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FE Mechanical practice questions

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Question 1Mathematics

A simply supported beam of length 6 m carries a uniformly distributed load of 4 kN/m over its entire span. What is the magnitude of the maximum bending moment, in kN·m?

  1. 36
  2. 24
  3. 18 Correct answer.
  4. 12

Explanation

For a simply supported beam with a uniform load w over span L, the maximum moment is wL²/8, occurring at midspan. Substituting w = 4 kN/m and L = 6 m gives 4(36)/8 = 18 kN·m. The reaction at each support is wL/2 = 12 kN, which is a force, not a moment.

Question 2Mathematics

A right triangle has legs of length 3 and 4. What is the sine of the angle opposite the leg of length 3?

  1. 0.60 Correct answer.
  2. 0.75
  3. 1.33
  4. 0.80

Explanation

The hypotenuse is the square root of (3² + 4²), which is 5. The sine of the angle opposite the side of length 3 equals opposite over hypotenuse, or 3/5 = 0.60.

Question 3Mathematics

Evaluate the definite integral of 2x with respect to x from x = 1 to x = 3.

  1. 9
  2. 4
  3. 8 Correct answer.
  4. 6

Explanation

The antiderivative of 2x is x². Evaluating x² at the upper limit and subtracting its value at the lower limit gives 3² − 1² = 9 − 1 = 8.

Question 4Mathematics

A vector has components (6, 8). What is its magnitude?

  1. 48
  2. 10 Correct answer.
  3. 7
  4. 14

Explanation

The magnitude of a vector with components a and b is the square root of (a² + b²). Here that is the square root of (36 + 64), which is the square root of 100, or 10.

Question 5Mathematics

In a standard normal distribution, approximately what fraction of observations lie within one standard deviation of the mean?

  1. 95 percent
  2. 68 percent Correct answer.
  3. 50 percent
  4. 99.7 percent

Explanation

The empirical rule states that about 68 percent of a normal distribution falls within one standard deviation of the mean. About 95 percent falls within two standard deviations and about 99.7 percent within three.

Question 6Mathematics

What is the derivative with respect to x of the function f(x) = 5x³?

  1. 3x²
  2. 15x³
  3. 15x² Correct answer.
  4. 5x²

Explanation

The power rule multiplies the coefficient by the exponent and then reduces the exponent by one. Applying it to 5x³ gives 5 times 3 times x², which is 15x².

Question 7Mathematics

Convert the angle 150 degrees to radians.

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

Explanation

To convert degrees to radians, multiply by π/180. Thus 150 times π/180 equals 150π/180, which simplifies to 5π/6.

Question 8Mathematics

A system of two linear equations is graphed as two straight lines. Under which condition does the system have infinitely many solutions?

  1. The two lines are perpendicular
  2. The two lines coincide Correct answer.
  3. The two lines intersect at exactly one point
  4. The two lines are parallel and distinct

Explanation

A linear system has infinitely many solutions when the equations describe the same line, so every point on that line satisfies both. Distinct intersecting lines give one solution, and distinct parallel lines give none.

Question 9Mathematics

A force of 200 N acts at an angle of 30 degrees above the horizontal. What is the horizontal component of the force, in N?

  1. 100
  2. 115
  3. 200
  4. 173 Correct answer.

Explanation

The horizontal component equals the magnitude times the cosine of the angle from the horizontal. Thus 200 cos 30° = 200(0.866) ≈ 173 N. The sine would give the vertical component.

Question 10Mathematics

A steel bar 2 m long is heated and its temperature rises by 50 °C. If the coefficient of linear thermal expansion is 12 × 10⁻⁶ per °C, what is the change in length, in mm?

  1. 2.4
  2. 12
  3. 1.2 Correct answer.
  4. 0.6

Explanation

Thermal expansion is ΔL = αLΔT. Substituting gives (12 × 10⁻⁶)(2 m)(50 °C) = 1.2 × 10⁻³ m, which equals 1.2 mm.

Question 11Mathematics

A simply supported beam of length 4 m carries a concentrated load of 10 kN at its midpoint. What is the maximum bending moment, in kN·m?

  1. 40
  2. 5
  3. 20
  4. 10 Correct answer.

Explanation

For a central point load P on a simply supported span L, the maximum moment is PL/4 at midspan. Substituting gives (10)(4)/4 = 10 kN·m.

Question 12Mathematics

What is the determinant of the 2×2 matrix with rows (3, 5) and (2, 4)?

  1. 12
  2. 22
  3. 7
  4. 2 Correct answer.

