Topics in Mathematics Trivia Quiz

Answer: 1/6

This can be solved without a calculator

Let x = 0.1666666... (equation 1)
10x = 1.6666666... (equation 2)
100x = 16.666666... (equation 3)

(equation 3) - (equation 2)
90x=15
x=15/90
x=1/6

Answer: -2

3y - 8 = 21y + 28
3y - 8 - 21y = 28
-18y - 8 = 28
-18y = 36
y = -2

Answer: 156

Using the formula S = (n-2) x 180 (S = interior angle sum, n = number of sides)
S = (15-2) x 180
S = 13 x 180
S = 2340
To find interior angle, use I = S/n (I = interior angle)
I = 2340/15
I = 156

Answer: 60, 90, 270, 300

2cos^2(x) - cosx = 0
cosx(2cosx - 1) = 0
cosx = 0 or cosx = 1/2
For cosx = 0
x = 90, 270
For cosx = 1/2
Basic angle = 60
Angles lies in Quadrants I and IV
x = 60, 300
Therefore: x = 60, 90, 270, 300

Answer: x = 3

If x = 3 was subbed into the equation, the denominator would then equal 0. This creates an undefined answer, therefore making x = 3 a vertical asymptote.

Answer: -1

If two lines are perpendicular, then they meet at right angles
For example: The two lines 3x+y+1=0 and x-3y-1=0 are perpendicular since the product of their gradients (-3 and 1/3) is equal to -1.

Answer: 30x^2+18x-7

Using the product rule
If y= f(x)g(x)
Then dy/dx = f'(x)g(x)+g'(x)f(x)
So if y = (2x+3)(5x^2-3x+1)
Then dy/dx
= 2(5x^2-3x+1) + (10x-3)(2x+3)
= 10x^2-6x+2+20x^2+30x-6x-9
= 30x^2+18x-7

Answer: a>0 and discriminant <0

If a quadratic is positive definite, then any x-value will give a positive solution. If a is greater than 0 and if the quadratic has no real solutions (proven if discriminant is less than 0), then the quadratic is positive definite.

Answer: y= -1

The directrix can be found using the general equation of a parabola
(x-h)^2=4a(y-k), with vertex (h,k), focus (h,a+k) and directrix y=k-a. The equation given shows that the vertex is (4,3) and that 4a=16
Solving 4a=16 gives a=4
Subbing this into the general form of the directrix gives
y=3-4
y=-1
Therefore, y=-1 is the equation of the directrix.

Answer: 3.169925001

This answer can be found using the change of base theorem. Log 9, base 2 can be changed into ln(9)/ln(2), which can then be solved on a calculator.

Answer: 12

This can be solved using the equation for the sum of an arithmetic series
S = (n/2)(2a+(n-1)d)
where n = number of terms,
S = sum of terms,
a = first term,
d = common difference between each tern
From the series given, a=2, d=9 and S=618, these can be subbed into the equation to give
618 = (n/2)(4+(n-1)9)
This can be re-arranged into a quadratic
1236 = n(4+9n-9)
1236 = n(9n-5)
1236 = 9n^2-5n
9n^2-5n-1236=0
Factorising and solving gives:
(9n+103)(n-12)=0
n = -103/9, n = 12
However in this case, n must be positive
Therefore: n only equals 12.

Answer: $4094.20

This can be found using the formula A = P(1+r)^n
Where P= Amount invested
r = rate of interest/time period
n = Number of years x time period
A = Final amount
From the information given
P = 2000, r= 0.01 and n= 72
Therefore A = 2000(1+0.01)^72
A=$4094.20

Answer: f'(x)=0 and f''(x)=0

A point of inflexion occurs when the second derivative of a function at a certain point is equal to 0. If the function is neither increasing or decreasing at this same point, then it is called a horizontal point of inflexion.

Answer: 28pi/3

The volume can be found using the formula
V= pi x integral of y^2dx from a to b
The curve x^2+y^2=9 is a circle, and can be re-arranged to give y^=9-x^2. This
can then be integrated from x=1 to x=3
This gives
= pi(9x-x^3/3)from 1 to 3
= pi((27-9)-(9-1/3))
=28pi/3

Answer: 5sin^4(x)cosx

Using the differentiation rules:
If y=sin f(x), then dy/dx = f'(x)cos f(x)
and if y= (f(x))^n, then dy/dy = n(f(x))^n-1 * f'(x)
So for y=sin^5(x)
y=(sinx)^5
dy/dx = 5(sinx)^4 * cosx
=5sin^4(x)cosx

Answer: (1/3)(e^6-1)

To find the area, the function y=e^3x needs to be integrated from x=0 to x=2
This gives:
A = (e^3x/3) from 0 to 2
A = e^6/3-e^0/3
A = (1/3)(e^6-1)

Answer: -1/4

To find the gradient of the tangent, we need to first find the derivative of the curve.
y=ln(x^3-5)
so dy/dx = 3x^2/(x^3-5)
sub in x=2 to find the gradient at this point
at x=2
dy/dx = 4
To find the gradient of the normal, we need to take the reciprocal of the gradient of the tangent.
dy/dx = -1/4
Therefore the gradient of the normal to the curve is -1/4

Answer: 101

To find this answer, we first need to find values for A and k in the given equation. Initially, there is 6000 bacteria.
Therefore, at t=0 and N=6000
6000=Ae^0
A=6000
After 8 hours, the number of bacteria has increased to 9000
Therefore, at t=8 and N=9000
9000=6000e^8k
e^8k=9000/6000
e^8k=3/2
8k=ln(3/2)
k=ln(3/2)/8
Therefore: k = 0.0507
We can now find when the total will equal 1000000 by solving the equation for t
Sub in N = 1000000
1000000=6000e^0.0507t
e^0.0507t=1000000/6000
e^0.0507t=500/3
0.0507t=ln(500/3)
t=ln(500/3)/0.0507
t=100.9072152
Therefore, there will be 1000000 bacteria after approximately 101 hours

Answer: 153

The velocity of a particle can also be known as the first derivative. To find displacement, we need to integrate the velocity with respect to t.
Therefore:
x = integral(3t^2+2t+1)dt
x= t^3+t^2+t+C
To find C, we use our already known information which is at t=0, x=-2
So
-2 = 0^3+0^2+0+C
C=-2
Therefore x=t^3+t^2+t-2
So to find displacement after five seconds, we sub t=5 into x
x=(5)^3+(5)^2+5-2
x=153cm

Answer: 2/893

We can work this answer out using the rule P(AB) = P(A) x P(B), where A and B are different events. In this case P(A) is the probability of winning first prize and P(B) is the probability of winning second prize.
P(A) = 5/95, P(B) = 4/94
Therefore P(AB) = 5/95 x 4/94
P(AB) = 2/893