Ify is expressed in terms of x, such that, for any x, a unique value of y is obtained, the expression ofy in terms of x is called a function of x.
Functions ofx are usually denoted by symbols of the form f (x) , g (x) , h(x), etc.
Some examples of functions:
• f (x) = 2x + 5
• f (x) = x2−2x + 1
• f (x) = 1 x
• f (x) =√ x + 1
• g (x) = x − 2 x2+3
• g (x) = 1
The idea of a function can be represented by a simple diagram as shown below:
𝑓 𝑥
Input 𝑥 Output 𝑦
Method of substitution in functions:
For any givenf (x), the value of f (a) is determined by substituting x = a on the right side of the equality.
Let us take an example:
f (x) = 2x + 1 Thus, we have:
• f (1) = 2 (1) + 1, i.e. we replace x with 1 on the right side of the equality
=> f (1) = 3
• f x2 = 2 x2 + 1 = 2x2+1
• f (2x + 1) = 2 (2x + 1) + 1 = 4x + 3
Domain and Range:
The Domain refers to the set of values ofx that can be used in the function.
The Range refers to the set of values off (x) obtained using the above values of x.
Let us take an example:
f (x) = x2−1
Let the set of values ofx to be used be {−1, 0, 1}.
Thus, we have:
• f (−1) = (−1)2−1 = 0
• f (0) = 02−1 = −1
• f (1) = 12−1 = 0
Here, Domain = {−1, 0, 1} and Range = {−1, 0}
To determine the Domain for a function, two things need to considered:
• For any term under a square-root or fourth-root, etc., the term should be non-negative.
For example: f (x) =√ x − 1 We have:x − 1 ≥ 0 => x ≥ 1 Thus, the Domain is: 1 ≤x < ∞
• For any term in the denominator, the term must be non-zero.
For example: f (x) = 2x x − 3 We have:x − 3 6= 0 => x 6= 3
Thus, the Domain is:x is any real number except 3
=> −∞ < x < 3 OR 3 < x < ∞
Let us take an example:
f (x) = x + 3
√2x−4−4 Thus, we have:
• 2x − 4 ≥ 0 => x ≥ 2 . . . (i)
• √
2x − 4 − 4 6= 0 =>√
2x − 4 6= 4 => 2x − 4 6= 16 => x 6= 10 . . . (ii)
Thus, the Domain: 2 ≤x < 10 OR 10 < x < ∞ Composite functions:
For any two functions f (x) and g (x), the functions defined as f (f (x)), f g (x), g g (x) and g (f (x)) are composite functions.
Let us take an example:
f (x) = 2x + 1 and g (x) = x2−1 Thus, we have:
• f (f (x)) = 2 (f (x)) + 1 = 2 (2x + 1) + 1 = 4x + 3
• f g (x) = 2 g (x) + 1 = 2 x2−1 + 1 = 2x2−1
• g g (x) = g (x)2+1 = x2+12
+1 =x4+2x2+2
• g (f (x)) = (f (x))2+1 = (2x + 1)2+1 = 4x2+4x + 2
Special case in a composite function:
Iff (x) and g (x) are two functions and it is observed that f g (x) = g (f (x)) = x, we have:
Input 𝑎 𝑓 𝑥 Output 𝑏 Input 𝑏 𝑔 𝑥 Output 𝑎
Thus, we have:
Iff (a) = b => g (b) = a
Note: Such functionsf (x) and g (x) are inverse functions of one another.
Let us take an example:
Iff (x) = (x + 1)3−1 andg (x) =√3
x + 1 − k, such that f g (x) = g (f (x)), what is the value of k?
The normal way of solving, by evaluating the composite functionsf g (x) and g (f (x)), is compli-cated. Instead, we use the above method:
f (1) = (1 + 1)3−1 = 7
=> g (7) = 1
=>√3
7 + 1 −k = 1 => 2 − k = 1
=> k = 1
Periodic function:
A functionf (x) is periodic if there exists a number n so that f (x + n) = f (x) for all x. Here, n is the period of the function.
