engalg1 mock finals a.y. 2016-2017 multiple choice...

ENGALG1 MOCK FINALS A.Y. 2016-2017

MULTIPLE CHOICE (Write E if the answer is not in the choices.)

1. What is the value of c + 2d – e if c = 3, d = –1, and e = 1?

(A) – 1 (B) 0 (C) 1 (D) 2

2. What is the multiplicative inverse of the additive inverse of e?

(A) 1

𝑒 (B) −

1

𝑒 (C) – 𝑒 (D) 𝑒

3. Factor out 16𝑦2 − 9𝑥2 completely.

(A) (4𝑥 + 3𝑦)(4𝑥 − 3𝑦) (B) (4𝑦 + 3𝑥)(3𝑥 − 4𝑦)

(C) (4𝑦 − 3𝑥)2 (D) (3𝑥 + 4𝑦)(4𝑦 − 3𝑥)

4. The shorter base of a trapezoid measures (x + y), while its longer base is thrice that length. If the

trapezoid is (x – y) units high, what is its area?

(A) 4𝑥2 − 4𝑦2 (B) 3𝑥2 − 3𝑦2 (C) 2𝑥2 − 2𝑦2 (D) 𝑥2 − 𝑦2

5. Simplify: 3x – 2{4 + (2x – 5)•(–2) – [5 – (3 ÷ 6 × 8x + 1) + 70]}

(A) 3x – 14 (B) 3x – 7 (C) 3(x – 6) (D) 3(x – 7)

6. By commutative property, [(𝑎 + 𝑏𝑥) + 𝑐𝑏][(𝑐 + 𝑎𝑥) + 𝑎𝑏)] is equal to:

(A) [𝑐𝑏 + (𝑎 + 𝑏𝑥)][𝑎𝑏 + (𝑐 + 𝑎𝑥)] (B) [𝑏𝑎 + (𝑐 + 𝑥𝑎)][𝑏𝑐 + (𝑎 + 𝑥𝑏)]

(C) [𝑏𝑐 + (𝑏𝑥 + 𝑎)][𝑏𝑎 + (𝑎𝑥 + 𝑐)] (D) [𝑏𝑐 + (𝑥𝑏 + 𝑎)][𝑏𝑎 + (𝑥𝑎 + 𝑐)]

7. Which of the following is true?

(A) A polynomial’s degree gives the number of real and non-real solutions it can maximally have.

(B) The number of terms a polynomial has can be a basis of its degree.

(C) A polynomial can have extraneous roots provided its degree is fractional.

(D) More than one of the above is true.

8. Rationalize the numerator of √5− 3

√7+2√3 .

(A) (√5− 3)(√7+2√3)

19 (B)

(√5− 3)(2√3−√7)

5

(C) (3−√5)(2√3−√7)

5 (D)

−4

(√7+2√3)(√5+3)

9. How many distinct factors does (x8 – 2x7 – 64x2 + 128x) have?

(A) 3 (B) 4 (C) 5 (D) 6


10. Which of the following statements is correct?

(A) √𝑥2 + 𝑦2 = 𝑥 + 𝑦 if at least one of x and y is 0, or x = y.

(B) For any a and b, √𝑎2 + 2𝑎𝑏 + 𝑏2 is numerically equal to a + b.

(C) f(b) = ln(−𝑏) and f(b) = 𝑒−𝑏 are inverses of each other.

(D) For all values of x, 𝑥4(𝑥−2)2

𝑥2+6𝑥+9 is greater than or equal to zero.

11. Find the restriction of (6𝑥2 + 36𝑥 + 48)1/2

(1−5𝑥)2/3.

