NOBCChE Science Bowl: MATH Set 1

Simplify the following without the use of negative exponents: (-24x^3y^5)/(10y^6z^2)

12x^3/5yz^2

What is the angle formed, in degrees, between the hour and minute hand of an analog clock at 10:30 AM?

135

Find the sum of all interior angles, in degrees, in a regular polygon having 100 sides:

17,640 (Solution: (100 - 2)(180) = 17,640º)

Giving your answer in standard form, what is the last term in the binomial expansion of the following: (x + y)^6

y^6

Simplify the following expression: x^(1/2)x^(2/5)

x^(9/10) (ACCEPT: ^10sq rt x^9)

Given f(x) = x^2 + 2x + 3, express f(x + 1) in standard polynomial form:

x^2 + 4x + 6

If $10,000 is placed in a savings account that pays 5% interest compounded annually, what is the total amount of money in the account at the end of 2 years, to the nearest dollar?

$11,025 (Solution: A = P[1 + (r/n)]nt; = $10,000[1 + (0.05/1)](1x2) = $11,025)

What are the coordinates of the vertex in the graph of the following equation: y = 4x^2 + 8x - 5

(-1, -9) (Solution: x = -b/2a = -8/2(4) = -1; y = 4(-1)^2 +8(-1) - 5 = -9)

Find the center of the circle given by the following equation: x^2 + y^2 + 4x + 16y + 3 = 0

(-2, -8)

If a point has polar coordinates (3, π), what are the rectangular coordinates?

(-3, 0) (Solution: x = 3 cos π = -3 and y = 3 sin π = 0)

Giving your answer as an ordered pair, find the solution of the system of the following 2 equations: x = 2/3y - 2 and x = y - 3:

(0, 3)

The point (3, 2) is reflected across the graph of y = 6 and the resulting point is then reflected across the graph of y = x. What are the coordinates of the final point?

(10, 3) (Solution: the 1st reflection (3,10); 2nd reflection interchanges the x and y coordinates)

What are the coordinates of the hole in the graph of the function, g(x) = (2x^2-8) / (x-2)

(2, 8) (Solution: simplifies to 2(x + 2); graph looks like a line except at the indicated hole)

Giving your answer in simplest radical form, rationalize the denominator of the following expression: (-3sqrt2)/(sqrt3 - sqrt5)

(3 sqrt6 + 3 sqrt10)/ 2 ( ACCEPT: 3/2 (sqrt6 + sqrt10)

Factor the following expression completely: 81x^2 + 180xy + 100y^2

(9x + 10y)^2

Factor the following expression completely over the integers: x^4 - 16

(x - 2)(x + 2)(x^2 + 4)

Add the following rational expressions and give your answer in fully factored form: (x/(x+2)) + (4x/(x-6))

(x(5x+2))/((x-6)(x+2))

Simplify the following rational expression by combining like terms, assuming x is not equal to 4: (x^2 -16) / (4x - x^2)

- (x-4)/x (ACCEPT: (x+4)/(-x) or (-(x-4))/x)

For the function f(x) =1/x, what is f'(5) (read as: f prime of 5)?

-1/25 (Solution: f ' (5) = -1/5^2 = -1/25)

What is the determinant of a 2-by-2 square matrix whose rows are 4, 2 and -3, -4?

-10 (Solution: D = ad - bc = 4(-4) - (-3)(2) = -10)

Solve the following inequality for x: |3 - 2x| ≤ 7 (read as: the absolute value of, 3 minus 2x, close absolute value, is less than or equal to 7)

-2 ≤ x ≤ 5 (ACCEPT: 5 ≥ x ≥ -2)

Giving your answer as a fraction reduced to lowest terms, solve the following equation for x: (2/3)x - (1/4) = (10/6)x + (1/2)

-3/4

What is the slope of a line perpendicular to the line given by the graph of the equation, -5x = -6y + 4

-6/5

Assuming that the natural log of 5 = 1.6, find, to the first decimal place, the natural log of 5^(1/3):

0.5 (Solution: (1/3)(ln 5) = 1.6/3 = 0.53)

Solve the following equation for x over the integers: x^2 = 3x

0; 3

Solve the following equation for all real values of x: (x + 3)2 = 16

1 AND -7

In the algebra of real-valued functions, give the name or number of all of the following 3 choices that the implied domain usually excludes: 1) numbers causing division by zero 2) numbers causing imaginary numbers in the range 3) numbers causing irrational numbers in the range

