Free Printable Summative in Pre Calculus

1.

According to the principle of mathematical induction, to prove a statement that is asserted about every natural number n, there are two things to prove. What is the first?

a)

The statement is true for n = 1.

b)

The statement is true for n = k.

c)

The statement is true for n = k+1.

2.

The second part (If the statement is true for n = k, then it will be true for its successor, k + 1) contains the induction assumption. What is it?

a)

If the statement is true for n = k, then it will be true for its successor, k + 1.

b)

The statement is true for n = k.

c)

The statement is true for n = k+1.

3.

The sum of the first n odd numbers is equal to the n^th square.
1 + 3 + 5 + 7 + . . . + (2n − 1) = n²

To prove this by mathematical induction, what will be the induction assumption?

a)

The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

b)

The statement is true for n = 1:
2x1 − 1 = 1²

c)

The statement is true for n = k + 1:
1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

4.

1 + 3 + 5 + 7 + . . . + (2n − 1) = n²
On the basis of this assumption,
[The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²]
What must we show?

a)

The statement is true for n = 1:
2x1 − 1 = 1²

b)

The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k²

c)

The statement is true for n = k + 1:
1 + 3 + 5 + 7 + . . . + (2k − 1) + (2k + 1) = (k + 1)²

5.

Let S(n) = 2n − 1. Evaluate:
a) S(k)
b) S(k + 1)

a)

a) S(k) = 2k − 1
b) S(k + 1) = 2n + 1

b)

a) S(k) = 2k + 1
b) S(k + 1) = 2k + 1

c)

a) S(k) = 2k − 1
b) S(k + 1) = 2k + 1

6.

When using mathematical induction to prove : $\sum_{i=1}^ni^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$ . In step #2, after you have made your assumption, what are you trying to prove? (What is your goal?)

a)

$\sum_{i=1}^{k+1}i^2=\frac{\left(k\right)\left(k+1\right)\left(2k+1\right)}{6}+\left(k+1\right)^2$

b)

$S_{k+1}=\frac{k\left(k+1\right)\left(2k+1\right)}{6}$

c)

$\sum_{i=1}^{k+1}i^2=\frac{\left(k+1\right)\left(k+2\right)\left(2k+3\right)}{6}$

d)

$S_{k+1}=\left(k+1\right)^2$

7.

Expand the expression using the Binomial Theorem.

(2x+5)⁴

a)

2x⁴ + 40x³+ 300x² + 1000x +625

b)

16x⁴+ 160x³ +600x² +1000x + 625

c)

16x⁴ + 1000x³ + 600x²+ 160x +625

8.

Expand the expression using the Binomial Theorem.
(2x+5)⁴

a)

2x⁴ + 40x³+ 300x² + 1000x +625

b)

16x⁴+ 160x³ +600x² +1000x + 625

c)

16x⁴ + 1000x³ + 600x²+ 160x +625

9.

What is row 7 of Pascal's Triangle?

a)

1, 5, 10, 5, 1

b)

1, 5, 10, 10, 5, 1

c)

1, 7, 21, 35, 35, 21, 7, 1

d)

1, 7, 21, 35, 21, 7, 1

10.

Choose the right Pascal's triangle

a)

b)

c)

d)

11.

a)

A

b)

B

c)

C

d)

D

12.

a)

b)

c)

d)

13.

a)

$-336x^2y^4$

b)

$3360x^6y^4$

c)

$-3360x^4y^6$

d)

$3360x^4y^6$

14.

a)

$252$

b)

$210$

c)

$45$

d)

$120$

15.

$Fourth\ of\ the\ \exp ansion\ \left(\frac{x}{2}+y\right)^7\ is$

a)

$\frac{25x^3y^4}{16}$

b)

$\frac{35x^4y^3}{16}$

c)

$\frac{70x^4y^3}{16}$

d)

$\frac{35x^4y^3}{32}$

16.

What is the reference angle for 125°?

a)

235°

b)

125°

c)

75°

d)

55°

17.

What is the reference angle for 275°?

a)

85°

b)

95°

c)

-85°

d)

-95°

18.

What is 5π/6 radians in degrees?

a)

180

b)

150

c)

360

d)

120

19.

What is 7π/4 radians in degrees?

a)

300

b)

135

c)

315

d)

45

20.

Tangent is...

a)

a. Opposite/hypotenuse

b)

b. sine/cosine

c)

c. neither a or b

d)

d. both a and b

21.

How many radians in a full circle?

a)

π/2

b)

π

c)

3π/2

d)

2π

22.

Angles are measured in which direction from standard position?

a)

Clockwise

b)

Counterclockwise

23.

The coordinates of the points on the unit circle are written in which order?

a)

$\left(\cos\theta,\ \sin\theta\right)$

b)

$\left(\sin\theta,\ \cos\theta\right)$

24.

The x-coordinate on the unit circle is equivalent to which value

a)

$\cos\theta$

b)

$\sin\theta$

25.

The y-coordinate on the unit circle is equivalent to which value

a)

$\cos\theta$

b)

$\sin\theta$

Summative in Pre Calculus

Total questions: 25

Worksheet time: 2hrs 5mins

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