(A) (−∞, −4] ∪ [−2,5) ∪ (5, ∞) (B) [−4, −2] ∪ {5}

(C) (−∞, −4] ∪ [−2,1

5 ) ∪ (

1

5, ∞) (D) [−4, −2] ∪ {

1

5}

12. What is the result when 2𝑥4 + 12𝑥2𝑦 + 18𝑦2 − 9𝑥2 − 27𝑦 + 4 is divided by 2𝑥2 + 6𝑦 − 8?

(A) 𝑥2 +3

2𝑦 −

1

2 (B) 𝑥2 −

3

2𝑦 −

1

2 (C) 𝑥2 − 3𝑦 −

1

2 (D) 𝑥2 + 3𝑦 −

1

2

13. At least one of the following statements is true. Choose among the following:

(A) 𝑎𝑏𝑐

= 𝑎(𝑏𝑐) ≠ (𝑎𝑏)

𝑐

(B) If 1

𝑧=

1

𝑥+

1

𝑦 , x and y are both greater than z for all non-negative x, y and z.

(C) The maximum value of (𝑎 +9

𝑎) where a ≠ 0 is 6.

(D) all of the above

14. If g(x) = 4−𝑥

3𝑥+2 and h(x) =

1

2𝑥−1, then what is (𝑔 ∘ ℎ)(1) − (ℎ ∘ 𝑔)(−1)?

(A) 2

3 (B)

−2

3 (C)

38

55 (D)

58

5

15. If k is an even number, q is an odd number and z is a prime number, then how many among the

following are always odd?

I. kq + z II. kz + q III. qz + k IV. kq + z

V. qk + z VI. zq + k VII. qz + k VIII. (z – k)q

(A) 2 (B) 3 (C) 4 (D) 5

16. If [(d + 2e) (d – 2f) – 2d (e – f)] 3/4 is a positive integer, then what can we be certain of the roots of the

polynomial ex2 + dx + f = 0, given d, e and f are also integers?


(A) One root is positive, one root is negative. (B) The roots of the equation are rational.

(C) The roots of the equation are real and repetitive. (D) All abovementioned choices are possible.

17. Which of the following is false about the graph of f(x) = 𝑥3+2𝑥2−4𝑥−8

2𝑥2−7𝑥+6 ?

(A) The range of its inverse includes y = 1

2 . (B) The graph is below the x – axis only if x <

3

2 .

(C) It has a hole at (2, –4). (D) It has vertical asymptote but no horizontal asymptote.

18. What is the inverse of 𝑦 = 2log(𝑥+3) 𝑒 ?

(A) 𝑦−1 = 𝑒1

log2 𝑥 − 3 (B) 𝑦−1 = 2ln 𝑥 − 3

(C) 𝑦−1 = 𝑒1

log𝑥 2 − 3 (D) 𝑦−1 = 2log𝑥 𝑒 − 3

19. Simplify:(𝑥4+9𝑥+81)(27−𝑥3)

−2(𝑥4−13𝑥2+36)

(−6−5𝑥−𝑥2)(6𝑥−8−𝑥2)(𝑥2−7𝑥+12)−1

(A) 1 – 6𝑥

𝑥2+3𝑥+9 (B) 1 +

6𝑥

𝑥2+3𝑥+9 (C)

6𝑥

𝑥2+3𝑥+9 – 1 (D) 1 +

6𝑥

𝑥2−3𝑥+9

20. What is the LCD of 4𝑥+4

𝑥3+3𝑥2+3𝑥+1,

6

𝑥+2−𝑥2 𝑎𝑛𝑑

𝑥2−7𝑥+10

𝑥3−3𝑥2+4 ?

(A) (x + 1)3(x – 2)2 (B) (x + 1)2(x – 2)2 (C) (x + 1)3(x – 2) (D) (x + 1)2(x – 2)

21. Which is not a property of the inverse, f –1(x), of f(x) = 𝑥+1

𝑥+5, 𝑥 ≠ 3 ?

(A) Its graph is increasing from x = 1 to positive infinity.

(B) Its range (possible values of y) is (– ∞, – 5) ∪ (−5, −3) ∪ (−3, ∞).

(C) For all x < 1

5 or x > 1, f –1(x) is negative.

(D) The vertical asymptote of f –1(x) is x = 1.

