math c4 practice

A-LEVEL STUDENTS

Math Edexcel Revision C4

Collected By Moustafa Sohdy

6/2/2011

Useful Edexcel C4 Questions From the Solomon Press

Solomon Press

PARTIAL FRACTIONS C4 Worksheet A 1 Find the values of the constants A and B in each identity.

a x − 8 ≡ A(x − 2) + B(x + 4) b 6x + 7 ≡ A(2x − 1) + B(x + 2) 2 Find the values of the constants A and B in each identity.

a 2( 1)( 3)x x+ +

≡ 1

Ax +

+ 3

Bx +

b 3( 1)x

x x−−

≡ Ax

+ 1

Bx −

c 1( 3)( 5)

xx x

+− −

≡ 3

Ax −

+ 5

Bx −

d 10(1 )(2 )

xx x+

+ − ≡

1A

x+ +

2B

x−

e 24 1

2x

x x−

+ − ≡

2A

x + +

1B

x − f 2

94 3

xx x

−− +

≡ 1

Ax −

+ 3

Bx −

3 Express in partial fractions

a 8( 1)( 3)x x− +

b 1( 2)( 3)

xx x

−+ +

c 10( 4)( 1)

xx x+ −

d 25 7xx x

++

e 22

5 4x

x x+

− + f 2

4 69

xx

+−

g 23 2

2 24x

x x+

− − h 2

3812

xx x−

− − i 4 5

(2 1)( 3)x

x x−

+ −

j 1 3(3 4)(2 1)

xx x

−+ +

k 21

3x

x x+

− l 2

52 3 2x x+ −

m 22( 5)

8 10 3x

x x+

+ − n 2

3 72 3

xx x

−− −

o 21 3

1 2x

x x−

− −

4 Find the values of the constants A, B and C in each identity.

a 3x2 + 17x − 32 ≡ A(x − 1)(x + 3) + B(x − 1)(x − 4) + C(x + 3)(x − 4)

b 14x + 2 ≡ A(x + 1)(x − 2) + B(x + 1)(3x − 1) + C(x − 2)(3x − 1)

c x2 + x + 12 ≡ A(x + 1)2 + B(x + 1)(x + 5) + C(x + 5)

d 4(5x2 + 4) ≡ A(2x + 1)2 + B(2x + 1)(x − 3) + C(x − 3) 5 Find the values of the constants A, B and C in each identity.

a 8 14( 2)( 1)( 3)

xx x x

+− + +

≡ 2

Ax −

+ 1

Bx +

+ 3

Cx +

b 22 6 20

( 1)( 2)( 6)x x

x x x− +

+ + − ≡

1A

x + +

2B

x + +

6C

x −

c 29 14

( 4)( 1)x

x x−

+ − ≡

4A

x + +

1B

x − + 2( 1)

Cx −

d 2

23 7 4

( 3)( 2)x x

x x− −

− − ≡

3A

x − +

2B

x − + 2( 2)

Cx −

Solomon Press


a 22 4

( 1)( 4)x

x x x+

− − b 2

9( 2)( 1)x x− +

c 2 11 21

(2 1)( 2)( 3)x x

x x x+ −

+ − −

d 210 9

( 4)( 3)x

x x+

− + e

2

24 5

( 1)( 2)x x

x x+ +

+ + f 2

16 2( 3)( 4)

xx x

−− −

g 22 9

( 3)(2 1)x

x x−

− − h

2

23 24 4( 1)( 4)

x xx x+ −+ −

i 2

3 29 2 12

6x x

x x x− −

+ −

j 2

3 25 3 20

4x xx x

+ −+

k 2

213 3

(2 3)( 1)x

x x−

+ − l

226( 1)( 3)( 5)

x xx x x

− −− + +

7 Find the values of the constants A, B and C in each identity.

a 2

( 2)( 6)x

x x− − ≡ A +

2B

x − +

6C

x −

b 2

22 94 5

x xx x

+ ++ −

≡ A + 1

Bx −

+ 5

Cx +

8 a Find the quotient and remainder obtained in dividing (x3 + 4x2 − 2) by (x2 + x − 2).

b Hence, express 3 2

24 2

2x xx x

+ −+ −

in partial fractions.


a 2 3

( 3)( 1)x

x x+

− + b

3 2

23 2

4x x x

x− − +

− c

2

22 7

6 8x x

x x+

+ +

d 3( 1)( 1)( 4)( 5)

x xx x

+ −− +

e 3 2

23 7 4

4 3x xx x

+ ++ +

f 2

24 7 52 7 3

x xx x

− +− +

g 2

22

2 3x

x x− − h

3 2

26 6 1

6 5x x x

x x− + +

− + i

3

29 27 23 4 4x xx x

− −− −

10 f(x) = 5( 1)(2 1)

xx x

+− +

.

a Express f(x) in partial fractions.

b Find the exact x-coordinates of the stationary points of the curve y = f(x).

