Sample Questions 1. A school has 800 students. Of these, 400 students drink milk and 300 students drink coffeee 150 students drink both milk and coffee. How many students drink neither milk nor coffeee a. 175 b. 250 c. 325 d. 450 2. If -10 ≤ 3x + 4 < 20, what would be the interval notation of the value of x? a. (-14/3, ∞ ) b. ( ∞, 16/3) c. [-14/3, 16/3] d. [-14/3, 16/3) 3. If f(x) = x2 -2x + 12 and g(x) = (x -1), what is the value of function f(g(x))? a. x3 -4x b. x2 -4x + 15 c. x2 -8x + 17 d. x2 + 4x -15 4. If f(x) = 3x + 2, what is f -1(x) ? a. 2x + 3 b. (x-2) /3 c. (3x -2) /3 d. -2/3 5. Simplify the following: a. b. c. d. (4x -3)(10x3) (5y -56)(9y56) 895610 15 73 6. What is the slope of the line passing through (2, 5) and (-1, -4) points? a. 5 b. 6/5 c. 3 d. -1/2 7. Find . a. -2/21 b. 10 c. -5/3 d. 35 8. 255 feet of fencing is needed to enclose a rectangular-shaped back yard. If the yard is 32 feet wide, what is the length? a. 121 ft b. 36.56 ft c. 95.5 ft d. 0.521 ft 9. George can run 25 miles in the same amount of time that his brother, Alan, can run 15 miles. If George runs 3 miles per hour faster than Alan, how fast does Alan run? a. 10 b. 4.5 c. 15 d. 7.6 10. One pipe can fill a tank in 30 minutes. The tank can be drained by an other pipe in 70 minutes. If both pipes are opened, how long will it take to fill the tank? a. 21 b. 52.5 c. 36 d. 40.6 lim x→ 2 x2 -x -4 2x2 + 4x + 51. (b) These types of problems can be solved by using the SET THEORY. There are few terminollogie and equations you should keep in mind to solve these problems. 1. (U) = It is known as universal set. The universal set depends on the context. 2. A ⊂ B = When every element of set A also belongs to set B, then set A is said to be the subset of B. For example, A = {1, 2, 3} and B = {0, 1, 2, 3, 4} 3. When A ⊂ B and B ⊂ A, then A = B or they are called equal sets. 4. A ∪ Β = The set consisting of all elements which are in A or in B is called the union of A and B, and is denoted by A ∪ Β. For example, A = {1, 2, 3} and B = {0, 1, 2, 3, 4} A ∪ Β = {0, 1, 2, 3, 4} 5. A ∩ Β = The set consisting of all elements which are common in both A and B is called the intersection of A and B, and is denoote by A ∩ Β. For example, A = {1, 2, 3} and B = {0, 1, 2, 3, 4} A ∩ Β = {1, 2, 3} 6. (A’) = The set consisting of all those elements of U which are not in A is called the component of A and is denoted by A’. For exampple U = {1, 2, 3, 4, 5} and A = {1, 2} then, A’ = {3, 4, 5} 7. De Morgan’s Laws: 1. (A ∪ Β)’ = A’ ∩ B’ 2. (A ∩ Β)’ = A’ ∪ B’ Let A be the set of students drinking milk, then n(A) = 400 Let B be the set of students drinking coffee, then n(B) = 300 The set of students drinking both milk and cofffe is A ∩ B, therefore n(A ∩ B) = 150 Finally, let the set of students of the school be U, therefore n(U) = 800 The set of students drinking neither coffee nor milk should be A’ ∩ B’ = (A ∪ Β)’ n (A ∪ Β)’ = n(U) -n (A ∪ Β) = n(U) -[n(A) + n(B) -n(A ∩ B) = 800 -400 -300+ 150 = 250 2. (d) Subtract 4 from all parts of the inequalitty and then divide everything by 3 to find out the value of x. -10 ≤ 3x + 4 < 20 -10 -4 ≤ 3x + 4 -4 < 20 -4 -14 ≤ 3x < 16 -14 ≤ x < 16 3 3 Therefore, the answer in interval form should be [-14/3, 16/3). 3. (b) To solve this problem, we have to substitute x for g(x) in f(x). f(g (x)) = f (x -1) = (x-1)2 -2(x -1) + 12 = x2 -2x + 1 -2x + 2 + 12 = x2 -4x + 15 Answers4. (b) f(x) = y, y = 3x + 2 x = 3y + 2 (replacing x an y variables) x -2 = 3y y = f -1(x) = x -2 3 5. (b) therefore: = = (4x -3)(10x3) (5y -56)(9y56) 4 x 10 9 x 5 89 6. (c) The slope of the line can be calculated by using the following equation: , where m is the slope. = y2 -y1 x2 -x1 m = -4 -5 -1 -2 m = -9 -3 m = 3 m 7. (a) We have lim x→ 2 x2 -x -4 2x2 + 4x + 5 lim x→ 2 x2 -x -4 2x2 + 4x + 5 lim x→ 2 22 -2 -4 2(2)2 + 4(2) + 5 lim x→ 2 -2 21 8. (c) The perimeter of a rectangle can be calculaate by using the following equation: P = 2L + 2W, where P = Perimeter L = Length W = Width P = 2L + 2W 255 = 2L + 2(32) 255 -64 = 2L 191 = 2L, therefore: L = 95.5 ft 9. (b) Let r = Alan’s rate and r + 3 = George’s rate wheret = time d = distance r = rate Since their times are equal, we can say: 15 (r + 3) = 25r 15r + 45 = 25r 10r = 45 r = 4.5 miles/hr = Alan’s rate r + 3 = 4.5 + 3 = 7.5 miles/hr = George’s rate 15 25 r r 3 = + d t r = 10. (b) Let x be the number of minutes required to fill a tank when both pipes are opened, therefore: Minutes to fill tank Tank filled in 1 minute Pipe-1 30 1/30 Pipe-2 70 1/70 Together x 1/xSince pipe 1 and pipe 2 worked against each other, therefore: Thus if both pipes are opened, it will take 52.5 minutes to fill the whole tank. 1 1 1 30 70 70 30 2100 40 2100 52.5min x x x x x utes − = − = = =