CSET Mathematics Exam (211, 212, 213)

1. Question 1 · Number and Quantity

   In the complex plane, the point representing 3 + 2i is rotated 90° counterclockwise about the point representing 1 + i. Which complex number represents the image?
   • A.-1 + 2i
   • B.2 - i
   • C.3i
   • D.-2 + 3i
   Show answer and explanation

   Answer: C. 3i

   Translate so the center is at the origin: (3 + 2i) - (1 + i) = 2 + i. Multiplying by i rotates 90° counterclockwise: i(2 + i) = -1 + 2i. Translating back gives (-1 + 2i) + (1 + i) = 3i. Rotating about the origin instead gives -2 + 3i, and multiplying by -i (a clockwise turn) gives 2 - i.

2. Question 2 · Algebra

   Which describes the solution set of (x + 2)(x - 1)^2(x - 3) < 0?
   • A.-2 < x < 3
   • B.-2 < x < 1
   • C.x < -2 or 1 < x < 3
   • D.-2 < x < 3 and x ≠ 1
   Show answer and explanation

   Answer: D. -2 < x < 3 and x ≠ 1

   The zeros are -2, 1, and 3. The factor (x - 1)^2 is positive except at x = 1, so the sign matches (x + 2)(x - 3), which is negative exactly for -2 < x < 3. At x = 1 the product equals 0, which does not satisfy the strict inequality, so 1 must be removed. Treating the double root as a sign change produces the incorrect set x < -2 or 1 < x < 3.

3. Question 3 · Geometry

   A triangle has sides of length 5, 9, and x, where x is an integer. For how many values of x is the triangle obtuse?
   • A.3
   • B.7
   • C.6
   • D.9
   Show answer and explanation

   Answer: C. 6

   The Triangle Inequality requires 4 < x < 14, so x is 5 through 13. The triangle is obtuse when the square of the longest side exceeds the sum of the squares of the other two: if x is longest, x^2 > 106 gives x = 11, 12, 13; if 9 is longest, 81 > 25 + x^2 gives x^2 < 56, so x = 5, 6, 7. That is 6 values; counting only the case where x is the longest side gives 3.

4. Question 4 · Probability and Statistics

   The letters of the word PARALLEL are arranged in a random order, with every distinct arrangement equally likely. What is the probability that the three L's are all next to each other?
   • A.1/28
   • B.1/56
   • C.3/28
   • D.3/56
   Show answer and explanation

   Answer: C. 3/28

   PARALLEL has 8 letters with A twice and L three times, so there are 8!/(2!·3!) = 3360 distinct arrangements. Treating LLL as one block leaves 6 units with A twice, giving 6!/2! = 360 arrangements. The probability is 360/3360 = 3/28. Equivalently, the L's fill 3 of the 8 positions, and 6 of the C(8, 3) = 56 equally likely position sets are consecutive; 1/56 counts only one of those 6 blocks.

5. Question 5 · Calculus

   What is the limit of (tan x - 1)/(x - π/4) as x → π/4?
   • A.sqrt(2)
   • B.1/2
   • C.1
   • D.2
   Show answer and explanation

   Answer: D. 2

   Since tan(π/4) = 1, the expression is the difference quotient (tan x - tan(π/4))/(x - π/4), so the limit is the derivative of tan x at π/4. That derivative is sec^2(π/4) = (sqrt(2))^2 = 2. Writing sec(π/4) without squaring gives sqrt(2).

6. Question 6 · Number and Quantity

   Which pair lists both complex numbers z that satisfy z^2 = -5 + 12i?
   • A.3 + 2i and -3 - 2i
   • B.3 - 2i and -3 + 2i
   • C.2 - 3i and -2 + 3i
   • D.2 + 3i and -2 - 3i
   Show answer and explanation

   Answer: D. 2 + 3i and -2 - 3i

   Write z = a + bi with a, b real: a^2 - b^2 = -5 and 2ab = 12. Also |z|^2 = |z^2| = 13, so a^2 + b^2 = 13; adding gives a^2 = 4 and b^2 = 9, and ab = 6 > 0 forces matching signs, so z = 2 + 3i or z = -2 - 3i. The pair 3 + 2i and -3 - 2i squares to 5 + 12i, with the sign of the real part reversed.

7. Question 7 · Algebra

   Which describes the solution set of x^4 - 5x^2 + 4 ≤ 0?
   • A.-2 ≤ x ≤ 2
   • B.-2 ≤ x ≤ -1 or 1 ≤ x ≤ 2
   • C.x ≤ -2 or -1 ≤ x ≤ 1 or x ≥ 2
   • D.1 ≤ x ≤ 4
   Show answer and explanation

   Answer: B. -2 ≤ x ≤ -1 or 1 ≤ x ≤ 2

   Factor: x^4 - 5x^2 + 4 = (x^2 - 1)(x^2 - 4), which is ≤ 0 exactly when 1 ≤ x^2 ≤ 4. That means 1 ≤ |x| ≤ 2, so -2 ≤ x ≤ -1 or 1 ≤ x ≤ 2. The interval 1 ≤ x ≤ 4 solves the inequality for u = x^2 but never converts back to x, and the three-piece set is where the polynomial is nonnegative.

8. Question 8 · Geometry

   What is the exact area of a regular octagon with side length 10?
   • A.200·sqrt(2)
   • B.100(1 + sqrt(2))
   • C.200(1 + sqrt(2))
   • D.400(1 + sqrt(2))
   Show answer and explanation

   Answer: C. 200(1 + sqrt(2))

   Each central angle is 45°, so the apothem is a = 5/tan(22.5°) = 5(1 + sqrt(2)), using tan(22.5°) = sqrt(2) - 1. The area is (1/2)·a·perimeter = (1/2)·5(1 + sqrt(2))·80 = 200(1 + sqrt(2)) ≈ 482.8. Forgetting the 1/2 gives 400(1 + sqrt(2)), and using a perimeter of only four sides gives 100(1 + sqrt(2)).

9. Question 9 · Probability and Statistics

   A committee of 5 is chosen from 7 women and 5 men. The committee must include at least 2 women and at least 2 men. Two particular men refuse to serve together.

   How many different committees are possible?
   • A.462
   • B.525
   • C.560
   • D.497
   Show answer and explanation

   Answer: A. 462

   Without the restriction, the count is C(7, 2)·C(5, 3) + C(7, 3)·C(5, 2) = 210 + 350 = 560. Committees that include the two men together: 2 women and 3 men gives 21·3 = 63, and 3 women and 2 men gives 35·1 = 35. Subtracting gives 560 - 98 = 462; subtracting only one of the two cases gives 497 or 525.

10. Question 10 · Calculus

    Suppose f is differentiable at x = a. What is the limit of [f(a + 3h) - f(a - h)]/h as h → 0?
    • A.4f'(a)
    • B.3f'(a)
    • C.2f'(a)
    • D.f'(a)
    Show answer and explanation

    Answer: A. 4f'(a)

    Add and subtract f(a): the quotient equals 3 · [f(a + 3h) - f(a)]/(3h) + [f(a - h) - f(a)]/(-h). Each bracketed quotient tends to f'(a), so the limit is 3f'(a) + f'(a) = 4f'(a). Subtracting the second piece instead of adding it gives 2f'(a). x: 1, 2, 3 f(x): 2, 1, 0 f'(x): 4, -3, 5 g(x): 3, 1, 2 g'(x): -2, 6, 7

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