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MATHEMATICS Focused Quiz 1: Basic Properties of Real Numbers

1. Which of the following laws of real numbers does (𝑧𝑧 + 12) + (−𝑧𝑧 − 12) = 0 illustrate?

a. Distributive law b. Commutative law of addition c. Additive inverse law d. Commutative law of multiplication e. Associative law of addition

2. Which of the following laws of real numbers does 8(𝑥𝑥 + 7) = 8𝑥𝑥 + 56 illustrate?

a. Distributive law b. Associative law of addition c. Commutative law of addition d. Associative law of multiplication e. Commutative law of multiplication

3. Which of the following numbers are an irrational number?

a. √2√2 b. √676 c. 0.75 d. √13313 e. 7√5

4. If 𝑎𝑎 + 𝑏𝑏 = 35, 𝑏𝑏 + 𝑐𝑐 = 15 and 𝑎𝑎 + 𝑐𝑐 = 25, then which of the following is true?

a. 𝑏𝑏 > 𝑐𝑐 > 𝑎𝑎 b. 𝑏𝑏 > 𝑎𝑎 > 𝑐𝑐 c. 𝑎𝑎 > 𝑐𝑐 > 𝑏𝑏 d. 𝑏𝑏 > 𝑐𝑐 > 𝑎𝑎 e. 𝑎𝑎 > 𝑏𝑏 > 𝑐𝑐

5. Which of the following equations illustrates the Commutative Law of Addition?

a. 157 × 4 = 4 × 157 b. 40 + (−40) = 0 c. 157 + 4 = 4 + 157 d. 2 × (157 + 4) = 2 × 157 + 2 × 4 e. 40 + 0 = 40

MATHEMATICS FOCUSED QUIZ 1 PAGE 2 OF 65 DEMIDEC ©2016

6. Which of the following numbers is divisible by both 2 or 3?

a. 700 b. 2770 c. 9999 d. 850 e. 1122

7. Convert 𝑥𝑥 = 1.272727 … to fractional form.

a. 127100

b. 1411

c. 1272710000

d. 99126

e. 12699

8. If 𝑎𝑎𝑏𝑏 = 5, 𝑏𝑏𝑐𝑐 = 211

𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑐𝑐 = 9, what is one possible value of 𝑎𝑎𝑏𝑏𝑐𝑐?

a. 1.85 b. 4.09 c. 8.18 d. 2.86 e. 4.50

9. Find two irrational numbers between √2 and √7.

a. √3 and √5 b. √6.25 and √5 c. √2 and √9 d. √3 and √9 e. √5 and √11

10. Find the greatest positive integer that will divide 398, 436 𝑎𝑎𝑎𝑎𝑎𝑎 542 leaving remainders 7, 11 and 15 respectively.

a. 25 b. 7 c. 17 d. 15 e. 77

MATHEMATICS Focused Quiz 2: Linear Equations

1. Solve for 𝑥𝑥: 4(5𝑥𝑥 − 2) = 16𝑥𝑥.

a. −4 b. 2 c. 4 d. −2 e. 8

2. Solve for 𝑦𝑦: 4(𝑦𝑦 + 5) = 6(𝑦𝑦 − 8).

a. −34 b. −14 c. 34 d. 24 e. 14

3. Adam sold his Macbook and accessories for $400. If he received seven times as much money for the Macbook as he did for the accessories, how much did he receive for the Macbook?

a. $450 b. $50 c. $300 d. $400 e. $350

4. Solve for 𝑚𝑚: 10𝑚𝑚 − 8 − 4𝑚𝑚 + 2 = 16𝑚𝑚 + 4

a. 1 b. −1 c. 2 d. −2 e. 4

5. The linear equation that converts Fahrenheit to Celsius is 𝐹𝐹 = �95� 𝐶𝐶 + 32 . If the temperature

is 95𝑜𝑜 𝐹𝐹, the temperature in Celsius is

a. 35° b. 60° c. 125° d. 85° e. 32°


6. 20 years from now Jim’s age will be 5 times his present age. Jim’s present age is

a. 10 𝑦𝑦𝑦𝑦𝑎𝑎𝑦𝑦𝑦𝑦 b. 5 𝑦𝑦𝑦𝑦𝑎𝑎𝑦𝑦𝑦𝑦 c. 15 𝑦𝑦𝑦𝑦𝑎𝑎𝑦𝑦𝑦𝑦 d. 20 𝑦𝑦𝑦𝑦𝑎𝑎𝑦𝑦𝑦𝑦 e. 8 𝑦𝑦𝑦𝑦𝑎𝑎𝑦𝑦𝑦𝑦

7. Two numbers sum to 60 and their ratio is 2 3� . Find the larger of the two numbers.

a. 20 b. 120 c. 40 d. 36 e. 24

8. The unequal side of an isosceles triangle is 5 𝑐𝑐𝑚𝑚 longer than its equal sides. If the perimeter of the triangle is 20 𝑐𝑐𝑚𝑚, find the length of the unequal side.

a. 10𝑐𝑐𝑚𝑚 b. 5𝑐𝑐𝑚𝑚 c. 15𝑐𝑐𝑚𝑚 d. 25𝑐𝑐𝑚𝑚 e. 18𝑐𝑐𝑚𝑚

9. In a classroom the number of boys exceeds the number of girls by 20. If the total number of students is 80, find the number of boys in the class.

a. 30 b. 50 c. 20 d. 10 e. 60

10. You have 10𝑙𝑙 of a solution at 25% concentration and want to obtain a 30% concentration. How much of a solution at 40% concentration should you add?

a. 10𝑙𝑙 b. 3𝑙𝑙 c. 4𝑙𝑙 d. 5.5𝑙𝑙 e. 5𝑙𝑙

MATHEMATICS Focused Quiz 3: Quadratic Equations

1. Find all possible values of 𝑥𝑥 if 𝑥𝑥2 − 3𝑥𝑥 = 10.

a. −2 and 5 b. 2 and 5 c. −2 and − 5 d. −5 only e. −2 only

2. Find 𝑚𝑚 if 8 is a root of the equation 2𝑥𝑥2 − 10𝑥𝑥 − 𝑚𝑚 = 0.

a. −24 b. 16 c. 48 d. −80 e. 128

3. Which of the following terms is a factor of 𝑥𝑥2 + 2𝑥𝑥 − 48 = 0?

a. (𝑥𝑥 + 6) b. (𝑥𝑥 + 8) c. (𝑥𝑥 − 8) d. (𝑥𝑥 + 7) e. (𝑥𝑥 − 7)

4. Find the discriminant of the quadratic equation 2𝑥𝑥2 + 7𝑥𝑥 = 4.

a. 81 b. −28 c. 49 d. 32 e. −8

5. Which of following terms is a solution of the equation 𝑥𝑥2 − 114𝑥𝑥 + 15

8= 0?

a. 32

b. −54

c. −158

d. 114

e. −37


6. Determine the value of 𝑝𝑝 for which the equation 4𝑥𝑥2 + 6 + 2𝑝𝑝 = 0 has a real root.

a. 𝑝𝑝 > 32

b. 𝑝𝑝 = 23

c. 𝑝𝑝 ≤ 12

d. p ≤ 98

e. 𝑝𝑝 ≥ 103

7. The sum of a number and its reciprocal is 103

. Find the numbers.

a. 103

and 310

b. 7 and 17

c. 5 and 15

d. 12

and 2

e. 3 and 13

8. Find the root(s) of the equation 2𝑥𝑥2 − 12𝑥𝑥 + 32 = 4𝑥𝑥.

a. 4 and 10 b. 4 only c. 16 and 10 d. 10 only e. 30 and 16

9. Which of the following quadratic equation has roots −23

and − 35 ?

a. 2𝑥𝑥2 + 3𝑥𝑥 + 15 = 0 b. 𝑥𝑥2 − 19𝑥𝑥 + 6 = 0 c. 5𝑥𝑥2 + 9𝑥𝑥 + 16 = 0 d. 15𝑥𝑥2 + 19𝑥𝑥 + 6 = 0 e. 5𝑥𝑥2 + 3𝑥𝑥 − 2 = 0

10. The sum of the ages of a father and his son is 50 years. Five years ago, the product of their ages was 175. How old is the son now?

a. 40 years b. 10 years c. 15 years d. 8 years e. 12 years

MATHEMATICS Focused Quiz 4: Adding and Subtracting Polynomials

1. Find the value of the polynomial 𝑓𝑓(𝑥𝑥) = 49𝑥𝑥3 + 18𝑥𝑥2 + 42𝑥𝑥 + 16 when 𝑥𝑥 = 3.

a. 745 b. 1323 c. 1388 d. 1483 e. 576

2. Find the degree of the polynomials 3𝑥𝑥7 + 9𝑥𝑥6 × 𝑥𝑥2 − 5𝑥𝑥4 × 𝑥𝑥 + 19𝑥𝑥3 − 12𝑥𝑥2 + 25𝑥𝑥 + 7 = 0.

a. 3 b. 7 c. 8 d. 6 e. 5

3. Fill the missing term: (8𝑥𝑥2 + 42𝑥𝑥 + 16) + (3𝑥𝑥2 − [? ]𝑥𝑥 + 6) = 11𝑥𝑥2 + 20𝑥𝑥 + 22.

a. 19 b. 62 c. 22 d. 12 e. 40

4. Find the sum of the polynomials (2𝑥𝑥3 + 5𝑥𝑥 + 5𝑥𝑥2 + 4) and (6𝑥𝑥3 + 5𝑥𝑥2 + 16).

a. 8𝑥𝑥2 + 5𝑥𝑥2 + 5𝑥𝑥 + 16 b. 8𝑥𝑥3 + 10𝑥𝑥2 + 5𝑥𝑥 + 20 c. 6𝑥𝑥3 + 8𝑥𝑥2 + 5𝑥𝑥 + 12 d. 8𝑥𝑥3 + 8𝑥𝑥2 + 5𝑥𝑥 − 20 e. 6𝑥𝑥3 + 6𝑥𝑥2 + 5𝑥𝑥 + 20

5. Which of the following expressions is equal to: (7𝑥𝑥 + 3𝑦𝑦) − (3𝑦𝑦 + 15) + (−7𝑥𝑥 + 12)?

a. 10𝑥𝑥 + 10𝑦𝑦 + 27 b. 𝑥𝑥 − 𝑦𝑦 = 3 c. 𝑥𝑥 = 3 d. 𝑥𝑥 + 𝑦𝑦 = 3 e. −3


6. Which of the following expressions is a cubic polynomial?

a. 8𝑥𝑥3 + 10𝑥𝑥2 + 5𝑥𝑥 + 20 b. 8𝑥𝑥4 + 10𝑥𝑥2 + 5𝑥𝑥 c. 8𝑥𝑥2 + 5𝑥𝑥 + 20 d. 8𝑥𝑥2 + 𝑥𝑥 + 27 e. 8𝑥𝑥5 + 10𝑥𝑥4 + 4𝑥𝑥3

7. Find the coefficient of 𝑥𝑥2 in 4𝑥𝑥3 + 𝑥𝑥2 − 4𝑐𝑐 = 0.

a. −2 b. 4 c. 2 d. 1 e. −4

8. Simplify �54𝑎𝑎 + 7𝑏𝑏� + (6.4𝑏𝑏 − 2𝑎𝑎) − �7

8𝑎𝑎 − 15𝑏𝑏�.

a. −0.15𝑎𝑎 + 10𝑏𝑏 b. 1.625𝑎𝑎 + 28.4𝑏𝑏 c. −0.15𝑎𝑎 + 10𝑏𝑏 d. 15.45𝑎𝑎 + 20𝑏𝑏 e. −1.625𝑎𝑎 + 28.4𝑏𝑏

9. Subtract (−4𝑥𝑥2 + 6𝑥𝑥 − 18) from (8𝑥𝑥2 − 12𝑥𝑥 − 18).

a. −12𝑥𝑥2 + 18𝑥𝑥 b. 12𝑥𝑥2 − 18𝑥𝑥 c. 4𝑥𝑥2 − 6𝑥𝑥 − 36 d. −4𝑥𝑥2 + 6𝑥𝑥 e. 2𝑥𝑥2 − 8𝑥𝑥 − 36

10. Find (5𝑥𝑥2 + 4𝑥𝑥3 + 14𝑥𝑥7 + 14) − (4𝑥𝑥3 + 5 − 13𝑥𝑥2).

a. 14𝑥𝑥7+8𝑥𝑥3+18𝑥𝑥2 + 19 b. 14𝑥𝑥7+8𝑥𝑥3+18𝑥𝑥2 + 9 c. 14𝑥𝑥7+18𝑥𝑥2 + 9 d. 14𝑥𝑥7+17𝑥𝑥3 − 9𝑥𝑥2 + 9 e. 14𝑥𝑥7+17𝑥𝑥3 + 9𝑥𝑥2 + 19

MATHEMATICS Focused Quiz 5: Multiplying and Dividing Polynomials

1. Expand (8𝑥𝑥2 − 8𝑥𝑥 − 14)(𝑥𝑥 + 3).

a. 4𝑥𝑥3 − 16𝑥𝑥2 + 38𝑥𝑥 − 14 b. 8𝑥𝑥3 + 16𝑥𝑥2 − 38𝑥𝑥 − 42 c. 18𝑥𝑥3 + 8𝑥𝑥2 − 36𝑥𝑥 − 12 d. 8𝑥𝑥3 + 12𝑥𝑥2 − 32𝑥𝑥 − 14 e. 12𝑥𝑥3 + 24𝑥𝑥2 − 38𝑥𝑥 − 42

