Mathematics and Statistics
8–12 of 110 itemsNCEES draws between 8 and 12 of the 110 from this area on any legal paper. Its outline lists 4 sub-topics; the bank holds 40 verified templates against them, which expand to 1,276 questions.
Sub-topics, from the NCEES outline
- AAnalytic geometry
- BSingle-variable calculus
- CVector operations
- DStatistics (e.g., distributions, mean, mode, standard deviation, confidence interval, regression and curve fitting)
5 worked problems
Real questions from the bank, at the depth the exam asks them. Pick an answer first: the reasoning opens when you do, along with why each wrong answer is tempting.
Question 1 of 5
A line passes through (6, 5) and (19, 9). Its y intercept is most nearly:
Line through the points (6, 5) and (19, 9), extended back to where it crosses the y axis. Show the answer and the reasoning
Why B
Given: (6, 5) and (19, 9). Relation: m = (y2 - y1)/(x2 - x1), then b = y1 - m x1, stepping back from the point to x = 0. Substitute: m = 4/13 = 0.3077; b = 5 - 0.3077(6). Result: b = 3.154.
Why A is tempting
Computed something resembling an x intercept.
Why C is tempting
Added the slope term. Working backwards from a point to x = 0 means subtracting m times x.
Why D is tempting
Mixed the y value of one point with the x value of the other.
Source NCEES FE Reference Handbook 10.6 — Mathematics; Engineering Probability and Statistics
Question 2 of 5
The value of the integral of 2x^2 - 4 from x = 3 to x = 4 is most nearly:
Show the answer and the reasoning
Why D
Given: integrand 2x^2 - 4, from x = 3 to x = 4. Relation: F(x) = 2x^3/3 - 4x; the integral is F(4) - F(3). Substitute: F(4) = 26.667, F(3) = 6.000. Result: integral = 20.67.
Why A is tempting
Reversed the limits.
Why B is tempting
Integrated x^2 as if it were x.
Why C is tempting
Differentiated instead of integrating.
Source NCEES FE Reference Handbook 10.6 — Mathematics; Engineering Probability and Statistics
Question 3 of 5
The scalar projection of A = 8i + 7j + 4k onto B = 6i - 2j is most nearly:
Show the answer and the reasoning
Why C
Given: A = 8i + 7j + 4k, projected onto B = 6i - 2j. Relation: scalar projection = (A dot B)/|B|, a length along B. Substitute: A dot B = 34, |B| = 6.3246. Result: projection = 5.376.
Why A is tempting
Divided by the squared magnitude, which gives the scalar multiplier for the vector projection rather than its length.
Why B is tempting
Divided by the magnitude of the projected vector rather than the one being projected onto.
Why D is tempting
Reported the dot product without normalising by the target direction.
Source NCEES FE Reference Handbook 10.6 — Mathematics; Engineering Probability and Statistics
Question 4 of 5
A least squares line is fitted to the points (3, 2), (5, 10) and (13, 26). Its slope is most nearly:
Three data points at (3, 2), (5, 10) and (13, 26), through which a least-squares line is fitted. Show the answer and the reasoning
Why C
Given: (3, 2), (5, 10), (13, 26); n = 3. Relation: m = (n sum(xy) - sum(x) sum(y))/(n sum(x^2) - (sum x)^2), which uses every point. Substitute: sum(x) = 21, sum(y) = 38, sum(xy) = 394, sum(x^2) = 203; m = (3(394) - 21(38))/(3(203) - 21^2). Result: m = 2.286.
Why A is tempting
Took the ratio of the sums.
Why B is tempting
Omitted the correction for the means, which only holds if both variables are centred.
Why D is tempting
Drew a line through the two extreme points, which ignores the middle observation entirely.
Source NCEES FE Reference Handbook 10.6 — Mathematics; Engineering Probability and Statistics
Question 5 of 5
Events A and B occur together with probability 0.1. Event A alone has probability 0.35 and event B alone 0.5. The probability of A given that B has occurred is most nearly:
Show the answer and the reasoning
Why C
Given: P(A and B) = 0.1, P(B) = 0.5. P(A) = 0.35 is not needed. Relation: P(A|B) = P(A and B)/P(B); knowing B occurred shrinks the sample space to B. Substitute: P(A|B) = 0.1/0.5. Result: P(A|B) = 0.200.
Why A is tempting
Reported the joint probability without conditioning.
Why B is tempting
Multiplied the marginals, which assumes independence.
Why D is tempting
Divided by the product of the marginals.
Source NCEES FE Reference Handbook 10.6 — Mathematics; Engineering Probability and Statistics
Four mistakes that cost the question
Each of these lands on an answer that is offered, so it costs the question outright. The minutes are our estimate of the time each one burns on top, against an average of 2.9 minutes a question (320 minutes for 110).
Costs the question and about 3 min
Doing calculus with the calculator in degree mode
Derivatives and integrals of trig functions assume radians: the derivative of sin x is cos x only when x is in radians. In degree mode the limit of sin(3x)/(4x) comes out near 0.013 instead of 0.75, off by the factor π/180. Switch to radians before any calculus item and back to degrees for bearings and geometry.
Costs the question and about 2 min
Dividing by n instead of n − 1 for a sample standard deviation
A sample standard deviation divides by n − 1; a population one by n. Calculators show both, and the population value is smaller by the square root of (n − 1)/n: about 7% low for seven readings, which is close enough to sit among the choices. Decide whether the data are a sample before reading the screen.
Costs the question and about 3 min
Taking a definite integral as the area when the curve crosses the axis
A definite integral nets the area below the axis against the area above it. When a question asks for the area bounded by a curve and the x axis, split the interval at every root and add the absolute values. Integrating straight across gives the net, which is smaller, and which is offered.
Costs the question and about 2 min
Leaving the square root of n out of a confidence interval
The half-width of a confidence interval on a mean is the multiplier times the standard error, s divided by the square root of n, not times s itself. With nine readings that drops a factor of three, and the tripled interval is one of the choices.
Questions about Mathematics and Statistics
Do I need to know calculus by heart for the FE Civil exam?
No. The FE Reference Handbook lists standard derivatives, integrals and series, so memorising tables is not the work. What it will not do is tell you that a rate of change means a derivative, or that an area bounded by a curve means splitting the integral at its roots. Practise turning the words of a question into an operation.
Which calculator settings matter for mathematics questions?
Angle mode first: radians for calculus, degrees for most geometry and surveying stems. Then know which of your statistics mode's two standard deviations is the sample one, and whether your model evaluates definite integrals and vector products. NCEES publishes the list of approved calculator models; check that yours is on the current one.
How much statistics is in Mathematics and Statistics?
It is one of four sub-topics, beside analytic geometry, single-variable calculus and vector operations, and it reaches past this area: standard deviation, confidence intervals and regression turn up in materials testing and traffic data too. Expect computations such as a mean, a sample standard deviation, a confidence interval or a regression slope, not derivations.
Is analytic geometry worth studying if it looks easy?
Yes, because it is fast. Slopes, intercepts, distances and conic sections are one-step items, and answering them in a minute banks time for the long questions elsewhere on the paper. The cost here is carelessness, such as a sign dropped from a negative slope, not difficulty.
How many study hours this area is worth · Using the handbook under the clock