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AEEE practice questions

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Question 1Mathematics

For the function f(x) = (x² − 1)/(x − 1) defined for x ≠ 1, what is the limit of f(x) as x approaches 1?

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

Explanation

Factor the numerator as (x − 1)(x + 1), so for x ≠ 1 the function equals x + 1. Substituting x = 1 into x + 1 gives 2. The removable discontinuity at x = 1 does not affect the limit.

Question 2Mathematics

A function f is continuous at x = a. Which statement must be true?

  1. f(a) = 0
  2. f(a) is defined and equals the limit of f(x) as x approaches a Correct answer.
  3. f has a local maximum at x = a
  4. f'(a) exists and is finite

Explanation

Continuity at a point requires three things: f(a) is defined, the two-sided limit exists, and the limit equals f(a). Differentiability is stronger and not implied, and no extremum or zero value is required.

Question 3Mathematics

What is the derivative of f(x) = x³ − 3x with respect to x?

  1. 3x² − 3 Correct answer.
  2. 3x³ − 3
  3. 3x² − 3x
  4. x² − 3

Explanation

Apply the power rule term by term: the derivative of x³ is 3x², and the derivative of −3x is −3. The constant multiple rule keeps the coefficient on the linear term. Hence f'(x) = 3x² − 3.

Question 4Mathematics

Evaluate the definite integral of 2x from x = 0 to x = 3.

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

Explanation

An antiderivative of 2x is x². Evaluating x² at the limits gives 3² − 0² = 9. This also matches the area of the triangle under the line y = 2x from 0 to 3.

Question 5Mathematics

The position of a particle moving along a line is s(t) = t² − 4t + 1, with s in metres and t in seconds. What is the velocity at t = 3 s?

  1. −1 m/s
  2. 2 m/s Correct answer.
  3. −4 m/s
  4. 6 m/s

Explanation

Velocity is the derivative of position: s'(t) = 2t − 4. Substituting t = 3 gives 2(3) − 4 = 2 m/s. The positive sign indicates motion in the positive direction.

Question 6Mathematics

What is the value of the determinant of the 2×2 matrix with rows (3, 4) and (1, 2)?

  1. 2 Correct answer.
  2. 5
  3. 6
  4. 10

Explanation

For a matrix with rows (a, b) and (c, d) the determinant is ad − bc. Here that is 3·2 − 4·1 = 6 − 4 = 2. A nonzero determinant shows the matrix is invertible.

Question 7Mathematics

If the roots of the quadratic equation x² − 5x + 6 = 0 are α and β, what is the value of α + β?

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

Explanation

For ax² + bx + c = 0 with roots α and β, the sum of roots equals −b/a. Here a = 1 and b = −5, so α + β = 5. The product would be c/a = 6.

Question 8Mathematics

The arithmetic sequence has first term 7 and common difference 4. What is the 10th term?

  1. 47
  2. 43 Correct answer.
  3. 36
  4. 40

Explanation

The nth term of an arithmetic sequence is a + (n − 1)d. With a = 7, d = 4 and n = 10, the term is 7 + 9·4 = 43. The index must be n − 1 because the first term already accounts for one step.

Question 9Mathematics

What is the value of the limit of (sin 5x)/x as x approaches 0?

  1. 0
  2. 1
  3. The limit does not exist
  4. 5 Correct answer.

Explanation

Use the standard limit sin θ/θ → 1 as θ → 0. Writing (sin 5x)/x as 5·(sin 5x)/(5x) shows the limit is 5·1 = 5. The inner argument must match the denominator for the standard result to apply.

Question 10Mathematics

The function f(x) = x³ − 3x² + 4 is defined on the whole real line. At which value of x does f have a local minimum?

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

Explanation

Setting f'(x) = 3x² − 6x = 3x(x − 2) to zero gives critical points x = 0 and x = 2. The second derivative 6x − 6 equals 6 at x = 2, which is positive, so x = 2 is a local minimum.

Question 11Mathematics

A curve is given parametrically by x = t², y = t³. What is dy/dx at t = 2?

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

Explanation

For parametric curves, dy/dx = (dy/dt)/(dx/dt). Here dy/dt = 3t² and dx/dt = 2t, so dy/dx = 3t/2. At t = 2 this equals 3.

Question 12Mathematics

Evaluate the integral of x·e^x with respect to x.

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

Explanation

Integrate by parts with u = x and dv = e^x dx, giving ∫x e^x dx = x e^x − ∫e^x dx = x e^x − e^x + C, which factors as (x − 1)e^x + C. Differentiating the result returns x e^x, confirming it.