Let us take an example:
Iff (x + 3) = f (x + 2) − f (x + 1), what is the value of n if f (1) = −f (1 + n)?
We have:f (x + 3) = f (x + 2) − f (x + 1) Substituting different values ofx:
• x = 0 : f (3) = f (2) − f (1) . . . (i)
• x = 1 : f (4) = f (3) − f (2) . . . (ii)
Adding (i) and (ii):
f (3) + f (4) = f (3) − f (1)
=> f (1) = −f (4)
=> f (1) = −f (1 + 3)
=> n = 3
Piece-wise functions:
Functions which have different expressions over different values ofx are piece-wise functions. Some examples are shown below:
• Modulus function:f (x) = |x|:
◦ f (x) = x, if x ≥ 0,
◦ f (x) = −x, if x < 0
The graph off (x) = |x| is shown:
X Y
O
• Greatest Integer Function:f (x) = [x]: It is a function that returns the greatest integer less than or equal tox. Thus, we have:
◦ [1.23]: The greatest integer less than or equal to 1.23, i.e. the greatest integer among 1, 0, −1, −2, · · · = 1
◦ [1]: The greatest integer less than or equal to 1, i.e. the greatest integer among 1, 0, −1, −2, · · · = 1
◦ [−1.23]: The greatest integer less than or equal to −1.23, i.e. the greatest integer among
−2, −3, −4, · · · = −2
◦ [−1]: The greatest integer less than or equal to −1, i.e. the greatest integer among −1, −2, −3, −4, · · · =
−1
• Least Integer Function: f (x) = {x}: It is a function that returns the least integer greater than or equal tox. Thus, we have:
◦ {1.23}: The least integer greater than or equal to 1.23, i.e. the least integer among 2, 3, 4, · · · = 2
◦ {1}: The least integer greater than or equal to 1, i.e. the least integer among 1, 2, 3, 4, · · · = 1
◦ {−1.23}: The least integer greater than or equal to −1.23, i.e. the least integer among
−1, 0, 1, 2, · · · = −1
◦ {−1}: The least integer greater than or equal to −1, i.e. the least integer among −1, 0, 1, 2, · · · =
−1
• Max-Min function:
◦ f (x) = max(a, b) implies that f (x) = a if a > b OR f (x) = b if b > a
◦ f (x) = min(a, b) implies that f (x) = a if a < b OR f (x) = b if b < a
Let us take an example:
Iff (x) = min(6x − 8, x2), for what integer values of x is f (x) = x2?
Sincef (x) = min(6x − 8, x2) = x2, we have:
x2< 6x − 8 => x2−6x + 8 < 0
=> (x − 2) (x − 4) < 0 => 2 < x < 4
Thus, the only integer value ofx = 3
Properties of graphs of functions:
• The graph off x+p is obtained by shifting the graph of f (x) by p units left
• The graph off x−p is obtained by shifting the graph of f (x) by p units right
• The graph off (x) +p is obtained by shifting the graph of f (x) by p units up
• The graph off (x) −p is obtained by shifting the graph of f (x) by p units down
• The graph off (−x) is obtained by reflecting the graph of f (x) about the Y-axis
• The graph of −f (x) is obtained by reflecting the graph of f (x) about the X-axis
Graphs of some quadratic functions:
• f (x) = x2:
X Y
• f (x) = x2+1:
X Y
0, 1
• f (x) = (x − 1)2:
X Y
1, 0
Additional solved problems:
=> f (a) = 4a
Since the answer options are constant values, the answer must be Option B.
(3) Are the following functions the same?
(A) f (x) = xx
(B) g (x) = x × x × x × x × · · · × x (x times)
Explanation:
(A) We havef (x) = xx Ifx is multiplied for x times, the result obtained is xx. Thus, apparently,f (x) and g (x) appear to be identical.
However, we are multiplyingx for x times, which makes sense only when x is a positive integer. (Multiplying 1
2for 1
2 times or multiplying −1 for −1 times makes no sense) For example:
g (3) = 3 × 3 × 3 (3 is multiplied for 3 times)
=27
Thus,g (x) is valid only for positive integer values of x.