22. In the simplified expansion of (2𝑥2 −3

4𝑥)

3

, only one term is a constant. What is its value?

(A) 9

4 (B)

9

8 (C)

27

4 (D)

27

8

23. If Q, R, S and T are different numbers such that p(Q) = p(R) = p(S) = p(T) = 0, where:

𝑝(𝑥) = 6𝑥4 − 17𝑥3 + 32𝑥2 + 9𝑥 − 6

Also note that the expression is factorable. Compute for Q × R × S × T.

(A) – 1 (B) 1 (C) – 6 (D) 6


24. Which of the following is NOT true?

(A) All functions are relations.

(B) All equations in the form y = 𝑏𝑒−𝑥 + 𝑐 have range y > c.

(C) To rationalize 5𝑥+𝑦

√𝑥−3√𝑦 , multiply √𝑥 + 3√𝑦 to it.

(D) The expression log𝑎 𝑏 has a unique value if a > 0, or a ≠ 1, and b > 0.

25. Which two values of x is the solution of the equation:

27(𝑥+8)(𝑥−1) (1

243)

6

=81(𝑥

2+5𝑥−24)

(1

95)

4 ?

(A) 1 and – 2 (B) – 1 and – 2 (C) 1 and 2 (D) – 1 and 2

26. Anton can finish a job alone 4 hours less than twice Sean does alone. If they work together and finish

the job in 7.5 hours, how many hours can Anton finish the job alone later than Sean does alone?

(A) 4 (B) 8 (C) 12 (D) 20

27. For what value(s) of x will satisfy the following equation:

3√1 − 𝑥 + √18(1 − 𝑥) + 2 = 3√5𝑥 + 5 − √3 − 27𝑥

(A) −1

9 𝑜𝑛𝑙𝑦 (B)

1

9 𝑜𝑛𝑙𝑦 (C)

−1

9 𝑎𝑛𝑑 0 (D)

1

9 𝑎𝑛𝑑

−1

9

28. What is the value of x if log5 6 = 2−𝑥

4𝑥+1 ?

(A) log(150)

4 log 6+log 5 (B)

log(150)

4 log 6−log 5 (C)

log(25

6)

4 log 6−log 5 (D)

log(25

6)

4 log 6+log 5

29. If p(x) = x8 + 3x4 – 4, then:

(A) p(11), p(12), p(13) and p(14) are all divisible by 25. (B) p(11), p(12) and p(13) are divisible by 61.

(C) p(11), p(12), p(13) and p(14) are all divisible by 29. (D) p(12), p(13) and p(14) are divisible by 51.

30. Find the value of 𝑎2 + 𝑏2 + 𝑐2 − 𝑎𝑏 − 𝑎𝑐 − 𝑏𝑐 if a = 990, b = 987, c = 992. Use your knowledge in

factoring. (Hint: it is not necessary to factor the expression completely.)

(A) 0 (B) – 9 (C) 9 (D) 19

31. Jaime and Jamie start at the same point in an oval track. Jaime runs counterclockwise at a speed of 32

m/s while Jamie runs clockwise at a speed of 5 m/s more than half of Jaime’s. After exactly an hour and 7

minutes, Jaime and Jamie meet again at some other point in the track. How long is a lap?


(A) 3.551 km (B) 3.591 km (C) 3.951 km (D) 2.881 km

32. 5 people can finish 5 pizzas in 5 minutes. Which of the following can be deduced from the statement?

(A) 10 people can finish 5 pizzas in 10 minutes. (B) 10 people can finish 10 pizzas in 10 minutes.

(C) The rate a person eating is 1 pizza/minute. (D) In 5 minutes, 2 pizzas can be finished by 2 people.

33. Ali plans to buy a blouse and a dress. She was not sure which one was discounted by 35%. If the

blouse was the discounted item, she would pay a total of 74 pesos. If the dress was the discounted item,

she would pay only 58 pesos. She finds out both were discounted, so how much should she pay in total?