Question 13Mathematics

A 3×3 matrix A satisfies A² = A and A is not the identity matrix. Which statement about A is necessarily true?

  1. A is invertible with inverse A
  2. A is the zero matrix
  3. The determinant of A equals 1
  4. The determinant of A equals 0 Correct answer.

Explanation

From A² = A we get A(A − I) = 0. If A were invertible, multiplying by A⁻¹ would force A = I, contradicting the hypothesis. Hence A is singular and det A = 0.

Question 14Mathematics

The system of equations x + 2y = 3 and 2x + 4y = 7 has which type of solution set?

  1. Exactly two solutions
  2. Exactly one solution
  3. No solution Correct answer.
  4. Infinitely many solutions

Explanation

The second equation is twice the left side of the first, but its right side 7 is not twice 3. The equations are therefore inconsistent, representing parallel lines. A linear system with two variables never has exactly two solutions.

Question 15Mathematics

What is the value of the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + …?

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

Explanation

The first term is a = 1 and the ratio is r = 1/3, with |r| < 1 so the series converges. Its sum is a/(1 − r) = 1/(1 − 1/3) = 3/2.

Question 16Mathematics

If z = 1 + i√3, what is the modulus |z| and the argument of z (in radians, principal value)?

  1. |z| = 4, arg z = π/3
  2. |z| = 2, arg z = π/3 Correct answer.
  3. |z| = 2, arg z = π/6
  4. |z| = √3, arg z = π/3

Explanation

The modulus is √(1² + (√3)²) = √4 = 2. The argument satisfies tan θ = √3/1 with the point in the first quadrant, so θ = π/3. Thus z = 2(cos π/3 + i sin π/3).

Question 17Mathematics

A fair die is rolled twice. What is the probability that the sum of the two outcomes equals 7?

  1. 1/12
  2. 7/36
  3. 1/9
  4. 1/6 Correct answer.

Explanation

There are 36 equally likely ordered outcomes. Sums of 7 occur for (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), six outcomes in total. The probability is therefore 6/36 = 1/6.

Question 18Mathematics

What is the coefficient of x⁵ in the binomial expansion of (1 + x)⁸?

  1. 56 Correct answer.
  2. 70
  3. 28
  4. 8

Explanation

The general term is C(8, k)x^k, so the coefficient of x⁵ is C(8, 5). By symmetry C(8, 5) = C(8, 3) = 56.

Question 19Mathematics

The curve y = x³ − 3x + 2 has a tangent line at x = 1. What is the slope of that tangent?

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

Explanation

The slope of the tangent is the derivative y' = 3x² − 3 evaluated at x = 1, giving 3 − 3 = 0. A zero slope means the tangent is horizontal, and indeed x = 1 is a stationary point here.

Question 20Mathematics

For the function f(x) = x⁴ − 4x³, find all x where the second derivative f''(x) equals zero.

  1. x = 0 and x = 2 Correct answer.
  2. x = 0 and x = 3
  3. x = 2 only
  4. x = 0 only

Explanation

First f'(x) = 4x³ − 12x², then f''(x) = 12x² − 24x = 12x(x − 2). Setting this to zero gives x = 0 and x = 2. These are candidate inflection points.

Question 21Mathematics

A particle's velocity is v(t) = 3t² − 6t for t ≥ 0, and its position at t = 0 is s(0) = 2. What is s(3)?

  1. 11 Correct answer.
  2. 9
  3. 2
  4. 29

Explanation

Integrate velocity: s(t) = t³ − 3t² + C. Using s(0) = 2 gives C = 2, so s(3) = 27 − 27 + 2 = 2 + 9 = 11. The constant of integration must be fixed by the initial condition.

Question 22Mathematics

Consider f(x) = x·sin(1/x) for x ≠ 0, with f(0) = 0. Which statement about f at x = 0 is correct?

  1. f is differentiable at 0 with f'(0) = 1
  2. f is discontinuous at 0
  3. f is differentiable at 0 with f'(0) = 0
  4. f is continuous but not differentiable at 0 Correct answer.

Explanation

Since |x sin(1/x)| ≤ |x|, the squeeze theorem gives f(x) → 0 = f(0), so f is continuous. The difference quotient [f(h) − f(0)]/h = sin(1/h) oscillates without a limit as h → 0, so f is not differentiable at 0.

Question 23Mathematics

Evaluate the definite integral of x²·ln x from x = 1 to x = e.

  1. e³/9
  2. (2e³ + 1)/9 Correct answer.
  3. (e³ − 1)/3
  4. (2e³ − 1)/9

Explanation

Integrate by parts with u = ln x and dv = x² dx, giving ∫x² ln x dx = (x³/3)ln x − x³/9. At x = e this is e³/3 − e³/9 = 2e³/9, and at x = 1 it is −1/9, so the definite integral is (2e³ + 1)/9.