1 AND 2

If y varies directly as x and z, and y = 12 when x = 4 and z = 5, find y when x = 10 and z = 20:

120 (Solution: y= kxz; 12 = k(4)(5); k = 3/5; y = 3/5(10)(20) = 120)

Simplify the following expression by combining like terms: 12.03A - 4.03B - 0.03(A - 40)

12A - 4.03B + 1.2

Cards numbered 1 through 10 lie on a table. If two cards are picked at random, without replacement, what is the probability, as a fraction in lowest terms, that the two cards will have values within one of each other?

1/5 (Solution: 9/(10 nCr 2) = 9/45 = 1/5)

Express 600º in radians as a reduced fraction in terms of π:

10π/3

Giving your answer in centimeters, what is the length of a side of a square whose diagonal measures 12 sqrt 2 centimeters?

12

If the shortest leg of a 30-60-90 right triangle is 6 inches, what is the measure, in inches, of the other 2 sides?

12 AND 6 sqrt 3

Factor the following expression completely: 12x^2 + 24x - 36

12(x + 3)(x - 1) (ACCEPT: 12(x - 1)(x + 3))

What is the area, in square feet, of a triangle whose perpendicular height is 20 feet with a base of 12 feet?

120 (Solution: A = ½ bh = ½ (12)(20) = 120 ft2)

Convert the angle, 18º18'36" (read as: 18 degrees, 18 minutes, 36 seconds) to degrees, to the second decimal place:

18.31 (Solution: 18 + 18/60 + 36/3600 = 18.31)

Find the prime factorization of 240:

2 × 2 × 2 × 2 × 3 × 5 (ACCEPT: 24 × 3 × 5)

Simplify the following, giving your answer in proper scientific notation: (6.0 × 10^12)(2.0 × 10^10)^2

2.4 × 10^33

Multiply the following 4 numbers and give your answer in scientific notation: 30,000 × 3,000 × 30 × 0.1

2.7 × 10^8

Find the inverse of the 2-by-2 matrix whose rows are 3, 5 and 7, -2. Give your answer in rows.

2/41, 5/41, 7/41, 3/41

What is the angular velocity, in radians per second, of a wheel with a radius of 250 centimeters and a surface velocity of 50 meters per second?

20 (Solution: Τ = v/r = 50m/s/2.5m = 20 rad/s)

Multiply the following complex numbers, giving your answer in standard a + bi form: (6 + 3i)(4 + i)

21 + 18i

What is the degree measure of the angle formed by 2 secants intersecting outside a circle if the measures of the intercepted arcs are 80º and 34º?

23 (Solution: ½(80º - 34º) = 23º)

Find the area, in square centimeters, of a rhombus with a diagonal of 6 centimeters and a side of 5 centimeters:

24 (Solution: A = ½ d1d2 = ½ (6)(8) = 24 cm2, (3:4:5 triangle))

Factor the following expression completely: 100x^2 - 25y^2

25(2x + y)(2x - y)

Find the sum of the first 10 terms of the arithmetic sequence whose first three terms are 7, 11, and 15:

250

Giving your answer as a proper fraction, what is the cube root of 125/343?

5/7

An object is launched straight up from the ground with an initial velocity of 176 feet per second. What is the height of the object above the ground, in feet, 2 seconds into the flight?

288 (Solution: P(t) = -16t^2 + 176t; P(2) = -16(4) + 352 = 288 ft)

Find the following product, giving your answer in standard form: (4x^3 + x)(7x^3 + 4x^2)

28x^6 + 16x^5 + 7x^4 + 4x^3 (ACCEPT: x^3(28x^3 + 16x^2 + 7x + 4))

In what quadrant does θ (read as: theta) terminate if sine θ is positive and secant θ is negative?

2nd (ACCEPT: 2)

What is the smaller of two integers whose sum is 19 and whose product is 48?

3 (Solution: 16 + 3 = 19; 16 × 3 = 48)

It takes 20 minutes for Janis to prepare lunch. When Kristen helps, it only takes 12 minutes. How many minutes would it take Kristen to prepare the same lunch if she worked alone?