2. Find the missing term: (𝑥𝑥 + 1)(𝑥𝑥4 + 3𝑥𝑥3 + 4𝑥𝑥2 − 8) = 𝑥𝑥5 + 4𝑥𝑥4 + [? ] + 4𝑥𝑥2 − 8𝑥𝑥 − 8 .

a. 3𝑥𝑥3 b. 4𝑥𝑥3 c. 3𝑥𝑥4 + 7𝑥𝑥3 d. 7𝑥𝑥3 e. 2𝑥𝑥5 + 5𝑥𝑥4

3. Expand (3𝑥𝑥3 + 6𝑥𝑥2 + 12)(6𝑥𝑥3 + 3𝑥𝑥 + 3).

a. 18𝑥𝑥6 + 36𝑥𝑥5 + 9𝑥𝑥4 + 99𝑥𝑥3 + 18𝑥𝑥2 + 36𝑥𝑥 + 36 b. 18𝑥𝑥5 + 18𝑥𝑥5 + 9𝑥𝑥4 + 72𝑥𝑥3 + 18𝑥𝑥2 + 36𝑥𝑥 + 36 c. 18𝑥𝑥6 + 36𝑥𝑥5 + 9𝑥𝑥3 + 99𝑥𝑥3 + 18𝑥𝑥2 + 9𝑥𝑥 + 18 d. 18𝑥𝑥5 + 30𝑥𝑥5 + 9𝑥𝑥4 + 18𝑥𝑥3 + 18𝑥𝑥2 + 20𝑥𝑥 + 36 e. 18𝑥𝑥6 + 36𝑥𝑥5 + 9𝑥𝑥4 + 9𝑥𝑥3 + 18𝑥𝑥2 + 16𝑥𝑥 + 36

4. Let 𝑃𝑃(𝑥𝑥) = (𝑥𝑥3 + 𝑥𝑥2 − 4𝑥𝑥 − 4) and 𝐾𝐾(𝑥𝑥) = (𝑥𝑥 − 2). Find the quotient when 𝑃𝑃(𝑥𝑥) is divided by 𝐾𝐾(𝑥𝑥) with no remainder.

a. 𝑥𝑥2 + 6𝑥𝑥 + 2 b. 𝑥𝑥2 + 6𝑥𝑥 + 4 c. 2𝑥𝑥2 + 3𝑥𝑥 + 2 d. 2𝑥𝑥2 + 2 e. 𝑥𝑥2 + 3𝑥𝑥 + 2

5. 𝑃𝑃(𝑥𝑥) = (𝑥𝑥3 + 3𝑥𝑥2 − 𝑘𝑘𝑥𝑥 + 4) divided by 𝐾𝐾(𝑥𝑥) = 𝑥𝑥 − 2, gives the remainder 𝑘𝑘. Find k.

a. 6 b. 8 c. 12 d. 4 e. 10


6. Find the product of (2 − 𝑥𝑥 + 𝑥𝑥2), (2 − 𝑥𝑥2 + 𝑥𝑥3) and (1 + 𝑥𝑥 + 𝑥𝑥2).

a. 𝑥𝑥7 + 𝑥𝑥6 − 2𝑥𝑥5 + 3𝑥𝑥4 + 3𝑥𝑥3 − 2𝑥𝑥2 − 2𝑥𝑥 − 4 b. 𝑥𝑥7 + 𝑥𝑥6 + 2𝑥𝑥5 + 2𝑥𝑥4 + 𝑥𝑥3 + 2𝑥𝑥2 + 2𝑥𝑥 c. 𝑥𝑥7 − 𝑥𝑥6 + 2𝑥𝑥5 + 𝑥𝑥4 + 3𝑥𝑥3 + 2𝑥𝑥2 + 2𝑥𝑥 d. 𝑥𝑥7 − 𝑥𝑥6 + 2𝑥𝑥5 + 𝑥𝑥4 + 𝑥𝑥3 + 2𝑥𝑥2 + 2𝑥𝑥 + 4 e. 𝑥𝑥7 + 𝑥𝑥6 − 2𝑥𝑥5 + 3𝑥𝑥4 + 3𝑥𝑥3 − 2𝑥𝑥2 + 2𝑥𝑥

7. Find the remainder when 𝑝𝑝(𝑥𝑥) = 4𝑥𝑥3 − 12𝑥𝑥2 + 14𝑥𝑥 − 3 is divided by 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 12.

a. 2 b. 1

2

c. 32

d. 4 e. 3

8. What must be added to the polynomial 𝑓𝑓(𝑥𝑥) = 𝑥𝑥5 + 𝑥𝑥4 + 3𝑥𝑥3 − 6𝑥𝑥2 − 4𝑥𝑥 + 8 so that the resulting polynomial is exactly divisible by 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 − 2?

a. −10 b. −18 c. 10 d. 18 e. −8

9. The polynomials 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥3 − 7𝑥𝑥2 + 7𝑥𝑥 − 2 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥3 − 2𝑎𝑎𝑥𝑥2 + 8𝑥𝑥 − 8 when divided by 𝑥𝑥 − 2 leave the same remainder. Find the value of 𝑎𝑎.

a. 5 b. 8 c. 1 d. 4 e. 2

10. What must be subtracted from 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 − 3𝑥𝑥2 − 8𝑥𝑥 to make it exactly divisible by 𝑔𝑔(𝑥𝑥) =2𝑥𝑥 + 1?

a. 2𝑥𝑥 + 10 b. 4𝑥𝑥 − 2 c. −6𝑥𝑥 d. 6𝑥𝑥 e. −4𝑥𝑥 + 2

MATHEMATICS Focused Quiz 6: Solving Polynomial Equations

1. Solve 𝑥𝑥3 − 6𝑥𝑥 = 0.

a. 𝑥𝑥 = ±√6 b. 𝑥𝑥 = 0, 𝑥𝑥 = ±√6 c. 𝑥𝑥 = 2, 𝑥𝑥 = ±√6 d. 𝑥𝑥 = 0, 𝑥𝑥 = ±2 e. 𝑥𝑥 = 3, 𝑥𝑥 = ±3

2. Solve −8𝑥𝑥3 + 12𝑥𝑥2 + 8𝑥𝑥 = 0.

a. 𝑥𝑥 = 0, 𝑥𝑥 = −12

, 𝑥𝑥 = ±2

b. 𝑥𝑥 = −12

, 𝑥𝑥 = 2 c. 𝑥𝑥 = 0, 𝑥𝑥 = −2, 𝑥𝑥 = 2 d. 𝑥𝑥 = 0, 𝑥𝑥 = −1

2, 𝑥𝑥 = 1

2

e. 𝑥𝑥 = 0, 𝑥𝑥 = −12

, 𝑥𝑥 = 2

3. Evaluate 𝑥𝑥4 − 34𝑥𝑥2 + 225 = 0.

a. 𝑥𝑥 = ±5 ; 𝑥𝑥 = ±3 b. 𝑥𝑥 = ±4 ; 𝑥𝑥 = ±25 c. 𝑥𝑥 = ±25 ; 𝑥𝑥 = ±3 d. 𝑥𝑥 = ±5 ; 𝑥𝑥 = ±25 e. 𝑥𝑥 = ±9 ; 𝑥𝑥 = ±12

4. Solve for polynomial function (𝑃𝑃(𝑥𝑥) = 0) when 𝑃𝑃(𝑥𝑥) = (𝑥𝑥2 + 4𝑥𝑥 − 3)(𝑥𝑥 + 1).

a. 𝑥𝑥 = −2 ± √7 ; 𝑥𝑥 = 1 b. 𝑥𝑥 = ±√7 ;𝑥𝑥 = ±1 c. 𝑥𝑥 = −2 ± √7 ; 𝑥𝑥 = −1 d. 𝑥𝑥 = ±1 e. 𝑥𝑥 = −2 ; 𝑥𝑥 = −1

5. Solve 𝑥𝑥5 − 13𝑥𝑥3 + 30𝑥𝑥 = 0.

a. 𝑥𝑥 = 0 ; 𝑥𝑥 = ±10 ; 𝑥𝑥 = ±3 b. 𝑥𝑥 = 0 ; 𝑥𝑥 = ±√3 c. 𝑥𝑥 = ±√10 ; 𝑥𝑥 = ±√3 d. 𝑥𝑥 = 0 ; 𝑥𝑥 = ±√5 ; 𝑥𝑥 = ±√10 e. 𝑥𝑥 = 0 ; 𝑥𝑥 = ±√10 ; 𝑥𝑥 = ±√3


6. Evaluate: 𝑥𝑥4 = 25𝑥𝑥2 − 144.

a. 𝑥𝑥 = ±4 ; 𝑥𝑥 = ±5 b. 𝑥𝑥 = ±4 ; 𝑥𝑥 = ±3 c. 𝑥𝑥 = ±5 ; 𝑥𝑥 = ±2 d. 𝑥𝑥 = ±5 ; 𝑥𝑥 = ±3 e. 𝑥𝑥 = ±4 ; 𝑥𝑥 = ±12

7. Solve: (𝑥𝑥2 + 8𝑥𝑥 + 2)( 𝑥𝑥2 − 16) = 0.

a. 𝑥𝑥 = −4 + √14 ; 𝑥𝑥 = ±2 b. 𝑥𝑥 = ±4 ; 𝑥𝑥 = ±2 c. 𝑥𝑥 = −4 ± √14 ; 𝑥𝑥 = ±2 d. 𝑥𝑥 = −4 ± √14 ; 𝑥𝑥 = ±4 e. 𝑥𝑥 = −4 ± √14

8. Evaluate: 4𝑥𝑥5 + 𝑥𝑥4 − 4𝑥𝑥 − 1 = 0.

a. 𝑥𝑥 = ±1 ; 𝑥𝑥 = ±𝑖𝑖 ; 𝑥𝑥 = −14

b. 𝑥𝑥 = −14

;𝒙𝒙 = ±𝒊𝒊

c. 𝑥𝑥 = ± 14

; 𝑥𝑥 = ±1 ; 𝑥𝑥 = ±𝑖𝑖 ;

d. 𝑥𝑥 = −14

; 𝑥𝑥 = ±1 e. 𝑥𝑥 = ±1 ; 𝑥𝑥 = ±𝑖𝑖 ;

9. Solve 50𝑥𝑥(𝑥𝑥2 + 3𝑥𝑥) = 4(2𝑥𝑥 + 6).

a. 𝑥𝑥 = −25

; 𝑥𝑥 = −3

b. 𝑥𝑥 = 25

; 𝑥𝑥 = −2

c. 𝑥𝑥 = ± 25𝑖𝑖; 𝑥𝑥 = −3

d. 𝑥𝑥 = ± 25

, 𝑥𝑥 = −2

e. 𝑥𝑥 = ± 25

; 𝑥𝑥 = −3

10. Evaluate 𝑃𝑃(𝑥𝑥) = (𝑥𝑥2 + 25)(𝑥𝑥 + 4).

a. 𝑥𝑥 = 5𝑖𝑖 ; 𝑥𝑥 = −4 b. 𝑥𝑥 = ±5 ; 𝑥𝑥 = −4 c. 𝑥𝑥 = ±5𝑖𝑖 ; 𝑥𝑥 = −4 d. 𝑥𝑥 = ±5 ; 𝑥𝑥 = −4 e. 𝑥𝑥 = 5𝑖𝑖 . 𝑥𝑥 = −4

MATHEMATICS Focused Quiz 7: Rational Root and Factor Theorems

1. If 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 − 13𝑥𝑥2 + 17𝑥𝑥 + 12, 𝑓𝑓(−3)=

a. −117 b. −210 c. −54 d. −198 e. −185

2. Which of the following expressions is a factor of both 𝑓𝑓(𝑥𝑥) = 𝑥𝑥10 − 1 and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥11 − 1?

a. (𝑥𝑥 − 1) b. (𝑥𝑥 − 3) c. (𝑥𝑥 + 1) d. (𝑥𝑥 − 2) e. (𝑥𝑥 + 2)

3. Which of the following expressions is a factor of 𝑝𝑝(𝑥𝑥) = 2𝑥𝑥3 − 9𝑥𝑥2 + 𝑥𝑥 + 12?

a. (3𝑥𝑥 − 2) b. (𝑥𝑥 − 1) c. (𝑥𝑥 + 2) d. (𝑥𝑥 − 2) e. (2𝑥𝑥 − 3)

4. Find the value of 𝑎𝑎 if (𝑥𝑥 − 𝑎𝑎) is a factor of 𝑝𝑝(𝑥𝑥) = 𝑥𝑥3 − 𝑎𝑎2𝑥𝑥 + 𝑥𝑥 + 2.

a. 3 b. −1 c. 5 d. −2 e. −3

5. The real roots of 𝑓𝑓(𝑥𝑥) = 𝑥𝑥4 − 81 are

a. ±√10 b. ±√13 c. ±√3 d. ±√17 e. ±√9


6. Find 𝑘𝑘 if �𝑥𝑥 − 43� is a root of the polynomial 𝑓𝑓(𝑥𝑥) = 6𝑥𝑥3 − 11𝑥𝑥2 + 𝑘𝑘𝑥𝑥 − 20.

a. 17 b. 25 c. 19 d. 23 e. 13

7. Given that (𝑥𝑥 + 2) and (𝑥𝑥 + 3) are factors of 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 + 𝑎𝑎𝑥𝑥2 + 7𝑥𝑥 − 𝑏𝑏, determine the values of 𝑎𝑎 and 𝑏𝑏.

a. 𝑎𝑎 = 9 and 𝑏𝑏 = 6 b. 𝑎𝑎 = 2 and 𝑏𝑏 = 3 c. 𝑎𝑎 = 6 and 𝑏𝑏 = 9 d. 𝑎𝑎 = 3 and 𝑏𝑏 = 6 e. 𝑎𝑎 = 3 and 𝑏𝑏 = 2