Thus,f (x) is not the same as g (x).
(4) If (2, 1) are the coordinates of a point on the graph of f (x), what would be the coordinates of that point for the function −f (x) + 1?
Explanation:
To modifyf (x) to −f (x) + 1, we follow the following steps:
• f (x) → −f (x): The graph is reflected about the X-axis. Thus, the Y-coordinate of the point would be negated.
Thus, the coordinates of −f (x) = (2, −1)
• −f (x) → −f (x) + 1: The graph is shifted ‘up’ by 1 unit. Thus, the Y-coordinate of th point would increase by ‘1’.
Thus, the final coordinates of −f (x) + 1 = (2, −1 + 1) = (2, 0)
In the GMAT, only two kinds of questions asked: Problem Solving and Data Sufficiency.
Problem Solving
Problem solving (PS) questions may not be new to you. You must have seen these types of questions in your school or college days. The format is as follows: There is a question stem and is followed by options, out of which, only one option is correct or is the best option that answers the question correctly.
PS questions measure your skill to solve numerical problems, interpret graphical data, and assess information. These questions present to you five options and no option is phrased as “None of these“.
Mostly the numeric options, unlike algebraic expressions, are presented in an ascending order from option A through E, occasionally in a descending order until there is a specific purpose not to do so.
Data Sufficiency
For most of you, Data Sufficiency (DS) may be a new format. The DS format is very unique to the GMAT exam. The format is as follows: There is a question stem followed by two statements, labeled statement (1) and statement (2). These statements contain additional information.
Your task is to use the additional information from each statement alone to answer the question. If none of the statements alone helps you answer the question, you must use the information from both the statements together. There may be questions which cannot be answered even after combining the additional information given in both the statements. Based on this, the question always follows standard five options which are always in a fixed order.
(A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient to answer the ques-tion asked.
(B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient to answer the ques-tion asked.
(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
(D) EACH statement ALONE is sufficient to answer the question asked.
(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
In the next chapters, you will find 150 GMAT-like quants questions. Best of luck!
Practice Questions
3.1 Problem Solving
1. The price of 19 chocolates and 21 pens is $29, while the price of 21 chocolates and 19 pens is
$31. What is the price of 1 chocolate?
(A) $0.50 (B) $1.00 (C) $1.25 (D) $1.50 (E) $2.00
Solve yourself:
2. Abe’s age is equal to the sum of the ages of his son and a 12-year old daughter. If Abe’s son is elder to Abe’s daughter, and the average age of Abe and his two children ten years ago was 20 years, what is Abe’ present age?
(A) 30 years (B) 33 years (C) 39 years (D) 45 years (E) 51 years
Solve yourself:
3. 3 apples, 3 guavas and 4 bananas, together cost $10. Also, 3 apples, 2 guavas and 4 bananas together cost $9. What is the total cost of 9 apples, 8 guavas and 12 bananas?
(A) 26 (B) 29 (C) 30 (D) 32 (E) 37
Solve yourself:
4. A person has a few cents and a few dollars such that the total amount isa dollars and b cents, whereb < 100. After spending $3.50, he was left with 2b dollars and 64 cents. What is the value of (a + b)?
(A) 14 (B) 28 (C) 32 (D) 46 (E) 64
Solve yourself:
5. In a fraction, if 4 is added to both numerator and denominator, the fraction increases by 1 8. If however, 2 is subtracted from both numerator and denominator, the fraction decreases by 1
4. What is the value of the original fraction?
(A) 7 8 (B) 3 4 (C) 1 2 (D) 1 4 (E) 3
16 Solve yourself:
6. If 7x − 2y = 12, 4x + y = 9 and 2x + 5y = K, what is the value of K?
(A) 9 (B) 10 (C) 11 (D) 12 (E) 13
Solve yourself:
7. If 2x + 3y = 7, 5x + 3y = 13 and x A = y
B = 1
C, whereA, B and C are positive integers and the greatest common divisor ofA, B and C is 1, what is the value of (A + B + C)?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 8
Solve yourself:
8. The sum of the digits of a two-digit number is 5. The ratio of 20 less than the number and 12 more than the number is 3
11. What is the product of the digits of the number?
(A) 0 (B) 1 (C) 4 (D) 5 (E) 6
Solve yourself:
9. x, y and z satisfy the following set of equations:
3x + 7y − 11z = 0 6x − y − 7z = 0 3x + y − kz = 0
What is the value ofk?
