exam questions

Maria invested $5,000 in a 3% annual rate CD, compounding continuously. What is the current worth after three years?

$5,471

Which of the following is the vertex of y=3|x−1|+2?

(1, 2)

Which of the following is the correct factorization of 6x2−7x+26x2-7x+2?

(3x−2)(2x−1)

If 𝑛,𝑚,and𝑘n,m,andk are non-negative integers, what is the total number of factors of 2^𝑚3^𝑛10^𝑘?

(𝑚+𝑘+1)(𝑛+1)(𝑘+1) We rewrite 2𝑚3𝑛10𝑘2m3n10k completely factored as 2𝑚3𝑛(2⋅5)𝑘=2𝑚+𝑘3𝑛5𝑘2m3n(2⋅5)k=2m+k3n5k. Each factor of 2𝑚+𝑘3𝑛5𝑘2m+k3n5k is the product of between 0 and 𝑚+𝑘m+k twos, between 0 and 𝑛n threes, and between 0 and 𝑘k fives. This yields (𝑚+𝑘+1)(m+k+1) possible products of twos, (𝑛+1)(n+1) possible products of threes, and (𝑘+1)(k+1) possible products of fives, so there are (𝑚+𝑘+1)(𝑛+1)(𝑘+1)(m+k+1)(n+1)(k+1) possible factors. The correct answer is (C).

Find the dot product < 0, 3 > • < 4, -2>

-6

Consider the following statement in choosing the correct answer below: ∑𝑗=1𝑛𝑗3=13+23+33+⋯+𝑛3=𝑛2(𝑛+1)24forall𝑛=1,2,3,...∑j=1nj3=13+23+33+⋯+n3=n2(n+1)24foralln=1,2,3,... In using the Principle of Mathematical Induction to prove the statement, one needs to show:

. that the statement is true for 𝑛=1n=1, and if the statement is true for 𝑛=𝑘n=k, where 𝑘≥1k≥1 then it is true for 𝑛=𝑘+1n=k+1. To prove the statement is true for all positive integers using the Principle of Mathematical Induction, one has to first prove the statement is true for 𝑛=1n=1; this is called the base case. Next one must prove that if the statement is true for 𝑛=𝑘n=k, where 𝑘k is a fixed but arbitrary number greater than 1, then the statement is true for 𝑛=𝑘+1n=k+1. This will mean that the statement is true for all positive integers. The correct answer is (C).

Convert 30 yards per hour into inches per second.

0.3 in/sec

A rectangle has length 10 feet and width 6 feet. It is enlarged to be a similar rectangle with length 18 feet. What is the new width?

10.8 feet

1 kilometer = cm

100,000 cm

1 KCAL

1000 cal

1 megabyte

1000 kilobytes

1 kilometer

1000 meters

1 liter

1000 milliliters

What is the Greatest Common Factor of 77 and 99?

11

Huntsville's population grows from 25,000 to 28,000. What is the percent increase in Huntsville's population?

12%

Juan has $20 and is saving $6 per week. Angel has $150 and is spending $4 per week. After how many weeks will they have the same amount of money?

13 weeks

A bacteria population is currently exactly one million and increasing at a linear rate of 15 per second. What will the population be in two weeks?

19,144,000

1000G

1kg

100CM

1m

If f(x)=x2−3f(x)=x2-3 and g(x)=2x+1g(x)=2x+1, which of the following represents f(g(2))f(g(2))?

22

In order to identify all the prime divisors of 578, a person needs to check all the primes less than or equal to 𝑝p, where 𝑝p equals what prime?

23 In order to identify all the prime divisors of 578, a person needs to check all the primes less than or equal to 𝑝p, where 𝑝p is the greatest prime less than or equal to 578‾‾‾‾√578. Factoring 578 we learn that 578=2⋅17⋅17578=2⋅17⋅17, so that 578‾‾‾‾√=172‾√≈17(1.4)=23.8578=172≈17(1.4)=23.8. Hence, one should check all primes less than or equal to 23. The correct answer is (B).

A class has three fewer girls than boys. If the ratio of girls to boys is four to five, how many total students are there in the class?

27

What is the sum of the coefficients in any row of Pascal's triangle, (a+b)n(a+b)n?

2^n

Find all the complex zeros of f(x)=x^3−3x^2+4x−12

3,2i,−2i

According to the Rational Root Theorem, which of the following could not be a root of the polynomial 𝑝(𝑥)=4𝑥3−𝑏𝑥2+3𝑥+2p(x)=4x3−bx2+3x+2 where 𝑏b is an integer?

