8)/2 = 4.

Question 20Mathematics

What is (1 + i)⁴ in simplest rectangular form?

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

Explanation

First square: (1 + i)² = 1 + 2i + i² = 2i. Squaring again gives (2i)² = 4i² = −4.

Question 21Mathematics

Which function satisfies dy/dx = 2y together with the condition y = 3 when x = 0?

  1. y = 3e^(2x) Correct answer.
  2. y = 2e^(3x)
  3. y = 3e^(−2x)
  4. y = 6x + 3

Explanation

Separating variables gives dy/y = 2 dx, so ln y = 2x + C and y = Ae^(2x). Applying y(0) = 3 gives A = 3, hence y = 3e^(2x).

Question 22Mathematics

What is the Laplace transform of the function e^(2t) sin 3t?

  1. (s − 2)/((s − 2)² + 9)
  2. 3/((s − 2)² + 9) Correct answer.
  3. 3/(s² + 9)
  4. 3/((s + 2)² + 9)

Explanation

Multiplication by e^(at) shifts the transform of the base function by a, replacing s with s − a. Since the transform of sin 3t is 3/(s² + 9), the result is 3/((s − 2)² + 9).

Question 23Mathematics

A committee of 3 is chosen at random from a group of 5 men and 4 women. What is the probability that all three members chosen are women?

  1. 1/21 Correct answer.
  2. 2/7
  3. 1/84
  4. 4/21

Explanation

The total number of 3-person committees is C(9,3) = 84, and the number made up entirely of women is C(4,3) = 4. The probability is therefore 4/84 = 1/21.

Question 24Mathematics

What is the maximum value of f(x) = 12x − x³ on the closed interval from 0 to 3?

  1. 0
  2. 16 Correct answer.
  3. 9
  4. 24

Explanation

Setting f′(x) = 12 − 3x² = 0 gives the critical point x = 2 inside the interval. Comparing f(0) = 0, f(2) = 16, and f(3) = 9, the largest value is 16.

Question 25Mathematics

Exam scores are normally distributed with mean 70 and standard deviation 8. What is the probability that a randomly chosen score exceeds 86?

  1. 0.0456
  2. 0.9772
  3. 0.0228 Correct answer.
  4. 0.1587

Explanation

The z-score is (86 − 70)/8 = 2. The area to the right of z = 2 under the standard normal curve is 0.0228.

Question 26Mathematics

What is the value of log₂ 32?

  1. 6
  2. 4
  3. 5 Correct answer.
  4. 8

Explanation

A logarithm answers the question of what exponent on the base gives the argument. Since 2⁵ = 32, log₂ 32 = 5.

Question 27Mathematics

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

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

Explanation

The modulus of a + bi is √(a² + b²). Here √(3² + (−4)²) = √25 = 5.

Question 28Mathematics

What is the value of the binomial coefficient ⁷C₃?

  1. 35 Correct answer.
  2. 42
  3. 28
  4. 21

Explanation

⁷C₃ = 7!/(3!·4!) = (7·6·5)/(3·2·1) = 35. It counts the 3-element subsets of a 7-element set.

Question 29Mathematics

What is the derivative of e^(3x) with respect to x?

  1. 3x·e^(3x)
  2. e^(3x)/3
  3. 3e^(3x) Correct answer.
  4. e^(3x)

Explanation

The chain rule requires multiplying by the derivative of the exponent. Since d/dx(3x) = 3, the derivative is 3e^(3x).

Question 30Mathematics

A line has slope 2/3. What is the slope of any line perpendicular to it?

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

Explanation

Perpendicular slopes are negative reciprocals, so their product is −1. The negative reciprocal of 2/3 is −3/2.

Question 31Mathematics

What is the value of sin 30° + cos 60°?

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

Explanation

sin 30° = 1/2 and cos 60° = 1/2 because the two angles are complementary. Their sum is therefore 1.

Question 32Mathematics

What is the sum of the first 20 positive integers?

  1. 220
  2. 190
  3. 200
  4. 210 Correct answer.

Explanation

An arithmetic series with first term 1 and last term 20 sums to n(n + 1)/2. Substituting n = 20 gives 20·21/2 = 210.

Question 33Mathematics

A geometric sequence begins 2, 6, 18, ... What is its fifth term?

  1. 162 Correct answer.
  2. 108
  3. 54
  4. 81

Explanation

The common ratio is 6/2 = 3, and the nth term is a₁r^(n−1). Thus a₅ = 2·3⁴ = 2·81 = 162.

Question 34Mathematics

Evaluate lim (x→0) [sin(5x)]/x.

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

Explanation

Rewrite the expression as 5·sin(5x)/(5x) and use the standard limit sin u/u → 1 as u → 0. The result is 5·1 = 5.

Question 35Mathematics

Evaluate ∫ x·e^(x²) dx.

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

Explanation

Let u = x², so du = 2x dx and x dx = du/2. The integral becomes (1/2)∫e^u du = (1/2)e^(x²) + C.