11 f(x) = 2(4 5)

( 1)( 2)x x

x x+

− +.

a Find the values of the constants A, B and C such that

f(x) = 1

Ax −

+ 2

Bx +

+ 2( 2)C

x +.

b Show that the tangent to the curve y = f(x) at the point where x = −1 has the equation

3x − 4y + 5 = 0.

C4 PARTIAL FRACTIONS Worksheet A continued

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PARTIAL FRACTIONS C4 Worksheet B 1 Given that

22(2 3)( 4)x x− +

≡ 2 3

Ax −

+ 4

Bx +

,

find the values of the constants A and B. (3) 2 Find the values of A, B and C such that

25

( 1)( 3)x

x x+

+ − ≡

1A

x + +

3B

x − + 2( 3)

Cx −

. (4)

3 Given that

2

24 16 72 9 4x xx x

− −− +

≡ A + 2 1

Bx −

+ 4

Cx −

,

find the values of the constants A, B and C. (4) 4 f(x) = 3x3 + 11x2 + 8x − 4.

a Fully factorise f(x). (4)

b Express 16f ( )

xx

+ in partial fractions. (4)

5 Given that

f(x) = 21

(2 1)x x −,

express f(x) in partial fractions. (4)

6 f(x) = 3 2

25 2 19

7 10x x x

x x+ − −

+ +.

Show that f(x) can be written in the form

f(x) = x + A + 2

Bx +

+ 5

Cx +

,

where A, B and C are integers to be found. (5) 7 The function f is defined by

f(x) = 24

1x −.

a Express f(x) in partial fractions. (3)

The function g is defined by

g(x) = 22 5

( 4)( 2)( 1)x x

x x x+ −

− − −.

b Express g(x) in partial fractions. (3)

c Hence, or otherwise, solve the equation f(x) = g(x). (5)

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SERIES C4 Worksheet A 1 Find the binomial expansion of each of the following in ascending powers of x up to and

including the term in x3, for | x | < 1.

a (1 + x)−1 b (1 + 12)x c 2(1 + x)−3 d (1 +

23)x

e 3 1 x− f 21

(1 )x+ g 4

14(1 )x−

h 31 x−

2 Expand each of the following in ascending powers of x up to and including the term in x3 and

state the set of values of x for which each expansion is valid.

a (1 + 122 )x b (1 − 3x)−1 c (1 −

124 )x − d (1 + 1

2 x)−3

e (1 − 136 )x f (1 + 1

4 x)−4 g (1 + 322 )x h (1 −

433 )x −

3 a Expand (1 − 122 )x , | x | < 1

2 , in ascending powers of x up to and including the term in x3.

b By substituting a suitable value of x in your expansion, find an estimate for 0.98

c Show that 0.98 = 710 2 and hence find the value of 2 correct to 8 significant figures.

4 Expand each of the following in ascending powers of x up to and including the term in x3 and

state the set of values of x for which each expansion is valid.

a (2 + x)−1 b (4 + 12)x c (3 − x)−3 d (9 +

123 )x

e (8 − 1324 )x f (4 − 3x)−1 g (4 +

126 )x − h (3 + 2x)−2

5 a Expand (1 + 2x)−1, | x | < 1


b Hence find the series expansion of 11 2

xx

−+

, | x | < 12 , in ascending powers of x up to and

including the term in x3. 6 Find the first four terms in the series expansion in ascending powers of x of each of the following

and state the set of values of x for which each expansion is valid.

a 1 31

xx

+−

b 22 1

(1 4 )x

x−

+ c 3

2xx

+−

d 11 2

xx

−+

7 a Express 2(1 )(1 2 )

xx x

−− −


b Hence find the series expansion of 2(1 )(1 2 )

xx x

−− −

in ascending powers of x up to and

including the term in x3 and state the set of values of x for which the expansion is valid. 8 By first expressing f(x) in partial fractions, find the series expansion of f(x) in ascending powers

of x up to and including the term in x3 and state the set of values of x for which it is valid.