4 The Rational Root Theorem tells us that if 𝑎a is a root of the polynomial 𝑝(𝑥)=4𝑥3−𝑏𝑥2+3𝑥+2p(x)=4x3−bx2+3x+2, then 𝑎a is of the form 𝑎=𝑐𝑑a=cd, where 𝑐c is a divisor of the constant term, 22, and 𝑑d is a divisor of the leading coefficient, 44. Thus 𝑐∈{±1,±2}c∈{±1,±2}, while 𝑑∈{±1,±2,±4}d∈{±1,±2,±4}. Hence, it follows that 𝑎=𝑐𝑑∈{±2,±1,±12,±14}a=cd∈{±2,±1,±12,±14}. So, (D) is the correct answer.

A ball is thrown straight up from the edge of a 15 foot balcony with initial velocity 40 feet/sec. How high will the ball travel?

40 feet

Turn the following sentence into an algebraic equation: "The tax on a $400 game system is $34." Let t represent tax as a decimal.

400t=34

Steve wants to cover a high school football field with sod. The field is 360 feet long and 160 feet wide. Sod can be purchased in squares in 1 foot increments from 1 foot wide up to 7 feet wide. What is the largest size squares Steve can purchase with which he can cover the field completely without any gaps or overhang?

5

Line 𝑚m is perpendicular to the line segment with endpoints 𝐴(3,−3)A(3,−3) and 𝐵(6,0)B(6,0). If line 𝑚m passes through the midpoint of 𝐴𝐵⎯⎯⎯⎯⎯⎯⎯⎯AB¯, what is the sum of the 𝑥x-coordinate of line 𝑚m's 𝑥x-intercept and the 𝑦y-coordinate of line 𝑚m's 𝑦y-intercept?

6 The containing line segment 𝐴𝐵AB has slope =−3 − 03 − 6=1=−3 − 03 − 6=1, hence line 𝑚m which is perpendicular to this line has slope −1−1. The midpoint of the line segment 𝐴𝐵AB has coordinates (3+62),(−3+02)=(92,−32)(3+62),(−3+02)=(92,−32). Using the point-slope formula for a line we obtain the equation of line 𝑚:𝑦−−32=−1(𝑥−92)m:y−−32=−1(x−92) or 𝑦=−𝑥+3y=−x+3. The x-intercept of line 𝑚m is (3,0)(3,0) and the 𝑦y-intercept is (0,3)(0,3). The sum of the 𝑥x-coordinate of the 𝑥x-intercept and the 𝑦y-coordinate of the 𝑦y-intercept if line 𝑚m is 6. The correct answer is (B).

Jill decides to move her fish from a 20 gallon aquarium to a 50 gallon aquarium. She added 3 tablespoons of salt to her old aquarium. If she wishes to maintain to same salinity level, how many tablespoons should she add to her new aquarium?

7.5

Which of the following is a vector that is orthogonal to the plane determined by the triangle with vertices 𝐴(3,0,−2),𝐵(−1,−2,1),𝐶(0,−4,−1)A(3,0,−2),B(−1,−2,1),C(0,−4,−1)?

<2,−1,2> The vector 𝑣→=𝐴𝐵−→−×𝐴𝐶−→−v→=AB→×AC→ is orthogonal to the plane determined by the three given vertices. Any vector parallel to 𝑣→v→ is also orthogonal to the determined plane; i.e., any non-zero multiple of 𝑣→v→ is orthogonal to the determined plane. Hence we must calculate 𝑣→=𝐴𝐵−→−×𝐴𝐶−→−v→=AB→×AC→: First we find the vectors: 𝐴𝐵−→−=<−4,−2,3>, 𝐴𝐶−→−=<−3,−4,1>AB→=<−4,−2,3>, AC→=<−3,−4,1>, therefore, 𝑣→=𝐴𝐵−→−×𝐴𝐶−→−=<(−2+12),−(−4+9),(16−6)> = <10,−5,10>.v→=AB→×AC→=<(−2+12),−(−4+9),(16−6)> = <10,−5,10>. Only <2,−1,2><2,−1,2> is a multiple of 𝑣→v→ so the correct answer is (B).

Which of the following statements is true?

According to the Fundamental Theorem of Algebra, the equation 8𝑥^4+2𝑥^5=5𝑥 must have five complex number solutions.

Which of the following describes the relationship between 45 and 51?

B. They are both composite numbers.

Why does the Vertical Line Test always reveal if a relation is a function?

Because a function cannot have multiple dependent values for a single independent value.

The equation below is an example of which axiom? 5 (7 + 3) = 5 (7) + 5 (3)

D. Distributive property

In what quadrant would the graph of vector u=<−3u=<-3, 2>lie2>lie?

II

Which of the following statements is true for the equation 𝑖𝑧3+27=0iz3+27=0?