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)(4) − (5)(2) = 12 − 10 = 2.

Question 13Mathematics

Evaluate the limit as x approaches 2 of (x² − 4)/(x − 2).

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

Explanation

Factoring the numerator gives (x − 2)(x + 2), so the expression simplifies to x + 2 for x not equal to 2. As x approaches 2, x + 2 approaches 4, so the limit is 4 even though the function is undefined at x = 2.

Question 14Mathematics

A sample of 10 measurements has a mean of 50 and a sample standard deviation of 4. What is the standard error of the mean?

  1. 1.26 Correct answer.
  2. 12.65
  3. 0.40
  4. 4.00

Explanation

The standard error of the mean equals the sample standard deviation divided by the square root of the sample size. Here that is 4 divided by the square root of 10, approximately 4/3.162 ≈ 1.26.

Question 15Mathematics

A right circular cylinder has a radius of 3 cm and a height of 10 cm. What is its total surface area, in cm², to the nearest whole number?

  1. 188
  2. 94
  3. 283
  4. 245 Correct answer.

Explanation

Total surface area equals 2πr² for the two ends plus 2πrh for the lateral surface. That is 2π(9) + 2π(3)(10) = 18π + 60π = 78π ≈ 245 cm².

Question 16Mathematics

The position of a particle is given by s(t) = 3t² − 12t + 5, where s is in meters and t in seconds. At what time is the particle momentarily at rest?

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

Explanation

Velocity is the derivative of position, so v(t) = 6t − 12. Setting velocity to zero gives 6t = 12, so t = 2 s. At that instant the particle reverses direction.

Question 17Mathematics

Two forces act at a point: 8 N directed east and 6 N directed north. What is the angle of the resultant measured from the east direction?

  1. 53.1 degrees
  2. 45.0 degrees
  3. 60.0 degrees
  4. 36.9 degrees Correct answer.

Explanation

The resultant has an east component of 8 N and a north component of 6 N. Its angle from east is the arctangent of 6/8, which is arctan 0.75 ≈ 36.9 degrees.

Question 18Mathematics

A simply supported beam of length 8 m has a concentrated load of 12 kN located 2 m from the left support. What is the reaction at the left support, in kN?

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

Explanation

Taking moments about the right support, the left reaction times 8 m equals the load times its distance from the right support, which is 6 m. Thus R_left = 12(6)/8 = 9 kN.

Question 19Mathematics

What is the value of the expression 5!/(3!·2!)?

  1. 10 Correct answer.
  2. 20
  3. 120
  4. 60

Explanation

This is the binomial coefficient for choosing 2 items from 5. Computing gives 120/(6·2) = 120/12 = 10.

Question 20Mathematics

A vector function r(t) = (2t, t²) describes a path. What is the unit tangent vector at t = 1?

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

Explanation

Differentiating gives r'(t) = (2, 2t), so at t = 1 the tangent is (2, 2). Its magnitude is 2√2, and dividing gives (1/√2, 1/√2), which has unit length.

Question 21Mathematics

A cantilever beam of length 3 m carries a uniformly distributed load of 5 kN/m over its full length. What is the magnitude of the fixed-end moment, in kN·m?

  1. 22.5 Correct answer.
  2. 45
  3. 7.5
  4. 11.25

Explanation

For a cantilever with a uniform load w over length L, the fixed-end moment is wL²/2. Substituting gives 5(9)/2 = 22.5 kN·m. This moment is larger than the midspan value for a simply supported beam of the same span because the cantilever carries the full load at one support.

Question 22Mathematics

Evaluate the integral of x·eˣ with respect to x from 0 to 1, to three decimal places.

  1. 0.718
  2. 0.500
  3. 1.718
  4. 1.000 Correct answer.

Explanation

Integration by parts with u = x and dv = eˣ dx gives x·eˣ − eˣ. Evaluating at x = 1 gives 0, and at x = 0 gives −1, so the definite integral is 0 − (−1) = 1.000.

Question 23Mathematics

In a normal distribution with mean 100 and standard deviation 15, what is the z-score of a value of 130?

  1. 3.00
  2. 0.30
  3. 1.50
  4. 2.00 Correct answer.

Explanation

The z-score is the difference between the value and the mean divided by the standard deviation. Here that is (130 − 100)/15 = 30/15 = 2.00.

Question 24Mathematics

A 2×2 matrix has eigenvalues 3 and 5. What are its determinant and trace, respectively?