a f(x) ≡ 4(1 )(1 3 )x x+ −

b f(x) ≡ 21 6

1 3 4x

x x−

+ − c f(x) ≡ 2

52 3 2x x− −

d f(x) ≡ 27 3

4 3x

x x−

− + e f(x) ≡ 2

3 5(1 3 )(1 )

xx x+

+ + f f(x) ≡

2

22 4

2 1x

x x+

+ −

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SERIES C4 Worksheet B

1 a Expand (1 − 12)x , | x | < 1, in ascending powers of x up to and including the term in x3.

b By substituting x = 0.01 in your expansion, find the value of 11 correct to 9 significant figures.

2 The series expansion of (1 + 128 )x , in ascending powers of x up to and including the term in x3, is

1 + 4x + ax2 + bx3, | x | < 18 .

a Find the values of the constants a and b.

b Use the expansion, with x = 0.01, to find the value of 3 to 5 decimal places.

3 a Expand (9 − 126 )x , | x | < 3

2 , in ascending powers of x up to and including the term in x3, simplifying the coefficient in each term.

b Use your expansion with a suitable value of x to find the value of 8.7 correct to 7 significant figures.

4 a Expand (1 + 136 )x , | x | < 1


b Use your expansion, with x = 0.004, to find the cube root of 2 correct to 7 significant figures. 5 a Expand (1 + 2x)−3 in ascending powers of x up to and including the term in x3 and state the set

of values of x for which the expansion is valid.

b Hence, or otherwise, find the series expansion in ascending powers of x up to and including

the term in x3 of 31 3

(1 2 )xx

++

.

6 Find the coefficient of x2 in the series expansion of 24 2

xx

+−

, | x | < 2.

7 a Find the values of A and B such that

22 11

1 5 4x

x x−

− + ≡

1A

x− +

1 4B

x−.

b Hence, find the series expansion of 22 11

1 5 4x

x x−

− + in ascending powers of x up to and including

the term in x3 and state the set of values of x for which the expansion is valid.

8 f(x) ≡ 24 17

(1 2 )(1 3 )x

x x−

+ −, | x | < 1

3 .

a Express f(x) in partial fractions.

b Hence, or otherwise, find the series expansion of f(x) in ascending powers of x up to and including the term in x3.

9 The first three terms in the expansion of (1 + ax)b, in ascending powers of x, for | ax | < 1, are

1 − 6x + 24x2.

a Find the values of the constants a and b.

b Find the coefficient of x3 in the expansion.

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SERIES C4 Worksheet C 1 a Expand (1 −

124 )x in ascending powers of x up to and including the term in x3 and state

the set of values of x for which the expansion is valid. (4)

b By substituting x = 0.01 in your expansion, find the value of 6 to 6 significant figures. (3)

2 f(x) ≡ 24

1 2 3x x+ −.


b Hence, or otherwise, find the series expansion of f(x) in ascending powers of x up to and including the term in x3 and state the set of values of x for which the expansion is valid. (5)

3 a Expand (2 − x)−2, | x | < 2, in ascending powers of x up to and including the term in x3. (4)

b Hence, find the coefficient of x3 in the series expansion of 23

(2 )xx

−−

. (2)

4 f(x) ≡ 23

41 x+

, 32− < x < 3

2 .

a Show that f( 110 ) = 15 . (2)

b Expand f(x) in ascending powers of x up to and including the term in x2. (3)

c Use your expansion to obtain an approximation for 15 , giving your answer as an exact, simplified fraction. (2)

d Show that 55633 is a more accurate approximation for 15 . (2)

5 a Expand (1 − 13)x , | x | < 1, in ascending powers of x up to and including the term in x2. (3)

b By substituting x = 10−3 in your expansion, find the cube root of 37 correct to 9 significant figures. (3)

6 The series expansion of (1 + 355 )x , in ascending powers of x up to and including the

term in x3, is

1 + 3x + px2 + qx3, | x | < 15 .

a Find the values of the constants p and q. (4)

b Use the expansion with a suitable value of x to find an approximate value for 35(1.1) . (2)

c Obtain the value of 35(1.1) from your calculator and hence find the percentage error in

your answer to part b. (2) 7 a Find the values of A, B and C such that

2

28 6

(1 )(2 )x

x x−

+ + ≡

1A

x+ +

2B

x+ + 2(2 )

