Texas A&M MPE1 – Core Concepts & Skills

Forty Q&A flashcards summarizing the principal skills, formulas and results appearing in the two-page Math Placement Exam practice set.

6 − 2√2

1) What is the rationalized form of 14 ⁄ (3 + √2)?

(x² + 3ax − 12x − 6a) ⁄ [2x(x + a)]

2) Simplify (x + 2a − 3)/(x + a) − (x + 6)/(2x).

(2x + 5y)/(x + 4y)

3) Reduce (6x² + 11xy − 10y²)/(3x² + 10xy − 8y²) to lowest terms.

Take the reciprocal of a^(3⁄4); i.e., a^(−3⁄4)=1⁄(a^(3⁄4)).

4) What does a negative fractional exponent such as a^(−3⁄4) tell you to do?

x = 25⁄4 (6.25)

5) Solve 5(x − 7) − 13(x − 7) − 6 = 0.

x = 24, y = 15, so x + y = 39

6) For the system −2x + 4y = 12 and 3x − 5y = −3, what is x + y?

$8,000 is at 6¾ % and $2,000 at 5½ %, so $6,000 more is at 6¾ %.

7) $10,000 earns $650 simple interest; part at 5½ %, part at 6¾ %. How much MORE is at 6¾ %?

Δy = (20a)/(3b) (increase if a,b>0).

8) In 2ax + 3by = 7c, if x decreases 10 units, what is Δy?

k = 11⁄2 (5.5)

9) Line through (2k+3, 4k−6) & (−2, 16) has slope 0. Find k.

V(x)=x(10−2x)(6−2x)

10a) Give the volume function for an open box cut from 10"×6" card by removing squares x×x.

S(x)=(10−2x)(6−2x)+2x(10−2x)+2x(6−2x)

10b) Give the surface-area (no lid) for the same box.

x ∈ (−∞,−16] ∪ (10,∞)

11) Solve (5x+2)/(x−10) ≥ 3.

(−∞,−3)∪(−3,−1]∪[4,∞)

12) Domain of f(x)=√(x²−3x−4)/(6x²−54).

(−∞,−1)∪(−1,0)∪[0,∞)

13) Domain of f(x)= { (2x²+13)/(x²−1) for x<0 ; (5x−26)/(x+2) for x≥0 }.

x = 5⁄3 only (x=−½ is excluded).

14) x-intercepts of (6x² − 7x − 5)/(4x² − 12x − 7).

Vertical: x = 7⁄2; Horizontal: y = 3⁄2.

15) Vertical & horizontal asymptotes of same function.

x-intercepts at x=−3,0,3; y-intercept (0,0).

16) x- and y-intercepts of f(x)=x³−9x.

[−5,1]

17a) Domain of f(x)=√(−x²−4x+5).

t > ¾

17b) Domain of g(t)=ln(4t−3).

All real except x=−3,−1,1

17c) Domain of h(x)=1/(x³+3x²−x−3).

(√3·x·√x) ⁄ 2 = (√3 x^{3⁄2})⁄2

18) Simplify 2√(x⁵)·√3 ⁄ (4x).

y = ½(x−4)² + 10

19) Write the equation after shifting f(x)=x² right 4, vertical shrink ½, then up 10.

x = 3

20) Solve log(x+2)+log(x−1)=1.

x²(4x²+1)^7(76x²+3)

21) Completely factor 3x²(4x²+1)^8 + 64x⁴(4x²+1)^7.

d = 18/ tan 41° ≈ 20.7 ft

22) How far from an 18-ft pole gives a 41° angle of elevation?

f(g(x)) = 2/(x+2)

23) Find f∘g for f(x)=x/(x+1), g(x)=2/x.

8/(x+1) − (yw)/[(z+2)(y−4)]

24) Simplify 8/(x+1) − [ y/(z+2) ÷ (y−4)/w ].

x = ln 3

25) Solve e^{2x} − 2e^{x} − 3 = 0.

y=7x−34, so y(−4)=−62

26) Equation of line through (5,1) slope 7; then find y when x=−4.

1⁄[√(h+6)+√6]

27) Simplify [f(2+h)−f(2)]/h for f(x)=√(x+4).

y^{16}

28) Simplify (x²y⁴)⁵(x³y)^{−3} ⁄ (xy).

³√(a^n)=a^{n/3} (for a ≥ 0).

29) What property lets you pull an exponent of 1⁄3 outside a cube root?

[x³+4x²−6x−4] ⁄ [(x−2)(x+1)(x+3)]

30) Combine x²/(x²−x−2) − 4/(x²+x−6)+x/(x²+4x+3).

Zero: x=−1⁄3; Vertical asymptote: x=−3⁄4; Horizontal asymptote: y=3⁄4; Hole at x=5.

31) Zeros & asymptotes of f(x)=3x²−14x−5 / 4x²−17x−15.

cos θ=−4√3⁄7

32) If θ in Quadrant II and sin θ=1⁄7, find cos θ.

½ ln x + (5⁄2) ln y − 4 ln(z+1)

33) Expand ln[√(xy⁵)/(z+1)⁴].

−2 − √3⁄3 = (−6−√3)/3

34) Evaluate sec (2π⁄3) − tan (π⁄6).

Area rises from 20 in² to 21.6 in², a gain of 1.6 in² (8 %).

35) A 5 in ×4 in rectangle’s length increases 8 %. Change in area?

−6

36) Find f(2)−f(−3) for f(x)= {x³+1 if x>1; 2x²−3 if x≤1}.

1 − sin θ

37) Simplify cos²θ ⁄ (1+sin θ).

2 − ½ log₄ 3 ≈ 1.604

38) Evaluate log₄[(16)/√3].

(b³ + a)/(a − b)

39) Simplify (1/(a−b)) ÷ (1/(b³+a)).

t≈73.4 h (since t=24·log₂(25⁄3))

40) 1200 bacteria double daily. Hours to reach 10,000?