8. Factorize 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 − 5𝑥𝑥2 − 4𝑥𝑥 + 20.

a. (𝑥𝑥 − 2)(𝑥𝑥 − 3)(𝑥𝑥 + 5) b. (𝑥𝑥 − 2)(𝑥𝑥 + 2)(𝑥𝑥 − 5) c. (𝑥𝑥 − 7)(𝑥𝑥 + 2)(𝑥𝑥 − 3) d. (𝑥𝑥 − 1)(𝑥𝑥 + 2)(𝑥𝑥 − 5) e. (𝑥𝑥 − 13)(𝑥𝑥 − 1)(𝑥𝑥 + 3)

9. Find the rational roots of the polynomial 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 + 3𝑥𝑥2 − 11𝑥𝑥 − 6.

a. 1, 3 and −13

b. 2,−2 and −14

c. −2,−1 and −12

d. 2,−3 and −12

e. 4,−1 and −13

10. Find the average of all the roots of 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 − 6𝑥𝑥2 + 11𝑥𝑥 − 6.

a. 2 b. 6 c. 4 d. 1 e. 5

MATHEMATICS Focused Quiz 8: Complex Numbers

1. Find the imaginary part of the complex number (2+𝑖𝑖)(1+𝑖𝑖)(1−2𝑖𝑖)

.

a. −12𝑖𝑖

b. 12𝑖𝑖

c. 52𝑖𝑖

d. −52𝑖𝑖

e. −32𝑖𝑖

2. If 𝑧𝑧 = 4 − 3𝑖𝑖, find the value of |𝑧𝑧|.

a. 5 b. √7 c. 2√3 d. 2 − √3 e. 2√7

3. Express (4+5𝑖𝑖)2

(2+3𝑖𝑖)2 in the form of 𝑎𝑎 + 𝑏𝑏𝑖𝑖.

a. 525169

+ 1625𝑖𝑖

b. 525169

+ 92169

𝑖𝑖

c. 525169

− 92169

𝑖𝑖

d. 2516− 92

169𝑖𝑖

e. 2516

+ 92169

𝑖𝑖

4. If = (3−𝑖𝑖)(2+3𝑖𝑖)(1−2𝑖𝑖)(2−𝑖𝑖)

, find the conjugate of 𝑧𝑧.

a. 75

+ 95𝑖𝑖

b. −75

+ 95𝑖𝑖

c. −75

± 95𝑖𝑖

d. −75− 9

5𝑖𝑖

e. 75− 9

5𝑖𝑖

5. 2√−9√−16 =

a. 24 b. −12 c. −48 d. −24 e. 48


6. 𝑖𝑖 + 𝑖𝑖3 + 𝑖𝑖5 + 𝑖𝑖7 + 𝑖𝑖9 =

a. −𝑖𝑖 b. 𝑖𝑖 c. −1 + 𝑖𝑖 d. 1 + 𝑖𝑖 e. 2𝑖𝑖

7. Find the least positive integer 𝑚𝑚 for which �1+𝑖𝑖1−𝑖𝑖�𝑚𝑚

= 1.

a. 1 b. 2 c. 3 d. 4 e. 5

8. If 𝑧𝑧1 = 4 − 𝑖𝑖, 𝑧𝑧2 = −4 + 𝑖𝑖, find the real portion of 1𝑧𝑧1𝑧𝑧2

.

a. − 15289

b. −28916

c. −16 d. − 8

289

e. − 117

9. Solve the equation 𝑥𝑥2 − 2𝑥𝑥 + 32

= 0.

a. √22

+ 𝑖𝑖

b. 1 ± √22𝑖𝑖

c. 1 ± 𝑖𝑖 d. √2

2+ √2𝑖𝑖

e. −1 ± √22𝑖𝑖

10. If 𝑧𝑧1 = 2 − 𝑖𝑖, 𝑧𝑧2 = 1 + 𝑖𝑖, then the value of �𝑧𝑧1+𝑧𝑧2+1𝑧𝑧1−𝑧𝑧2+1

� is

a. √2 b. 2√3 c. 3√2 d. 2 e. 2√2

MATHEMATICS Focused Quiz 9: Functions

1. Given the real valued functions 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = 1𝑥𝑥 , (𝑓𝑓 + 𝑔𝑔)𝑥𝑥 =

a. 2x b. 1

𝑥𝑥

c. 𝑥𝑥 + 1𝑥𝑥

d. 𝑥𝑥 − 1𝑥𝑥

e. 2𝑥𝑥 + 1𝑥𝑥

2. Find the domain of the function (𝑥𝑥) = 𝑥𝑥1+𝑥𝑥2

.

a. All real numbers less than one b. All real numbers greater than or equal to one c. All real numbers greater than one d. All real numbers e. All real numbers greater than two

3. Given 𝑐𝑐 is a non-zero real number and a real valued function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥𝑐𝑐, then 𝑥𝑥 =

a. 𝑐𝑐2𝑓𝑓 b. 𝑐𝑐𝑓𝑓 c. �1

𝑐𝑐� 𝑓𝑓

d. � 1𝑐𝑐2� 𝑓𝑓

e. 2𝑐𝑐𝑓𝑓

4. Find the range of the function 𝑓𝑓(𝑥𝑥) = |𝑥𝑥 − 3|.

a. All real numbers greater than or equal to three b. All real numbers greater than or equal to zero c. All real numbers greater than zero d. All real numbers e. All real numbers greater than two

5. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥𝑥𝑥2+1

, find 𝑓𝑓�𝑓𝑓(2)�.

a. 1029

b. 1425

c. 2925

d. 425

e. 25


6. If 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥2 − 5𝑥𝑥 + 11, find 𝑓𝑓(𝑥𝑥 − 1).

a. 2𝑥𝑥2 − 11𝑥𝑥 + 5 b. 𝑥𝑥2 + 9𝑥𝑥 + 11 c. 3𝑥𝑥2 − 11𝑥𝑥 + 19 d. 3𝑥𝑥2 + 11𝑥𝑥 − 11 e. 2𝑥𝑥2 − 9𝑥𝑥 + 19

7. If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2, find the value of 𝑓𝑓(1.1)−𝑓𝑓(1)(1.1)−1

.

a. 0.8 b. 1.7 c. 4.2 d. 2.1 e. 3.6

8. Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 − 4 and (𝑥𝑥) = �𝑥𝑥2−16𝑥𝑥+4

, 𝑥𝑥 ≠ 4𝛼𝛼, 𝑥𝑥 = −4

. Find 𝛼𝛼 such that 𝑓𝑓(𝑥𝑥) = 𝑔𝑔(𝑥𝑥) 𝑓𝑓𝑓𝑓𝑦𝑦 𝑎𝑎𝑙𝑙𝑙𝑙 𝑥𝑥.

a. −8 b. 4 c. 8 d. 2 e. −4

9. Three real valued functions are defined by 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 − 3, 𝑔𝑔(𝑥𝑥) = 1𝑥𝑥 , and ℎ(𝑥𝑥) = 2𝑥𝑥2 + 3. Find

2𝑓𝑓 + 4𝑔𝑔 − ℎ at 𝑥𝑥 = 4.

a. −20 b. 25 c. −30 d. 35 e. −32

10. Given a real valued function 𝑓𝑓(𝑥𝑥) = �3𝑥𝑥 − 2, 𝑥𝑥 < 0

1, 𝑥𝑥 = 04𝑥𝑥 + 1, 𝑥𝑥 > 0

, evaluate 𝑓𝑓(1)+𝑓𝑓(0)𝑓𝑓(−1)+𝑓𝑓(2)

.

a. 37

b. 10 c. 3

2

d. −5 e. 1

2

MATHEMATICS Focused Quiz 10: Graphing Linear Functions

1. Which of the following lines passes through all quadrants except the third (lower left)?

a. 𝑦𝑦 = 4𝑥𝑥 + 5 b. 𝑦𝑦 = 4𝑥𝑥 c. 𝑦𝑦 = −4𝑥𝑥 + 5 d. 𝑦𝑦 = 4𝑥𝑥 − 5 e. 𝑦𝑦 = −4𝑥𝑥 − 5

2. Which of the following equations could this graph represent?

a. 𝑥𝑥 = 5𝑦𝑦 − 10 b. 𝑦𝑦 = −10𝑥𝑥 − 5 c. 𝑦𝑦 = 6𝑥𝑥 − 10 d. 𝑦𝑦 = 2𝑥𝑥 + 2 e. 𝑦𝑦 = −6

3. If no horizontal line meets the graph of a function at multiple points, the function is

a. one-to-one b. many-to-one c. not one-to-one d. many-to-many e. none of these

y

x


4. Which of the following functions is both odd and even? (Note: A function 𝑓𝑓(𝑥𝑥) is odd if 𝑓𝑓(−𝑥𝑥) = −𝑓𝑓(𝑥𝑥) for all 𝑥𝑥 and even if 𝑓𝑓(𝑥𝑥) = 𝑓𝑓(−𝑥𝑥) for all 𝑥𝑥.)

a.

x

y

b.

x

y

c.

x

y

d.

x

y

e.

x

y


5. Find the 𝑥𝑥 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑦𝑦𝑦𝑦𝑐𝑐𝑦𝑦𝑝𝑝𝑖𝑖 and 𝑦𝑦 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑦𝑦𝑦𝑦𝑐𝑐𝑦𝑦𝑝𝑝𝑖𝑖 of 12𝑥𝑥 + 6𝑦𝑦 = 60.

a. x − intercept is at (0,5) and y − intercept is at (10,0) b. x − intercept is at (5,0) and y − intercept is at (0,10) c. x − intercept is at (4,0) and y − intercept is at (0,8) d. x − intercept is at (5,0) and y − intercept is at (0,5) e. x − intercept is at (10,0) and y − intercept is at (0,5)

6. Which of the following statements is false?

a. A vertical line cannot meet the graph of a linear function in more than one point. b. 𝑓𝑓 is a real valued function of a real variable iff 𝐷𝐷𝑓𝑓 ⊂ ℝ and also 𝑅𝑅𝑓𝑓 ⊂ ℝ. c. If no horizontal line meets the graph of a function in more than one point, then the

function is one-one, otherwise it is many-one d. Many-to-one quadratic functions have a discriminant smaller than 0. e. 𝑓𝑓 is said to be even function if 𝑓𝑓(𝑥𝑥) = 𝑓𝑓(−𝑥𝑥) for all 𝑥𝑥.

7. Find the intercepts of the graph of the equation y = - 2x – 6.

a. x-intercept is −5 and y-intercept is 6 b. x-intercept is −3 and y-intercept is −6 c. x-intercept is −2 and y-intercept is −3 e. x-intercept is 5 and y-intercept is −6

8. Find the slope of the following graph.

a. −1 b. −2 c. −5 d. −7 e. −4

y

x 0 1

7


9. Find the range of the function represented in the graph below.

a. [0,∞) b. (4,6) c. [4,6] d. (−2,2) e. [0,2]

10. Which of the following function is represented in the graph below?

a. 𝑓𝑓(𝑥𝑥) = �𝑥𝑥 + 1, 𝑥𝑥 < 22𝑥𝑥 − 1, 𝑥𝑥 ≥ 2

b. 𝑓𝑓(𝑥𝑥) = �𝑥𝑥 − 1, 0 ≤ 𝑥𝑥 < 1

1, 𝑥𝑥 = 1𝑥𝑥 + 1, 1 < 𝑥𝑥 ≤ 2

c. 𝑓𝑓(𝑥𝑥) = �𝑥𝑥, 𝑥𝑥 ≤ 0𝑥𝑥2, 𝑥𝑥 > 0

d. 𝑓𝑓(𝑥𝑥) = �3 − 𝑥𝑥, 𝑥𝑥 > 11, 𝑥𝑥 = 12𝑥𝑥, 𝑥𝑥 < 1

e. 𝑓𝑓(𝑥𝑥) = �1, 𝑥𝑥 ≥ 1𝑥𝑥, − 1 < 𝑥𝑥 < 1−1, 𝑥𝑥 ≤ −1

0

-1 2

1

2

1

-1

-2

-2

y

x

Y=1

Y=-1

MATHEMATICS Focused Quiz 11: Graphing Quadratic Functions and Polynomials


a. The 𝑥𝑥 − 𝑖𝑖𝑎𝑎𝑖𝑖𝑦𝑦𝑦𝑦𝑐𝑐𝑦𝑦𝑝𝑝𝑖𝑖 of the parabola shows the value of x when value of 𝑥𝑥 = 0. b. The parabola of a quadratic function is concave upwards if 𝑎𝑎 > 0. c. The parabola of a quadratic function is concave downwards if 𝑎𝑎 < 0. d. The axis of symmetry separates a parabola into two equal halves. e. The graph of a quadratic function is known as a parabola.

2. Which of the following is the equation of the graph shown below?

a. (𝑥𝑥 + 2)(𝑥𝑥 − 5) b. (𝑥𝑥 + 3)(𝑥𝑥 − 4) c. (𝑥𝑥 + 3)(𝑥𝑥 − 2) d. (𝑥𝑥 + 8)(𝑥𝑥 − 5) e. (𝑥𝑥 + 8)(𝑥𝑥 − 4)


a. The x-intercept refers to the point of the parabola where it passes through the x-axis. b. A linear function’s parabola has two axial intercepts. c. Linear and quadratic functions are examples of polynomial functions. d. A parabola graph which opens upwards is known as concave upward. e. The behavior of a parabola is determined by the behavior of the function’s zero degree.