30 (Solution: 12/20 + 12/x = 1, x = 30 min)

Find the numerical value of the geometric mean of the following 4 values: 2^1; 2^3; 2^8; 2^8

32 (DO NOT ACCEPT: 25) (Solution: 1 + 3 + 8 + 8 = 20/4 = 5; 25 = 32)

Evaluate the following expression when a = 1/3, b = 4, c = -1, and d = -5: 2[3a - (b + c) - 4d]

36

Giving your answer in meters squared, find the total surface area of a right prism whose base is a right triangle with legs of length 3 meters and 4 meters and whose altitude is 2 meters:

36 (Solution: 3:4:5; Pbase = 3+4+5 = 12m; A = ½bh = 6 m^2; TA = LA + 2B = (12)(2) + 2(6) = 36 m^2)

Arrange the following 4 radicals in order of increasing value: 6sqrt12; 6sqrt63; 3sqrt2; 4sqrt8

3sqrt2; 6sqrt12; 4sqrt8; 6sqrt63

Solve the following equation for x: -2|3 + (x/2)| = -10 (read as: negative 2 times the absolute value of 3 plus, open parenthesis, x divided by 2, close parenthesis, close absolute value, equals negative 10)

4 AND -16

Solve the following equation for all real values of X: (1/8)x^2 - (1/4)x - 1 = 0

4 OR -2

Solve the following equation for x: 2^7 x 2^(x^2-2) = 2^21

4 OR -4 (ACCEPT: x=±4) (Solution: 7 + x^2 - 2 = 21; x = 4 and -4)

A jar of 100 marbles contains only 2 colors of marbles, red and blue. If there are 13 red marbles for every 12 blue marbles, how many blue marbles are in the jar?

48 (Solution: 12/25 = B/100, B = 48)

Find the area, in centimeters squared and in simplest radical form, of a regular or equilateral triangle whose circumcircle has a radius of 8 centimeters:

48 sqrt3 (Solution: apothem = 4; perimeter = (8 sqrt3)(3); A = ½(4)(24 sqrt3) = 48 sqrt3 cm^2)

Divide the following and give your answer in scientific notation: (8E7)/(4E4)^(1/2)

4E5

There are 10 people on a bus who, when ranked from youngest to oldest, all differ from those on the bus closest in age to them by the same number of years. If the youngest is 6 and the oldest is 51, what is the common difference in their age?

5 (Solution: 6-11-16-21-26-31-36-41-46-51)

Giving your answer in inches and in simplest radical form, if the hypotenuse of a 45-45-90º triangle is 10 inches, find the length of the other sides:

5 sqrt 2 (ACCEPT: BOTH 5sqrt 2)

Find 2 mean proportionals between the numbers 3 and 24:

6 AND 12 (Solution: 3r^3 = 24, r = 2)

In a circle with diameter AB, if the radius is 6 centimeters, find the length in centimeters, giving your answer in terms of π, of the arc AB:

6π (Solution: AB = ½(2πr) = 6π cm)

If the area of a regular pentagon is 280 square centimeters, what is the length, in centimeters, of each side if the apothem is 16 centimeters?

7 (Solution: A = ½ ap; 280 = ½ (16)(p); p = 35/5 = 7 cm)

Giving your answer in millimeters squared, find the lateral surface area of a right pentagonal prism whose altitude is 12 millimeters and base has sides of length 8, 10, 12, 14, and 16 millimeters, respectively:

720 (Solution: LA = (P)(h) = (8+10+12+14+16)(12) = 720 mm^2)

A flagpole casts a 100-foot long shadow. If a 6-foot tall person standing at the base of the pole casts an 8-foot shadow, how many feet tall is the flagpole?

75 (Solution: (100 × 6)/8 = 75 ft)

How many solutions are there for the equation, cosine 4x = 1/2, for the interval 0 ≤ x < 2π?

8 (Solution: 4 cycles, 2 solutions per cycle)

A circle has a diameter of 32 meters. Find the degree measure of the central angle of a sector of the circle if its arc length measures 8π meters:

90 (Solution: arc/circumference = nº/360º; n = 90º or (8π/2π16)(360º) = 90º)

Giving your answers in terms of π and in inches squared, find the total surface area of a closed right circular cone whose altitude is 8 inches and radius is 6 inches:

96π (Solution: l62 = 862 + 6^2, l = 10 in; LSA = πrl = π(6)(10) = 60π; SA = LA + B = 60π + π(6)^2 = 96π in^2)

Giving your answer in terms of π and in inches, what is the arc-length of a semi-circle whose diameter is 18 inches?

