Am Paper 1
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NAME: __________________________ (
)
CLASS: ______
READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all questions. Write your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of a scientific calculator is expected, where appropriate. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together. The number of marks is given in the brackets [ ] at the end of each question or part question. The total number of the marks for this paper is 80
This question paper consists of 5 printed pages. [Turn over
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 3of5
Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, b b 2 4ac x . 2a Binomial expansion n n n (a b) n a n a n1 b a n 2 b 2 a n r b r b n , 1 2 r n n! n(n 1)...(n r 1) where n is a positive integer and . r! r r!(n r )! 2. TRIGONOMETRY Identities
sin 2 A cos 2 A 1 sec 2 A 1 tan 2 A cos ec 2 A 1 cot 2 A sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B sin A sin B tan( A B )
tan A tan B 1 tan A tan B
sin 2A = 2sin A cos A
cos 2 A cos 2 A sin 2 A 2 cos 2 A 1 1 2 sin 2A tan 2 A sin A sin B 2 sin
2 tan A 1 tan 2 A
1 1 ( A B ) cos ( A B ) 2 2
1 1 sin A sin B 2 cos ( A B ) sin ( A B ) 2 2
1 1 cos A cos B 2 cos ( A B) cos ( A B) 2 2 1 1 cos A cos B 2 sin ( A B) sin ( A B) 2 2 Formulae for ABC a b c . sin A sin B sin C a2 = b2 + c2 2bc cos A. CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 4of5
= 1.
2.
1 ab sin C. 2
It is given that ( x 1) is a factor of f ( x) x 3 px 2 qx 12 where p and q are constants and f(x) has a stationary value at x 4 . (i)
Find the value of p and of q.
[4]
(ii)
Factorise f(x) completely and solve f ( x ) 0
[3]
(iii)
Hence find the values of y such that ( y 3)3 p( y 3)2 qy 36 .
[2]
1 2 3 1 Given that A ,B and ABX I , find 3 4 0 1 (i) the matrix AB, (ii)
the matrix X.
[4]
8
3.
(a)
(b)
4.
2 Write down and simplify the first four terms in the expansion of x 2 in x descending powers of x. 2 Hence find the coefficient of x10 in the expansion of (2 x 3 5)( x 2 )8 . x
[4]
In the binomial expansion of (3 kx 2 )n , where n 3 and k is a constant, the coefficient of x 4 is twice the coefficient of x 2 . Express k in terms of n. [3]
A curve is such that its gradient function is given by
3 . Given that the tangent of the (2 x a )3
1 curve at ( , 9) is perpendicular to the line 3 y 27 x 4 , find 2 (a) the value of a,
5.
(b)
the equation of the curve.
(a)
Find all the values of x between 0 and 360 inclusive for which 4 tan 2 x 4 cos x
(b)
[5]
Find all the values of x between 0 and 5 for which 2cos2 x 6sin x cos x 1
[4]
[4]
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 5of5
6.
7.
A circle, C whose equation is given by x 2 y 2 2ax 2 y 20 0 where a 0 has radius 5 units. (a) Find the value of a. (b)
Find the equation of another circle which passes through the point ( 7, 2) and has the same centre as C.
(c)
the value(s) of k if the line y k meets the circle, C.
(a)
Prove that
(b)
Given that cos A
(cos sin )(tan
[8]
1 ) cos ec sec tan
[3]
5 and 180 A 270 , evaluate, without using tables or a 13
calculator, (i)
sec (A)
(ii)
cot (2 A)
(iii)
sin
8.
Given that
9.
(a)
5
1
A 2
[5]
f ( x)dx 8 , evaluate
5
3
1
f ( x)dx ( 3
1 f ( x ))dx . 3x
[2]
Solve the simultaneous equations 3x 3 y 84 0.2(5x 2 ) 5 y
[4]
10.
log 2 ( x 2) log 2 ( x 1) 3
(b)
Solve the equation
(a)
The table below shows experimental values of two variables x and y which are known to be related by the equation y a (1 x )b where a and b are constants.
x y
1 5.2
2 7.2
3 9.1
4 10.9
[4]
5 12.6
By drawing a suitable straight line graph, use your graph to estimate (i) the value of a and of b, (ii) the value of x when y 100.9 .