4. Find the y-intercept of the graph: 𝑓𝑓(𝑥𝑥) = 7𝑥𝑥2 − 5𝑥𝑥 − 2.

a. (0,5) b. (0,−2) c. (0,−3) d. (0,2) e. (0,−5)

(-8,0) (5,0)

Y

X


5. The concave portion of the graph of which of the following graphs faces towards the left?

a. 𝑦𝑦2 = 23𝑥𝑥 b. 𝑦𝑦2 = √3

2𝑥𝑥

c. 𝑦𝑦2 = 48𝑦𝑦 d. 𝑥𝑥2 = −4𝑎𝑎𝑦𝑦 e. 𝑦𝑦2 = −√8𝑥𝑥

6. Find the x-intercept of the graph: 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 − 2𝑥𝑥 − 15.

a. (−5,0) and (−2,0) b. (3,0) and (−3,0) c. (5,0) and (−3,0) d. (5,0) and (2,0) e. (8,0) and (−2,0)

7. Find the maximum point of the graph 𝑓𝑓(𝑥𝑥) = 5𝑥𝑥2 − 20𝑥𝑥 − 13.

a. (2,−33) b. (2,−20) c. (0,−13) d. (1,−28) e. (3,−2)

8. Find the axis of symmetry of the graph of 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 − 18𝑥𝑥 − 208.

a. 1 b. 18 c. 5 d. 9 e. 15


9. Find the minimum possible degree of the polynomial in the graph shown below.

a. 3 b. 9 c. 7 d. 11 e. 5

10. Find the vertex of the quadratic function 𝑓𝑓(𝑥𝑥) = 8𝑥𝑥2 − 26𝑥𝑥 + 15.

a. (4.324, 6.345) b. (−1.625, 1) c. (3.857, 7.237) d. (5.23, 9.37) e. (1.625,−6.125)

Y

X

MATHEMATICS Focused Quiz 12: Graphing Exponential and Logarithmic Functions

1. Which of the following statements about the logarithmic function 𝑓𝑓(𝑥𝑥) = log𝑏𝑏 𝑥𝑥 is not true?

a. Logarithmic function 𝑓𝑓(𝑥𝑥) = log𝑏𝑏 𝑥𝑥 is the inverse of the exponential function 𝑔𝑔(𝑥𝑥) =𝑏𝑏𝑥𝑥.

b. Function is continuous and is one-to-one c. Domain is the set of all positive real number. d. Range is the set of all real numbers. e. The graph intersects the x-axis at (0,0).

2. The graph of 𝑦𝑦 = 4𝑥𝑥+5 − 7 is asymptotical to the line

a. 𝑦𝑦 = −7 b. 𝑥𝑥 = −7 c. 𝑥𝑥 = 7 d. 𝑦𝑦 = 7 e. 𝑥𝑥 = −4

3. Which of the following statements about the exponential function 𝑓𝑓(𝑥𝑥) = 𝑏𝑏𝑥𝑥 is false?

a. The domain is the set of all real numbers. b. The range is set of all positive real numbers. c. The graph of 𝑓𝑓(𝑥𝑥) always has a horizontal asymptote at the y-axis. d. If 0 < 𝑏𝑏 < 1, the graph of 𝑓𝑓(𝑥𝑥) = 𝑏𝑏𝑥𝑥 will decrease as x increases. e. If 𝑏𝑏 > 1, the graph of 𝑓𝑓(𝑥𝑥) = 𝑏𝑏𝑥𝑥 will increase as x increases.

4. Find the y-intercept of the graph 𝑦𝑦 = 5𝑥𝑥−2.

a. 0.12 b. 0.3 c. 0.4 d. 0.35 e. 0.04

5. Find the horizontal asymptote of 𝑓𝑓(𝑥𝑥) = �45�𝑥𝑥− 5.

a. 𝑥𝑥 = −5 b. 𝑦𝑦 = −5 c. 𝑥𝑥 = 5 d. 𝑦𝑦 = 5 e. 𝑦𝑦 = −4


6. The graph of 𝑦𝑦 = −(82𝑥𝑥)

a. passes through the origin. b. intersects both the x-axis and y-axis. c. intersects the x-axis only. d. intersects the y-axis only. e. intersects neither x-axis or y-axis.

7. If 𝑓𝑓(𝑥𝑥) = �23�𝑥𝑥

and ℎ(𝑥𝑥) is the reflection of 𝑓𝑓(𝑥𝑥) in the line 𝑥𝑥 = 𝑦𝑦, find ℎ(𝑥𝑥).

a. log23(𝑦𝑦)

b. −�23�𝑥𝑥

c. ln𝑦𝑦(23)

d. log23(𝑥𝑥)

e. (𝑦𝑦)325

8. If 𝑓𝑓(𝑥𝑥) = 5𝑥𝑥 and ℎ(𝑥𝑥) is the reflection of 𝑓𝑓(𝑥𝑥) with respect to the y-axis, find ℎ(𝑥𝑥).

a. (2)𝑥𝑥 b. (0.5)𝑥𝑥 c. (0.2)𝑥𝑥 d. −(0.2)−𝑥𝑥 e. (0.4)−𝑥𝑥

9. Which of the following is not true about the function 𝑓𝑓(𝑥𝑥) = −8ln(𝑥𝑥 − 9)?

a. The domain of 𝑓𝑓 is the set of all 𝑥𝑥 values 𝑥𝑥 > 9. b. The range of 𝑓𝑓 is the set (−𝑖𝑖𝑎𝑎𝑓𝑓, +𝑖𝑖𝑎𝑎𝑓𝑓) c. The vertical asymptote is obtained by solving 𝑥𝑥 − 4 = 0. d. There is no 𝑦𝑦-intercept. e. The 𝑥𝑥-intercept is at (9,0).

10. Which of the following could be the equation of the graph below?

a. �23�𝑥𝑥− 2

b. 2𝑥𝑥−1 − 2

c. �23�𝑥𝑥−2

d. 2𝑥𝑥 + 2 e. �3

4�𝑥𝑥

y

x

(0,3)

MATHEMATICS Focused Quiz 13: Transformation of Graphs

1. For 𝑐𝑐 > 0, to obtain the graph of 𝑓𝑓(𝑥𝑥 − 𝑐𝑐) shifts the graph of 𝑓𝑓(𝑥𝑥)

a. right 𝑐𝑐 units b. left 𝑐𝑐 units c. upward 𝑐𝑐 units d. downward 𝑐𝑐 units e. downward (𝑐𝑐 − 1) units

2. Find the equation of the graph of 𝑦𝑦 = √𝑥𝑥 transformed horizontally 20 units to the right.

a. 𝑦𝑦 = √𝑥𝑥 + 20 b. 𝑦𝑦 = −√𝑥𝑥 − 20 c. 𝑦𝑦 = √𝑥𝑥 − 20 d. 𝑦𝑦 = √𝑥𝑥 + 20 e. A 𝑦𝑦 = √𝑥𝑥 + 20

3. For 𝑐𝑐 > 1, to obtain the graph of 1𝑐𝑐𝑓𝑓(𝑥𝑥) compress the graph of 𝑓𝑓(𝑥𝑥)

a. vertically by a factor of 1𝑐𝑐2

b. horizontally by a factor of 𝑐𝑐2 c. vertically by a factor of 𝑐𝑐 d. vertically by a factor of 𝑐𝑐2 e. horizontally by a factor of 𝑐𝑐

4. Which of the following shifts would make 𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 5)4 − 21 an even function?

a. 21 units up b. 21 units down c. 5 units left d. 5 units right e. both (c) and (d)

5. Which of the following shifts transforms 𝑦𝑦 = 𝑥𝑥3 to the graph of 𝑦𝑦 + 2 = 𝑥𝑥3?

a. a vertical shift by 2 units upwards b. a horizontal shift by a factor of 1

2

c. a horizontal shift 2 units leftwards d. a horizontal shift by a factor of 2 e. a vertical shift by 2 units downwards


6. If a reflection of a graph in the line 𝑦𝑦 = −𝑥𝑥 occurs, which of the following changes occurs?

a. (𝑥𝑥,𝑦𝑦) → (𝑥𝑥,−𝑦𝑦) b. (𝑥𝑥,𝑦𝑦) → (−𝑦𝑦,−𝑥𝑥) c. (𝑥𝑥,𝑦𝑦) → (𝑥𝑥𝑦𝑦, 𝑥𝑥) d. (𝑥𝑥,𝑦𝑦) → (𝑦𝑦, 𝑥𝑥) e. (𝑥𝑥,𝑦𝑦) → (−𝑥𝑥,−𝑦𝑦)

7. Find the equation of the reflection of 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) = 8|𝑥𝑥 + 12| across the y-axis.

a. 𝑦𝑦 = 8|𝑥𝑥 + 12| b. 𝑦𝑦 = 4|𝑥𝑥 + 12| c. 𝑦𝑦 = 4|𝑥𝑥 − 12| d. 𝑦𝑦 = 8|12 − 𝑥𝑥| e. 𝑦𝑦 = |𝑥𝑥 + 12|

8. Find the equation of the reflection of 𝑓𝑓(𝑥𝑥) = √𝑥𝑥 + 4 − 5 across the y-axis.

a. √𝑥𝑥 + 4 − 5 b. √4 − 𝑥𝑥 − 5 c. √𝑥𝑥 + 4 + 5 d. √4 − 𝑥𝑥 + 5 e. √2𝑥𝑥 + 4 − 5

9. Find the equation of the reflection of 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥3 − 5𝑥𝑥2 − 12𝑥𝑥 + 5 across the x-axis.

a. 2𝑥𝑥3 − 5𝑥𝑥2 − 12𝑥𝑥 + 5 b. −2𝑥𝑥3 − 5𝑥𝑥2 + 12𝑥𝑥 + 5 c. 2𝑥𝑥3 + 5𝑥𝑥2 + 12𝑥𝑥 − 5 d. −4𝑥𝑥3 − 5𝑥𝑥2 − 12𝑥𝑥 − 5 e. −2𝑥𝑥3 + 5𝑥𝑥2 + 12𝑥𝑥 − 5

10. Find the equation of the reflection of 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥−1𝑥𝑥2+4

across the x-axis.

a. 2𝑥𝑥−1𝑥𝑥2+4

b. −2𝑥𝑥−1−𝑥𝑥2+4

c. −2𝑥𝑥+1𝑥𝑥2+4

d. −2𝑥𝑥−1−𝑥𝑥2−4

e. 2𝑥𝑥−1𝑥𝑥2−4

MATHEMATICS Focused Quiz 14: Rational Equations

1. Find the domain of 7𝑥𝑥(𝑥𝑥−1)(𝑥𝑥2−7).

a. 𝑥𝑥 ∈ ℝ b. 𝑥𝑥 ∈ ℝ,≠ 1 and ± √7 c. 𝑥𝑥 ∈ ℝ,≠ 1 d. 𝑥𝑥 ∈ ℝ,≠ ±√7 e. 𝑥𝑥 ∈ ℝ,≠ 1 and 7

2. For all 𝑥𝑥 ≠ 0, 12𝑥𝑥

+ 14𝑥𝑥

+ 16𝑥𝑥

+ 18𝑥𝑥

=

a. 58𝑥𝑥

b. 512𝑥𝑥

c. 812𝑥𝑥

d. 2524𝑥𝑥

e. 76𝑥𝑥

3. Find the domain of 𝑥𝑥+3𝑥𝑥2+3𝑥𝑥−15

.

a. 𝑥𝑥 ∈ ℝ,≠ 2 and 8 b. 𝑥𝑥 ∈ ℝ,≠ 2 and 5 c. 𝑥𝑥 ∈ ℝ d. 𝑥𝑥 ∈ ℝ,≠ 2 and − 8 e. 𝑥𝑥 ∈ ℝ,≠ 2 and − 5

4. If 3(𝑥𝑥+1)4𝑥𝑥

− 3(𝑥𝑥−1)8𝑥𝑥

= 12 , 𝑥𝑥 =

a. 5 b. 9 c. 7 d. 3 e. 12

5. If 78𝑎𝑎− 7

8𝑎𝑎2= 7

32, then 𝑎𝑎 =

a. 2 b. 4 c. 3 d. 7 e. 14


6. Solve 2 = 12− 1

2−𝑥𝑥 for x.

a. 𝑥𝑥 = 16

𝑓𝑓𝑦𝑦 𝑥𝑥 = 13

b. 𝑥𝑥 = 15

c. No real solution d. 𝑥𝑥 = 1

4 𝑓𝑓𝑦𝑦 𝑥𝑥 = 1

5

e. 𝑥𝑥 = ∞

7. Find the number where the sum of one-third of the number and the number’s reciprocal is the same as 13 divided by that number.

a. ±2 b. ±5 c. ±3 d. ±4 e. ±6

8. If 𝑥𝑥+1𝑥𝑥+2

+ 𝑥𝑥+3𝑥𝑥+4

= 23 , 𝑥𝑥 =

a. 1 or −72

b. 3 or −54

c. −5 or 52

d. 4 or −72

e. −2 or −43

9. If 2𝑟𝑟+10𝑟𝑟2+𝑟𝑟

= 2𝑟𝑟2+𝑟𝑟

− 2𝑟𝑟−12𝑟𝑟+1

, then the value of 𝑦𝑦 is

a. ±4 b. 2 or 1 c. 3 or ± 1 d. −4 or 2 e. 4 or 1

10. Solve 30𝑚𝑚𝑚𝑚2−6𝑚𝑚−16

= 3𝑚𝑚+12𝑚𝑚+2

for m.

a. −1 or 12 b. −2 or 4 c. −2 or 16 d. −1 or 8 e. −4 or 16

MATHEMATICS Focused Quiz 15: Exponential Equations

1. 255 × 260 − 297 × 218 =

a. 8 b. 4 c. 2 d. 0 e. 1

2. If 𝑥𝑥 = 2 and 𝑦𝑦 = 3, find the value of �1𝑥𝑥

+ 1𝑦𝑦�𝑥𝑥

.

a. 2536

b. 56

c. 256

d. 1533

e. 13

3. Find the value of 16×2𝑥𝑥+1−4×2𝑥𝑥

16×2𝑥𝑥+2−2×2𝑥𝑥+2.

a. 34

b. 18

c. 12

d. 516

e. 732

4. Solve 27𝑥𝑥 = 93𝑥𝑥

for x.

