CLT Math Practice
If A and B are positive integers and A is a factor of B, which of the following MUST be false? A) a is greater than b. B) b=ak for some integer k. C) b is a multiple of a. D) b/a is an integer
A Factor of an integer- Any other integer that divides it evenly (no remainders). A) a is greater than b. = FALSE, in this case the factor is equal. B) b=ak for some integer k. = true C) b is a multiple of a. = true D) b/a is an integer. = true
Line x=3 is reflected over y-axis. What is the resulting line? A) x=−3 B) x=3 C) x=1/3 D) y=3
A Reflection across the y-axis results in x=-3
1, 1/4, ?, 1/16, 1/25, 1/36... What is the third term? A) 1/9 B) 1/10 C) 1/12 D) 1/14
A They're all consecutive perfect squares. 2(2)=4 3(3)=9 4(4)=16 5(5)=25 6(6)=36... 1/9 is the third term.
How many integers from 1 to 300 (inclusive) satisfy the following two conditions? 1. It is evenly divisible by 15. 2. None of its digits are odd. A) 2 B) 3 C) 5 D) 7
A Thinking about condition 1, there are 20 multiples of 15 between 1 and 300. Thinking about condition 2, notice that the multiples of 15 alternate ending in 5 or 0. Therefore the odd multiples of 15, which all end in 5, can be ignored. You need only really think about the multiples of 30 from 1 to 300, of which there are 10 and they are easy to calculate: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300. Of these, only 60 and 240 do not contain an odd digit.
16 round tables set up for 126 guests. 1 table is left with 2 open spaces. How many chairs are at each table? A) 7 chairs B) 8 chairs C) 9 chairs D) 10 chairs
B 126 + 2= 128 chairs total. 128/16= 8 chairs at each table.
If "cube" is defined x"cubed"y= 2x - 3y what is the equivalent to (-2)"cubed"2 ? A) −12 B) −10 C) −2 D) 2
B 2(-2) - 3(2)= -4 - 6= -10
27 summer interships, 42 part time jobs. 3 summer interships have filled all their positions. What is the ratio of companies offereing summer internships to companies offering part time jobs? A) 3 : 5 B) 4 : 7 C) 5 : 7 D) 9 : 14
B 27 - 3= 24 summer internships. 24:42 24/6= 4 42/6= 7 4:7 is the ratio.
Both circles have a center at A, and point B is the intersection of AC. AB is 4 units and BC is 2 units. What is the difference in circumference between the longer circle and smaller circle? A) 2π units B) 4π units C) 8π units D) 12π units
B AB= 4 (the radius) AC= AB + BC 6= 4 + 2 Calculate their circumferences (check math equations for reference) C= 2pieR AC= 2pie(6) AC= 12pie BC= 2pie (4) BC= 8pie 12pie - 8pie= 4pie.
Which is NOT a solution of the given equation? x^3 - 9x = 16x A) 0 B) 1 C) 5 D) All of the above are solutions of the given equation.
B Just plug in the given choices. 1^3 - 9(1) = 16(1) 1 - 9 = 16 -8 = 16 1 is not a solution to the equation.
Two rectangles are similar. Which of the following must be true? A) Both rectangles are squares. B) The interior angles of each rectangle add up to 360°. C) One of the rectangles has a larger area than the other rectangle. D) The lengths of the diagonals of the rectangles are equivalent.
B The interior angles of any quadrilateral add to 360°, so this would be true even if the rectangles were not similar. A- false C- it is possible for similar polygons to have the same area. D- unless the rectangles happen to be congruent, their diagonals, which can be calculated as a function of their side lengths using the Pythagorean theorem, will not be equal.
A store owner tells his clerk to raise all the prices by 10% each Wednesday, keep Wednesday's prices the same on Thursday, and decrease the prices by 10% each Friday. At the end of the first week, what is the price, ignoring sales tax, of a shirt that was originally $100? A) $98 B) $99 C) $100 D) $101
B The original price of the shirt is $100. On Wednesday, the price will increase by 10% (0.10 ⋅ $100 = $10) the new price will be $110. On Friday, the price will drop by 10% (0.10 ⋅ $110 = $11) $110 - $11= $99 the final price at the end of the week will be $99.
