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数学工程代考 IFYME004代写

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数学工程代考

IFYME004 Mathematics (Engineering) Examination Exemplar

数学工程代考 The line with equation y = x + 12 intersects the line withequation y = 2 − 4x at point P.Find the coordinates of point P.  You must show your working.

INSTRUCTIONS TO STUDENTS  数学工程代考

Answer ALL questions

The marks for each question are indicated in square brackets [ ].

  • A formula booklet and graph paper will be provided.
  • An approved calculator may be used in the examination.
  • Show ALL workings in your answer booklet.
  • Examination materials must not be removed from the examination booklet.

DO NOT OPEN THIS QUESTION PAPER UNTIL INSTRUCTED BY THE INVIGILATOR

Question 1  数学工程代考

The table below shows the values of for some values of x given to 3 significant figures where appropriate.

x 0 0.2 0.4 0.6 0.8 1
1

1 + x2

 

1

 

0.962

 

0.862

 

0.735

 

0.610

 

0.5

Use the trapezium rule with five intervals to find an approximate value of

[ 3 ]

Question 2

a)The line with equation y = x + 12 intersects the line withequation y = 2 − 4x at point P.

Find the coordinates of point P.  You must show your working. [ 3 ]

b)Point  A  lies  at  (2p, −q)   and  point  B  lies  at (5p,  8q). A line which is perpendicular to AB has gradient

Express p  in terms of q. [ 3 ]

Question 3

a)A bag holds 8 red sweets, 4 blue sweets and 6 yellow sweets.

Three sweets are drawn, one after the other with no replacement.

Find the probability that all three sweets are red. Give your answer as a decimal to 3 significant figures.

In this question, 1 mark will be given for the correct use of significant figures.[ 4 ]  数学工程代考

b)The masses of melons can be assumed to follow a Normal distribution with standard deviation 40 grams.

A sample of 30 melons has a mean mass of 820 grams.

Find a 99% confidence interval for the mean mass of all the melons. [ 2 ]

Question 4  数学工程代考

A curve has equation y =3x3 + 7x2 − 28x − 32.

a)Use the Factor Theorem to show that (x + 4) is a factorof 3x3 + 7x2  − 28x − 32.[ 2 ]

(b)Divide 3×3 + 7x2− 28x − 32 by (x + 4) and hence factorise the expression completely. [ 3 ]

数学工程代考
数学工程代考

Question 5

a)In the expansion of(2 + cx)6, the coefficient of the x3 term is times larger than the coefficient of the x4 term.

Find the value of c. [ 3 ]  数学工程代考

b)The 5th term of an arithmetic series is one more than the common difference.

If a is the first term and d is the common difference, show that and hence find an expression for the sum of the first 37 terms of the series,giving your answer in terms of a.[ 3 ]

(c)A geometric series has first term 240 and common ratio

Show that the sum of the series cannot exceed 275. [ 2 ]

Question 6

a)

数学工程代考
数学工程代考

Figure 1

Figure 1 shows triangle PQR with angle P = 60o, angle Q = 50o and QR = 12 cm.

Find the length of PR and hence find the area of triangle PQR. [ 4 ]

b) Solve 2cos2θ=1 (0 ≤ 8 ≤ 2n)

Give your answers as exact multiples of π. [ 4 ]

Question 7  数学工程代考

Line  l1  has equation r = (2i − j + 7k) + h(−i + 2j − 3k) where h is a scalar.

Line  l2  has equation r = (i + 3j − 2k) + µ(2i − 5j + 9k) where µ is a scalar.

(a)Show that lines l1and l2 intersect and find the coordinates of their point of intersection.[ 6 ]

(b)Find the acute angle between lines l1and l2. [ 3 ]

You are given point P(4, −5, 13) lies on line l1 and point Q(3, −2, 7) lies on line l2.

(c)Find the unit vector [ 2 ]

Question 8

数学工程代考
数学工程代考

Figure 2

The shape in figure 2 shows a rectangular piece of wood with a smaller rectangle cut from one of the ends. All measurements are in cm.

The perimeter of the shape is 246 cm.

a)Express y in terms of x and hence show that the area of the shape, A, is given by

A = 615x − 41x2   [ 5 ]

(b)Use calculus to find the value of x  which gives themaximum area.  [ 3 ]

(c)Confirm that your value of x  gives a maximum.[ 3 ]

Question 9

A circle has equation x2 + y2− 12x + 2y − 63 = 0.

a)Find the radius of the circle and the coordinates of its centre.[ 4 ]

b)Find in terms of x and y. [ 3 ]  数学工程代考

(c)Confirm that point A(14, 5) lies on the circle and find the equation of the normal to the circle at point A.

Give your answer in the form  ax + by + c  = 0 where a, b  and c are integers. [ 4 ]

Question 10

数学工程代考
数学工程代考

Figure 3

Figure 3 shows the curve y =  − 1 and the line y = 2x + 2. The curve crosses the x− axis at (2, 0) and meets the line y = 2x + 2 at (  , 3).

Find the area, which is shaded on the diagram, that is bounded by the line

y = 2x + 2, the curve y =  − 1 and both axes.

Give your answer in the form ln where a and b are integers.[ 6 ]

All working must be shown. Just giving the answer, even the correct one,  will score no marks if this working is not seen.

(b)Resolve into partial fractions. [ 4 ]

Question 11

a) Show that sin 2A   where A is acute. [ 3 ]

b) Solve  3 cos 2θ − cos θ + 1=0 (0 ≤  θ ≤ 360°) [ 5 ]

c) Differentiate

Question 12  数学工程代考

Functions ƒ, g and ℎ are defined as follows:

数学工程代考
数学工程代考

Figure 4

Figure 4 shows a sketch of y = ƒ(x). The curve crosses the x − axis at(−4, 0) and (6,0); and the y − axis at (0, −4).

a)Draw a sketch of y = ƒ(2x).  This must not be drawn on graph paper.  On  your sketch, show clearly the coordinates where the graph crossesthe x − and y − axes.

Explain why y = ƒ(x) has no inverse. [ 4 ]

b)Find an expression for g–1(x). State its domainand  [ 3 ]

c)Solve the equation  ℎ[g(x)]=  [ 3 ]

Question 13 

a)Find the values of n if

数学工程代考
数学工程代考

[ 3 ]

b)Solve the equation  log5(5×3  + 3×2) − 3 log5x= 2 (x > 0)

All working must be shown.[ 4 ]

c)The variables p and t are connected by the formula p =e2t− 7et + 8t +1.

Find and hence find the exact value of t  when  = 2. [ 5 ]

Question 14

a)Evaluate

数学工程代考
数学工程代考

All working must be shown. Just giving the answer, even the correct one,will score no marks if this working is not seen. [ 5 ]

b)Find

数学工程代考
数学工程代考

[ 3 ]

c)Solve the differential equation  数学工程代考

subject to y = when x = 0.

Give your answer in the form y = ƒ(x). [ 4 ]

数学工程代考
数学工程代考

 

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