(A) 39 pesos (B) 42 pesos (C) 52 pesos (D) 56 pesos

34. Find all values of u that satisfies the inequality:

|𝑢2 + 2𝑢 − 2| < 𝑢 − 2

(A) (– ∞, – 4) ∪ (1, ∞) (B) (–1, 0) (C) (– ∞, – 4) ∪ (−1,0) ∪ (1, ∞) (D) ∅

35. Solve the following simultaneous equations.

{ 2𝑥 + 3𝑦 − 3𝑧 = −17−𝑥 + 5𝑦 − 𝑧 = −14−2𝑥 − 4𝑦 + 𝑧 = 17

Calculate for: (x – 3)2 + (y + 4)2 + (z – 2)2.

(A) 25 (B) 26 (C) 27 (D) 29

PROBLEM SOLVING

(1) A family of four, two children and both parents, went to a grand cafe where the entrance fee per

person is age-dependent. It is posted that 90 – (3/2) x was the entrance fee to pay, where x is the age of the

person. Altogether they paid 117 pesos for the entrance fee.

The mother said to the older child, who was half her age, "Nak, exactly an integer number of years ago –

the day before your sister was born – Dad, you and I went here and paid the same total entrance fee." The

older sibling then revealed, "I just want you to notice too that, in two years, Dad will be twice my age."

What are their ages?

(2) Ralph likes the taste of coke and sprite together. He poured a certain amount of dilute coke and a

certain amount of dilute sprite to fill a 500-mL container.

He wanted to know the concentration of coke and sprite in the mixture, so he prepared 250-mL of it to

add to 50-mL pure coke and the other 250-mL of it to add to 50-mL pure sprite. Before pouring the first

portion of the mixture to the pure coke, he spilled 30-mL of the 250-mL portion, and then he also spilled

10-mL of the second 250-mL portion for the pure sprite. He now had two new mixtures. After a test, he

found out that the mixture with pure sprite has 18 more mL of coke than mL of sprite in the other mixture,

and it also has 11.6 more mL of water than mL of water in the other.


If the original dilute sprite was 30% sprite, what was the % coke in the original dilute coke?

(3) Two 120 m long trains enter a tunnel in the same direction. At the initial opening of the tunnel, the

front of train A is in line with the rear of train B. After 2.5 seconds, the front of train B is in line with the

rear of train A at the final opening of the tunnel. The faster train has thrice the speed of the other.

(a) Which is the faster train?

(b) What is the length of the tunnel?

(c) After how many seconds after the faster train has completely left the tunnel does the slower train

completely leave the tunnel?

(4) 1280 workers are assigned to build a mansion. After every 80 minutes, 1/4 of the present workers

leave the construction. 6 hours after the start of construction, the task is finished. How many hours earlier

should the task have been finished if nobody left the construction from the start?

(5) Find all possible values of x that will satisfy the equation:

log2(8𝑥−3) + 1 − log4

3

√9

16= {log𝑥−1[1 + 𝑥(𝑥 − 2)]}

2 − log𝑥 4

(6) Give the real solutions for the following systems of equation:

{𝑥2 − 4𝑥 + 𝑦2 − 2𝑦 = 5

−4𝑥 + (𝑦 + 1)2 − 13 = 0

(7) Solve the inequality:

|𝑥2 − 𝑥 − 1| > 3 − 2|𝑥 − 1|

***

Prepared by:

Sephi Liclican

Cary Chan

Jerome Garces

ANSWER KEY

1. B

2. B

3. D

4. C

5. C

6. A, B, C, and D

7. A

8. D

9. C

10. E, none of the above

11. E, (−4, −2) ∪ {1

5}

12. D

13. A

14. C

15. A

16. B

17. C

18. A

19. A

20. D

21. C

22. D

23. A

24. C

25. D

26. B

27. B

28. D

29. A

30. D

31. A

32. D

33. C

34. D

35. C

(1) A family of four, two children and both parents, went to a grand cafe where the entrance fee per

person is age-dependent. It is posted that 90 – (3/2) x was the entrance fee to pay, where x is the age of the

person. Altogether they paid 117 pesos for the entrance fee.