9π (Solution: C = πd; ½(18π) = 9π)

Give the resulting vector for the cross product of vector <0,0,1> with vector <1,0,0>:

<0,-1,0>

The curve defined by the equation, Ax^2 + Bx + Cy^2 + Dy + E =0, can be an ellipse if which of the constants are positive?

A AND C

By words or number, name all of the following 3 statements that are TRUE for the function, f(x) = -3x2 - 2x - 2 = 0: 1) there are no real zeros 2) the graph is a parabola opening downward 3) the graph has no x-intercepts

ALL

Giving your answer in standard form, give the equation of the circle with center at (0, 0) and diameter of 6:

x^2 + y^2 = 9

Which of the following points is closest to the point (1, -2): W) (3, -5) X) (0, 3) Y) (3, 0) Z) (-5, -5)

ANSWER: Y) (3, 0)

Consider triangle ABC, where the measure of angle A = 20º, B =117º, and C = 43º. Arrange the 3 sides in order of increasing length:

BC; AB; AC (Solution: the longer side is opposite the greater angle)

Simplify the following rational expression: (5y - 3x^2) / (9y^2 + 5x)

CANNOT BE FURTHER SIMPLIFIED (ACCEPT: (5y - 3x^2) / (9y^2 + 5x) or THE SAME)

Identify the type of conic section represented by the equation, x^2 + y^2 + 10y + 16 = 0:

CIRCLE (Solution: x^ + (y + 5)^2 = 9)

What term BEST describes 2 angles with 90º as the sum of their measurements?

COMPLEMENTARY

Find the cotangent and cosecant, respectively, of a 90º angle:

COTAN = 0; COSECANT = 1

Identify the type of conic section represented by the equation, x^2 - 16y^2 = 64:

HYPERBOLA

The graph of the polar equation, r = 5 - 5 cosine θ, is a cardioid. In which two quadrants is most of the area of this cardioid?

II AND III (ACCEPT: 2 AND 3 or SECOND AND THIRD)

What is the specific name for a common tangent that intersects the segment joining the centers of 2 circles?

INTERNAL COMMON TANGENT (ACCEPT: INTERNAL or INTERNAL TANGENT)

What are the leading coefficient and degree, respectively, of the following polynomial: 3x^2 - 6x^5 + 7

LEADING COEFFICIENT = -6; DEGREE = 5

Giving your answer as up, down, left, or right, the graph of the equation x = 5y - 8y2 - 1 opens in what direction?

LEFT

What physical property of neutrinos, which was confirmed in the past decade, contradicts earlier assumptions that neutrinos travel at the speed of light?

MASS (ACCEPT: NEUTRINOS HAVE MASS)

The terminal side of angle θ (read as: theta) in standard position passes through the point (-5, 12). Find the sine, cosine and tangent, in fractional form:

SIN = 12/13; COS = -5/13; TAN = -12/5

What is the name for the test that can be used to visually determine whether or not a relation defined by a graph represents a function?

VERTICAL LINE TEST (ACCEPT: VERTICAL LINE)

Which of the following is a second quadrant angle in standard position: W) 500º X) -180º Y) -680º Z) 930º

W) 500º

Which of the following is a quadratic equation with a root of multiplicity of 2: W) 9x^2 - 30x^2 + 25 = 0 X) x^2 + 2x - 15 = 0 Y) 6x^2 - 11x - 2 = 0 Z) 4x^2 - 1 = 0

W) 9x^2 - 30x^2 + 25 = 0 (Solution: x = 5/3)

If triangle ABD is congruent with triangle EFG, which of the following MUST be true: W) side AB is congruent to side EF X) side AD is congruent to side FG Y) measure of angle B = measure of angle G Z) measure of angle A = measure of angle G

W) SIDE AB IS CONGRUENT TO SIDE EF

A data set has a mean of 10 and variance of 10. If every set element is multiplied by 2, which of the following is TRUE of the new mean and variance: W) mean is 20, variance is 40 X) mean is 20, variance is 10 Y) mean is 20, variance is 20 Z) mean is 10, variance is 20

