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Quiz about Extension 1 Mathematics
Quiz about Extension 1 Mathematics

Extension 1 Mathematics Trivia Quiz


Test yourself on the topics of Extension 1 math. You will need a pen, paper and a calculator for this quiz. Good luck.

A multiple-choice quiz by dialga483. Estimated time: 6 mins.
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Author
dialga483
Time
6 mins
Type
Multiple Choice
Quiz #
364,985
Updated
Dec 03 21
# Qns
20
Difficulty
Tough
Avg Score
11 / 20
Plays
197
-
Question 1 of 20
1. Harder graphs of real functions

What is the equation of the horizontal asymptote of the function x^2/(x^2-9)?
Hint


Question 2 of 20
2. Circle Geometry

The cyclic quadrilateral ABCD lies inside a circle. The line BC is extended to the point E, creating the line BE and the exterior angle DCE. Angle DCE is equal to which interior angle of the quadrilateral?
Hint


Question 3 of 20
3. Further trigonometry

In first quadrant, if sinx = 4/7, then what is the exact value of sin(2x)?
Hint


Question 4 of 20
4. Angle between two lines

What is the size of the angle between the two lines 3x-2y+1=0 and x-3y=0 (answer in degrees and minutes)
Hint


Question 5 of 20
5. Dividing an interval

The point P(x,y) divides A(6,-2) and B(-7,5) internally in the ratio 3:4. What are the values of x and y?
Hint


Question 6 of 20
6. Parametric equations

What is the equation of the normal to the parabola x^2=4ay at the point P(2ap,ap^2)?
Hint


Question 7 of 20
7. Polynomials

If a polynomial function has a zero from a cubic factor, what happens to the function when it reaches the x-axis at that point? Pick the MOST SPECIFIC answer that applies.
Hint


Question 8 of 20
8. Iteration methods

By using x=4.5, what is the approximation for the positive root of the function
x^2-4x-1 when Newton's method is applied once?
Hint


Question 9 of 20
9. Integration by substitution

Using the substitution u=x^2-1, what is the value of the integral of
2xsqrt(x^2-1)dx from x=1 to x=2?
Hint


Question 10 of 20
10. Integration of sin^2(x) and cos^2(x)

What is the integral of sin^2(x)cos^2(x)dx?
Hint


Question 11 of 20
11. Inverse Functions

If the function f(x)=x^2-4x has a restricted domain of x>2, then what is the inverse function of f(x)?
Hint


Question 12 of 20
12. Inverse trigonometric functions

The curve y=3/(sqrt(x^2+4)) is rotated about the x-axis between x=0 and x=2. What is the volume of solid generated? (in units^3)
Hint


Question 13 of 20
13. Permutations and Combinations

How many 6 letter words can be formed using the letters of the word PRESSES?
Hint


Question 14 of 20
14. Rates involving multiple variables

A spherical metal ball is heated so that its radius is expanding at the constant rate of 0.04mm/second. At what rate will its volume be increasing when the radius is 3.4mm? (in mm^3/sec)
Hint


Question 15 of 20
15. Harder exponential growth and decay

In a certain town, the growth in population is given by N=125+Ae^kt. If the population is initially 25650 and after 5 years is 31100, what is the population after 8 years?
Hint


Question 16 of 20
16. Rectilinear Motion

The acceleration of a particle is given by a=6x^2-4x-3, where x is the displacement. What is the exact velocity when the particle is 2cm from the origin if initially, the particle is at the origin and has velocity 3cm/s?
Hint


Question 17 of 20
17. Simple Harmonic Motion

A particle moves in a straight line with acceleration given by a=-16x. If initial displacement is 3m and initial velocity is 12m/s, what is the equation of the displacement of the particle over time t?
Hint


Question 18 of 20
18. Binomial Theorem

The sum of the coefficients of (a+x)^n is always equal to what?
Hint


Question 19 of 20
19. Projectile Motion

Which equation represents the greatest height of a projectile?
Hint


Question 20 of 20
20. Binomial Probability

If I throw 5 dice in a game of Yahtzee, what is the probability of throwing 3 sixes?
Hint



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Quiz Answer Key and Fun Facts
1. Harder graphs of real functions What is the equation of the horizontal asymptote of the function x^2/(x^2-9)?

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.
2. Circle Geometry The cyclic quadrilateral ABCD lies inside a circle. The line BC is extended to the point E, creating the line BE and the exterior angle DCE. Angle DCE is equal to which interior angle of the quadrilateral?

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.
3. Further trigonometry In first quadrant, if sinx = 4/7, then what is the exact value of sin(2x)?

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
4. Angle between two lines What is the size of the angle between the two lines 3x-2y+1=0 and x-3y=0 (answer in degrees and minutes)

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
5. Dividing an interval The point P(x,y) divides A(6,-2) and B(-7,5) internally in the ratio 3:4. What are the values of x and y?

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
6. Parametric equations What is the equation of the normal to the parabola x^2=4ay at the point P(2ap,ap^2)?

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
7. Polynomials If a polynomial function has a zero from a cubic factor, what happens to the function when it reaches the x-axis at that point? Pick the MOST SPECIFIC answer that applies.

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.
8. Iteration methods By using x=4.5, what is the approximation for the positive root of the function x^2-4x-1 when Newton's method is applied once?

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
9. Integration by substitution Using the substitution u=x^2-1, what is the value of the integral of 2xsqrt(x^2-1)dx from x=1 to x=2?

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)
10. Integration of sin^2(x) and cos^2(x) What is the integral of sin^2(x)cos^2(x)dx?

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
11. Inverse Functions If the function f(x)=x^2-4x has a restricted domain of x>2, then what is the inverse function of f(x)?

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)
12. Inverse trigonometric functions The curve y=3/(sqrt(x^2+4)) is rotated about the x-axis between x=0 and x=2. What is the volume of solid generated? (in units^3)

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
13. Permutations and Combinations How many 6 letter words can be formed using the letters of the word PRESSES?

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
14. Rates involving multiple variables A spherical metal ball is heated so that its radius is expanding at the constant rate of 0.04mm/second. At what rate will its volume be increasing when the radius is 3.4mm? (in mm^3/sec)

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
15. Harder exponential growth and decay In a certain town, the growth in population is given by N=125+Ae^kt. If the population is initially 25650 and after 5 years is 31100, what is the population after 8 years?

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
16. Rectilinear Motion The acceleration of a particle is given by a=6x^2-4x-3, where x is the displacement. What is the exact velocity when the particle is 2cm from the origin if initially, the particle is at the origin and has velocity 3cm/s?

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
17. Simple Harmonic Motion A particle moves in a straight line with acceleration given by a=-16x. If initial displacement is 3m and initial velocity is 12m/s, what is the equation of the displacement of the particle over time t?

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.
18. Binomial Theorem The sum of the coefficients of (a+x)^n is always equal to what?

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
19. Projectile Motion Which equation represents the greatest height of a projectile?

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
20. Binomial Probability If I throw 5 dice in a game of Yahtzee, what is the probability of throwing 3 sixes?

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
Source: Author dialga483

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