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## The First Quarterly Exam

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**The First Quarterly Exam**El primero examen trimestal**Question #1**• For the function , find Your answer is a**Question #2**• Which is the correct recursive formula for the sequence?{-2, 1, 4, 7, … } • A recursive function has two parts • The first term • The function, doing something to the previous term • The first term, u1 = -2 • The function adds 3 to each term, so un = un-1 + 3 • The answer is b**Question #3**• Select the correct description of the sequence{-12, -17, -22, -27, -32, …} • The sequence is arithmetic, because we’re adding a -5 to each term (arithmetic = add) • The answer is b**Question #4Option a**• Find the sum of • Use the calculator – that’s why I got them • sum seq(function, variable, start, end, increment) • sum seq(4x + 3, x, 1, 32, 1) = 2208 • The answer is a**Question #4Option b**• Use the partial sum formula because you’re stubborn • Find u1 and uk • u1 = 4(1) + 3 = 7 • u32 = 4(32) + 3 = 131 • Use the first formula • The answer is still a**Question #5Option A**• Find the kth partial sum of the arithmetic sequence {un} with a common difference dk = 14, u1 = -1, d=6 • To use the calculator, we need a function • That’s achieved by using the explicit form • un = u1 + (n-1)(d) = -1 + (n-1)(6) = -1 + 6n – 6 = 6n – 7 • Use the calculator • sum seq(6x – 7, x, 1, 14, 1) = 532 • The answer is d**Question #5Option B**• Use the partial sum formula, particularly the 2nd partial sum formula. Remember your order of operation… • The answer is still d**Question #6**• Which best describes the relationship between the line through E and F and the line through G and H?E = (-8, -5), F = (-5, -1) and G = (-1, 2), H = (-5, 5) • Find the slope of each line • Because the slopes are inverse reciprocals (flip the fraction, flip the sign), the two lines are perpendicular. • The answer is b**Question #7**• Find an equation for the line satisfying the given conditions.y-intercept 6 and slope • You’ve got slope intercept form, so plug in the slope and the intercept • Your answer is d**Question #8**• Find the common ratio for geometric sequence 10(5)n-1 • The common ratio is the number that is multiplying the function again and again • That number is 5, and I don’t know how to explain that any more simply. • Your answer is d**Question #9Option A**• Solve by completing the square: • x2 + 3x – 10 = 0 • Use the quadratic formula. It always works. • a = 1, b = 3, c = -10 • The answer is c**Question #9Option B**• Solve by completing the square: • x2 + 3x – 10 = 0 • Turns out this one can be factored • Find two numbers that multiply to get ac: -10 • That add together to get b: 3 • Those numbers are -2 and 5 • Factor • (x2 – 2x) + (5x – 10) = 0 • x(x – 2) + 5(x – 2) = 0 • (x + 5)(x – 2) = 0 • x + 5 = 0 or x – 2 = 0 • x = -5 or x = 2 • The answer is b**Question #9Option C**• Solve by completing the square: • x2 + 3x – 10 = 0 • Sure, complete the squareThe answer is still c**Question #9Option D**• Plug in for x • If both answers equal 0, you’ve got a solution • (2)2 + 3(2) – 10 = 0 • 4 + 6 – 10 = 0 • So, 2 is an answer • (-2) 2 + 3(-2) – 10 = 0 • 4 – 6 – 10 ≠ 0 • So -2 isn’t an answer • Check, 2 and -5 both work • For the last time, the answer is c**Question #10**• Solve by taking the square root of both sides4(x-2)2 - 252 = 0 • Get the squared term by itself The answer is d**Question #11**• Determine the nature of the roots:4x2 + 32x + 64 = 0 • Use the discriminate to determine the number of real roots • Because the discriminate equals 0, there is one real root, and the answer is b**Question #12**• Solve the equation5x = 3x2 + 1 • Get everything to equal 0 and use the Quadratic Equation • The answer is d**Question #13**• If {un} is an arithmetic sequence with u1=4 and u2=5.6 • Find the common difference • Subtract u1 from u2 to find d • d = 5.6 – 4 = 1.6 • Write the system as a recursive function • Recursive functions have two parts, starting point and a function that uses the previous term (Just like problem #2) • u1 = 4 and un = un-1 + 1.6 • Give the first eight terms of the sequence • Put ‘4’ into the calculator, hit enter • Put ‘Ans + 1.6’, and keep hitting enter to get the rest of the terms • 4, 5.6, 7.2, 8.8, 10.4, 12, 13.6, 15.2 • Graph the sequence • See the answer sheet, but in short. The first term (4) has an x value of 1 and a y value of 4; the second term (5.6) has an x value of 2 and a y value of 5.6, etc.