(A) 1 (B) 4 3 (C) 2 (D) 7 3 (E) 5
Solve yourself:
10. A group of children have a number of pens, such that each child has at least one pen. If one of the children, Ann, takes 1 pen from each of the other, the number of pens with her would be thrice the number of children in the group. If the total number of pens among the children is 42, which of the following could be the number of children in the group, so that it can be ensured that Ann has the greatest number of pens?
I. 5 II. 9 III. 15
(A) Only I (B) Only II (C) Only III (D) Both I and II
(E) Both II and III
Solve yourself:
11. A ball is thrown from the top of a building. The distance, in feet, covered by the ball int seconds after it was dropped is given by 15t2. If the distance covered by the ball in thetthsecond after it was dropped was 225 feet. What is the value oft?
(A) 3 (B) 4 (C) 7 (D) 8 (E) 10
Solve yourself:
12. A ball is thrown up from a height of 3 feet above the ground. The distance, in feet, of the ball from the ground is given byh = 3 + 24t − 4t2, wheret = time in seconds. What is the maximum height above the ground reached by the ball?
(A) 32 feet (B) 33 feet (C) 35 feet (D) 36 feet (E) 39 feet
Solve yourself:
13. The number of units sold,N, of a new product is expected to follow the relation: N = 120 − P , whereP is the selling price per unit.
If the cost of manufacturing any number of units of the new product is constant, and equal to
$2000, what should be the maximum selling price of each unit so that there is neither profit nor loss?
(A) $20 (B) $60 (C) $80 (D) $100
(E) $110
Solve yourself:
14. What is the minimum positive integer value ofp so that x2−px + 8p = 0 has real and unequal roots?
(A) 12 (B) 24 (C) 27 (D) 32 (E) 33
Solve yourself:
15. If one of the solutions ofx2−px + 12 = 0 is x = 3
2, what is the value ofp?
(A) 3 2 (B) 8 (C) 19
2 (D) 57
4 (E) 15
16. Ify = x2+kx + l intersects the X-axis at (4, 0) and the Y-axis at (0, 64), what is the value of k?
(A) −80 (B) −40 (C) −20 (D) 20
(E) 80
Solve yourself:
17. The inside dimensions of a rectangular steel frame, as shown in the diagram below, having uniform width of x inches, are 12 inches by 8 inches. If the area of the frame is 44 square inches, what is the total perimeter of the frame?
12 𝑥 8
𝑥
(A) 98 inches (B) 49 inches (C) 48 inches (D) 24 inches (E) 8 inches
Solve yourself:
18. A roller-coaster track is designed in the form of a parabolic arch, whose height, in feet, above the ground is given as h = −kd (d − 20), where k is a positive number and d represents the distance along the length of the track measured from the left-most point where the arch starts.
If the arch reaches to a maximum height of 30 feet, what is the value ofk?
(A) 3 10 (B) 2
5 (C) 3 4 (D) 4 5 (E) 3 2
Solve yourself:
19. Forf (x) = x2+bx + c, f (1) = 0. If f (3) = 2f (5), what is the value of k such that f (k) = 0, k 6= 1?
(A) 10 3 (B) 37
7 (C) 17
3 (D) 7
(E) 19 2 Solve yourself:
20. If the roots of the fourth degree equationx4−2x3+x2+x + 3 = 0 are p, q, r and s, what is the value of 2 +p
2 +q (2 + r ) (2 + s)?