a. 18

b. 12

c. 118

d. 13

e. 19


5. Solve 5𝑥𝑥−3 × 32𝑥𝑥−8 = 225 for x.

a. 10 b. 125 c. 25 d. 5 e. 15

6. Solve for 𝑥𝑥, if 3𝑥𝑥𝑦𝑦𝑥𝑥 + 𝑥𝑥2𝑦𝑦𝑥𝑥 = 0

a. 𝑥𝑥 = 0 or 𝑥𝑥 = −3 b. 𝑥𝑥 = 1 or 𝑥𝑥 = 3 c. 𝑥𝑥 = 0 or 𝑥𝑥 = 3 d. 𝑥𝑥 = 3 or 𝑥𝑥 = −3 e. 𝑥𝑥 = 1 or 𝑥𝑥 = −2

7. Solve 10(8𝑦𝑦2𝑥𝑥 − 3)3 = 1250 for 𝑥𝑥.

a. 1 b. 3 c. 0 d. 5 e. 2

8. Find 𝑥𝑥 if 22𝑥𝑥2−3𝑥𝑥−8 = 18.

a. 𝑥𝑥 = −3, 5 b. 𝑥𝑥 = −2, 3 c. 𝑥𝑥 = −1, 2 d. 𝑥𝑥 = 0, 2 e. 𝑥𝑥 = −1, 2.5

9. Solve for 𝑥𝑥: 3𝑦𝑦2𝑥𝑥+4 = 5

a. 2.568 b. −1.745 c. 1.667 d. −0.593 e. −1.563

10. Emma puts $2000 into an account that earns interest rate of 12%. In what length of time will she have $4000 in the account if the interest is compounded 6 times a year? The formula for the amount (𝐴𝐴) when a principal (𝑃𝑃) is compounded m times a year is given by 𝐴𝐴 =

𝑃𝑃 �1 + 𝑟𝑟𝑚𝑚�𝑡𝑡𝑚𝑚

, where R is the annual interest rate and 𝑖𝑖 is the time in years.

a. 3.834 years b. 2.417 years c. 7.454 years d. 5.834 years e. 4.367 years

MATHEMATICS Focused Quiz 16: Logarithmic Equations

1. Solve for 𝑥𝑥 if log𝑥𝑥 64 = 32.

a. 128 b. 64 c. 16 d. 42.667 e. 8


a. log𝑎𝑎 1 = 0 b. log𝑎𝑎 𝑚𝑚𝑎𝑎 = log𝑎𝑎 𝑚𝑚 − log𝑎𝑎 𝑎𝑎 c. log𝑎𝑎

𝑚𝑚𝑛𝑛

= log𝑎𝑎 𝑚𝑚 − log𝑎𝑎 𝑎𝑎 d. log𝑎𝑎 𝑎𝑎 = 1, where 𝑎𝑎 is any positive integer except 1 e. log𝑎𝑎 𝑚𝑚𝑛𝑛 = 𝑎𝑎 log𝑎𝑎 𝑚𝑚

3. log8−log2log32

=

a. 632

b. 4 c. 1

8

d. 25

e. 32

4. If log3 𝑥𝑥 = 𝑎𝑎, find 81𝑎𝑎−1 in terms of 𝑥𝑥.

a. 𝑥𝑥4

81

b. 2𝑥𝑥9

c. 𝑥𝑥2

9

d. 𝑥𝑥81

e. 𝑥𝑥3

27

5. If log 7 − log 2 + log 16 − 2log 3 − log 745

= 1 + log𝑎𝑎, find 𝑎𝑎.

a. 8 b. 20 c. 4 d. 10 e. 5


6. If the value of log2(log2(log2 𝑥𝑥)) = 1, then the value of 𝑥𝑥 is

a. 8 b. 32 c. 4 d. 16 e. 64

7. If log𝑎𝑎𝑏𝑏−𝑐𝑐

= log𝑏𝑏𝑐𝑐−𝑎𝑎

= log𝑐𝑐𝑎𝑎−𝑏𝑏

, find the value of 𝑎𝑎𝑎𝑎 × 𝑏𝑏𝑏𝑏 × 𝑐𝑐𝑐𝑐 .

a. 14 b. 1 c. 3 d. 6 e. 16

8. Find the value of 𝑥𝑥, if log 𝑥𝑥 − log(2𝑥𝑥 − 1) = 1.

a. 78

b. 3029

c. 35

d. 12

e. 1019

9. Solve 3log𝑥𝑥- 2log𝑥𝑥 = 2log𝑥𝑥+1 − 3log𝑥𝑥−1 for x.

a. 100 b. 10 c. 2 d. 12 e. 3

10. Solve log2 𝑥𝑥 + log4 𝑥𝑥 + log16 𝑥𝑥 = 214

for x.

a. 3 b. 4 c. 8 d. 21 e. 84

MATHEMATICS Focused Quiz 17: Radical Equations

1. Find 𝑥𝑥 if 2√𝑥𝑥 − 4 = 10.

a. 10 b. 49 c. 14 d. 12 e. 7

2. If √3𝑥𝑥 − 53 = 4, then 𝑥𝑥 is

a. 23 b. 64 c. 12 d. 69 e. 35

3. Find 𝑥𝑥 if 2√𝑥𝑥√𝑥𝑥 − 7 = 24.

a. −9 b. 12 c. 7 d. −5 e. 16

4. If √𝑥𝑥 + 4 + √8 − 𝑥𝑥 = √12 , then 𝑥𝑥 is

a. 4 or 12 b. 3 or 8 c. −4 or 8 d. 4 or 2 e. −8 or 12

5. Find 𝑥𝑥 if √𝑥𝑥 + √4𝑥𝑥 = 6.

a. 4 b. 9 c. 5 d. 36 e. 6


6. Solve for 𝑦𝑦 if 1 + �2𝑦𝑦 + 3 = 6.

a. 2 b. 22 c. 5 d. 11 e. 8

7. Solve for 𝑥𝑥 if �5 − √𝑥𝑥 + �5 + √𝑥𝑥 = √12.

a. −2√6 b. 24 c. 2√6 d. 25 e. 3√2

8. Find 𝑥𝑥 if √𝑥𝑥 − 5 = −6.

a. −6 b. 6 c. −5 d. 5 e. No solution

9. Solve for 𝑥𝑥: if √𝑥𝑥 + 7 + 2 = √3 − 𝑥𝑥.

a. 2 only b. −6 and 2 c. −6 only d. −4 only e. −4 and 6

10. If √5𝑥𝑥2 − 54 = −√2𝑥𝑥, then 𝑥𝑥 is

a. −3√2 b. −√2 c. 3√2 d. −2√2 e. 2√2

MATHEMATICS Focused Quiz 18: Linear Inequalities

1. Solve the inequality 2𝑥𝑥 − 3 < 5.

a. All the numbers smaller than 8. b. All the numbers smaller than 4. c. All the numbers smaller than or equal to 2. d. All the numbers smaller than or equal to 4. e. All the numbers smaller than 5

2. If 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 are real numbers such that 𝑎𝑎 > 𝑏𝑏, 𝑐𝑐 > 0, then

a. 𝑎𝑎𝑐𝑐 < 𝑏𝑏𝑐𝑐 b. 𝑎𝑎𝑐𝑐 ≥ 𝑏𝑏𝑐𝑐 c. 𝑎𝑎𝑐𝑐 > 𝑏𝑏𝑐𝑐 d. 𝑎𝑎𝑐𝑐 ≤ 𝑏𝑏𝑐𝑐 e. none of the above

3. Find the set of solutions for the inequality 5𝑥𝑥2

+ 3𝑥𝑥4≥ 39

4 .

a. All numbers greater than or equal to 3 b. All numbers greater than or equal to 13 c. All numbers greater than 3 d. All numbers greater than 13 e. All numbers greater than or equal to 39

4. If 𝑥𝑥 < 5, then

a. −𝑥𝑥 < −5 b. −𝑥𝑥 ≤ −5 c. −𝑥𝑥 ≥ −5 d. −𝑥𝑥 > −5 e. 𝑥𝑥 > −5

5. Solve the inequality 3(𝑥𝑥−2)5

≥ 5(2−𝑥𝑥)3

.

a. All numbers greater than or equal to 2 b. All numbers greater than or equal to 3 c. All numbers greater than 2 d. All numbers greater than 3 e. All numbers greater than or equal to 5


6. Find the solution set of the linear inequality represented below.

a. 𝑥𝑥 ≤ 0 b. 𝑥𝑥 ≤ 7

2

c. 𝑥𝑥 < 0 d. 𝑥𝑥 > 7

2

e. 𝑥𝑥 ≤ 3

7. Solve the inequality −|2𝑥𝑥 − 1| ≤ 3.

a. −1 ≤ 𝑥𝑥 ≤ 2 b. 2 ≤ 𝑥𝑥 ≤ 4 c. 1 < 𝑥𝑥 < 2 d. −3 ≤ 𝑥𝑥 ≤ 4 e. 2 < 𝑥𝑥 < 4

8. The length of rectangle is three times the length of breadth. If the minimum perimeter of the rectangle is 160𝑐𝑐𝑚𝑚, then

a. breadth > 20𝑐𝑐𝑚𝑚 b. length < 20𝑐𝑐𝑚𝑚 c. breadth ≥ 20𝑐𝑐𝑚𝑚 d. length ≤ 20𝑐𝑐𝑚𝑚 e. breadth < 10𝑐𝑐𝑚𝑚

9. Solve the following system of inequalities: 2𝑥𝑥−35

< 1−𝑥𝑥3

< 3+4𝑥𝑥2

.

a. 𝑥𝑥 < −12

b. 𝑥𝑥 > 12

c. 𝑥𝑥 > −14

d. 𝑥𝑥 > −12

e. 𝑥𝑥 > 14

10. To receive ‘A’ in a course, one must obtain an average of 90 marks or more in five examinations (graded out of 100.) If Edward’s marks in first four examinations are 87, 92, 94 and 95, find the minimum mark that Edward must score for the last test to obtain an ‘A’.

a. 92 b. 82 c. 87 d. 55 e. 90

7/2 X

MATHEMATICS Focused Quiz 19: Quadratic Inequalities

1. Find the determinant of the quadratic inequality 1𝑥𝑥− 1

𝑥𝑥−2< 3, 𝑥𝑥 ≠ 0,2.

a. 12 b. 24 c. 36 d. 48 e. 6

2. Solve the inequality: 𝑥𝑥2 − 81 < 0

a. −1 < 𝑥𝑥 < 3 b. −3 < 𝑥𝑥 < 3 c. −9 < 𝑥𝑥 < −3 d. −9 < 𝑥𝑥 < 9 e. 1 < 𝑥𝑥 < 9

3. Solve the inequality: 𝑥𝑥(𝑥𝑥 − 7) ≤ −12

a. 3 ≤ 𝑥𝑥 ≤ 4 b. 1 ≤ 𝑥𝑥 ≤ 4 c. 𝑥𝑥 ≤ 3 and 𝑥𝑥 ≥ 4 d. 3 > 𝑥𝑥 > 4 e. −1 ≤ 𝑥𝑥 ≤ 3

4. Solve the inequality: 𝑥𝑥2 + 2𝑥𝑥 > 24

a. 𝑥𝑥 < −1 and 𝑥𝑥 > 4 b. −6 ≤ 𝑥𝑥 ≤ 1 c. 𝑥𝑥 < −6 or 𝑥𝑥 > 4 d. 𝑥𝑥 < −5 or 𝑥𝑥 > 2 e. 𝑥𝑥 < 1 or 𝑥𝑥 > 3

5. Solve the inequality: 𝑥𝑥2 + 4𝑥𝑥 < −10

a. −2 < 𝑥𝑥 < 4 b. 𝑥𝑥 < −6 c. 𝑥𝑥 ≥ 4 d. 𝑥𝑥 < −4 e. No solution


6. Solve the inequality: 𝑥𝑥2 + 10𝑥𝑥 < −9

a. 3 < 𝑥𝑥 < 9 b. −9 < 𝑥𝑥 < −1 c. −1 < 𝑥𝑥 < 1 d. −4 < 𝑥𝑥 < 4 e. 2 < 𝑥𝑥 < 6

7. Solve the inequality: 𝑥𝑥2 + 8𝑥𝑥 + 7 > 0

a. x < −3 or x > −7 b. x < 3 or x > 5 c. A x > −7 or x < 1 d. x < −7 or x > −1 e. x < −1 or x > −3

8. Solve the inequality: : 𝑥𝑥2 + 10𝑥𝑥 + 25 > 0

a. 𝑥𝑥 ≠ −5 b. 𝑥𝑥 < −5 c. 𝑥𝑥 > −5 d. 𝑥𝑥 = −5 e. 𝑥𝑥 > −1

9. A motor boat whose speed is 18 𝑘𝑘𝑚𝑚/ℎ in still water takes 1 hour more to go 24 𝑘𝑘𝑚𝑚 upstream than to return downstream to the same spot. Find the speed of the stream.

a. 4 km/h b. 12 km/h c. 6 km/h d. 15 km/h e. 20 km/h

10. Solve the inequality: 𝑥𝑥2 + 2𝑥𝑥 + 1 < 0

a. 𝑥𝑥 < −1 b. 𝑥𝑥 ≤ −2 c. 𝑥𝑥 > −2 d. 𝑥𝑥 > 4 e. No solution

MATHEMATICS Focused Quiz 20: Coordinate Geometry: Points, Lines, and Circles

1. Find the expression for the distance of the point (𝑥𝑥,𝑦𝑦) from the origin (0,0).

a. �𝑥𝑥2 + 2𝑦𝑦2 b. �2𝑥𝑥2 + 𝑦𝑦2 c. �𝑥𝑥2 + 𝑦𝑦2 d. �2𝑥𝑥2 + 2𝑦𝑦2 e. �2𝑥𝑥2 + 4𝑦𝑦2

2. 𝐾𝐾𝐾𝐾 is a straight line of 13 𝑢𝑢𝑎𝑎𝑖𝑖𝑖𝑖𝑦𝑦. If 𝐾𝐾 has the coordinates (2,5) and 𝐾𝐾 has the coordinates (𝑥𝑥,−7), find the possible values of 𝑥𝑥.

a. 7 or −3 b. 5 or −5 c. 2 or −5 d. 7 or −12 e. 5 or −3

3. Find the points on the x-axis 5 units distance from the points (5,−4)?

a. (0, 0) or (2, 0) b. (1, 0) or (16, 0) c. (4, 0) or (5, 0) d. (2, 0) or (8, 0) e. (2, 2) or (8, 4)

4. The mid-point of the line joining (2𝑎𝑎, 4) and (−2, 2𝑏𝑏) is (1, 2𝑎𝑎 + 1). Find the values of 𝑎𝑎 and 𝑏𝑏.

a. 𝑎𝑎 = 0 and 𝑏𝑏 = 7. b. 𝑎𝑎 = 2 and 𝑏𝑏 = 3. c. 𝑎𝑎 = 1 and 𝑏𝑏 = −6. d. 𝑎𝑎 = −1 and 𝑏𝑏 = 3. e. 𝑎𝑎 = 2 and 𝑏𝑏 = 8.