The mother said to the older child, who was half her age, "Nak, exactly an integer number of years ago –

the day before your sister was born – Dad, you and I went here and paid the same total entrance fee." The

older sibling then revealed, "I just want you to notice too that, in two years, Dad will be twice my age."

What are their ages?

Solution:

Let the ages of the younger sibling, older sibling, mother and father be A, B, C and D, respectively.

Generate the following equations from the clues:

Altogether they paid 117 pesos for the entrance fee:

[90 – (3/2) A] + [90 – (3/2) B] + [90 – (3/2) C] + [90 – (3/2) D] = 117

360 – (3/2) (A + B + C + D) = 117

A + B + C + D = 162 …. {1}

The mother said to the older child, who was half her age:

B = (1/2) C or C = 2B …. {2}

ANSWER KEY

Exactly an integer number of years ago – the day before your sister was born – Dad, you and I went here

and paid the same total entrance fee:

[90 – (3/2)(B – A)] + [90 – (3/2)(C – A)] + [90 – (3/2)(D – A)] = 117

270 – (3/2) (B + C + D – 3A) = 117

B + C + D – 3A = 102 …. {3}

[Older sibling] In two years, Dad will be twice my age:

2 (B + 2) = D + 2 2B + 4 = D + 2

2B + 2 = D …. {4}

Use equations {1} and {3} to solve for A:

A + B + C + D = 162

– B + C + D – 3A = 102

4A = 60; A = 15

Use A = 15, equations {2} and {4} on equation {1}:

15 + B + (2B) + (2B + 2) = 162

5B = 145; B = 29; C = 2(29) = 58; D = 2(29) + 2 = 60

Answer: younger sibling – 15, older sibling – 29,

mother – 58, father – 60

(2) Ralph likes the taste of coke and sprite together. He poured a certain amount of dilute coke and a

certain amount of dilute sprite to fill a 500-mL container.

He wanted to know the concentration of coke and sprite in the mixture, so he prepared 250-mL of it to

add to 50-mL pure coke and the other 250-mL of it to add to 50-mL pure sprite. Before pouring the first

portion of the mixture to the pure coke, he spilled 30-mL of the 250-mL portion, and then he also spilled

10-mL of the second 250-mL portion for the pure sprite. After a test, he found out that the mixture with

pure sprite has 18 more mL of coke than mL of sprite in the other mixture, and it also has 11.6 more mL

of water than mL of water in the other.

If the original dilute sprite was 30% sprite, what was the % coke in the original dilute coke?

Solution:

Note that the concentrations of both 250-mL potions are the same, as well as the amount (in mL) of coke

and sprite in both. Let X and Y be the amount of coke and sprite, respectively, in each 250-mL portion.

New mixture with first 250-mL portion (spilled 30-mL; added to 50-mL of pure coke):

total volume: 250 mL portion – 30 mL spilled + 50 mL coke = 270 mL

Amount of coke: 𝑋 (250−30

250) + 50; Amount of sprite: 𝑌 (

250−30

250)

New mixture with second 250-mL portion (spilled 10-mL; added to 50-mL of pure sprite):

ANSWER KEY

total volume: 250 mL portion – 10 mL spilled + 50 mL sprite = 290 mL

Amount of coke: 𝑋 (250−10

250); Amount of sprite: 𝑌 (

250−10

250) + 50 total volume: 290 mL

The mixture with pure sprite has 18 more mL of coke than mL of sprite in the other mixture:

𝑋 (250−10

250) = 𝑌 (

250−30

250) + 18; 𝑋 (

24

25) = 𝑌 (

22

25) + 18 …. {1}

The mixture with pure sprite has 11.6 more mL of water than mL of water in the other:

