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p4 Practice Paper 15

The document contains a series of mathematical questions covering topics such as calculus, vector equations, and differential equations. It includes problems related to evaluating integrals, expanding functions as series, finding tangents to curves, and solving differential equations. Each question requires specific calculations and proofs, with some parts asking for simplified answers or coordinates of points.

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45 views3 pages

p4 Practice Paper 15

The document contains a series of mathematical questions covering topics such as calculus, vector equations, and differential equations. It includes problems related to evaluating integrals, expanding functions as series, finding tangents to curves, and solving differential equations. Each question requires specific calculations and proofs, with some parts asking for simplified answers or coordinates of points.

Uploaded by

abrarjay
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Question 1 P4 Practice paper 15

30 A B
----•--+--
(x+3)(9-2x) x+3 9-2x ·

a) Determine the value of each of the constants A and B . (3)

b) Evaluate

r' 30
J1 (x+3)(9-2x) dx •

giving the answer as a single simplified natural logarithm. (5)

Quesdon 2

20
/ (x) = :}
4+2x
• Ix! < 2 .

a) Expand / ( x) as an infinite series. up and including the tenn in x3 . (S)

b) By substituting x = .!. in the above expansion. show that (3)


12

Ji,-.2.45.

Quatioa3
A curve C has implicit equation

x 2 +4.zy+2y2 =1.

a) Show that ...

L dy =- x+2y (5)
dx 2x+2y

lL ... the equation of the tangent to the curve at P(I.I) is (2)

3x+4y=1 .

1be tangent to the curve at the point Q is parallel to the tangent to the curve at P .

b) Fmd the coordinates of Q. (6)


Question4
Liquid is pouring into a container al the constant rate of 30 cm3s-• .

1be container is initially empty and when the height of the liquid in the container is
h cm the volume of the liquid, V cm3 • is given by

2
V =36h .

a) Find the rate at which the height of the liquid in the container is rising whc-n
the height of the liquid ~aches 3 cm . (4)

b) Determine the rate al which the height of the liquid in the container is risina
12.5 minutes after the liquid started pouring in. (6)

QuestiCNI 5
The points with coordinates A ( 3.O. 3) and B ( 4. -1. S) are given.

a) Find a vector equation of the straight line /1 that passes through A and B . ( l)

The straight line 12 has equation

r =51 + IOJ+4k +µ(l+JJ+k).

where µ is a scalar parameter.

b) Show that /1 and /2 are perpendicular. (2)

c) Show further that /1 and /2 intersect al some point P and fmd its coordinates.
(5)
The point E is on the /1 •

A circle with centre at E is drawn so that it cuts /2 at the points C and D .

d) Given that the coordinates of C are ( o.-5. -I) . find the coordinates of D . ( l)
Question 6
A machine is used to produce waves in the swimming pool of a water theme parlc.

Let x cm be the height of the wave produced above a certain level in the pool. and
suppose it can be modelled by the differential equation

dx = 2xsin 2t • t ~ 0 •
dt

where t is the time in seconds.

When t=O. x=6 .

a) Solve the differential equation to show

X = 6el-cos2l. (8)

b) Fmd the maximum height of the wave. (2)

Question 7

11IC figure above shows the curve C with paramclric equations

x=Scost, y=3sin2t. os,s!:.


2

11IC curve meets the x axis at the origin O and at the point P .

a) Find the value of r at O and at P . (3)

11IC fmite region R bounded by C and the x axis is revolved by 2K radians in the
x axis forming a solid of revolution S •

b) Show that the volume of S is given by the integral

1
Jr
It
2 3 2
180sin 1 cos t dt. (4)

c) By using the substitution u = cost • or otherwise, find the volume of S . (6)

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