Question 36Mathematics

For A = (2, 1, 0) and B = (1, 3, 0), what is |A × B|, the magnitude of the cross product?

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

Explanation

The cross product is (1·0 − 0·3, 0·1 − 2·0, 2·3 − 1·1) = (0, 0, 5). Its magnitude is 5, equal to the area of the parallelogram formed by A and B.

Question 37Mathematics

For two events A and B, P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. What is P(A | B)?

  1. 0.4
  2. 0.2
  3. 0.1
  4. 0.5 Correct answer.

Explanation

Conditional probability is defined as P(A | B) = P(A ∩ B)/P(B). Dividing 0.2 by 0.4 gives 0.5.

Question 38Mathematics

The ellipse x²/9 + y²/16 = 1 has its foci at which points?

  1. (0, ±7)
  2. (0, ±√7) Correct answer.
  3. (±7, 0)
  4. (±√7, 0)

Explanation

The larger denominator 16 lies under y², so the major axis is vertical with a² = 16 and b² = 9. Then c² = a² − b² = 7, giving foci (0, ±√7).

Question 39Mathematics

Solve the system 2x + y = 7 and x − y = 2.

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

Explanation

Adding the two equations eliminates y and gives 3x = 9, so x = 3. Substituting back gives 6 + y = 7, so y = 1.

Question 40Mathematics

What is the variance of the data set 2, 4, 6, 8, 10?

  1. 8 Correct answer.
  2. 10
  3. 4
  4. 6

Explanation

The mean is 6, and the squared deviations are 16, 4, 0, 4, 16. Their average is 40/5 = 8, which is the variance.

Question 41Mathematics

Which of the following infinite series converges?

  1. Σ 1/√n
  2. Σ 1/n^(1/3)
  3. Σ 1/n
  4. Σ 1/n² Correct answer.

Explanation

A p-series Σ 1/n^p converges exactly when p > 1. Only the exponent 2 exceeds 1, so that is the convergent series.

Question 42Mathematics

In the expansion of (1 + 2x)⁶, what is the coefficient of x²?

  1. 30
  2. 120
  3. 15
  4. 60 Correct answer.

Explanation

The general term is ⁶C_k (2x)^k. For k = 2 this gives ⁶C₂·4 = 15·4 = 60.

Question 43Mathematics

What is the angle between the vectors (1, 1, 0) and (1, 0, 0)?

  1. 45° Correct answer.
  2. 60°
  3. 30°
  4. 90°

Explanation

The dot product is 1 and both vectors have magnitude √2 and 1 respectively. So cos θ = 1/√2, giving θ = 45°.

Question 44Mathematics

Evaluate ∫ from 0 to 1 of dx/(1 + x²).

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

Explanation

The antiderivative is arctan x. Evaluating from 0 to 1 gives arctan 1 − arctan 0 = π/4.

Question 45Mathematics

If 1/(x² − 1) = A/(x − 1) + B/(x + 1), what is the value of A?

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

Explanation

Multiplying through gives 1 = A(x + 1) + B(x − 1). Setting x = 1 yields 1 = 2A, so A = 1/2.

Question 46Mathematics

Evaluate the improper integral ∫ from 1 to ∞ of x^(−2) dx.

  1. 1/2
  2. 2
  3. The integral diverges
  4. 1 Correct answer.

Explanation

Integrating gives [−1/x] from 1 to b, which equals 1 − 1/b. Taking the limit as b → ∞ leaves 1, so the integral converges.

Question 47Mathematics

What is the inverse Laplace transform of 1/(s² + 4s + 5)?

  1. e^(2t) sin t
  2. e^(−t) sin 2t
  3. e^(−2t) cos t
  4. e^(−2t) sin t Correct answer.

Explanation

Completing the square gives (s + 2)² + 1, which matches the shifted form 1/((s − a)² + ω²). With a = −2 and ω = 1, the inverse is e^(−2t) sin t.

Question 48Mathematics

Solve y″ − 3y′ + 2y = 0 subject to y(0) = 1 and y′(0) = 0.

  1. y = e^t − e^(2t)
  2. y = 2e^(2t) − e^t
  3. y = 2e^t − e^(2t) Correct answer.
  4. y = e^t + e^(2t)

Explanation

The characteristic equation r² − 3r + 2 = 0 has roots 1 and 2, so y = Ae^t + Be^(2t). The conditions give A + B = 1 and A + 2B = 0, hence A = 2 and B = −1.

Question 49Mathematics

What is the area of the region enclosed by the polar curve r = 2 cos θ?

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

Explanation

The curve r = 2cos θ traces a circle of radius 1 centered at (1, 0) in the plane. Its enclosed area is therefore π·1² = π, consistent with the formula (1/2)∫r² dθ over one full trace.

Question 50Mathematics

What are the eigenvalues of the matrix whose rows are (2, 1) and (1, 2)?

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

Explanation

Solve det(A − λI) = (2 − λ)² − 1 = 0, which factors as (1 − λ)(3 − λ) = 0. The eigenvalues are therefore 1 and 3.