Since the sprite-mixture has 11.6 more mL of water than mL of water in the other, while it also has 18

more mL of coke than mL of sprite in the other. Additionally, its total volume is 20 mL (290 mL – 270

mL) more than the other. Therefore, its net difference of mL of sprite to the mL of coke in the other is:

11.6 mL + 18 mL – 20 mL = 9.6 mL (more coke in the other than in the sprite–mixture)

𝑋 (250−30

250) + 50 = 𝑌 (

250−10

250) + 50 + 9.6; 𝑋 (

22

25) = 𝑌 (

24

25) + 9.6 …. {2}

Adding {1} and {2}:

𝑋 (46

25) = 𝑌 (

46

25) + 27.6; 𝑋 − 𝑌 = 15

Use 𝑋 − 𝑌 = 15 on either {1} or {2} we get 𝑋 = 60 𝑎𝑛𝑑 𝑌 = 45.

In the original mixture, the amounts of coke and sprite are 2𝑋 and 2𝑌, respectively, or 120 mL of coke

and 90 mL of sprite. Because the amount of sprite is 30% of the original dilute sprite solution:

dilute sprite solution = 90

30% = 300 mL

The total volume of the original solution was 500 mL, leaving the dilute coke solution to be 200 mL (500

mL – 300 mL), where there is 120 mL of coke. Therefore the % coke is:

% coke = 120

200 𝑥 100% = 60% coke

Answer: 60% coke

(3) Two 120 m long trains enter a tunnel in the same direction. At the initial opening of the tunnel, the

front of train A is in line with the rear of train B. After 2.5 seconds, the front of train B is in line with the

rear of train A at the final opening of the tunnel. The faster train has thrice the speed of the other.

(a) Which is the faster train?

(b) What is the length of the tunnel?

(c) After how many seconds after the faster train has completely left the tunnel does the slower train

completely leave the tunnel?

Solution:

Illustration:

ANSWER KEY

Let the distance travelled by A be dA and distance travelled by B be dB in 2.5 seconds, so:

dA = dB + 240

Using v = d/t, let the speed of A be vA and speed of B be vB. With t = 2.5 seconds,

vA (2.5) = vB (2.5) + 240

But vA is thrice vB so: 3vB (2.5) = vB (2.5) + 240; vB = 48 m/s; vA = 3(48) = 144 m/s

(a) The faster train is train A.

(b) The length of the tunnel is x + 120. But x = dB = vB (2.5), and vB = 48 m/s, so the tunnel is 240 m.

(c) The distance to be travelled by train B is x which is 120 m. So, the time needed is:

120 𝑚

𝑣𝐵=

120 𝑚

48 m/s= 5 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

Answer: (a) Train A, (b) 240 m, (c) 5 secs

(4) 1280 workers are assigned to build a mansion. After every 80 minutes, 1/4 of the present workers

leave the construction. 6 hours after the start of construction, the task is finished. How many hours earlier

should the task have been finished if nobody left the construction from the start?

Solution:

Let x be the rate a worker helps in the construction. Using work = (rate) (time), the total work (W) can be

solved. Time will be expressed in hours, so 80 minutes converts to 4/3 hours. In 6 hours, there will be 4

80-min intervals and a last 40-min interval, with only ¾ of the workers left after each 80-min interval:

𝑊 = [1280𝑥] (4

3) + [(

3

4) 1280𝑥] (

4

3) + [(

3

4)

2

1280𝑥] (4

3) + [(

3

4)

3

1280𝑥] (4

3) + [(

3

4)

4

1280𝑥] (2

3)

𝑊 = 1280𝑥 [4

3+ 1 +

3

4+

9

16+

27

128]

𝑊 =14810

3𝑥

If nobody left from the start:

𝑊 = [1280𝑥](𝑡) 14810

3𝑥 = [1280𝑥](𝑡)

ANSWER KEY

𝑡 = 1481

384≈ 3.857 ℎ𝑟𝑠

Hours earlier:

6 −1481

384=

823

384≈ 2.143 ℎ𝑟𝑠 𝑒𝑎𝑟𝑙𝑖𝑒𝑟

Answer: 823/384 hours or 2.143 hours earlier

(5) Find all possible values of x that will satisfy the equation:

log2(8𝑥−3) + 1 − log4

3

√9

16= {log𝑥−1[1 + 𝑥(𝑥 − 2)]}

2 − log𝑥 4

Solution:

log2(8) + log2(𝑥−3) + 1 − log4

3

3

4= [log𝑥−1(𝑥

2 − 2𝑥 + 1)]2 − log𝑥 22

log2(23) − 3log2(𝑥) + 1 − log4

3(

4

3)

−1

= [log𝑥−1(𝑥 − 1)2]2 − 2 log𝑥 2

3 − 3log2(𝑥) + 1 − (−1) = [2]2 − 2 log𝑥 2

5 − 3log2(𝑥) = 4 − 2 log𝑥 2

2 log𝑥 2 − 3log2(𝑥) + 1 = 0

Note that log𝑥 2 =log 2

log 𝑥 and log2 𝑥 =

log 𝑥

log 2, which are reciprocals of each other, so we let one be A, and

the other as 1/A. Use log𝑥 2 as A.

2(𝐴) − 3 (1

𝐴) + 1 = 0

2(𝐴2) − 3 + 𝐴 = 0

(2𝐴 + 3)(𝐴 − 1) = 0

𝐴 = −3

2; 𝐴 = 1

log𝑥 2 = −3

2; log𝑥 2 = 1

2 = 𝑥−32; 2 = 𝑥1

𝑥 = 2−23; 𝑥 = 2

ANSWER KEY

Testing the values back to the equation, they are both restricted because the base (𝑥 − 1) should be

greater than 0, or not equal to 1. With x as either 2−2

3 or 2, the equation is not satisfied.

Answer: no value of x

(6) Give the real solutions for the following systems of equation:

{𝑥2 − 4𝑥 + 𝑦2 − 2𝑦 = 5

−4𝑥 + (𝑦 + 1)2 − 13 = 0

Solution:

Subtracting the second equation from the first equation, we can obtain that:

𝑥2 − 4𝑥 + 𝑦2 − 2𝑦 = 5

– −4𝑥 + (𝑦 + 1)2 − 13 = 0

𝑥2 − 4𝑦 = −7

4𝑦 = 𝑥2 + 7

𝑦 =1

4𝑥2 +

7

4

Substitute the resulting equation to the first equation:

𝑥2 − 4𝑥 + 𝑦2 − 2𝑦 = 5

𝑥2 − 4𝑥 + (1

4𝑥2 +

7

4)

2

− 2 (1

4𝑥2 +

7

4) = 5

𝑥2 − 4𝑥 + (1

16𝑥4 +

7

8𝑥2 +

49

16) + (−

1

2𝑥2 −

7

2) = 5

1

16𝑥4 +

11

8𝑥2 − 4𝑥 −

87

16= 0

𝑥4 + 22𝑥2 − 64𝑥 − 87 = 0

By synthetic division, we get that the values of x are as follows:

(𝑥2 + 2𝑥 + 29)(𝑥 + 1)(𝑥 − 3) = 0

𝑥 = −1 𝑎𝑛𝑑 3 𝑜𝑛𝑙𝑦

(no real roots for 𝑥2 + 2𝑥 + 29)

Using 𝑦 =1

4𝑥2 +

7

4 , the values of y are, respectively:

𝑦 = 2 𝑎𝑛𝑑 4

Answer: (−𝟏, 2), (3, 4)

(7) Solve the inequality:

|𝑥2 − 𝑥 − 1| > 3 − 2|𝑥 − 1|

Answer: x < 0 or x > 2

(– ∞, 𝟎) ∪ (𝟐, ∞)

Solution (picture):

ANSWER KEY

engalg1 mock finals a.y. 2016-2017 multiple choice...

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