It has three distinct complex solutions, with none being a real number. The Fundamental Theorem of Algebra tells us that there are three complex solutions to the equation 𝑖𝑧3+27=0iz3+27=0, not necessarily distinct; some might be real numbers. The equation is equivalent to 𝑖3𝑧3+27𝑖2=0i3z3+27i2=0 or (𝑖𝑧)3−(3)3=0(iz)3−(3)3=0. The left side of the equation is a difference of two cubes. Thus, the left side can be factored to obtain the equation (𝑖𝑧−3)((𝑖𝑧)2+3𝑖𝑧+9)=0(iz−3)((iz)2+3iz+9)=0. By the Zero Product Property 𝑧=3𝑖=−3𝑖z=3i=−3i is a solution, and also we can solve (𝑖𝑧)2+3𝑖𝑧+9=0(iz)2+3iz+9=0 or 𝑧2−3𝑖𝑧−9=0z2−3iz−9=0 using the quadratic formula: 𝑧=3𝑖±9𝑖2+36‾‾‾‾‾‾‾‾√2=3𝑖±27‾‾‾√2=3𝑖±33‾√2.z=3i±9i2+362=3i±272=3i±332. Hence, there are three distinct complex solutions, none which are real. The correct answer is (A).

Which of the following is true regarding the Euclidean Algorithm for finding a greatest common divisor between two integers?

It works without having to check which numbers are prime

Which of the following would be the next step in completing the proof by mathematical induction? Prove: 2 + 4 + 6 + 8 + ... + 2n = n (n + 1) Student work: Step 1: When n = 1, the formula is valid because 2 = 1 (1 + 1) Step 2: Assume the formula is true for k: 2 + 4 + 6 + ... + 2k = k (k + 1) Next Step:

Show 2 + 4 + 6 + ...2 (k+1) = (k + 1) (k + 1 + 1)

Which of the following is a group property that 𝑋=RX=R, the set of real numbers, with binary operation 𝛽(𝑥,𝑦)=(𝑥+𝑦)2β(x,y)=(x+y)2, satisfies?

The Commutative Property

Which of the following statements is false? A. The function 𝑦=3𝑒−10𝑥y=3e−10x is a one-to-one function. B. The function 𝑦=(𝑥+1)3−4y=(x+1)3−4 is an onto function. C. The graphs of 2𝑥−5=42x−5=4 and 5𝑥−2𝑦=65x−2y=6 intersect at exactly one point. D. The domain of the composition 𝑓∘𝑔f∘g, where 𝑓(𝑥)=1𝑥−2′f(x)=1x−2′ and 𝑔(𝑥)=𝑥2−1‾‾‾‾‾‾√g(x)=x2−1, is set (1,5‾√)∪(5‾√,∞)(1,5)∪(5,∞).

The domain of the composition 𝑓∘𝑔f∘g, where 𝑓(𝑥)=1𝑥−2′f(x)=1x−2′ and 𝑔(𝑥)=𝑥2−1‾‾‾‾‾‾√g(x)=x2−1, is set (1,5‾√)∪(5‾√,∞)(1,5)∪(5,∞). The function 𝑦=3𝑒−10𝑥y=3e−10x is a strictly decreasing function and hence must be a one-to-one function. The graph of the function 𝑦=(𝑥+1)3−4y=(x+1)3−4 is the graph of the function 𝑦=𝑥3y=x3 translated 1 unit to the left and 4 units down. The range of the function 𝑦=𝑥3y=x3 is (−∞,+∞)(−∞,+∞) and so is onto function. It follows that the translated function is also onto.The domain of the composition 𝑓∘𝑔,f∘g, where 𝑓(𝑥)=1𝑥−2f(x)=1x−2, and 𝑔(𝑥)=𝑥2−1‾‾‾‾‾‾√g(x)=x2−1, is the set of all reals, 𝑥x, in the domain of 𝑔g such that 𝑔(𝑥)g(x) is in the domain of 𝑓f. So, the domain of 𝑔g is the set of all 𝑥x,such that 𝑥2−1≥0x2−1≥0; this is the set (−∞,−1]∪[1,+∞)(−∞,−1]∪[1,+∞). We must eliminate any 𝑥x such that 𝑓(𝑥)f(x) is not defined; this would be any 𝑥x for which 𝑔(𝑥)=2g(x)=2.So we solve this equation:𝑥2−1‾‾‾‾‾‾√=2⇔𝑥2−1=4⇔𝑥2=5⇔𝑥=±5‾√x2−1=2⇔x2−1=4⇔x2=5⇔x=±5. Hence the domain of 𝑓⋅𝑔is(−∞,−5‾√)∪(−5‾√,−1]∪[1,5‾√)∪(5‾√,∞)f⋅gis(−∞,−5)∪(−5,−1]∪[1,5)∪(5,∞).The graphs of 2𝑥−5𝑦=42x−5y=4 and 5𝑥−2𝑦=65x−2y=6 intersect at exactly one point because their slopes are 2525 and 5252, respectively. From this we know the lines are not parallel.(D) is the correct answer.