Question 24Mathematics

The matrix A has rows (2, 1) and (1, 2). Which statement about its eigenvalues is correct?

  1. Both eigenvalues equal 2
  2. The eigenvalues are complex conjugates
  3. The eigenvalues are 1 and 3 Correct answer.
  4. The eigenvalues are 2 and 4

Explanation

The characteristic equation is det(A − λI) = (2 − λ)² − 1 = 0, so (2 − λ)² = 1 and λ = 2 ± 1, giving 1 and 3. Their sum 4 equals the trace and their product 3 equals the determinant, confirming the result.

Question 25Mathematics

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

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

Explanation

Multiply numerator and denominator by (1 + cos x) to get (sin² x)/(x²(1 + cos x)). Since (sin x)/x → 1 and 1 + cos x → 2, the limit is 1/2. Alternatively, the Taylor expansion cos x = 1 − x²/2 + … gives the same value.

Question 26Mathematics

A vector has components (3, 4) in the plane. What is its magnitude?

  1. 25
  2. 7
  3. 12
  4. 5 Correct answer.

Explanation

The magnitude of a vector (a, b) is √(a² + b²). Here √(9 + 16) = √25 = 5.

Question 27Mathematics

For the complex number z = 2 − 2i, what is the value of z·z̄ (z times its conjugate)?

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

Explanation

Multiplying by the conjugate gives the squared modulus: (2 − 2i)(2 + 2i) = 4 + 4 = 8. In general z·z̄ = |z|².

Question 28Mathematics

What is the derivative of y = sin(3x) with respect to x?

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

Explanation

The chain rule multiplies the outer derivative cos(3x) by the inner derivative 3. Hence dy/dx = 3cos(3x).

Question 29Mathematics

A bag contains 4 red and 6 blue balls. One ball is drawn at random. What is the probability that it is red?

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

Explanation

There are 4 favourable outcomes out of 10 total, so the probability is 4/10 = 2/5.

Question 30Mathematics

What is the value of the integral of 1/x with respect to x, for x > 0?

  1. e^x + C
  2. −1/x² + C
  3. x²/2 + C
  4. ln x + C Correct answer.

Explanation

The antiderivative of 1/x on the positive reals is the natural logarithm ln x, plus an arbitrary constant. Differentiating ln x returns 1/x.

Question 31Mathematics

The 3×3 identity matrix I is multiplied on the right by any 3×3 matrix B. What is the result?

  1. The zero matrix
  2. 3B
  3. B Correct answer.
  4. I

Explanation

The identity matrix satisfies IB = BI = B for every conformable matrix B. Multiplication by I leaves the matrix unchanged.

Question 32Mathematics

What is the value of cos(π/3)?

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

Explanation

The angle π/3 radians equals 60 degrees, and cos 60° = 1/2. This is a standard exact value.

Question 33Mathematics

What is the equation of the circle with centre (2, −1) and radius 3?

  1. (x − 2)² + (y − 1)² = 9
  2. (x + 2)² + (y − 1)² = 9
  3. (x − 2)² + (y + 1)² = 9 Correct answer.
  4. (x − 2)² + (y + 1)² = 3

Explanation

A circle with centre (h, k) and radius r has equation (x − h)² + (y − k)² = r². Substituting h = 2, k = −1, r = 3 gives (x − 2)² + (y + 1)² = 9.

Question 34Mathematics

What is the sum of the first 20 positive integers, 1 + 2 + … + 20?

  1. 200
  2. 210 Correct answer.
  3. 190
  4. 420

Explanation

The sum of the first n positive integers is n(n + 1)/2. With n = 20 this gives 20·21/2 = 210.

Question 35Mathematics

The polynomial p(x) = x³ − 4x² + x + 6 has x = 2 as a root. What is the sum of the other two roots?

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

Explanation

By Vieta's formulas the sum of all three roots equals 4, the negative of the x² coefficient. Subtracting the known root 2 leaves 2 for the other two.

Question 36Mathematics

A function f is defined by f(x) = (x² − 9)/(x − 3) for x ≠ 3. Which value must be assigned to f(3) to make f continuous at x = 3?

  1. 0
  2. No value can make f continuous there
  3. 6 Correct answer.
  4. 3

Explanation

Factorising the numerator gives (x − 3)(x + 3), so f(x) = x + 3 for x ≠ 3. The limit at x = 3 is therefore 6, and setting f(3) = 6 removes the removable discontinuity.