W) mean is 20, variance is 40

Which of the following is the equation for the tangent line for y = x^4 at the point (1, 1): W) y = 4x - 3 X) y = 4x - 1 Y) y = 4x + 1 Z) y = 4x + 3

W) y = 4x - 3 (Solution: using the derivative, y' = 4x^3, the slope of the tangent line at x = 1 is 4. The line must pass through (1, 1) so y - 1 = 4(x - 1))

What is the product of the following 2 values: (1) the greatest common divisor of 7 and 14; and (2) the least common multiple of 7 and 14: W) 21 X) 49 Y) 98 Z) 196

Y) 98 (Solution: the product will be the same as the product of the two numbers, 7 and 14)

At x = −1/2, the graph of y = x^3 is: W) increasing and concave up X) increasing and concave down Y) decreasing and concave up Z) decreasing and concave down

X) INCREASING AND CONCAVE DOWN (Solution: at the indicated value the 1st derivative is positive (implying increasing) and the 2nd derivative is negative (implying concave down))

The graph of the relation y = -sqrt(25 -x^2) is a: W) circle X) semi-circle Y) parabola Z) straight line

X) SEMI-CIRCLE

Find the x and y intercepts of the line passing through the point (4, 8) that is perpendicular to the line 3x - 6y - 12 = 0:

X-INTERCEPT = (8, 0) (ACCEPT: 8); Y-INTERCEPT = (0, 16) (ACCEPT: 16)

Which of the following represents the inverse function of f(x) = x^3 + 5: W) (x - 5)^3 X) x^3 - 5 Y) (x-5)^(1/3) Z) (1/(x-5))^3

Y) (x-5)^(1/3)

Which of the following is NOT a Pythagorean triple: W) 15, 20, 25 X) 10, 24, 26 Y) 16, 24, 30 Z) 33, 44, 55

Y) 16, 24, 30

If e^(x/5) = 30, which of the following is the value of x, assuming that the natural log of 2 = 0.7, and the natural log of 15 = 2.7: W) 15 X) 16 Y) 17 Z) 18

Y) 17 (Solution: x/5 = ln 15 + ln 2 = 3.4; x/5 = 3.4; x = 17)

Which of the following numbers is evenly divisible by both 11 and 3: W) 7791 X) 7553 Y) 5181 Z) 8769

Y) 5181

Which of the following is NOT true of matrices: W) a square matrix that has an inverse is called invertible X) a matrix does not have to have an inverse Y) a matrix that does not have an inverse is called non-singular Z) a matrix must be square in order to have an inverse

Y) A MATRIX THAT DOES NOT HAVE AN INVERSE IS CALLED NON-SINGULAR (Solution: it's called singular)

Which of the following is NOT true: W) all central angles in a regular polygon are equal X) irregular polygons are not considered to have centers and have no central angles Y) the circumcircle of a regular polygon is the circle that passes through at least one vertex Z) all regular polygons are convex

Y) THE CIRCUMCIRCLE OF A REGULAR POLYGON IS THE CIRCLE THAT PASSES THROUGH AT LEAST ONE VERTEX

Which of the following is TRUE about any two successive terms in the Fibonacci sequence: W) their product is a Fibonacci number X) they are either both odd or both even Y) they are relatively prime Z) their quotient is the golden ratio

Y) they are relatively prime

One-fifth of 0.04% is equal to: W) 8 × 10-2 X) 8 × 10-3 Y) 8 × 10-4 Z) 8 × 10-5

Z) 8 × 10-5 (Solution: (0.2)(0.0004) = 0.00008)

The graph of r = 3 secant (θ) (read as: r = 3 secant theta) is a: W) parabola X) circle Y) hyperbola Z) line

Z) line

Simplify the following expression, where a is not equal to zero if b = 0 or if b = -1: (a^b^2)(a^b)

a^(b^2 +b)

Simplify i^17 (read as: i to the 17th power):

i (Solution: i^17 = i^1 = i)

Of the following 5 functions identify all that are NOT differentiable for all real numbers: | x | (read as: absolute value of x); cosine(x); tangent(x); e^x; the greatest integer function

| x |; TAN(x); THE GREATEST INTEGER FUNCTION