(A) −37 (B) −9 (C) 9 (D) 37
(E) 40
Solve yourself:
21. Which of the following is the correct solution of the inequalityx + 2 x ≤3?
I. x > 2 II. 1 ≤x ≤ 2
III. x < 0
(A) Only I (B) Only II (C) Only III (D) Both I and III
(E) Both II and III
Solve yourself:
22. Which of the following is the correct solution of the inequality: −1< x + 3 x + 7 < 1?
I. x < 7 II. x < −5 III. x > −5
(A) Only I (B) Only II (C) Only III (D) Both I and II
(E) Both I and III
Solve yourself:
23. What is the smallest integer value ofx which satisfies the inequality x − 3
x2−9x + 18> 1 2? (A) 5
(B) 6 (C) 7 (D) 8 (E) 9
Solve yourself:
24. Which of the following is the correct solution of the inequalityx3−9x5> 0?
I. 0< x < 1 3 II. −1
3 < x < 0 III. x < −1
3 (A) Only I
(B) Only II (C) Only III (D) Both I and II
(E) Both I and III
Solve yourself:
25. If −2 ≤x ≤ 3, −6 ≤ y ≤ 4, −1 ≤ a ≤ 2 and −3 ≤ b ≤ 1, which of the following is the correct range of values of (a + b) x + y?
(A) −32 ≤ (a + b) x + y ≤ 21 (B) −28 ≤ (a + b) x + y ≤ 21 (C) −24 ≤ (a + b) x + y ≤ 32 (D) −24 ≤ (a + b) x + y ≤ 21 (E) −28 ≤ (a + b) x + y ≤ 32
Solve yourself:
26. Ifx and y are integers, such that 2 ≤ x ≤ 10 and −12 ≤ y ≤ −5, what is the product of the maximum possible value of x
y and the minimum possible value of y x? (A) −1
(B) −2 3 (C) 2
3 (D) 1 (E) 6
Solve yourself:
27. A dealer in electronic goods spends $15000 on a certain model of TV sets and a certain model of DVD players. The price of each TV set is $360 and the price of each DVD player is $240. If he does not wish to purchase greater than 42 items, what is the minimum number of total items that the dealer can purchase?
(A) 23 (B) 36 (C) 37 (D) 41 (E) 42
Solve yourself:
28. A burger store sells chicken burgers, priced at $8 per burger and vegetable burgers, priced at $5 per burger. If potato fries are also ordered along with a burger, there is a discount of $1 on the burger irrespective of the type of burger. The total revenue from selling burgers on a particular day was not greater than $1110, and potato fries were ordered along with a burger 50 times. If the total number of burgers purchased is at least 160, what is the maximum number of chicken burgers that were sold?
(A) 41 (B) 58 (C) 120 (D) 159
(E) 169
Solve yourself:
29. The people working in a company X are planning a trip, for which they want to use sedans and minivans. Each sedan can accommodate 4 people, while each minivan can accommodate 8 people. The rent applicable for each sedan and each minivan is $60 and $80, respectively. The total amount available for transportation is $600. If the sedans and minivans are always filled to capacity, and the number of people interested for the trip is at least 48, what is the minimum number of minivans necessary for the trip?
(A) 2 (B) 3 (C) 5 (D) 6 (E) 8
Solve yourself:
30. Gift boxes, to be distributed among school children, contain pencils and erasers. Each gift box can hold a maximum of 12 items. If each box must have at least 3 more pencils than the number of erasers, what is the maximum number of erasers in a gift box?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
Solve yourself:
31. A cylinder has to be designed such that its diameter is 3 inches, allowing for a margin of error of 1 inch. The height of the cylinder has to be 10 inches, allowing for a margin of error of 1 inch.
What is the difference between the maximum and the minimum volumes of the cylinder?
(A) 9π (B) 32π (C) 35π (D) 97
2π (E) 140π
Solve yourself:
32. Ify = |x + 3| + |x − 4| + |x − 7|, what is the minimum value of y?
(A) 2 (B) 7 (C) 10 (D) 13 (E) 17
Solve yourself:
33. How many integer values ofx satisfy the inequality
x2−15 ≤6?