  1. 15 and 2
  2. 15 and 8 Correct answer.
  3. 8 and 15
  4. 2 and 8

Explanation

For any square matrix, the determinant equals the product of its eigenvalues and the trace equals their sum. Thus the determinant is 15 and the trace is 8.

Question 25Mathematics

A projectile is launched at 40 m/s at an angle of 30 degrees above the horizontal over level ground. Neglecting air resistance and using g = 9.81 m/s², what is the horizontal range, in meters?

  1. 71
  2. 141 Correct answer.
  3. 163
  4. 82

Explanation

Range for a projectile over level ground is v² sin(2θ)/g. Substituting gives (1600)(sin 60°)/9.81 = (1600)(0.866)/9.81 ≈ 141 m.

Question 26Mathematics

For a function f(x) that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), with f(a) = f(b), Rolle's theorem guarantees the existence of at least one number c in (a, b) such that which condition holds?

  1. f'(c) = 0 Correct answer.
  2. f'(c) = [f(b) − f(a)] / (b − a)
  3. f(c) = 0
  4. f''(c) = 0

Explanation

Rolle's theorem is a special case of the mean value theorem where the function values at the endpoints are equal. Because f(a) = f(b), the average rate of change over [a, b] is zero, so the theorem guarantees a point c where the instantaneous rate of change f'(c) equals zero. This means the tangent is horizontal at some interior point.

Question 27Mathematics

Evaluate the definite integral ∫₀¹ (2x + 1) dx.

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

Explanation

Find an antiderivative: ∫(2x + 1) dx = x² + x + C. Evaluate from 0 to 1: (1² + 1) − (0² + 0) = 2. Thus, the value of the definite integral is 2.

Question 28Mathematics

The position of a particle moving along a straight line is given by s(t) = t³ − 6t² + 9t + 2, where s is in meters and t is in seconds. At t = 2 seconds, what is the acceleration of the particle, in m/s²?

  1. 6
  2. −6
  3. 12
  4. 0 Correct answer.

Explanation

Acceleration is the second derivative of position with respect to time. First, velocity v(t) = s'(t) = 3t² − 12t + 9. Then acceleration a(t) = v'(t) = 6t − 12. At t = 2, a(2) = 6(2) − 12 = 0 m/s².

Question 29Mathematics

A tank initially contains 100 liters of water with 10 kg of dissolved salt. Brine containing 0.2 kg of salt per liter flows into the tank at a rate of 5 L/min, and the well-mixed solution flows out at the same rate. Set up the differential equation for the amount of salt Q(t) in kg at time t in minutes. Which differential equation correctly models this system?

  1. dQ/dt = 1 − Q/20 Correct answer.
  2. dQ/dt = 1 − Q/100
  3. dQ/dt = 1 − Q/10
  4. dQ/dt = 0.2 − Q/20

Explanation

The rate of salt entering is (0.2 kg/L)(5 L/min) = 1 kg/min. The concentration in the tank is Q/100 kg/L, and the outflow rate is 5 L/min, so salt leaves at (Q/100)(5) = Q/20 kg/min. Therefore, the differential equation is dQ/dt = 1 − Q/20.

Question 30Probability and Statistics

A tensile test specimen has a yield strength that is normally distributed with mean 350 MPa and standard deviation 25 MPa. What is the probability that a randomly selected specimen yields below 325 MPa?

  1. 0.0228
  2. 0.3413
  3. 0.1587 Correct answer.
  4. 0.8413

Explanation

Convert to a standard normal score: z = (325 - 350)/25 = -1.0. The standard normal table gives the cumulative probability below z = -1.0 as 0.1587, so about 16 percent of specimens fall below 325 MPa.

Question 31Probability and Statistics

In a manufacturing process the number of surface defects per casting follows a Poisson distribution with a mean of 2.0 defects per casting. What is the probability that a casting has exactly zero defects?

  1. 0.1353 Correct answer.
  2. 0.2707
  3. 0.4060
  4. 0.5000

Explanation

The Poisson probability of exactly k events is P(k) = e^(-lambda) * lambda^k / k!. With lambda = 2.0 and k = 0 this reduces to e^(-2.0) = 0.1353.

Question 32Probability and Statistics

A sample of 16 steel rods has a mean diameter of 25.02 mm and a sample standard deviation of 0.08 mm. Which distribution is appropriate for constructing a confidence interval on the true mean diameter?