6 14.2
[7]
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 6of5
(b)
The variables x and y are related by the equation y m x
n . When the graph of x
y 1 against is drawn, a straight line is obtained which has a gradient of 5 and x x passes through the point (2,18) . Find (i) the value of m and of n, (ii) the positive value of y when x x 216 . [4]
11.
r h
The diagram above shows a container consisting of a hemisphere of radius r cm and a cylinder of the same radius and height h cm. The volume of the container is 1215 cm3 . 1215 2r . r2 3
(i)
Show that h
(ii)
Show that the total surface area, A cm 2 , of the container is given by 5 2 2430 A r . 3 r
(iii)
Find the value of r for which A is stationary.
(iv)
Hence find the stationary value of A and determine whether it is maximum or minimum .
(v)
Using the value of r in (iii), find the rate of decrease of the radius when the rate of increase in the height is 0.02 cms 1 . [10]
End of Paper
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 7of5
Answers: 1(i) p 5, q 8 3 1 2(i) 1 9 3(a) -336 4(a) a = 4
5(a) x 0,101.5,258.5,360 6(a) a = 2 7(b)(i) 2
3 5
(ii)
( x 1)( x 6)( x 2) x 1,6,2
1 6 1 2 12 (b) k n 1 (b) 3 11 y 8 2 4(2 x 4) 12 (b) x 0.161,1.73,3.30,4.87 (b) ( x 2) 2 ( y 1) 2 34 119 (ii) 120
(iii) y 4,9,1
1 (ii) 6 3 2
(c) 4 k 6 (iii)
3 13
8. 7.63 9(a) x 4, y 1 10(a)(i) a 3.02, b 0.8
(b) x 1.68 (ii) x 2.31
(b)(i) m 8, n 5
11(iii) 9
(iv) A 1270, minimum
(v) 0.005 cm/s
(ii) y 47
1 6
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 8of5
)
CLASS: ______
READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all questions. Write your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of a scientific calculator is expected, where appropriate. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together. The number of marks is given in the brackets [ ] at the end of each question or part question. The total number of the marks for this paper is 80
This question paper consists of 5 printed pages. [Turn over
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 3of5
Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, b b 2 4ac x . 2a Binomial expansion n n n (a b) n a n a n1 b a n 2 b 2 a n r b r b n , 1 2 r n n! n(n 1)...(n r 1) where n is a positive integer and . r! r r!(n r )! 2. TRIGONOMETRY Identities
sin 2 A cos 2 A 1 sec 2 A 1 tan 2 A cos ec 2 A 1 cot 2 A sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B sin A sin B tan( A B )
tan A tan B 1 tan A tan B
sin 2A = 2sin A cos A
cos 2 A cos 2 A sin 2 A 2 cos 2 A 1 1 2 sin 2A tan 2 A sin A sin B 2 sin
2 tan A 1 tan 2 A
1 1 ( A B ) cos ( A B ) 2 2
1 1 sin A sin B 2 cos ( A B ) sin ( A B ) 2 2
1 1 cos A cos B 2 cos ( A B) cos ( A B) 2 2 1 1 cos A cos B 2 sin ( A B) sin ( A B) 2 2 Formulae for ABC a b c . sin A sin B sin C a2 = b2 + c2 2bc cos A. CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 4of5
= 1.
2.
1 ab sin C. 2
It is given that ( x 1) is a factor of f ( x) x 3 px 2 qx 12 where p and q are constants and f(x) has a stationary value at x 4 . (i)
Find the value of p and of q.
[4]
(ii)
Factorise f(x) completely and solve f ( x ) 0
[3]
(iii)
Hence find the values of y such that ( y 3)3 p( y 3)2 qy 36 .
[2]
1 2 3 1 Given that A ,B and ABX I , find 3 4 0 1 (i) the matrix AB, (ii)
the matrix X.
[4]
8
3.
(a)
(b)
4.
