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Cambridge Prelim MX1 Textbook Marathon/Q&A (1 Viewer)

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

So we know x degree is xpi/180 rad

Hence find:

Derivative of (tan (x degrees + 45 degrees))
Convert everything to radians and then just differentiate with the chain rule.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I have converted everything to radians. Do I then add the fractions??
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I have converted everything to radians. Do I then add the fractions??
No, the 45 degrees is just a constant, don't bother adding it.

Once it's all converted to radians, the argument of the tan function will be a linear function of x. So to differentiate tan of a linear function of x, the answer is sec2 of that linear function of x, times the derivative of that linear function of x (which is just the coefficient of the x, i.e. π/180).
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Do I need to use product rule?? I tried using the chain rule and it is a big mess and getting me nowhere.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

If y = sin x

prove

i) y' = sin(pi/2 + x)

ii) y'' = sin ( pi +x )

iii) y''' = sin (3pi/2 + x )

Hence deduce an expression for y^n
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I was able to do the first and second derivative but am struggling with the 3rd.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

PQ is a diameter on the given circle and S is a point on the circumference. T is the point on PQ such that PS = PT . Let Angle SPT = alpha

a) Show that the are A of triangle SPT is A = 1/2 d^2 cos^2a sina, where d is the diameter of the circle.

b) Hence show that the maximum are of triangle SPT as S varies on the circle is 1/9 d^2 root3 units^2


I can do part a), unsure how to approach b) ?
 

kawaiipotato

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

PQ is a diameter on the given circle and S is a point on the circumference. T is the point on PQ such that PS = PT . Let Angle SPT = alpha

a) Show that the are A of triangle SPT is A = 1/2 d^2 cos^2a sina, where d is the diameter of the circle.

b) Hence show that the maximum are of triangle SPT as S varies on the circle is 1/9 d^2 root3 units^2


I can do part a), unsure how to approach b) ?






















 
Last edited:

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

A straight line passes through the point 2,1 and has positive x and y int at P and Q respectivley. Let Angle ORQ = a where O is the origin.

So the A of the Triange = (2tan a + 1 )^2 / 2tan a

Now how do I show that this area is maximised when tan a = 1/2
 

kawaiipotato

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

A straight line passes through the point 2,1 and has positive x and y int at P and Q respectivley. Let Angle ORQ = a where O is the origin.

So the A of the Triange = (2tan a + 1 )^2 / 2tan a

Now how do I show that this area is maximised when tan a = 1/2






























 

Paradoxica

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

A straight line passes through the point 2,1 and has positive x and y int at P and Q respectivley. Let Angle ORQ = a where O is the origin.

So the A of the Triange = (2tan a + 1 )^2 / 2tan a

Now how do I show that this area is maximised when tan a = 1/2
make the substitution x = tan a
all relevant quantities are strictly non-negative
2x + 1 >= 2 sqrt(2x) (easy to prove am-gm for n=2)
declare equality and hence solve for x.

beaten by kawaiipotato
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Use mathematical induction to prove that for all positive integers n:

a + (a+ d) + (a + 2d) + ... + ( a + ( n - 1 )d) = 1/2 . n (2a + (n - 1)d)
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

SOlve integral of (16 - x^2)^1/2 dx using the substitution x = 4sinE
 

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