  1. F distribution with 15 and 15 degrees of freedom
  2. Chi-square distribution
  3. Student t distribution with 15 degrees of freedom Correct answer.
  4. Standard normal distribution

Explanation

When the population standard deviation is unknown and estimated from a small sample, the sample mean follows a Student t distribution with n - 1 degrees of freedom. Here n = 16, so 15 degrees of freedom applies.

Question 33Probability and Statistics

A quality engineer records the number of rejected parts per shift for 40 shifts. Which graphical tool is best for displaying the frequency distribution of this discrete count data?

  1. Scatter diagram
  2. Pareto chart
  3. Histogram Correct answer.
  4. Control chart

Explanation

A histogram displays how often values or ranges of values occur and is the standard tool for showing a frequency distribution of measured or counted data. Scatter diagrams show relationships between two variables, and control charts track a process over time against limits.

Question 34Probability and Statistics

A random variable X takes the values 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3 respectively. What is the expected value of X?

  1. 2.0
  2. 2.5
  3. 1.8
  4. 2.1 Correct answer.

Explanation

The expected value is the probability-weighted sum: E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1.0 + 0.9 = 2.1.

Question 35Probability and Statistics

Which statement correctly describes a Type I error in hypothesis testing?

  1. Failing to reject a true null hypothesis
  2. Rejecting a false null hypothesis
  3. Rejecting a true null hypothesis Correct answer.
  4. Failing to reject a false null hypothesis

Explanation

A Type I error occurs when a true null hypothesis is rejected, and its probability is controlled by the significance level alpha. A Type II error is the opposite failure, failing to reject a false null hypothesis.

Question 36Probability and Statistics

The diameters of a batch of shafts are normally distributed with mean 30.00 mm and standard deviation 0.10 mm. What proportion of shafts has a diameter between 30.00 mm and 30.15 mm?

  1. 0.3413
  2. 0.4987
  3. 0.4332 Correct answer.
  4. 0.1500

Explanation

The upper limit corresponds to z = (30.15 - 30.00)/0.10 = 1.5. The area between the mean and z = 1.5 is 0.4332, which is the requested proportion.

Question 37Probability and Statistics

A sample of 25 measurements yields a sample variance of 16. What is the standard error of the sample mean?

  1. 4.00
  2. 0.64
  3. 0.16
  4. 0.80 Correct answer.

Explanation

The standard error of the mean is s/sqrt(n). With s = sqrt(16) = 4 and n = 25, the standard error is 4/5 = 0.80.

Question 38Probability and Statistics

A process produces parts whose length is normally distributed with mean 50 mm and standard deviation 2 mm. A part is rejected if its length is below 46 mm. What is the probability of rejection?

  1. 0.0228 Correct answer.
  2. 0.0455
  3. 0.1587
  4. 0.0013

Explanation

The rejection threshold is z = (46 - 50)/2 = -2.0. The standard normal cumulative probability below -2.0 is 0.0228, so about 2.3 percent of parts are rejected.

Question 39Probability and Statistics

Two independent assembly operations each have a probability of 0.05 of producing a defect. What is the probability that a unit passing through both operations has at least one defect?

  1. 0.9500
  2. 0.1000
  3. 0.0975 Correct answer.
  4. 0.0025

Explanation

For independent events, the probability of no defect in either operation is (0.95)(0.95) = 0.9025. The probability of at least one defect is 1 - 0.9025 = 0.0975.

Question 40Probability and Statistics

A sample of 10 bearings has a mean clearance of 0.250 mm and a sample standard deviation of 0.010 mm. What is the 95 percent two-sided confidence interval for the mean clearance? Use t(0.025, 9) = 2.262.

  1. 0.2404 to 0.2596 mm
  2. 0.2429 to 0.2571 mm Correct answer.
  3. 0.2480 to 0.2520 mm
  4. 0.2438 to 0.2562 mm

Explanation

The margin of error is t * s/sqrt(n) = 2.262 * 0.010/sqrt(10) = 2.262 * 0.003162 = 0.00715. Adding and subtracting this from 0.250 gives roughly 0.2429 to 0.2571 mm.

Question 41Probability and Statistics

A sample of 100 parts has a sample mean of 12.4 N and a known population standard deviation of 0.8 N. Test the null hypothesis that the true mean is 12.2 N against a two-sided alternative at the 0.05 significance level. What is the test statistic and conclusion?

  1. z = 0.25; fail to reject the null hypothesis
  2. z = 2.50; reject the null hypothesis Correct answer.
  3. z = 1.96; fail to reject the null hypothesis
  4. z = 2.50; fail to reject the null hypothesis

Explanation

The test statistic is z = (12.4 - 12.2)/(0.8/sqrt(100)) = 0.2/0.08 = 2.50. Since 2.50 exceeds the critical value of 1.96 for a two-sided test at alpha = 0.05, the null hypothesis is rejected.