(A) Two (B) Four (C) Five (D) Seven
(E) Nine
Solve yourself:
34. How many integer values ofx satisfy the inequality |x − 3| + |x − 9| ≤ 8?
(A) Three (B) Seven (C) Nine (D) Ten
(E) Twelve
Solve yourself:
35. What is the sum of the possible values ofx if |x + 15| = 3 |x − 15|?
(A) 7.5 (B) 22.5 (C) 30.0 (D) 37.5 (E) 52.5
Solve yourself:
36. Ifx and y are non-negative integers such that x + y
+ x − y
=6, how many possible pairs of solutions of x, y exist?
(A) Four (B) Five (C) Seven (D) Nine
(E) Twelve
Solve yourself:
37. How many integer values ofx satisfy the inequality
38. What is the greatest value ofx that satisfies the inequality
40. How many integer values ofx exist such that |x − 1| > |x| + 3?
(A) None (B) One (C) Two (D) Four
(E) Five
Solve yourself:
41. f (x) = 2x4−3x2+1 has rootsp, q, r and s. What is the value of p + q + r + s?
(A) −√ 2 (B) 0 (C) 1 2 (D) 3 2 (E) 2
Solve yourself:
42. At how many points do the graphs off (x) = x2+4 andg (x) = x2
2 +3 intersect?
(A) None (B) One (C) Two (D) Three
(E) Four
Solve yourself:
43. Letp and q be positive integers and the function f p, q is defined as shown:
• f p, q = 0, if p < q
• f p, q = f p − q, q + 1 if p ≥ q
Iff p, q = 3, which of the following CANNOT be true?
I. p = 7, q = 2 II. p = 10, q = 3 III. p = 17, q = 6
(A) Only I (B) Only II (C) Only III (D) Both I and II
(E) I, II and III
Solve yourself:
44. Letp and q be positive integers and the function f p, q is defined as shown:
• f p, q = f p, q − 1 + p if q ≥ 1
• f p, q = 0 if q = 0
Iff p, q = 12, which of the following are correct?
I. p = 6, q = 2 II. p = 4, q = 3 III. p = 6, q = 3
(A) Only I (B) Only II (C) Only III (D) Both I and II
(E) I, II and III
Solve yourself:
45. Letp and q be positive integers and the function f p, q is defined as shown:
• f p, q = p if p < q
• f p, q = f p − q, q if p > q
• f p, q = 0 if p = q
Iff p, q = 7, which of the following is true?
I. p = 12, q = 5 II. p = 25, q = 8 III. p = 27, q = 20
(A) Only I (B) Only II (C) Only III (D) Both I and II
(E) I, II and III
Solve yourself:
46. Letp and q be two positive integers. If f p, q = p
p + q andg p, q = p + q
q , which of the following options is correct?
(A) f p, q g p, q =
q p (B) f p, q
g p, q = pq p + q (C) f p, q
g p, q = p + q
pq
(D) f p, q × g p, q = p2q2 (E) f p, q × g p, q =p
q Solve yourself:
47. For all positive integer values ofx, let f (x) =
2x x + 1
, where [x] denotes the greatest integer less than or equal tox.
What is the value off (1) + f (2) + f (3) + · · · + f (100)?
(A) 11 (B) 99 (C) 100 (D) 101 (E) 202
Solve yourself:
48. For all positive integer values of x, if f (x) = 1
x and g (x) = x
x2+1, what is the value of f g (x) × g (f (x))?
(A) −1 (B) 0 (C) 1 (D) 3 2 (E) 2
Solve yourself:
49. For all non-negative numbersx, let f (x) = −x2+8 andg (x) = x + 2. For how many integer values ofx is f (x) ≥ 4 × g (x)?
(A) None (B) One (C) Two (D) Three
(E) Four
Solve yourself:
50. Ifp and q are the roots of x2+4x − 12 = 0, and f (x) =
x 4
, where [x] denotes the greatest integer less than or equal tox, what is the maximum value of f p − f q?
(A) −2 (B) −1 (C) 1 (D) 3 2 (E) 2
Solve yourself:
51. The graph of a quadratic functionf (x) shown below is symmetric about the line x = 4. If f (1) = 9, what is the value of f (7)?
