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Integration

Question 1No Calculator

[Maximum mark: 8]



a) Show that (48x)2=1664x+64x2.


b) Hence, find the value of 30(48x)2dx.


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Question 2No Calculator

[Maximum mark: 7]



Cosnider the curve y=f(x) where gradient f(x)=9x24x+1. Point Q(2,14) lies on f(x).


a) Find the equation of the tangent to f(x) at point Q in the form y=ax+b.


b) Find the equation of f(x).

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Question 3 Register

[Maximum mark: 10]



Consider the function y=(2x+1)(x2) presented on the graph below.



a) Write down an integral to find the area of a.


b) Find a.


c) Ratio c is measured as c=b2a. Find c.

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Question 4 Register

[Maximum mark: 12]



The function f(x) has the gradient function f(x)=kx+5. The graph of the function f(x) is presented below:



a) Find the equation of f(x).


b) Hence, find the area of f(x) in the interval 0<x<2.


c) Hence, find the equation of the tangent to f(x) at point x=1.


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Question 5 Register

[Maximum mark: 11]



Consider the function f(x)=2x3+6x2 presented on the graph below:



a) Find the equation of the tangent of f(x) at x=3.


b) Find the shaded area A.


Consider the following triangle, with vertices at A(1,0), B(5,0), and C(3,y):



c) Find the y-coordinate of point C, such that the area of the triangle ABC is the same as that of the shaded region A.

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Question 6 Register

[Maximum mark: 7]



The expression 4z+20z can be written as 4+20zp.


a) Find p.


b) Hence, find the value of 1614z+20zdz

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Question 7 Premium

[Maximum mark: 9]



Consider two functions:

f(x)=x2+3x11

g(x)=x25x5



a) Identify which function is red and which is green.


b) They have two points of interesection with x-coordinates a and b, such that a>b. Find a and b.


c) Find the area enclosed by the two functions.


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Question 8 Premium

[Maximum mark: 6]



Find the value of 2π0cos2(x4).


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Question 9 Premium

[Maximum mark: 7]



Consider a fuction f(x)=5x7x2+3 which passes through the point Q(1,3). Find the equation for f(x).


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Question 10 Premium

[Maximum mark: 14]



Consider the function f(x)=4cos2(x)2sin2(x)+2, for 0xπ.


a) Find the root(s) of the equation f(x)=0.


b) The derivative of f(x) can be written in the form a×sin(bx). Find a and b.


c) Hence, find the coordinates of the points of f(x) when f(x)=0.


d) Find the value of π04cos2(x)2sin2(x)+2.

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Question 11 Premium

[Maximum mark: 17]



Consider the function g(x)=3x+4


a) Find g1(x).


b) The graphs of f(x) and f1(x) have two points of intersection. Find the coordinates of those two points.


c) Find g(x).


d) Find the value of x for which g(x) and g1(x) have the same gradient.


e) Find the area enclosed by the positive x-axis, positive y-axis, g(x), g1(x).

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