Question 42Probability and Statistics

A random sample of 400 customers shows that 240 prefer a new design. What is the standard error of the sample proportion?

  1. 0.0006
  2. 0.6000
  3. 0.0010
  4. 0.0245 Correct answer.

Explanation

The sample proportion is 240/400 = 0.60. The standard error is sqrt[p(1-p)/n] = sqrt[(0.60)(0.40)/400] = sqrt(0.0006) = 0.0245.

Question 43Probability and Statistics

A system consists of three independent components in series, each with a reliability of 0.95. What is the reliability of the system?

  1. 0.9000
  2. 0.8574 Correct answer.
  3. 0.9975
  4. 0.9500

Explanation

For components in series, the system works only if all components work, so the reliability is the product of the individual reliabilities. Here 0.95^3 = 0.8574.

Question 44Probability and Statistics

A sample of 20 welds has a sample variance of 0.0045 mm squared. A chi-square test of the null hypothesis that the true variance is 0.0030 mm squared gives a test statistic of (n-1)s^2/sigma0^2. What is the value of the test statistic?

  1. 19.00
  2. 30.00
  3. 0.0150
  4. 28.50 Correct answer.

Explanation

The chi-square statistic is (n - 1)s^2 / sigma0^2 = 19 * 0.0045 / 0.0030 = 0.0855/0.0030 = 28.50.

Question 45Probability and Statistics

A process is monitored with a control chart for the mean. Which situation indicates that the process mean has likely shifted and warrants investigation?

  1. Two consecutive points lie close to the center line
  2. One point falls slightly inside the upper control limit
  3. Points alternate above and below the center line
  4. Nine consecutive points fall on the same side of the center line Correct answer.

Explanation

A run of nine consecutive points on one side of the center line is a standard run rule indicating a sustained shift in the process mean. Isolated points inside the limits and alternating patterns are consistent with common-cause variation.

Question 46Probability and Statistics

A random sample of 50 measurements is drawn from a population with mean 100 and variance 225. What is the probability that the sample mean exceeds 103?

  1. 0.0793 Correct answer.
  2. 0.0228
  3. 0.4207
  4. 0.1587

Explanation

The standard error is sqrt(225/50) = sqrt(4.5) = 2.121. The z score for 103 is (103 - 100)/2.121 = 1.414, and the upper-tail probability is about 0.0793.

Question 47Probability and Statistics

A sample of 12 bolts has a mean tensile strength of 82.0 kN and a sample standard deviation of 4.0 kN. Test the null hypothesis that the true mean is 80.0 kN against a two-sided alternative. What is the computed t statistic?

  1. 2.00
  2. 1.50
  3. 1.73 Correct answer.
  4. 0.50

Explanation

The t statistic is (xbar - mu0)/(s/sqrt(n)) = (82.0 - 80.0)/(4.0/sqrt(12)) = 2.0/1.1547 = 1.73.

Question 48Probability and Statistics

A sample of 8 measurements has a mean of 20.0 and a sample standard deviation of 2.0. A 90 percent two-sided confidence interval uses t(0.05, 7) = 1.895. What is the margin of error?

  1. 0.71
  2. 1.34 Correct answer.
  3. 1.06
  4. 1.90

Explanation

The margin of error is t * s/sqrt(n) = 1.895 * 2.0/sqrt(8) = 1.895 * 0.7071 = 1.34.

Question 49Probability and Statistics

A regression of tool wear on cutting speed yields a slope of 0.45 with a standard error of 0.10 based on 22 observations. What is the computed t statistic for testing whether the slope differs from zero?

  1. 4.50 Correct answer.
  2. 0.22
  3. 0.45
  4. 2.10

Explanation

The t statistic for a regression slope is the estimate divided by its standard error: 0.45/0.10 = 4.50. With 20 degrees of freedom this is highly significant.

Question 50Probability and Statistics

A process has a mean of 100 and a standard deviation of 5. The specification limits are 90 and 110. What is the process capability index Cp?

  1. 1.00
  2. 0.67 Correct answer.
  3. 2.00
  4. 1.33

Explanation

Cp is the specification width divided by six standard deviations: (110 - 90)/(6 * 5) = 20/30 = 0.67. A Cp below one indicates the process spread exceeds the specification width.