If 𝐴A is an 𝑛×𝑛n×n matrix that is not invertible, which of the following statements must be true?

The reduced row-echelon form of 𝐴A has a row of zeroes. If 𝐴A is an 𝑛×𝑛n×n matrix that is not invertible, then its determinant is zero and the reduced row-echelon form of 𝐴A has a row of zeroes.𝐴𝑋=𝐵AX=B is consistent for every 𝑛×1n×1 matrix 𝐵B in the case that 𝐴A is invertible (in fact it has exactly one solution when 𝐴A has an inverse); in the case that 𝐴A is not invertible, the corresponding system has either no solutions or infinitely many solutions.The correct answer is (B).

The set of integers forms a ring. Select the answer that is also valid.

The set of integers is not a field because there is no multiplicative inverse.

Which of the following properties is illustrated by: If (3x−1)(x+2)=0(3x-1)(x+2)=0, then x=13x=13 or x=−2

Zero product property

If a, b, and abab are natural numbers and 1 is the smallest natural number, then what must be true?

a≥b

Which is an equivalent expression for 2log4_3+log4_2

log4_18

Roses and carnations are on sale. You can buy a bouquet of 6 carnations and 5 roses for $23. Or you can get 2 carnations and 1 rose for $6. How much does a rose cost? How much does a carnation cost?

rose $2.50, carnation $1.75

Solve the inequality x2−6x>7

x<−1x<-1 or x>7

Solve the equation for x. √14−x =x−2

x=5

Which inequality represents the graph provided?

y<2x

The graph below illustrates the location of two points, A and B. What is the equation for line AB?

y=−2x−2

Which of the following set of vectors in R3 is linearly independent?

{<1,2,1>,<1,0,−1>,<1,1,1>} Two of the sets of vectors are clearly linearly dependent. The set of vectors in (C) contains the zero vector, hence, these vectors are linearly dependent. There are 4 vectors in the set in (D), thus, since the number of vectors in the set exceeds the dimension (which here is three) the vectors are linearly dependent.To see if the vectors in (A) are linearly independent, we look for constants 𝑐1,𝑐2,𝑐3c1,c2,c3 such that 𝑐1<1,2,1>+𝑐2<1,0,−1>+𝑐3<1,1,1>=<0,0,0>c1<1,2,1>+c2<1,0,−1>+c3<1,1,1>=<0,0,0>. This is equivalent to solving the system of equations: ⎧⎩⎨⎪⎪𝑐1+𝑐2+𝑐3=02𝑐1+𝑐3=0𝑐1−𝑐2+𝑐3=0{c1+c2+c3=02c1+c3=0c1−c2+c3=0 To this end, we perform Guass-Jordan Elimination on the following matrix:⎡⎣⎢⎢⎢121 1 0−1111000⎤⎦⎥⎥⎥[1 1102 0101−110]→−2𝑅1+𝑅2→𝑅2 −𝑅1+𝑅3→𝑅3→−2R1+R2→R2 −R1+R3→R3→⎡⎣⎢⎢⎢1001−2−21−10000⎤⎦⎥⎥⎥→→[11100−2−100−200]→−12𝑅2→𝑅2−1𝑅2+𝑅1→𝑅12𝑅2+𝑅3→𝑅3−12R2→R2−1R2+R1→R12R2+R3→R3→⎡⎣⎢⎢⎢⎢10001012122000⎤⎦⎥⎥⎥⎥→→[10120011200020]→ 12𝑅3→𝑅3−12𝑅3+𝑅1→𝑅1−12𝑅3+𝑅2→𝑅2 12R3→R3−12R3+R1→R1−12R3+R2→R2→⎡⎣⎢⎢⎢100010001000⎤⎦⎥⎥⎥→[100001000010].This indicates the solution is 𝑐1=𝑐2=𝑐3=0c1=c2=c3=0. Hence the vectors in the collection (A) are linearly independent.We note that another way to see that a collection is linearly independent is to take the determinant of the coefficient matrix. We do that here to show that (B) represents a linear dependent collection of vectors; this is because if the determinant of the coefficient matrix is zero then there are infinitely many solutions for the homogeneous equation and if the determinant is non-zero this means there is exactly one solution, the trivial solution as in (A).The coefficient matrix for (B) is 𝐴=⎡⎣⎢⎢111 1−1 2314⎤⎦⎥⎥A=[1 131−111 24]. The determinant of 𝐴A is (−4+1+6)−(−3+2+4)=0(−4+1+6)−(−3+2+4)=0. Thus the vectors in (B) are linearly dependent.The correct answer is (A).

2−5i / 7+5i

−11/74 − 45/74 i

What is the magnitude of the sum of vector a=(6,3,−1)a=(6,3,-1) and vector b=(−2,2,4)b=(-2,2,4)?

√50