5. If 2𝑥𝑥 − 3𝑦𝑦 + 5 = 0 and 𝑝𝑝𝑥𝑥 + 6𝑦𝑦 + 7 = 0 are parallel lines, find the value of 𝑝𝑝.

a. 5 b. 6 c. −3 d. 2 e. −4


6. Point P divides the line between 𝐴𝐴(4,−5) and 𝐵𝐵(4,5) such that 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

= 25 . Find P’s coordinates.

a. (2, 9) b. (4, 0) c. (8, 0) d. (4,−1) e. (5, 8)

7. 𝐴𝐴(6, 1),𝐵𝐵(8, 2),𝐶𝐶(9, 4) and 𝐷𝐷(𝑝𝑝, 3) are the vertices of a parallelogram ABCD. Find 𝑝𝑝.

a. 7 b. 15 c. 8 d. 2.5 e. 23

8. The equation of the straight line is 3𝑥𝑥 − 3𝑦𝑦 − 7 = 0. Find the slope of the line.

a. 7 b. −3 c. 1 d. 4 e. −2

9. Find the equation of the perpendicular line from the point (-1, 2) onto the line 𝐴𝐴(1,4) and 𝐵𝐵(2,3).

a. 𝑥𝑥 − 2𝑦𝑦 + 5 = 0 b. 𝑥𝑥 − 𝑦𝑦 + 3 = 0 c. 2𝑥𝑥 − 2𝑦𝑦 + 1 = 0 d. 2𝑥𝑥 − 𝑦𝑦 + 1 = 0 e. 𝑥𝑥 + 𝑦𝑦 + 3 = 0

10. If the points 𝐴𝐴(4,3) and 𝐵𝐵(𝑥𝑥, 5) are on the circle with centre 𝑂𝑂(2,3), find the value of 𝑥𝑥.

a. 8 b. 5 c. 6 d. 2 e. 3

MATHEMATICS Focused Quiz 21: Coordinate Geometry Problems

1. Find the coordinates of the centroid of a triangle with vertices (0,6), (8,12) and (8,0).

a. �166

, 3� b. (8, 9) c. (4,5) d. �16

3, 6�

e. �203

, 163� 𝑔𝑔

2. Find the equation of the perpendicular bisector of the line segment joining (7,1) and (3,5).

a. 𝑦𝑦 = 2𝑥𝑥 − 4 b. 𝑦𝑦 = 𝑥𝑥 − 2 c. 3𝑦𝑦 = 2𝑥𝑥 − 2 d. 𝑦𝑦 = 5𝑥𝑥 − 3 e. 2𝑦𝑦 = 3𝑥𝑥 − 2

3. The coordinates of P and Q are (−8,𝑎𝑎) and (2,𝑎𝑎 + 8). The mid point of PQ is (−3, 4). Find a.

a. 3 b. 8 c. 1 d. 5 e. 4

4. Find the lengths of the medians of a ∆𝐴𝐴𝐵𝐵𝐶𝐶 whose vertices are 𝐴𝐴(1,−3),𝐵𝐵(5,3) and 𝐶𝐶(3,−1).

a. 3, 5 and √10. b. 5, 8 and 10. c. 5, 5 and √10 d. 4, 5 and 8. e. 5, 9 and √7.

5. If 𝐴𝐴(−3,0),𝐵𝐵(1,−3) and 𝐶𝐶(4,1) are the vertices of an isosceles right angled triangle at B, find the area of the triangle.

a. 12.5 sq. units b. 15 sq. units c. 22 sq. units d. 17.5 sq. units e. 25 sq. units


6. Two opposite vertices of a square are 𝐴𝐴(−1, 2) and 𝐶𝐶(3, 2). Find the coordinates of the other two vertices?

a. (−2,0) and (−2,4) b. (1,2) and (1,4) c. (0,−2) and (0,4) d. (1,0) and (1,4) e. (4,0) and (1,−2)

7. The coordinates of one end of a diameter of a circle are (5,0) and the coordinates of the centre circle are (2,−2). Find the coordinates of the other end of the diameter.

a. (1,−4) b. (−1,−3) c. (−3,−4) d. (1,2) e. (−3,5)

8. Find the x-coordinate of the center of the circle passing through the points 𝑂𝑂(0,0),𝐴𝐴(−2,1) and 𝐵𝐵(−3,2).

a. 38

b. 12

c. 112

d. 35

e. 32

9. 𝐴𝐴(3,2) and 𝐵𝐵(−2,1) are two vertices of ∆𝐴𝐴𝐵𝐵𝐶𝐶 whose centroid 𝑃𝑃 has coordinates �53

, −13�. Find

the coordinates of the third vertex 𝐶𝐶 of the triangle.

a. (4,−4) b. (5,−2) c. (3,1) d. (5,−1) e. (2,−2)

10. The points 𝑃𝑃(2,−1),𝑄𝑄(3,4),𝑅𝑅(−2,3) and 𝑆𝑆(−3,−2) are the vertices of a

a. Square b. Rhombus c. Rectangle d. Trapezoid e. Parallelogram

MATHEMATICS Focused Quiz 22: Sine and Tangent Functions for Acute Angles

1. Which of the following expressions is false?

a. tan𝜃𝜃 = 𝑃𝑃𝑃𝑃𝑟𝑟𝐴𝐴𝑃𝑃𝑛𝑛𝑃𝑃𝑖𝑖𝑐𝑐𝑃𝑃𝑃𝑃𝑎𝑎𝑟𝑟𝐴𝐴𝑎𝑎𝐵𝐵𝑃𝑃

b. sin𝜃𝜃 = 𝑃𝑃𝑃𝑃𝑟𝑟𝐴𝐴𝑃𝑃𝑛𝑛𝑃𝑃𝑖𝑖𝑐𝑐𝑃𝑃𝑃𝑃𝑎𝑎𝑟𝑟ℎ𝑦𝑦𝐴𝐴𝑜𝑜𝑡𝑡𝑃𝑃𝑛𝑛𝑃𝑃𝐵𝐵𝑃𝑃

c. cot 𝜃𝜃 = 𝐴𝐴𝑎𝑎𝐵𝐵𝑃𝑃𝑃𝑃𝑃𝑃𝑟𝑟𝐴𝐴𝑃𝑃𝑛𝑛𝑃𝑃𝑖𝑖𝑐𝑐𝑃𝑃𝑃𝑃𝑎𝑎𝑟𝑟

d. sec 𝜃𝜃 = 𝐻𝐻𝑦𝑦𝐴𝐴𝑜𝑜𝑡𝑡𝑃𝑃𝑛𝑛𝑃𝑃𝐵𝐵𝑃𝑃𝑃𝑃𝑃𝑃𝑟𝑟𝐴𝐴𝑃𝑃𝑛𝑛𝑃𝑃𝑖𝑖𝑐𝑐𝑃𝑃𝑃𝑃𝑎𝑎𝑟𝑟

e. cos 𝜃𝜃 = 𝐴𝐴𝑎𝑎𝐵𝐵𝑃𝑃𝐻𝐻𝑦𝑦𝐴𝐴𝑜𝑜𝑡𝑡𝑃𝑃𝑛𝑛𝑃𝑃𝐵𝐵𝑃𝑃

2. In a Δ𝐴𝐴𝐵𝐵𝐶𝐶, if ∠𝐴𝐴 = 90°, find sin2 𝐵𝐵 + sin2 𝐶𝐶.

a. 3 b. 2 c. 0.6 d. 1 e. 4

3. If 4 sin2 𝜃𝜃 − 1 = 0 and 𝜃𝜃 is an acute angle, find the value of sin 3𝜃𝜃.

a. 3 b. 2 c. 5 d. 4 e. 1

4. Find tan 𝑥𝑥° for the given triangle.

a. 1√3

b. √32

c. 2 d. √3 e. 1

5. Δ𝐴𝐴𝐵𝐵𝐶𝐶 has ∠𝐴𝐴 = 55°,𝐴𝐴𝐵𝐵 = 8 and 𝐴𝐴𝐶𝐶 = 6. Find the approximate area of the triangle.

a. 17.36 b. 19.66 c. 15.70 d. 14.50 e. 12.32

B x°

A

C

2 √3


6. If 3𝜃𝜃 is an acute angle and tan 3𝜃𝜃 − √3 = 0, find the value of 𝜃𝜃.

a. 40° b. 30° c. 60° d. 45° e. 20°

7. Find the length of BC in the right angled triangle below.

a. 7𝑐𝑐𝑚𝑚 b. 3𝑐𝑐𝑚𝑚 c. 2𝑐𝑐𝑚𝑚 d. 5𝑐𝑐𝑚𝑚 e. 4.50𝑐𝑐𝑚𝑚

8. Isosceles triangle ABC has ∠𝐵𝐵𝐴𝐴𝐶𝐶 = 42°, unequal side 𝐵𝐵𝐶𝐶 = 5 𝑐𝑐𝑚𝑚, and area ≈ 19.948 𝑐𝑐𝑚𝑚2. Find the ratio 𝐴𝐴𝑟𝑟𝑃𝑃𝑎𝑎

𝑃𝑃𝑃𝑃𝑟𝑟𝑖𝑖𝑚𝑚𝑃𝑃𝑡𝑡𝑃𝑃𝑟𝑟.

a. 1.25 b. 1.50 c. 3.75 d. 2.50 e. 4.50

9. ∠𝐸𝐸𝐴𝐴𝐹𝐹 = 17°, ∠𝐶𝐶𝐸𝐸𝐷𝐷 = 75°,𝐸𝐸𝐹𝐹 = 5 𝑎𝑎𝑎𝑎𝑎𝑎 𝐷𝐷𝐸𝐸 = 7. Find the approximate area of triangle ACE.

a. 70.873 b. 82.145 c. 116.082 d. 213.723 e. 122.55

10. If sin𝛼𝛼 = 35, evaluate 1−tan𝛼𝛼

1+tan𝛼𝛼.

a. 13

b. 17

c. 43

d. 27

e. 12

30◦ 4cm 60◦

B C

A

D

MATHEMATICS Focused Quiz 23: Cosine and Cotangent Functions for Acute Angles

1. If sin 54° csc(90° − 𝜃𝜃) = 1 , find the value of 𝜃𝜃, 0° < 𝜃𝜃 < 90°.

a. 16° b. 36° c. 24° d. 42° e. 38°

2. If 40° + 𝑥𝑥 is an acute angle and cos(40° + 𝑥𝑥) = sin 30 °, find the value of 𝑥𝑥.

a. 35° b. 15° c. 10° d. 20° e. 50°


a. As 𝜃𝜃 increases from 0° to 90°, sin𝜃𝜃 increases while cos𝜃𝜃 decreases. b. sin2 𝜃𝜃 + cos2 𝜃𝜃 = 1 c. 1 + tan2 𝜃𝜃 = sec2 𝜃𝜃 d. 1 + cot2 𝜃𝜃 = csc2 𝜃𝜃 e. cos 0° + cot 0° = 1

4. If 4 sin2 𝜃𝜃 − 3 = 0 and 𝜃𝜃 is an acute angle, find the value of cos 0.5𝜃𝜃.

a. 32

b. 57

c. 14

d. 34

e. √32

5. Δ𝐴𝐴𝐵𝐵𝐶𝐶 is a right angled triangle. Use cos 𝑥𝑥° to find the length AB.

a. 2 b. 3 c. 1 d. 8 e. 5

B x

A

C

2 √3


6. In the right-angled triangle below, QR = 5 cm and PR is 1 cm longer than PQ. Evaluate sec𝑃𝑃.

a. 1312

b. 32

c. 57

d. 1115

e. 12

7. Evaluate: sin(90°−𝛼𝛼) sin𝛼𝛼tan𝛼𝛼

+ cos(90°−𝛼𝛼) cos𝛼𝛼cot𝛼𝛼

a. 2 b. 4 c. 5 d. 1 e. 6

8. If CD = 70, find AB.

a. 83.911 b. 91.378 c. 73.975 d. 21.445 e. 67.545

9. If 𝜃𝜃 is an acute angle and sin𝜃𝜃 = cos 𝜃𝜃, evaluate 2 sin2 𝜃𝜃 − 3 cos2 𝜃𝜃 + 12

cot2 𝜃𝜃.

a. 4 b. 2 c. 8 d. 1 e. 0

10. If tan𝜃𝜃 = 1√5

, find the value of 𝑐𝑐𝑦𝑦𝑐𝑐2 𝜃𝜃 − 𝑦𝑦𝑦𝑦𝑐𝑐2 𝜃𝜃.

a. 1 b. 25

4

c. 245

d. 2 e. 30

7

P

Q R 5cm

C

D

A B

25° 50°

MATHEMATICS Focused Quiz 24: Relations among Trigonometric Functions


A. tan 𝑥𝑥 = sin𝑥𝑥cos𝑥𝑥

B. tan 𝑥𝑥 cot 𝑥𝑥 = 1 C. tan2 𝑥𝑥 − sec2 𝑥𝑥 = 0 D. 1 + cot2 𝑥𝑥 = csc2 𝑥𝑥 E. sin2 𝑥𝑥 + cos2 𝑥𝑥 = 1

2. Find the value of tan2 𝑥𝑥 − sin2 𝑥𝑥.

A. cos2 𝑥𝑥 sin2 𝑥𝑥 B. cos2 𝑥𝑥 C. sec2 𝑥𝑥 cos2 𝑥𝑥 D. tan2 𝑥𝑥 sin2 𝑥𝑥 E. tan2 𝑥𝑥

3. Evaluate sin𝑥𝑥1+cos𝑥𝑥

+ 1+cos𝑥𝑥sin𝑥𝑥

.

