Vectors
Trig
Probability
Sequences and Series
Miscellaneous
100
Given P (-5, 11, -8) and Q (-4, 9, -5), find PQ.
PQ = (1, -2, 3)
100
In triangle ABC, angle A is 34 degrees, side [BC] is 16 and side [AB] is 9. Find angle C.
(sin 34 / 16 ) = (sin C / 9) 9 sin34 = 16 sinC sinC = .3145 C = 18.33
100
In how many ways can a president, vice-president and treasurer be chosen from a 33-person class?
33 * 32 * 31 = 32736
100
Let an = 3 - 2n Write down the first three terms.
a1 = 1 a2 = -1 a = -3
100
17 20 21 18 20 20 20 18 19 Using the data, find the mean, median and mode.
17 18 18 19 20 20 20 20 21 mean = 173/9 = 19.22 median = 20 mode = 20
200
The position of vectors of points P and Q ( 3, 2, 1 ) and (-2, 2, -4 ) respectively. The origin is at O. Find a)the angle POQ; b)the area of the triangle OPQ
a) dot product: (3)(-2) + (2)(2) + (1)(-4) = -6 magnitude: √(3^2 + 2^2 +1^2) = √14 √(-2^2 + 2^2 + -4^2) = √24 cos θ = -6 / (√140)(√24) θ = 109.1 b)-10i + 10j + 10k magnitude:√300 Area = √300 / 2
200
Given that π/2 ≤ θ ≤ π and that cosθ = -12/13 a) Find sinθ b) Find tanθ
sinθ = 5/13 tanθ = 5/-12
200
Find the number of distinguishable permutations of the letters in CREAMY AVOCADO
13! / 2! 2! 3! = 259459200
200
Consider the infinite geometric sequence 25, 5, 1, 0.2,... a) Find the common ratio b) Find an expression for the nth term
a) 25/5 = 1/5 = r b) an = 25(1/50)^(n-1)
200
Let f(x) = 2x^2 - 12x +5 a) express f(x) in the form f(x) = 2(x - h)^2 - k b) Find the vertex. c)Find the axis of symmetry
a) 2x^2 - 12x + = -5 2(x^2 - 6x + 9) = -5 + 18 2(x -3) =13 2(x -3) - 13 = 0 b) (3, -13) c) x = 3
300
Given u = 2i + 3j - 4k v = 2i - 3j + 4k Find the angle between u and v
cos θ = -21 / (√29)(√29) θ = 136.4
300
A farmer owns a triangular field ABC. One side of the triangle [AC}, is 104 m, a second side [AB], is 65 m and the angle between these two sides is 60 degrees.
a^2 = 104^2 + 65^2 - 2(104)(65)cos60 a^2 = 8281 a = 91
300
A game has spinner with three equal parts: red, green and blue. You spin the spinner six times. What is the chance of getting exactly 2 reds?
6C2 (.33)^2 (.67)^4 = .0411
300
Find the sum of the infinite series. 5(4)^(n-1)
no sum since r>1
300
Weights of paper discarded by households each week are normally distributed with a mean of 9.4 lb and a standard deviation of 4.2 lb. Find the weight that separates the bottom 33% from the top 67%.
inversenorm (.33, 9.4, 4.2) 7.55
400
Consider the points A (1, 3, -17) and B (6, -7, 8) which lie on the line L. Find thee equation of the line L giving the answer in parametric form.
x = 1 +5e y = 3 -10e z = 17 +25e x = 6 +5e y = -7 -10e z = 8 +25e
400
Given that 2sin^2θ + sinθ - 1 = 0, find the two values for sinθ.
(2sinθ - 1)(sinθ + 1) sinθ = 1/2 sinθ = -1 θ = 60 and 270
400
At RBVHS, 10% of the student body has seen the movie Lincoln. A random sample of 50 students were chosen. Find the probability that more than one person has seen the movie.
50C0 (.1)^0 (.9)^50 + 50C1 (.1)^1 (.9)^49 = .034 1 - .034 = .966
400
u21= -37 and u4 = -3 Find the common difference.
a) -37 = a1 +20d -3 = a1 +3d a1 = -3 -3d -37 = -3 -3d +20d -34 = 17d d = -2
400
Replacement times for iPods are normally distributed with a mean of 7.1 years and a standard deviation of 1.4 years. Find the probability that a randomly selected iPod will have a replacement less than 8.0 year.
normalcdf (-100,000, 8, 7.1, 1.4) .739
500
v = 2i - 3j + 4k L1 = 31i - 42j - 30k + s(-7i +10j +3k) L2 is the line through A ( 2, -1, -3) and in the direction of v. a)Write down the equation of L2. b) L1 and L2 intersect at point M. Find the coordinates of M.
a) r = 2i - 1j - 3k + f(2i - 3j + 4k) b)2 +2f = 31 - 7s 3(2f + 7s = 29) -1 - 3f = -42 +10s 2(-3f - 10s = -41) 6f +21s = 87 -6f -20s = -82 s = 5 (-4, 8 -15)
500
Solve the equation 6sinxcosx = 3/2, for π /4 ≤ x ≤ π/2
3(2sinxcosx) = 3/2 2sinxcosx = sin2x 3(sin2x) = 3/2 sin2x = 3/2 2x = π/6 2x = 5π/6 x =5π/12
500
Probability of getting 3 of a kind from a 52-card deck. SET-UP ONLY
13C1 4C3 12C2 4C1 4C1 / 52C2
500
Sum of the first 25 terms of an arithmetic sequence is 1850. Common difference is 6. Find the first term.
an = a1 + 6(25 - 1) an = a1 + 144 1850 = 12.5 ( a1 + a1 +144) 1850 = 12.5 (2a1 + 144) 1850 = 25a1 + 1800 50 = 25a1 a1 = 2
500
Proof by induction -2 + 3 + 8 + .... 5n - 7 = n(5n - 9) / 2
1) -2 = -2 2) -2 + 3 + 8 + .... 5k - 7 = k(5k - 9) / 2 3)k(5k - 9) / 2 + 5(k + 1) - 7 = (k+1)(5(k+1) - 9) / 2 5k^2 - 9k / 2 + 5k + 5 -7 5k^2 - 9k / 2 + 5k - 2 5k^2 - 9k / 2 + 10k - 4 / 2 5k^2 - 9k + 10k - 4 /2 5k^2 + k - 4 / 2 (5k - 4)(k + 1) / 2 (5k + 5 - 9)(k + 1) / 2 (k + 1)(5(k + 1) - 9) / 2
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