Question 37Mathematics

What is the value of the determinant of the 3×3 matrix whose rows are (1, 2, 3), (0, 1, 4) and (5, 6, 0)?

  1. 0
  2. −1
  3. 1 Correct answer.
  4. −49

Explanation

Expanding along the first row: 1·(1·0 − 4·6) − 2·(0·0 − 4·5) + 3·(0·6 − 1·5) = −24 + 40 − 15 = 1.

Question 38Mathematics

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

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

Explanation

Implicit differentiation gives 2x + 2y·dy/dx = 0, so dy/dx = −x/y. At (3, 4) the slope is −3/4.

Question 39Mathematics

What is the value of the integral of cos²x with respect to x over one full period, that is from 0 to π?

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

Explanation

Using cos²x = (1 + cos 2x)/2, the integral over [0, π] is [x/2 + sin 2x/4] evaluated, giving π/2. The average value of cos²x over a period is 1/2.

Question 40Mathematics

A geometric sequence has first term 2 and common ratio −1/2. What is the sum to infinity of the series?

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

Explanation

A geometric series converges when |r| < 1, and its sum is a/(1 − r). Here 2/(1 + 1/2) = 2/(3/2) = 4/3.

Question 41Mathematics

If tan θ = 3/4 and θ lies in the first quadrant, what is the value of sin 2θ?

  1. 7/25
  2. 12/25
  3. 24/7
  4. 24/25 Correct answer.

Explanation

With tan θ = 3/4 in the first quadrant, sin θ = 3/5 and cos θ = 4/5. Then sin 2θ = 2 sin θ cos θ = 2·(3/5)·(4/5) = 24/25.

Question 42Mathematics

Two events A and B satisfy P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. What is P(A ∪ B)?

  1. 0.9
  2. 0.1
  3. 0.7 Correct answer.
  4. 0.2

Explanation

The addition rule gives P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.5 + 0.4 − 0.2 = 0.7.

Question 43Mathematics

What is the value of the limit of (e^x − 1 − x)/x² as x approaches 0?

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

Explanation

Expanding e^x = 1 + x + x²/2 + …, the numerator becomes x²/2 plus higher-order terms. Dividing by x² and letting x → 0 leaves 1/2.

Question 44Mathematics

The plane vectors a = (2, −1) and b = (1, 3) are given. What is the value of the dot product a·b?

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

Explanation

The dot product is the sum of componentwise products: 2·1 + (−1)·3 = 2 − 3 = −1.

Question 45Mathematics

A fair coin is tossed 6 times. What is the probability of obtaining exactly 2 heads?

  1. 1/3
  2. 15/64 Correct answer.
  3. 20/64
  4. 6/64

Explanation

The number of heads follows a binomial distribution with n = 6 and p = 1/2. The probability is C(6,2)/2⁶ = 15/64.

Question 46Mathematics

The matrix A has rows (1, 2) and (2, 1). What are its eigenvalues?

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

Explanation

The characteristic equation is (1 − λ)² − 4 = 0, so 1 − λ = ±2, giving λ = 3 and λ = −1. These also match the trace 2 and determinant −3.

Question 47Mathematics

For what value of k does the system x + y = 2 and kx + 2y = 5 have no solution?

  1. k = 1
  2. k = 5
  3. k = 2 Correct answer.
  4. k = 0

Explanation

No solution occurs when the coefficient rows are proportional but the constants are not. The rows (1, 1) and (k, 2) are proportional only when k = 2, and then the right sides 2 and 5 disagree.

Question 48Mathematics

A function satisfies f'(x) = 3x² − 6x and f(0) = 1. What is the value of f(2)?

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

Explanation

Integrating gives f(x) = x³ − 3x² + C, and f(0) = 1 forces C = 1. Then f(2) = 8 − 12 + 1 = −3.

Question 49Mathematics

The cubic f(x) = x³ + ax + b has a local maximum value of 2 at x = −1. What are the values of a and b?

  1. a = −3, b = 0 Correct answer.
  2. a = 3, b = 4
  3. a = −3, b = 4
  4. a = −1, b = 2

Explanation

A stationary point at x = −1 requires f'(−1) = 3 + a = 0, so a = −3. Then f(−1) = −1 + 3 + b = 2 gives b = 0, and the second derivative −6 < 0 confirms a maximum.

Question 50Mathematics

A complex number z satisfies z + 2z̄ = 6 + 3i. What is z?

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

Explanation

Write z = x + iy, so z + 2z̄ = 3x − iy. Matching real and imaginary parts gives 3x = 6 and −y = 3, hence x = 2 and y = −3, so z = 2 − 3i.