A. 2 cos𝑥𝑥sin𝑥𝑥(1+cos𝑥𝑥)

B. 2 cos 𝑥𝑥 C. 2

sin𝑥𝑥(1+cos𝑥𝑥)

D. sin 𝑥𝑥 E. 2 csc 𝑥𝑥

4. Evaluate sec𝜃𝜃(1 − sin 𝜃𝜃)(sec𝜃𝜃 + tan𝜃𝜃).

A. 1 B. 4 C. 3 D. 5 E. 2

5. If sin𝜃𝜃cos𝜃𝜃

= 85, find sec2 𝜃𝜃.

A. 6425

B. 8925

C. −2925

D. 85

E. 645


6. Evaluate (1 + cot𝐴𝐴 − csc𝐴𝐴)(1 + tan𝐴𝐴 + sec𝐴𝐴).

A. 1 B. 4 C. 3 D. 5 E. 2

7. If sin𝜃𝜃 = 2129

and 𝜃𝜃 lies in the second quadrant, find the value of sec𝜃𝜃 + tan 𝜃𝜃.

a. 2029

b. 12

c. −52

d. 25

e. −2029

8. Evaluate �sin27°cos63°

�2

+ �cos63°sin27°

�2

A. 1 B. 4 C. 3 D. 5 E. 2

9. Find the value of cot2 𝐴𝐴(sec𝐴𝐴−1)(1+sin𝐴𝐴)

+ sec2 𝐴𝐴(sin𝐴𝐴−1)(1+sec𝐴𝐴)

.

A. 1 B. cot A C. sec A D. 0 E. tan A

10. Evaluate (sec𝐴𝐴 − tan𝐴𝐴)2.

A. 1−sin𝐴𝐴1+sin𝐴𝐴

B. 1−sin𝐴𝐴1+tan𝐴𝐴

C. 1−cos𝐴𝐴1+sin𝐴𝐴

D. 1+tan𝐴𝐴1−cot𝐴𝐴

E. 1−cos𝐴𝐴1−sin𝐴𝐴

MATHEMATICS Focused Quiz 25: Trigonometric Functions of Special Angles

1. Evaluate sin�−𝜋𝜋2�

cos�−𝜋𝜋4�

a. −2√2 b. −√2 c. √2 d. 2√2 e. −1

2. If 𝜃𝜃 = 30°, find the value of sin2𝜃𝜃cos2𝜃𝜃

.

a. 1√2

b. 2 c. √3

2

d. √3 e. 1

2

3. If 𝑓𝑓(𝑥𝑥) = sin(−𝜃𝜃)csc(𝜃𝜃) cot(−𝜃𝜃)sec𝜃𝜃 cos(−𝜃𝜃)

, find the reciprocal of 𝑓𝑓(𝑥𝑥).

a. cot(𝜃𝜃) b. tan(𝜃𝜃) c. 1

cos(𝜃𝜃) d. cot(𝜃𝜃) cos(𝜃𝜃) e. tan(𝜃𝜃) sin(𝜃𝜃)

4. Evaluate sin𝜃𝜃 tan𝜃𝜃1−cos(−𝜃𝜃)

.

a. sec 𝜃𝜃 b. 1+cos(𝜃𝜃)

sin(𝜃𝜃) c. sec 𝜃𝜃 + 1 d. tan𝜃𝜃 e. tan𝜃𝜃 + 1

5. If tan𝜑𝜑 + cot𝜑𝜑 = 2, then find the value of tan100 𝜑𝜑 + cot100 𝜑𝜑.

a. 4 b. 2 c. 1

2

d. 1 e. 100


6. Find the value of 43

tan2 30° + sin2 60° − 3 cos2 60° + 34

tan2 60° − 2 tan2 45° .

a. 49

b. 94

c. 34

d. 2536

e. 8136

7. Using the formula tan 2𝜃𝜃 = 2 tan𝜃𝜃1−tan2 𝜃𝜃

, find the value of tan 15 °.

a. −√3 ± 2 b. √3 + √2 c. −√3 + 2√2 d. 2√2 ± 1 e. −2√3 + 4

8. Evaluate 4(sin2 30° + cos2 60°) − 3(cos2 45° + sin2 90°).

a. −9 b. −4 c. 8 d. −3 e. 6

9. Find the value of sin(−330°) cos(−300°) + cos(30°) sin(−240°).

a. 3 b. √3

2

c. 12

d. 32

e. 1

10. Solve tan 2𝜃𝜃 = tan 2𝜃𝜃

for 𝜃𝜃.

a. 𝜃𝜃 = 𝑛𝑛𝑛𝑛±√𝑛𝑛2𝑛𝑛2+42

,𝑎𝑎𝑛𝑛𝑛𝑛

b. 𝜃𝜃 = 𝑛𝑛±√𝑛𝑛2+164


c. 𝜃𝜃 = 𝑛𝑛𝑛𝑛±√𝑛𝑛2𝑛𝑛2+164


d. 𝜃𝜃 = 3𝑛𝑛𝑛𝑛±√𝑛𝑛2𝑛𝑛2+162


e. 𝜃𝜃 = 5𝑛𝑛𝑛𝑛±√𝑛𝑛2𝑛𝑛2+168


MATHEMATICS Focused Quiz 26: Trigonometric Identities

1. Which of the following expressions is equivalent to 2 sin𝐴𝐴 cos𝐵𝐵?

a. tan2A b. sin(𝐴𝐴 + 𝐵𝐵) + sin(𝐴𝐴 − 𝐵𝐵) c. cot(𝐴𝐴 + 𝐵𝐵) d. tan(𝐴𝐴 + 𝐵𝐵) e. cos(𝐴𝐴 + 𝐵𝐵) + cos(𝐴𝐴 − 𝐵𝐵)

2. Simplify 1 + tan𝐴𝐴 tan 𝐴𝐴2 .

a. 1+tan𝐴𝐴 b. tan𝐴𝐴 c. cos𝐴𝐴 d. sec𝐴𝐴 e. sin𝐴𝐴

3. Simplify sin 51° + cos 81°.

a. cos 21° b. cos 30° c. sin 21° d. cos 51° e. sin 51°

4. Find the value of sin2 𝑛𝑛8

+ sin2 3𝑛𝑛8

+ sin2 5𝑛𝑛8

+ sin2 7𝑛𝑛8

.

a. 0 b. 4 c. 2 d. 6 e. 1

5. Evaluate cos(−1710°).

a. 0 b. 1

2

c. 2 d. 1

3

e. 1


6. If tan(𝐴𝐴 + 𝐵𝐵) = 𝑝𝑝 and tan(𝐴𝐴 − 𝐵𝐵) = 𝑞𝑞, then find the value of tan 2𝐴𝐴.

a. 𝐴𝐴+𝑞𝑞1+2𝐴𝐴𝑞𝑞

b. 𝐴𝐴𝑞𝑞

c. 2𝑝𝑝𝑞𝑞 d. 𝐴𝐴+𝑞𝑞

1−𝐴𝐴𝑞𝑞

e. 1

7. Simplify 1−tan2 𝐴𝐴

1+tan2 𝐴𝐴 .

a. cos 2𝐴𝐴 b. tan 2𝐴𝐴 c. sin𝐴𝐴 d. sin 2𝐴𝐴 e. cos𝐴𝐴

8. If tan𝐴𝐴 = 𝑎𝑎𝑎𝑎+1

, and tan𝐵𝐵 = 12𝑎𝑎+1

, find 𝐴𝐴 + 𝐵𝐵.

a. 0 b. 1

2

c. 2 d. 1

3

e. 1

9. Given tan 𝑛𝑛8

> 0, find its value.

a. √2 − √3 b. √2 c. √2 − 1 d. 1 e. √3 − 1

10. Simplify sin 𝑛𝑛18

+ sin 𝑛𝑛9

+ sin 2𝑛𝑛9

+ sin 5𝑛𝑛18

.

a. sin 7𝑛𝑛18

+ sin 4𝑛𝑛9

b. 1 c. cos 𝑛𝑛

6+ cos 3𝑛𝑛

7

d. cos 𝑛𝑛9

+ sin 𝑛𝑛9

e. 0

MATHEMATICS Focused Quiz 27: Graphs of Trigonometric Functions


a. The graph of a tangent repeats itself every 180°. b. Cosine is negative in the 2nd and 3rd quadrants. c. The graph of sine is continuous (i.e. has no breaks). d. Tangent is negative in the 2nd and 4th quadrants. e. Sine is positive in 1st and 4th quadrant.

2. Find the period of the function 𝑦𝑦 = sin 4𝑥𝑥.

a. 𝑛𝑛4

b. 3𝑛𝑛4

c. 𝑛𝑛

3

d. 𝑛𝑛2

e. 2𝑛𝑛3

3. Find the aptitude of the function 𝑓𝑓(𝑥𝑥) = 5 − 4 sin(3𝑥𝑥 − 4) ?

a. 4 b. 5 c. 2 d. 8 e. 16


a. A periodic function has a constant real p such that 𝑓𝑓(𝑥𝑥 + 𝑝𝑝) = 𝑓𝑓(𝑥𝑥) for all 𝑥𝑥𝑛𝑛𝐷𝐷𝑓𝑓 . b. Both sin 𝑥𝑥 and cos 𝑥𝑥 have a periodic interval of 2𝜋𝜋. c. The function 𝑓𝑓(𝑥𝑥) = tan(3𝑥𝑥 + 𝑛𝑛

4) has a periodic interval of 𝑛𝑛

3.

d. The functions sec 𝑥𝑥 and csc 𝑥𝑥 have a periodic interval of 2𝜋𝜋. e. The functions tan 𝑥𝑥 and cot 𝑥𝑥 have a periodic interval of peri 2𝜋𝜋.

5. Which of the following values is an asymptote of the function 𝑓𝑓(𝑥𝑥) = 2 csc 𝑥𝑥 ?

a. 2𝜋𝜋 b. 𝜋𝜋 c. 𝑛𝑛

2

d. 3𝜋𝜋 e. 3𝑛𝑛

4


6. Which of the following equations has an amplitude of 5, period of 2𝜋𝜋, and a maximum of 7?

a. 𝑓𝑓(𝑥𝑥) = 3 sin �𝑥𝑥 + 𝑛𝑛2� − 1

b. 𝑓𝑓(𝑥𝑥) = 5 sin �𝑥𝑥 − 𝑛𝑛2� − 2

c. 𝑓𝑓(𝑥𝑥) = −5 sin �𝑥𝑥 − 𝑛𝑛2�+ 1

d. 𝑓𝑓(𝑥𝑥) = 5 sin �𝑥𝑥 + 𝑛𝑛2� + 2

e. 𝑓𝑓(𝑥𝑥) = −3 sin �𝑥𝑥 − 𝑛𝑛2� − 2

7. Which of the following equations could describe the following graph?

a. 3 sin 2𝑥𝑥 b. −3 cos 2𝑥𝑥 c. 3 sin 𝑥𝑥 d. cos 𝑥𝑥 e. sin 3𝑥𝑥

8. Find the period of 𝑓𝑓(𝑥𝑥) = cos �𝑥𝑥2

+ 𝑛𝑛4�.

a. 4𝜋𝜋 b. 2𝜋𝜋 c. 8𝜋𝜋 d. 𝜋𝜋 e. 6𝜋𝜋

9. Which of the following equations has an amplitude of 2𝜋𝜋 and a period of 3?

a. 𝑓𝑓(𝑥𝑥) = 2𝜋𝜋 sin �3𝑛𝑛𝑥𝑥4�

b. 𝑓𝑓(𝑥𝑥) = −2𝜋𝜋 sin �𝑛𝑛𝑥𝑥2�

c. 𝑓𝑓(𝑥𝑥) = 2𝜋𝜋 sin �2𝑛𝑛𝑥𝑥3�

d. 𝑓𝑓(𝑥𝑥) = 2𝜋𝜋 sin(3𝜋𝜋𝑥𝑥) e. 𝑓𝑓(𝑥𝑥) = −2𝜋𝜋 sin �2𝑛𝑛𝑥𝑥

3�

10. Which of the following equations has amplitude of 3, period of 2𝑛𝑛3

and a minimum of 0?

a. 2 �−1 + sin 32𝑥𝑥�

b. 2(−1 + sin 3𝑥𝑥) c. 3(1 + sin 3𝑥𝑥) d. −3(−1 + sin 3𝑥𝑥) e. −3 �1 + sin 3

2𝑥𝑥�

Y

X

3 --

-3 --

π⁄2 π⁄4

3π/4 π

0

MATHEMATICS Focused Quiz 28: Inverse Trigonometric Functions

1. Evaluate arcsin (sin 𝑛𝑛3

).