2 Write down and simplify the first four terms in the expansion of x 2 in x descending powers of x. 2 Hence find the coefficient of x10 in the expansion of (2 x 3 5)( x 2 )8 . x
[4]
In the binomial expansion of (3 kx 2 )n , where n 3 and k is a constant, the coefficient of x 4 is twice the coefficient of x 2 . Express k in terms of n. [3]
A curve is such that its gradient function is given by
3 . Given that the tangent of the (2 x a )3
1 curve at ( , 9) is perpendicular to the line 3 y 27 x 4 , find 2 (a) the value of a,
5.
(b)
the equation of the curve.
(a)
Find all the values of x between 0 and 360 inclusive for which 4 tan 2 x 4 cos x
(b)
[5]
Find all the values of x between 0 and 5 for which 2cos2 x 6sin x cos x 1
[4]
[4]
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 5of5
6.
7.
A circle, C whose equation is given by x 2 y 2 2ax 2 y 20 0 where a 0 has radius 5 units. (a) Find the value of a. (b)
Find the equation of another circle which passes through the point ( 7, 2) and has the same centre as C.
(c)
the value(s) of k if the line y k meets the circle, C.
(a)
Prove that
(b)
Given that cos A
(cos sin )(tan
[8]
1 ) cos ec sec tan
[3]
5 and 180 A 270 , evaluate, without using tables or a 13
calculator, (i)
sec (A)
(ii)
cot (2 A)
(iii)
sin
8.
Given that
9.
(a)
5
1
A 2
[5]
f ( x)dx 8 , evaluate
5
3
1
f ( x)dx ( 3
1 f ( x ))dx . 3x
[2]
Solve the simultaneous equations 3x 3 y 84 0.2(5x 2 ) 5 y
[4]
10.
log 2 ( x 2) log 2 ( x 1) 3
(b)
Solve the equation
(a)
The table below shows experimental values of two variables x and y which are known to be related by the equation y a (1 x )b where a and b are constants.
x y
1 5.2
2 7.2
3 9.1
4 10.9
[4]
5 12.6
By drawing a suitable straight line graph, use your graph to estimate (i) the value of a and of b, (ii) the value of x when y 100.9 .
6 14.2
[7]
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 6of5
(b)
The variables x and y are related by the equation y m x
n . When the graph of x
y 1 against is drawn, a straight line is obtained which has a gradient of 5 and x x passes through the point (2,18) . Find (i) the value of m and of n, (ii) the positive value of y when x x 216 . [4]
11.
r h
The diagram above shows a container consisting of a hemisphere of radius r cm and a cylinder of the same radius and height h cm. The volume of the container is 1215 cm3 . 1215 2r . r2 3
(i)
Show that h
(ii)
Show that the total surface area, A cm 2 , of the container is given by 5 2 2430 A r . 3 r
(iii)
Find the value of r for which A is stationary.
(iv)
Hence find the stationary value of A and determine whether it is maximum or minimum .
(v)
Using the value of r in (iii), find the rate of decrease of the radius when the rate of increase in the height is 0.02 cms 1 . [10]
End of Paper
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 7of5
Answers: 1(i) p 5, q 8 3 1 2(i) 1 9 3(a) -336 4(a) a = 4
5(a) x 0,101.5,258.5,360 6(a) a = 2 7(b)(i) 2
3 5
(ii)
( x 1)( x 6)( x 2) x 1,6,2
1 6 1 2 12 (b) k n 1 (b) 3 11 y 8 2 4(2 x 4) 12 (b) x 0.161,1.73,3.30,4.87 (b) ( x 2) 2 ( y 1) 2 34 119 (ii) 120
(iii) y 4,9,1
1 (ii) 6 3 2
(c) 4 k 6 (iii)
3 13
8. 7.63 9(a) x 4, y 1 10(a)(i) a 3.02, b 0.8
(b) x 1.68 (ii) x 2.31
(b)(i) m 8, n 5
11(iii) 9
(iv) A 1270, minimum
(v) 0.005 cm/s
(ii) y 47
1 6
CSS/Preliminary Examination 2009/Sec 4E/5N/A Math P1/JT/Page 8of5
