Extension 1 Mathematics Trivia Quiz

Answer: y = 1

To find the limit, every term of the function must be divided by the highest power of x in the function.
For example: For the function f(x) = x^2/(x^2-9)
Dividing by the highest power of x gives,
= (x^2/x^2)/((x^2/x^2)-(9/x^2))
= 1/(1-0)
= 1
Therefore the horizontal asymptote of the function is y=1.

Answer: Angle DAC

In a cyclic quadrilateral, both pairs of opposite angles are supplementary. In this case, angle DAC is supplementary to angle BCD. Since point C lies on BE, this means that angle BCD is also supplementary to exterior angles DCE. Therefore this makes angle DCE equal to angle DAC.

Answer: 8sqrt33/49

If sinx=4/7, then the opposite side to x is equal to 4 and the hypotenuse is equal to 7. Using pythagoras theorem, the adjacent side is equal to sqrt33.
Therefore, cosx=sqrt33/7
Using the rule sin2x = 2sinxcosx
sin2x = 2 * 4/7 * sqrt33/7
sin2x = 8sqrt33/49

Answer: 37 degrees, 52 minutes

To find the angle, we need to first find the the gradients of both lines by rearranging them in the form y=mx+b, where m=gradient.
So for equation 1, 3x-2y+1=0
2y=3x+1
y=(3/2)x+1/2
Therefore m1 = 3/2
For equation 2, x-3y=0
3y=x
y=(1/3)x
Therefore m2 = 1/3
Now using the rule Tanx = abs((m1-m2)/(1+(m1)(m2))) abs = absolute value
So, Tanx = abs(((3/2)-(1/3))/(1+(3/2)(1/3)))
Tanx = abs(7/9)
Therefore x= 37.87498365
Which in degrees and minutes is, 37 degrees and 52 minutes

Answer: x=3/7, y=1

To divide an interval AB internally in a ratio at a certain point, we use the formulas:
x = (mx2+nx1)/(m+n), y = (my2+ny1)/(m+n)
where co-ordinates of points A and B are (x1,y1) and (x2,y2) respectively
and where m:n is the given ratio.
So for this example
x = (3(-7)+4(6))/(3+4)
x = (-21+24)/7
x = 3/7
y = (3(5)+4(-2))/(3+4)
y = (15-8)/7
y = 7/7
y = 1
Therefore, at point P, x = 3/7 and y = 1

Answer: x+py=2ap+ap^3

To find the equation of the normal, we first need to differentiate the given parabola.
So x^2=4ay
y=x^2/4a
dy/dx = 2x/4a
dy/dx = x/2a
at x = 2ap
m = 2ap/2a
m = p
Since we need to find the equation of the normal, we take the reciprocal of the gradient
Therefore, m = -1/p
Now with the general form of the equation of a line, we can find the equation of the normal:
y-y1=m(x-x1)
y-ap^2=(-1/p)(x-2ap)
py-ap^3=-1(x-2ap)
py-ap^3=-x+2ap
x+py=2ap+ap^3
Therefore, the equation of the normal is x+py=2ap+ap^3

Answer: Crosses the axis with a change in concavity

In a polynomial function, a zero from a linear factor will cross the axis and continue in the same direction. A zero from a square factor will touch the axis. In a function, a curve cannot continue in the opposite direction if it crosses the axis.

Answer: 4.25

The formula for Newton's method is a=x-f(x)/f'(x).
First we find the value of f(x) at x=4.5
So, f(4.5) = (4.5)^2-4(4.5)-1
f(4.5)=1.25
Now we need to differentiate the function and find the value of the derivative at x=4.5
So f'(x) = 2x-4
f'(4.5) = 2(4.5)-4
f'(4.5) = 5
Now we can find our approximation
x = 4.5 - f(4.5)/f'(4.5)
x = 4.5 - 1.25/5
x = 4.25

Answer: 2sqrt3

To solve this question, we need to first differentiate our given substitution, since we won't be able to integrate in terms of x.
So, u=x^2-1
du/dx=2x
Therefore, du=2xdx
We need to now find our new pair of limits
At x=1, u=0
At x=2, u=3
We can now change our given integral so it's in terms of u.
So 2xsqrt(x^2-1)dx from x=1 to x=2 can be changed into
Sqrt(u)du from u=0 to u=3
We can now find the definite integral of the function.
So integrating gives,
2sqrt(u^3)/3 from u=0 to u=3
2sqrt(27)/3 - 0
6sqrt(3)/3
=2sqrt(3)

Answer: (1/8)x-(1/32)sin4x+C

To integrate this function, it needs to be changed into something simpler.
Using the rule sin2x=2sinxcosx, sin^2(2x) = 4sin^2(x)cos^2(x)
Therefore sin^2(x)cos^2(x)=(1/4)sin^2(2x)
This can be made even simpler:
(1/4)sin^2(2x)=(1/4)*integral of (1/2-(1/2)cos(4x))
Integrating this gives
(1/4)((1/2)x-(1/8)sin4x)+C
=(1/8)x-(1/32)sin4x+C

Answer: y=2+sqrt(x+4)

