G r a d e 11 P r e -C a l C u l u s M a t h e M at i C s … › k12 › dl › iso ›...

G r a d e 1 1 P r e - C a l C u l u s M a t h e M a t i C s ( 3 0 s )

Midterm Practice Exam

M i d t e r m P r a c t i c e E x a m 3 of 25

G r a d e 1 1 P r e - C a l C u l u s M a t h e M a t i C s

Midterm Practice Exam

Name: ___________________________________

Student Number: ___________________________

Attending q Non-Attending q

Phone Number: ____________________________

Address: _________________________________

__________________________________________

__________________________________________

Instructions

The midterm examination will be weighted as follows: Modules 1–4 100%The format of the examination will be as follows: Module 1: Sequences and Series 24 marks Module 2: Factoring and Rational Expressions 23 marks Module 3: Quadratic Functions 25 marks Module 4: Solving Rational and Quadratic Equations 28 marksTime allowed: 2.5 hours

Note: You are allowed to bring the following to the exam: pencils (2 or 3 of each), blank paper, a ruler, a scientific or graphing calculator, and your Midterm Exam Resource Sheet. Your Midterm Exam Resource Sheet must be handed in with the exam. You will receive your Midterm Exam Resource Sheet back from your tutor/marker with the next module work that is submitted for marking.

Show all calculations and formulas used. Include units where appropriate. Clearly state your final answer.

For Marker’s Use Only

Date: _______________________________

Final Mark: ________ /100 = ________ %

Comments:

G r a d e 1 1 P r e - C a l c u l u s M a t h e m a t i c s4 of 25


Name:

Answer all questions to the best of your ability. Show all your work.

Module 1: Sequences and Series (24 marks)

1. For each sequence below, indicate whether it is geometric, arithmetic, or neither. If it is geometric, state the value of r (the common ratio). If it is arithmetic, state the value of d (the common difference). (1 mark each, for a total of 3 marks)

a) 81 9 1

19

, , , , . . .

b) 1, 6, 9, 10, . . .

c) 1, 4, 7, 10, . . .

2. Explain the relationship between arithmetic sequences and linear functions. (2 marks)


3. Write the defining linear function of the following arithmetic sequence. (2 marks) 3, 8, 13, . . .

4. Which term of the arithmetic sequence -4, 2, 8, . . . is the number 170? (3 marks)


Name:

5. Find the sum of the first 125 terms defined by t1 = 4, tn = tn – 1 + 7. (3 marks)

6. Find the total distance travelled by a ball in coming to rest, if it is dropped from a height

of 12 m and it rebounds 23

of its previous height every time it hits the ground (4 marks)


7. Consider the geometric sequence –1, 2, –4, . . .a) Write the function for the geometric sequence. (2 marks)

b) Using this function, find t14 of the sequence. (1 mark)

c) Which term of the geometric sequence is the number –4096? (2 marks)


Name:

8. If possible, find the value of

64341

1

=

∞−

∑k

k

. (2 marks)


Module 2: Factoring and Rational Expressions (23 marks)

1. Factor the following expressions completely. a) x2y – 7xy + 10y (2 marks)

b) 4x2 + 12x + 9 (2 marks)

c) 36x2 – 64y2 (1 mark)

d) 25(x – 4)2 – 49(y – 5)2 (3 marks)


Name:

2. Without factoring, determine whether the binomial (x – 1) is a factor of the following polynomial. Explain. (2 marks) 2x2 – x – 1

3. Create an equivalent rational expression for the following rational expression. Explain how you know the rational expression you created is equivalent to the original rational expression. State the non-permissible values of your rational expression. Do not simplify your answer. (2 marks)

xx 2

2


4. Perform the indicated operation and simplify your answer. State the non-permissible values.

a)x

x xx

x x+

+ −+

− −

16 22

2

2 (4 marks)

b)x xx x

x xx x

2

2

2

226

122 2 40

+ −

− −×

+ −

− − (3 marks)


Name:

c)2 84

2 86

2

2

2

3 2 2x

xx xx x

xx x

−÷

− −

+×

− − (3 marks)

5. Create a rational expression with the following non-permissible values. Do not simplify your answer. (1 mark) –3 and –5


Module 3: Quadratic Functions (25 marks)

1. Given the graph below, answer the following questions.

y

x

2

2

2 2

a) What are the coordinates of the vertex? (1 mark)

b) What is the range? (1 mark)

c) What is the equation of the axis of symmetry? (1 mark)

d) State whether the graph has a maximum or a minimum value and what that value is. (1 mark)

e) Write a quadratic function in vertex form to represent this graph. (2 marks)


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2. Match each equation to its corresponding graph. Place (a), (b), (c), and (d) next to its corresponding graph below. (1 mark each, for a total of 4 marks)a) y = 2(x – 2)2 – 2

b) y x=− −( ) +

12

2 22

c) y x=− +( ) +

12

2 22

d) y = 2(x + 2)2 – 2

______

x

y

0

2

2

_____

x

y

0

2

2

______

x

y

0 2

2

______

x

y

0

2

2

3. For what value of k is the expression x2 – 11x + k a perfect square trinomial? (1 mark)


4. Consider the parabola y = 3x2 + 12x – 36.a) Write the function in vertex form by completing the square. (2 marks)

b) Find the coordinates of the vertex. (1 mark)

c) Find the y-intercept. (1 mark)

d) State the domain. (1 mark)

e) Write an equation for the axis of symmetry. (1 mark)


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f) Sketch the graph. (2 marks)

g) Find the x-intercepts from the graph. (1 mark)


5. A projectile is shot straight up from a height of 40 m with an initial velocity of 30 m/s. Its height in metres after t seconds is given by h(t) = 40 + 30t – 5t2. (1.5 marks each, for a total of 3 marks)a) After how many seconds does the projectile reach its maximum height?

b) Find the maximum height above the ground that the ball reaches.

6. Consider the following quadratic functions. Determine how many x-intercepts the corresponding graph has by considering the values of a and q. (1 mark each, for a total of 2 marks)a) y = –2(x + 3)2 – 15

b) y = 9(x + 2)2


Name:

Module 4: Solving Rational and Quadratic Equations (28 marks)

1. Find the roots of the following equations. Explain which method you used to solve each equation and why. a) x2 – 9x + 18 = 0 (2 marks)

b) x2 – 5x – 1 = 0 (3 marks)


2. Use the quadratic formula to find the roots of 2x2 + 6x – 1 = 0. (2 marks)

3. Solve x2 – x – 3 = 0 by completing the square. (3 marks)


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4. Solve (x – 7)2 = 36 by taking square roots. (2 marks)

5. Solve (x – 2)2 – 1 = 0 by graphing. (2 marks)


6. Solve for x. State the non-permissible values of the rational equation. (4 marks)

xx

xx

−+

− =+( )−

23

34 3

2


Name:

7. The discriminant of the quadratic equation 4x2 + 2x – k = 0 is 84. (1 mark each, for a total of 2 marks)a) Find the value of k.

b) State the nature of the roots without solving for the roots.


8. Tia can ride her bicycle 1 km/h faster than Denzel. They begin at the same spot and at the same time, travelling in opposite directions. After travelling for the same length of time, they stop and realize that Tia has travelled 21 km and Denzel has travelled 19 km. How fast was Denzel travelling? (4 marks)

Recall: Distance = Rate ´ Time, or Time =

DistanceRate


Name:

9. Andrea can mow a lawn by herself in 4 hours. Samantha can mow the same lawn in 3 hours. If the girls work together, how long will it take to mow this lawn? (4 marks)

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