4, 0 O X
Y
(A) 3 (B) 5 (C) 6 (D) 9 (E) 12
Solve yourself:
52. In the graph shown below, the functionsf (x) = x2−6x + 8, which intersects the X-axis at x = 2 andx = 4 and g (x) = x − 2 are shown. For how many integer values of x is g (x) ≥ f (x)?
X Y
2, 0 4, 0
𝑓 𝑥
𝑔 𝑥
O
(A) One (B) Two (C) Three (D) Four
(E) Five
Solve yourself:
53. The graphs off (x) and g(x) are shown below. Which of the options is correct?
15
10
5
1 2 3 O
𝑓 𝑥 𝑔 𝑥
X Y
(A) f (x) = 5x (B) f (x) = x2+4 (C) g (x) = (x + 1)2−4 (D) g (x) = (x − 1)2+4
(E) f (x) − g (x) = 2x + 1
Solve yourself:
54. What is the area of the triangle bounded by the linesf (x) = x −2, g (x) = 2x −3 and the Y-axis, as shaded in the graph below?
𝑓 𝑥 𝑔 𝑥
X Y
O 1 2
−1
−2
−3
(A) 1 4
(B) 1 2 (C) 1 (D) 2 (E) 3 2
Solve yourself:
55. The graph of a quadratic functionf (x) = ax2+bx + c shown below intersects the X-axis at (2, 0) and (4, 0) and intersects the Y-axis at (0, 4). What is the value of (a + b + c)?
2 4 4
O X Y
(A) 1 2 (B) 3 2 (C) 3 (D) 4 (E) 6
Solve yourself:
56. f (x) is a function which satisfies 2f (x) + f (−x) = 2x + 1. What is the value of f (1)?
(A) 1 (B) 5 4 (C) 4 3 (D) 7 3 (E) 3
Solve yourself:
57. f (x) is a function such that f (2x − 1) = x2+3x. What is the value of f (5)?
(A) 3 (B) 15 (C) 18 (D) 25 (E) 40
Solve yourself:
58. Iff (x) = ax2+bx + c and f (x + 1) = f (x) + x + 1, what is the value of a?
(A) −1 (B) −1 2 (C) 0 (D) 1 2 (E) 1
Solve yourself:
59. Letx and y be positive integers and the function f (a, b) is defined as shown:
• f (a, b) = a if a ≤ b
• f (a, b) = b if a > b
Ifa = 6x − 8 and b = x2, for how many integer values ofx will f (a, b) = b?
(A) None (B) One (C) Two (D) Three
(E) Four
Solve yourself:
60. Iff (x) = |x − 1|, g (x) = x + a and f g (−3) = 2, what is the sum of the possible values of a?
(A) 2 (B) 6 (C) 8 (D) 9 (E) 10
Solve yourself:
61. For what value ofn the equation below have no possible solution?
4 (2x − 1) + 3 (x − 2) = n (x + 2) − 3 (x + 1) (A) −4
(B) −7 2 (C) 7
2 (D) 7 (E) 14
Solve yourself:
62. If the equationx2−px + 12 = 0 has exactly one root common with the equation x2−6x + 9 = 0, what is the value ofp?
(A) −10 (B) −7 (C) 4 (D) 7 (E) 10
Solve yourself:
63. What is the value of
(0.23 + 0.52)2−
0.232+0.522
0.752−0.292
?
(A) 0.3 (B) 0.4 (C) 0.5 (D) 2.0 (E) 4.0
Solve yourself:
64. Ifp and q, p > q are the roots of x2−x − 12 = 0, the roots of the equation x2−50x + 49 = 0 are
(A) 2p and 3q (B) p2andq2
(C) p + q and p − q
(D) p + q2
and p − q2
(E) p2+q2 and p2−q2
Solve yourself:
65. Ifa and b are non-zero integers such that a2+2b2+2a + b = 0 and a2+b2=2ab, what is the value of (a + b)?
(A) −2 (B) −1 (C) 0 (D) 1 (E) 2
Solve yourself:
66. Ifx + y = 2 and z = x2+y2, what is the minimum value ofz?
(A) 1 2 (B) 1 (C) 2 (D) 4 (E) 6
Solve yourself:
67. If q and 2q are the roots of the equation k x2−x + x + 1 = 0, where q 6= 0, which of the following is a correct relation betweenk and q?