To find an inverse function, we first swap the x and y variables in the equation.
So, let y=f(x)
y=x^2-4x
x=y^2-4y
By completing the square
x=y^2-4y+4-4
x+4=(y-2)^2
sqrt(x+4)=y-2
y=2+sqrt(x+4)
Therefore, the inverse function of f(x) is 2+sqrt(x+4)

Answer: 9pi^2/8

To find the volume about the x-axis, we use the formula V=pi*integral of y^2dx.
Since y=3/(sqrt(x^2+4))
y^2=9/(x^2+4)
Therefore, V=pi*integral of 9/(x^2+4)dx from x=0 to x=2
V=9pi/2 * integral of 2/(x^2+4) from x=0 to x=2
V=9pi/2 * (arctan(x/2)) from x=0 to x=2
V=9pi/2 * (arctan(1)-arctan(0))
V=9pi/2 * pi/4
V=9pi^2/8

Answer: 420

To find the answer, we need to consider each case where a certain letter is left out.
Leave out S
=6!/(2!*2!)
=180
Leave out E
=6!/3!
=120
Leave out P
=6!/(3!*2!)
=60
Leave out R
=6!/(3!*2!)
=60
Therefore, the total number of ways
= 180+120+60+60
= 420

Answer: 5.81

To solve this question, we first need to work out what dr/dt and dV/dr are equal to. From the given information, dr/dt=0.04 since it is a constant rate. dV/dr can be found through the volume of a sphere.
Since V = (4/3)pi(r)^3
dV/dr = 4pi(r)^2
Therefore,
dV/dt = dV/dr * dr/dt
dV/dt = 0.16pi(r)^2
at r=3.4
dV/dt = 0.16pi(3.4)^2
=5.810689772
=5.81

Answer: 34913

To find the population, we first need to find values for A and k in the given equation. From the given information, at t=0, N=25650
25650=125+Ae^0
A=25525
at t=5, N=31100
31100=125+25650e^5k
25650e^5k=30975
e^5k=1.213516161
5k=ln(1.213516161)
k=ln(1.213516161)/5
k=0.03870441269
k=0.0387
Now we can use t=8
N=125+25525e^0.0387(8)
N=34912.50825
N=34913

Answer: sqrt(13) cm/s

To find an equation for velocity, we can integrate the acceleration, using the
rule a=(d/dx)(1/2)v^2.
So (1/2)v^2=integral(6x^2-4x-3)dx
(1/2)v^2=2x^3-2x^2-3x+C
Using the information given, at t=0, v=3
(1/2)(3)^2=2(0)^3-2(0)^2-3(0)+C
C=4.5
Therefore (1/2)v^2=2x^3-2x^2-3x+4.5
v^2=4x^3-4x^2-6x+9
v=sqrt(4x^3-4x^2-6x+9)
since velocity is positive, at x=2
v=sqrt(4(2)^3-4(2)^2-6(2)+9)
v=sqrt(13)cm/s

Answer: x=3sqrt(2)cos(4t-pi/4)

To find the equation, we use the general form of an equation in simple harmonic motion which is x=acos(nt+O)
From the information given, we can use the rule a=-n^2(x) to work out that n=4
So, x=acos(4t+O)
Using given information for displacement, at t=0, x=3
So acos(O)=3 (equation 1)
We can differentiate x to find velocity
So, v=-4asin(4t+O)
Using given information for velocity, at t=0, v=12
So, -4asin(O)=12
-asin(O)=3 (equation 2)
We can now solve the equations by using (equation 2/equation 1)
So, -asin(O)/acos(O)= 3/3
tan(O)=-1
Therefore O = -pi/4
sub into equation 1
acos(-pi/4)=3
a/sqrt(2)=3
Therefore a=3sqrt(2)
Now we can put all the values into the equation for simple harmonic motion
Therefore x=3sqrt(2)cos(4t-pi/4) is the equation for displacement.

Answer: 2^n

The coefficients of a binomial expansion come from the rows of Pascal's triangle.
For example (a+x)^2
= a^2+2ax+b^2
The coefficients add up to 4, which is equal to 2^2

Answer: y=(v^2sin^2(O))/2g

The greatest height occurs when vertical component at initial velocity is equal to 0. The vertical component is -gt+Vsin(O)
So, -gt+Vsin(O)=0
gt=Vsin(O)
Therefore, t=Vsin(O)/g
This can then be subbed into the vertical component for displacement.
This is -(1/2)gt^2+Vtsin(O)
So, y=-(1/2)g(Vsin(O)/g)^2+V(Vsin(O)/g)sin(O)
y=(-gV^2sin^2(O))/2g^2 + (2V^2sin^2(O))/2g
y=(-V^2sin^2(O))/2g + (2V^2sin^2(O))/2g
Therefore y=(V^2sin^2(O))/2g is the equation for greatest height

Answer: 125/3888

We can solve this question using the rule P(r)=nCr * p^r * q^n-r,
So, p=1/6, q=5/6, r=3 and n=5
Therefore, P(3 sixes) = 5C3 * (1/6)^3 * (5/6)^2
= 125/3888