a. 𝜋𝜋 b. 𝑛𝑛

3

c. 𝑛𝑛2

d. 𝑛𝑛

4

e. 𝑛𝑛6

2. Evaluate tan−1(−1).

a. 4 b. 2 c. −1 d. 1 e. −2

3. Find the domain of arccos(2𝑥𝑥 − 1).

a. [0,1] b. [−1,1] c. (0,2) d. (0,1) e. (−1,1)

4. Evaluate arcsin �cos �arcsin �− √32��.

a. 𝑛𝑛4

b. 𝑛𝑛

2

c. 𝑛𝑛6

d. 2𝑛𝑛3

e. 𝜋𝜋

5. Evaluate cos−1 �12� + 2 sin−1 �1

2�.

a. 𝑛𝑛4

b. 𝑛𝑛

2

c. 𝑛𝑛6

d. 𝜋𝜋 e. 2𝑛𝑛

3


6. If arcsin 𝑥𝑥 = 𝑦𝑦, then

a. 0 ≤ 𝑦𝑦 ≤ 𝜋𝜋 b. −𝑛𝑛

2≤ 𝑦𝑦 ≤ 𝑛𝑛

2

c. 0 < 𝑦𝑦 < 𝜋𝜋 d. −𝑛𝑛

2< 𝑦𝑦 < 𝑛𝑛

2

e. 0 ≤ 𝑦𝑦 ≤ 𝑛𝑛2

7. Evaluate cot �arcsin 35�.

a. 43

b. 35

c. 23

d. 45

e. 13

8. Solve sin �arcsin 15

+ arccos 𝑥𝑥� = 1 for x.

a. 13

b. 𝑛𝑛2

c. 43

d. 15

e. 35

9. Evaluate cos �arcsin 513�.

a. 1213

b. 113

c. 513

d. 23

e. 1213

10. Find the minimum value of n for which tan−1 𝑛𝑛4

> 𝑛𝑛4

,𝑎𝑎 𝑛𝑛 Ν.

a. 3.14 b. 2 c. 4 d. 1 e. 1.07

MATHEMATICS Focused Quiz 29: Trigonometric Equations

1. Find the solution of sec𝑥𝑥 = 2.

a. x = 5π3

and 3π4

b. x = 2π3

and π4

c. x = π3

and 5π3

d. x = π3

and 4π3

e. x = π2

and 2π3

2. Find the smallest value of 𝜃𝜃 satisfying the equation tan �23𝜃𝜃� = √3.

a. 𝑛𝑛2

b. 𝑛𝑛

3

c. 𝑛𝑛4

d. 𝑛𝑛

6

e. 𝑛𝑛12

3. If cos 𝑥𝑥 = 𝑘𝑘 has exactly one solution in [0, 2𝜋𝜋], find 𝑘𝑘.

a. 5 b. −3 c. 2 d. −1 e. 1

4. Solve sin2 𝜃𝜃 = sin2 𝛼𝛼 ,𝛼𝛼𝑛𝑛ℝ.

a. 𝜃𝜃 = 𝑎𝑎𝜋𝜋 ± 2𝛼𝛼,𝑎𝑎 𝑛𝑛 ℤ b. 𝜃𝜃 = 𝑎𝑎𝜋𝜋 ± 𝛼𝛼, 𝑎𝑎 𝑛𝑛 ℤ c. 𝜃𝜃 = 2𝑎𝑎𝜋𝜋 ± 3𝛼𝛼,𝑎𝑎 𝑛𝑛 ℤ d. 𝜃𝜃 = 𝑎𝑎 𝑛𝑛

2± 𝛼𝛼,𝑎𝑎 𝑛𝑛 ℤ

e. 𝜃𝜃 = 3𝑎𝑎𝜋𝜋 ± 4𝛼𝛼,𝑎𝑎 𝑛𝑛 ℤ

5. Find the solution set of the equation (2 cos 𝜃𝜃+1)(4 cos𝜃𝜃 + 5) = 0 in the interval [0, 2𝜋𝜋].

a. 𝜃𝜃 = 2𝑛𝑛3

, 𝑛𝑛2

b. 𝜃𝜃 = 𝑛𝑛3

, 5𝑛𝑛2

c. 𝜃𝜃 = 3𝑛𝑛2

, 5𝑛𝑛2

d. 𝜃𝜃 = 𝑛𝑛3

, 2𝑛𝑛3

e. 𝜃𝜃 = 2𝑛𝑛3

, 4𝑛𝑛3


6. Find the smallest 𝜃𝜃 satisfying the equation √3(cot𝜃𝜃 + tan𝜃𝜃) = 4.

a. 𝑛𝑛6

b. 𝑛𝑛

12

c. 𝑛𝑛4

d. 𝜋𝜋 e. 𝑛𝑛

2

7. Find the number of points of intersection of the curves 2𝑦𝑦 = −1 and 𝑦𝑦 = csch 𝑥𝑥.

a. 4 b. 6 c. 2 d. 0 e. 1

8. Solve 7 cos2 𝜃𝜃 + 3 sin2 𝜃𝜃 = 4.

a. 𝜃𝜃 = 2𝑎𝑎𝜋𝜋 ± 𝑛𝑛3

,𝑎𝑎 𝑛𝑛 ℤ b. 𝜃𝜃 = 𝑎𝑎𝜋𝜋 ± 𝑛𝑛

3, 𝑎𝑎 𝑛𝑛 ℤ

c. 𝜃𝜃 = 𝑎𝑎 𝑛𝑛4

± 𝑛𝑛2

,𝑎𝑎 𝑛𝑛 ℤ d. 𝜃𝜃 = 𝑎𝑎 𝑛𝑛

2± 𝑛𝑛

3,𝑎𝑎 𝑛𝑛 ℤ

e. 𝜃𝜃 = 𝑎𝑎 𝑛𝑛6

± 𝑛𝑛3

,𝑎𝑎 𝑛𝑛 ℤ

9. Solve sin 3𝛼𝛼 = 4 sin𝛼𝛼 sin(𝑥𝑥 + 𝛼𝛼) sin(𝑥𝑥 − 𝛼𝛼),𝛼𝛼 ≠ 𝑎𝑎𝜋𝜋,𝑎𝑎 𝑛𝑛 ℤ.

a. 𝑥𝑥 = 2𝑎𝑎𝜋𝜋 ± 𝑛𝑛3

,𝑎𝑎 𝑛𝑛 ℤ b. 𝑥𝑥 = 𝑎𝑎 𝑛𝑛

3± 𝑛𝑛


c. 𝑥𝑥 = 𝑎𝑎 𝑛𝑛2

± 𝑛𝑛3

,𝑎𝑎 𝑛𝑛 ℤ d. 𝑥𝑥 = 4 𝑎𝑎𝜋𝜋 ± 𝑛𝑛


e. 𝑥𝑥 = 𝑎𝑎𝜋𝜋 ± 𝑛𝑛3

, 𝑎𝑎 𝑛𝑛 ℤ

10. How many solutions satisfy the equation sin2 𝜃𝜃 − cos 𝜃𝜃 = 14 in [0, 2𝜋𝜋]?

a. 1 b. 4 c. 2 d. 5 e. 3

MATHEMATICS Focused Quiz 30: The Law of Sines and Cosines

1. Which of the following statements about a ∆𝐴𝐴𝐵𝐵𝐶𝐶 is false?

a. 𝑎𝑎sin𝐴𝐴

= 𝑏𝑏sin𝐴𝐴

= 𝑐𝑐sin𝐶𝐶

b. cos𝐵𝐵 = 𝑐𝑐2+𝑎𝑎2−𝑏𝑏2

2𝑎𝑎𝑐𝑐

c. cos𝐶𝐶 = 𝑐𝑐2+𝑎𝑎2−𝑐𝑐2

2𝑎𝑎𝑐𝑐

d. cos𝐴𝐴 = 𝑐𝑐2+𝑏𝑏2−𝑎𝑎2

2𝑏𝑏𝑐𝑐

e. sin(𝐵𝐵 + 𝐶𝐶) = sin𝐴𝐴

2. In a ∆𝐴𝐴𝐵𝐵𝐶𝐶, if 𝑎𝑎 = 2, 𝑏𝑏 = 3 and sin𝐴𝐴 = 23

, find ∠𝐵𝐵.

a. 90° b. 45° c. 30° d. 60° e. 125°

3. In a ∆𝐴𝐴𝐵𝐵𝐶𝐶, if 𝑎𝑎 = 3, 𝑏𝑏 = 5 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 7, find cos𝐵𝐵.

a. 3314

b. 1314

c. −12

d. 1114

e. 3340

4. Find the area of the triangle ∆𝐴𝐴𝐵𝐵𝐶𝐶 in which 𝑎𝑎 = 1, 𝑏𝑏 = 2 and ∠𝑐𝑐 = 60°.

a. 6 sq units b. √3

4 sq units

c. 2 sq units d. 1

2 sq units

e. √32

sq units


5. In a ∆𝐴𝐴𝐵𝐵𝐶𝐶, if cos𝐴𝐴 = sin𝐴𝐴2sin𝐶𝐶

, then:

a. 𝑐𝑐 = 2𝑎𝑎 b. 𝑐𝑐 = 𝑎𝑎

4

c. 𝑐𝑐 = 𝑎𝑎 d. 𝑐𝑐 = 𝑎𝑎

2

e. 3𝑐𝑐 = 𝑎𝑎

6. In a ∆𝐴𝐴𝐵𝐵𝐶𝐶, if 𝑏𝑏 = 20, 𝑐𝑐 = 21, sin𝐴𝐴 = 35, find a.

a. 13 b. 19 c. 7 d. 9 e. 21

7. If the sides of a ∆𝐴𝐴𝐵𝐵𝐶𝐶, if 𝑎𝑎 = 4, 𝑏𝑏 = 6 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 = 8, evaluate 4 cos𝐵𝐵 + 3 cos𝐶𝐶.

a. 7 b. 5 c. 4 d. 2 e. 3

8. If a triangle has angles with degrees in the ratio 1: 2: 3, find the ratio of the lengths of its sides.

a. 1: 2: 3 b. 1:√3: 2 c. 1: 3: 5 d. 1:√3: √5 e. 2: 3: 5

9. In ∆𝐴𝐴𝐵𝐵𝐶𝐶, 2∠𝐵𝐵 = ∠𝐴𝐴 + ∠𝐶𝐶 and 𝑏𝑏: 𝑐𝑐 = √3:√2. Find ∠𝐶𝐶.

a. 60° b. 90° c. 45° d. 125° e. 75°

10. In a ∆𝐴𝐴𝐵𝐵𝐶𝐶, cos𝐴𝐴𝑎𝑎

= cos𝐴𝐴𝑏𝑏

= cos𝐶𝐶𝑐𝑐

. Find angles A, B and C.

a. 𝐴𝐴 = 30°,𝐵𝐵 = 60°,𝐶𝐶 = 90° b. 𝐴𝐴 = 30°,𝐵𝐵 = 90°,𝐶𝐶 = 60° c. 𝐴𝐴 = 𝐵𝐵 = 𝐶𝐶 = 60° d. 𝐴𝐴 = 45°,𝐵𝐵 = 45°,𝐶𝐶 = 90° e. 𝐴𝐴 = 90°,𝐵𝐵 = 60°,𝐶𝐶 = 30°

MATHEMATICS Focused Quiz 31: Radians

1. Express an angle of 𝜋𝜋 radian in degrees.

a. 90 b. 180 c. 180

𝑛𝑛

d. 360 e. None of these

2. The length of an arc of a circle of radius ‘𝑦𝑦’ which subtends an angle of 1 radian at the centre of the circle is

a. 𝑦𝑦 b. 𝑟𝑟

2

c. 2𝑟𝑟𝑛𝑛

d. 𝑛𝑛

2

e. 2𝑦𝑦

3. Find the radian measure of an angle of 330°.

a. 11𝑛𝑛2

b. 𝑛𝑛

18

c. 𝑛𝑛6

d. 11𝑛𝑛6

e. 5𝑛𝑛9

4. Find the degree measure of an angle of 6 radian. Take 𝜋𝜋 = 227

.

a. 240°38′ b. 343°38′11′′ c. 43°18′8′′ d. 240°28′16′′ e. 343°38′

5. The angles of a triangle are in ratio 1: 2: 3. Find the measure of the smallest angle in radians.

a. 2𝑛𝑛3

b. 𝑛𝑛

4

c. 𝜋𝜋 d. 𝑛𝑛

2

e. 𝑛𝑛6


6. A wheel makes 180 revolutions in one minute. Through how many radians does it turn in one second?

a. 𝜋𝜋 b. 2𝜋𝜋 c. 6𝜋𝜋 d. 3𝜋𝜋 e. 4𝜋𝜋

7. If the radian measure of an angle 𝛼𝛼 is �−𝑛𝑛3� radian, then the degree measure of 𝛼𝛼 is

a. −60° b. 30° c. −120° d. 60° e. 150°

8. A clock’s large hand is 42 cm long. How far does its extremity move in 20 minutes?

a. 42𝑐𝑐𝑚𝑚 b. 68𝑐𝑐𝑚𝑚 c. 50𝑐𝑐𝑚𝑚 d. 33𝑐𝑐𝑚𝑚 e. 88𝑐𝑐𝑚𝑚

9. What is the range of tan(sin(4𝑥𝑥 + 4))?

a. [tan(2), tan(4)] b. [tan(−1), tan 1] c. [0, tan 1] d. [tan(−1), tan 2] e. [tan(−2), tan 1]

10. Find the radian measure of 5°37′30′′.

a. 𝑛𝑛15

b. 𝑛𝑛

4

c. 2𝑛𝑛3

d. 𝑛𝑛

32

e. 45𝑛𝑛8

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MATHEMATICS Focused Quiz 1: Basic Properties of Real …€¦ · 7. Convert 𝑥𝑥= 1.272727… to fractional form. a. 127 100. b. 14 11. c. ... Focused Quiz 3: ... MATHEMATICS