(A) k = q 2 (B) k = 1
1 + 3q (C) k = 2q − 1 (D) k = 1
2q2 (E) k = q2+1
Solve yourself:
68. f (x) is a quadratic polynomial such that f (1) = 1 and f (2) = 2. If f (3) = 5, what is the value off (0)?
(A) −6 (B) −2 (C) 0 (D) 1 (E) 2
Solve yourself:
69. For all positive integersx > 2, let f (x) be defined as
f (x) = (−1)x×f (x − 1) × f (x − 2). If f (1) = −f (2) = 1, what is the value of f (23)?
(A) −1 (B) 0 (C) 1 (D) 3 (E) 7
Solve yourself:
70. For all values ofx, let f (x) =2x + 1
x − 2 . Iff p = q, what is the value of f q?
(A) −3 2 (B) 1
3 (C) 1 (D) p (E) pq
Solve yourself:
71. What is the area bounded byf (x) = |x − 1| − x, the X and Y axes?
(A) 1 4 (B) 1 2 (C) 1 (D) 2 (E) 3 2
Solve yourself:
72. Iff (x) = ax2+bx such that f (1) = f (−1) + 2, what is the value of f (3) − f (−3)?
(A) 1 (B) 2 (C) 4 (D) 6 (E) 8
Solve yourself:
73. Ifx = 1 +√
3, what is the value of x2−2x − 2?
(A) −2 (B) −√ 3 (C) 0 (D) 1 (E) √ 3
Solve yourself:
74. Ifx is a positive integer such that 24
√x+10
√x=26
√x, what is the value ofx?
(A) 0 (B) 1 (C) 2 (D) 4 (E) 6
Solve yourself:
75. The functionf (x) = 2x2−7x + 6 is positive for all values of x except when x lies in a particular range. How many integer values ofx lie within that range?
(A) None (B) One (C) Two (D) Three
(E) Four
Solve yourself:
76. Which of the following is the correct range of values ofx such that
12 −x 3
≤2?
(A) 2 ≤x ≤ 6 (B) 4 ≤x ≤ 10 (C) 8 ≤x ≤ 16 (D) 6 ≤x ≤ 18 (E) 4 ≤x ≤ 24
Solve yourself:
77. How many integer values ofx satisfy x2−4 |x| + 3 ≤ 0?
(A) Two (B) Three (C) Four (D) Six
(E) Eight
Solve yourself:
78. What is the maximum value of
12
x2−12x + 40
? (A) 1
(B) 2 (C) 3 (D) 4
(E) 6
Solve yourself:
79. Ifa and b are distinct roots of x2−px − q = 0, which of the following denotes the value of (a + b)?
(A) p (B) q (C) p + q (D) p − q
(E) pq
Solve yourself:
80. Let [x] denote the greatest integer value less than or equal to x for all positive integers x, such that 1 ≤x ≤ 4. Which of the following expressions denotes the value of
x + 1 2
+
x + 2 4
+
x + 4 8
? (A) 2x − 2
(B) 2x − 1 (C) x (D) x + 1
(E) x + 2
Solve yourself:
81. Iff (x) = x3−kx2+2x, and f (−x) = −f (x), what is the value of f (1 − k)?
(A) −3 (B) 0 (C) 1 (D) 3 (E) 4
Solve yourself:
82. The functionf is defined as f (x) = ax2+bx + c. If f (2) = f (3) = 0, and f (4) = 2, what is the value of(a + b + c)?
(A) 1 (B) 2 (C) 8 (D) 11
(E) 12
Solve yourself:
83. Running at their respective constant rates, machine X takes 2 days longer to producew widgets
83. Running at their respective constant rates, machine X takes 2 days longer to producew widgets