CTMUA 2021

Question 1

Sum of the first n terms of an arithmetic sequence

Answer 1

0.5n(2a + d[n - 1])

Question 2

Sum of the first n terms of a geometric sequence

Answer 2

a(r^n - 1)/(r - 1)

Question 3

Turning point of ax^2 + bx + c

Answer 3

x = -b/2a
y = c - (b^2/4a)

Question 4

log(a) + log(b)

log(a^b)

Answer 4

log(ab)

blog(a)

Question 5

Base change formula

Answer 5

loga(x) = logb(x)/logb(a)
loga(x) = ln(x)/ln(a) (Base is bottom)

Question 6

X is sufficient for Y

Answer 6

All X are Y

Being a horse is a sufficient condition for being a mammal, as all horses are mammals

Question 7

X is necessary for Y

Answer 7

All Y are X

Being male is a necessary condition for being a father, as all fathers are men

Question 8

Common mistakes in proofs

Answer 8

Squaring both sides introduces erroneous solutions
Dividing by zero
Squaring an inequality does not work
Taking some function of both sides of an inequality does not work
Flip inequality when multiplying by a negative number

Question 9

Contrapositive

Answer 9

If A, then B

If not A, then not B

Question 10

Converse

Answer 10

If A, then B

If B, then A

Question 11

Remainder theorem for polynomials

Answer 11

When f(x) is divided by (x-a), the remainder is f(a)

Question 12

What causes the trapezium rule to overestimate the area under the curve?

Answer 12

If f’(x) increasing, or f’‘(x) > 0

Question 13

What causes the trapezium rule to underestimate the area under the curve?

Answer 13

If f’(x) decreasing, or f’‘(x) < 0