If n>0 and n^y/ n^1/3= 1/ n^3, what is the value of y? A) −10/3 B) −8/3 C) 8/3 D) 10/3
B Use the properties of exponents to simplify n^y/ n^1/3= 1/n^3 n^y -1/3= n-^3 Since both sides have the same base, set their exponents as equal and solve for y. y - 1/3= -3 y -3 + 1/3 (multiply 3 by 3) = -9/3 + 1/3= -8/3
For biology class, you are calculating how much growth medium it will take to refill the class set of agar plates. Each agar plate is a cylinder with a base that has a radius of 5 cm and a height of 2 cm. There are 30 agar plates in the class set. To refill them, you need enough growth medium to fill each plate halfway. How much growth medium, in cm3, will you need? A) 50π cm3 B) 500π cm3 C) 750π cm3 D) 1500π cm3
C -To calculate the amount of growth medium needed in total, first calculate the amount of growth medium needed for each plate. -Find the area of the circular base & multiple by height. The height of each dish is 2 cm, only fill the dishes halfway, and calculate using a height of 1 cm. V= r^2 * h V= (5)^2 * 1 V= 25 The volume of growth medium needed for each plate is 25 cm3. 30 plates in total, multiply by 30 to get the total amount of growth medium needed 25 cm3 * 30 3 * 25= 75 * 10 = 750 cm3
A class of 12 has to divide into teams of two. How many different two-person combinations are possible in this group of 12? A) 12 B) 28 C) 66 D) 190
C 12(12)= 144 144-12= 132 = 132/2 =66
In the diagram below, ΔABC ~ ΔDEF∆. What is the area of ΔDEF∆? AB and BC= 6 AC= 8 A) 12√5 B) 12√7 C) 18√5 D) 72
C Divide the lengths of AC (8) and DF (12) 12/8= 1.5 is the scale factor. Calculate the lenght of sides of DE and EF. DE = 1.5(AB) =1.5(6)= 9 EF= 1.5(BC) =1.5(6)= 9 calculate the height of DEF using the Pythagorean Theorem. DF is 12, cut in half, 6. h^2 + 6^2= 9^2 h^2 + 36= 81 h^2=45 h2 = √45 = √9 ⋅ 5 = √9 ⋅ √5 = 3√5 Use the height to calculate the area of DEF A= 1/2 bh A=1/2 (12)(3√5) A=(6)(3√5) A=18√5
Which value does NOT yield a true statement in the inequa|lity below? | x - 5 | > 0 A) x=−5 B) x=0 C) x=5 D) x=6
C Plug in the multiple choices given. Brackets are always positive. x= 5 | 5 - 5 | > 0 | 0 | > 0 Wrong example: x=0 | 0 - 5 | > 0 | 5 | > 0
Rectangle LMNO is graphed in the (x,y)-coordinate plane such that none of its sides are parallel to either axis. If the slopes of all four sides of LMNO were multiplied together, which of the following would be the result? A) −1 B) 0 C) 1 D) Undefined
C Rectangles in this context as having two sets of parallel sides, with the non-parallel sides being perpendicular to each other. Suppose side LM has slope m; side NO will also have slope m since parallel lines have equal slopes. Sides MN and OL, which are perpendicular to LM and NO, will have the opposite reciprocal slope, −1/m. m * m * -1/m * -1/m = m^2 * 1/m^2 m^2/m^2 = 1.
The average of the elements in a set of consecutive integers is 13.5. If the largest element in the set is 15, what is the smallest element in the set? A) 10 B) 11 C) 12 D) 13
C Since it is known that 15 is one of the extremes in this set, use a simple average formula to find the other: A= x + e/ 2 13.5= x + 15/2 cancel the 2s 2(13.5)= x + 15 27= x + 15 12= x
What is the sum of the solutions of the following equation? 0= −3x2 + 3x + 126 A) −13 B) −1 C) 1 D) 13
C To find the sum of the solutions, first find the solutions. Begin by simplifying the quadratic by factoring out the greatest common factor. 0 = −3x^2 + 3x + 126 (divide all by -3) 0= x^2 - x - 42 (switch to subtraction to factor out the negative) Factoring method. 6 * 7= 42 The solutions of this quadratic are −6an d 7; their sum is 1.
What is the volume of a cube with a surface area of 24 inches^2? A) 4 inches^3 B) 6 inches^3 C) 8 inches^3 D) 12 inches^3
C To find the volume of the cube, work backwards from the given surface area to calculate the side length of the cube: A= 6s^2 24= 6s^2 4= s^2 2=s The side length of the cube is 2 inches. V= s^3 V= 2^3 V= 8 inches^3
For angle θ in the standard position, given that tan θ is positive and cos θ is negative, in which quadrant does (θ) lie? A) Quadrant I B) Quadrant II C) Quadrant III D) Quadrant IV
C cos θ negative is in quadrants 2 and 3 (left of y-coordinate) tan θ positive is in quadrants 1 and 3. So, quadrant 3. cos θ= x-coordinate sin θ= y-coordinate tan θ= sin θ/ cos θ
Which of the following is NOT equivalent to csc 5π/2? A) 1/ √1 − cos 2(5π/2) B) sin^2(5π/2) + cos^2(5π/2) / sin 5π/2 C) cotπ D) All of the above are equivalent to csc5π2csc5π2.
C cot π = cos π/ sin π = −1/0 -1/0 is undefined.
Which function is undefined at θ= π/2? A) y=sin θ B) y=cos θ C) y=tan θ D) y=cot θ
C sin π/2 = 1 and cos π/2 = 0 tangent can be defined as sinθ/cosθ sin π/2 / cos π/2 = 1/0 This value is undefined due to division by 0. (Its reciprocal, the cotangent, would be defined, with a value of 0/1 = 0.)
∠1 and ∠2 are supplementary angles, and ∠2 and ∠3 are complementary angles. Given that ∠3 and ∠4 are vertical angles and ∠4 Is 40°, what is the measure of ∠1? A) 40° B) 50° C) 130° D) 140°
C supplementary angles' measures add up to 180°, complementary angles' measures add up to 90°, and vertical angles have the same measures. m∠4 = 40° ° and it is vertical angles with ∠3, then m∠3 = 40° m∠2 + m∠3 = 90° m∠2 + 40° = 90° ∠2 = 90° − 40° = 50° m∠1 + 50° = 180° 180-50= 130°
Which of the following lines is parallel to the line x=y A) x=5 B) y=−x C) y=1/2x D) y=x−3
D Lines with the same slope and different y-intercepts are parallel Could be rewritten as y=x, so the slope is 1. A's slope is undefined because there's no y.. B's slope is -1 C's slope is 1/2 D's slope is 1 like the line given.