**Question #14**• For the geometric sequence with u1=3 and u2=12 • Find the common ratio • Divide u2 by u1 to find r • r = 12/3=4 • Write the system as a recursive function • Recursive functions have two parts, starting point and a function that uses the previous term (Just like problem #2) • u1 = 3 and un = un-1(4) • List the first four terms of the sequence • Put ‘3’ into the calculator, hit enter • Put ‘Ans • 4’, and keep hitting enter to get the rest of the terms • 3, 12, 48, 192 • Graph the sequence • See the answer sheet, but in short. The first term (3) has an x value of 1 and a y value of 3; the second term (12) has an x value of 2 and a y value of 12, etc.**Question #15**• Solve the equationx2 – 6x + 7 = 0 • Use the Quadratic Equation**Question #16Option A**• Find the mean, median, and mode for the set of numbers:1, 21, 21, 21, 18, 23, 13, 10 • We break out the OneVar function • Store the data as a list [2nd, subtract key] • {1, 21, 21, 21, 18, 23, 13, 10} D • Receive our data back as confirmation • OneVar [Alpha] D • is the mean (16) • Push down to get the median (19.5) • The answer is d (The mode is 21)**Question #16Option B**• 1, 21, 21, 21, 18, 23, 13, 10 • Rearrange the data in numerical order. The middle term(s) is/are the median • 1, 10, 13, 18, 21, 21, 21, 23 • (18 + 21)/2 = 39/2 = 19.5 • The mode is obviously 21 • The answer is still d**Question #17**• Describe the shape • Recap: • Skewed left graphs have a short left side(the left is screwed) • Skewed right graphs have a short right side(the right is screwed) • Uniform graphs all have the same data(uniforms are all the same) • Symmetric graphs look like a mirror(symmetry, reflection) • The answer is a**Question #18Option A**• Find the population standard deviation of the data set70, 58, 70, 43, 58, 55, 58, 68 • Use the ONEVAR function again • Store the data as a list [2nd, subtract key] • {70, 58, 70, 43, 58, 55, 58, 68} D • Receive our data back as confirmation • ONEVAR [ALPHA] D • Push down to get the population standard deviation (σx) ≈ 8.58778 ≈ 8.59 • The answer is b**Question #18Option B**• Data set: 70, 58, 70, 43, 58, 55, 58, 68 • Find the mean of the data set • (70 + 58 + 70 + 43 + 58 + 55 + 58 + 68) / 8 = 60 • Find the distances from the mean • Square them and add them together • 102 + 22 + 102 + 172 + 22 + 52 + 22 + 82 = 590 • For population standard distribution, take the average of the distance • 590 / 8 = 73.75 • Take the square root of that value • The answer, again, is b**Question #19**• In a clinical trial, a drug used to as caused side effects in 6% of patients who took it. Three patients were selected at random. Find the probability that all had side effects. • 0.06 probability for each having side effects • P(all three having SE) = 0.063 = 0.000216 • The answer is b**Question #20**• 5 yellow, 7 red, and 6 green marbles. • Two marbles are drawn. • Replacement occurs. • A random variable assigned to number of green marbles. • What is the probability that the random variable has an output of 2? • The only time you’d get a random variable of 2 is when you get 2 green marbles. • The probability of drawing a green marble is 6/18 • P(2 green) = (6/18)(6/18) = 1/9 • The answer is c**Question #21**• 2 yellow, 6 red, and 5 green marbles. • Two marbles are drawn. • Replacement occurs. • Random variable assigned to number of red marbles. • Calculate the expected value of the random variable. • We need to figure out all possibilities of red marbles (2 red, 1 red & 1 non-red, 0 red) • 2 red = (6/13)(6/13) = 36/169 • 0 red = (7/13)(7/13) = 49/169 • 1 red = everything else = 1 - 36/169 - 49/169 = 84/169 • Expected value = sum of each random variable multiplied by its probability • (2)(36/169) + (1)(84/169) + (0)(49/169) = 0.92 • The answer is b**Question #22**• 18 students. How many ways can the students who go first, second, and third be chosen? • Order matters, so we’re using Permutations • 18P3 = 4896 • The answer is b**Question #23**• What’s not right about this picture… • Each of the lines/boxes represents 25% of the data • A is true as it spans both boxes • B is true, as the range is the max value – min value • C is liar. Only half the data is greater than 65: 1 box and the right whisker • D is true, as the left side of the box represents Q1, the median of the lower half • The answer is c**Question #24**• Spin a spinner 5 times • Red = 17%; Blue = 22%; Green = 17%; Yellow = 44% • What is the probability all five will be red? • Take red probability and multiply by itself five times • (0.17)5≈ 0.000141 ≈ 0.01% • What is the probability that none of the outcomes will be yellow? • The probability of not yellow is 1 – P(yellow) • 1 – 0.44 = 0.56 • Take that probability and multiply by itself five times • (0.56)5≈ 0.0550 ≈ 5.5%**Question #25**• Find the expected value of the random variable with the given probability distribution. • Multiply each outcome by its probability and add them all together • (47)(0.05) + (23)(0.06) + (79)(0.29) + (58)(0.23) + (82)(0.37) • 70.32