Cx+

. (4)

b Hence find the series expansion of 2

28 6

(1 )(2 )x

x x−

+ +, | x | < 1, in ascending powers of x up

to and including the term in x3, simplifying each coefficient. (7)

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8 a Expand (1 −

122 )x , | x | < 1

2 , in ascending powers of x up to and including the term in x2. (3)

b By substituting x = 0.0008 in your expansion, find the square root of 39 correct to 7 significant figures. (4)

9 a Find the series expansion of (1 + 138 )x , | x | < 1

8 , in ascending powers of x up to and including the term in x2, simplifying each term. (3)

b Find the exact fraction k such that

3 5 = k 3 1.08 (2)

c Hence, use your answer to part a together with a suitable value of x to obtain an estimate for 3 5 , giving your answer to 4 significant figures. (3)

10 f(x) ≡ 264 3x

x x− +, | x | < 1.


b Show that for small values of x,

f(x) ≈ 2x + 83 x2 + 26

9 x3. (5)

11 a Find the binomial expansion of (4 + 12)x in ascending powers of x up to and including

the term in x2 and state the set of values of x for which the expansion is valid. (4)

b By substituting x = 120 in your expansion, find an estimate for 5 , giving your

answer to 9 significant figures. (3)

c Obtain the value of 5 from your calculator and hence comment on the accuracy of the estimate found in part b. (2)

12 a Expand (1 +

122 )x − , | x | < 1

2 , in ascending powers of x up to and including the term in x3. (4)

b Hence, show that for small values of x,

2 51 2

xx

−+

≈ 2 − 7x + 8x2 − 252 x3. (3)

c Solve the equation

2 51 2

xx

−+

= 3 . (3)

d Use your answers to parts b and c to find an approximate value for 3 . (2) 13 a Expand (1 + x)−1, | x | < 1, in ascending powers of x up to and including the term in x3. (2)

b Hence, write down the first four terms in the expansion in ascending powers of x of (1 + bx)−1, where b is a constant, for | bx | < 1. (1)

Given that in the series expansion of

11

axbx

++

, | bx | < 1,

the coefficient of x is −4 and the coefficient of x2 is 12,

c find the values of the constants a and b, (5)

d find the coefficient of x3 in the expansion. (2)

C4 SERIES Worksheet C continued

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VECTORS C4 Worksheet A 1

G H I D E F A B C The diagram shows three sets of equally-spaced parallel lines.

Given that AC = p and that AD = q, express the following vectors in terms of p and q.

a CA b AG c AB d DF e HE f AF

g AH h DC i CG j IA k EC l IB 2 B O

C

In the quadrilateral shown, OA = u, AB = v and OC = w.

Find expressions in terms of u, v and w for

a OB b AC c CB 3 A B D C E F H G

The diagram shows a cuboid.

Given that AB = p, AD = q and AE = r, find expressions in terms of p, q and r for

a BC b AF c DE d AG e GB f BH 4 R S O T The diagram shows parallelogram ORST.

Given that OR = a + 2b and that OT = a − 2b,

a find expressions in terms of a and b for

i OS ii TR

Given also that OA = a and that OB = b,

b copy the diagram and show the positions of the points A and B.

A

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5 A

C D O B

The diagram shows triangle OAB in which OA = a and OB = b.

The points C and D are the mid-points of OA and AB respectively.

a Find and simplify expressions in terms of a and b for

i OC ii AB iii AD iv OD v CD

b Explain what your expression for CD tells you about OB and CD . 6 Given that vectors p and q are not parallel, state whether or not each of the following pairs of

vectors are parallel.

a 2p and 3p b (p + 2q) and (2p − 4q) c (3p − q) and (p − 13 q)

d (p − 2q) and (4q − 2p) e ( 34 p + q) and (6p + 8q) f (2q − 3p) and ( 3

2 q − p)

7 The points O, A, B and C are such that OA = 4m, OB = 4m + 2n and OC = 2m + 3n, where m and n are non-parallel vectors.

a Find an expression for BC in terms of m and n.

The point M is the mid-point of OC.

b Show that AM is parallel to BC.

8 The points O, A, B and C are such that OA = 6u − 4v, OB = 3u − v and OC = v − 3u, where u and v are non-parallel vectors.

The point M is the mid-point of OA and the point N is the point on AB such that AN : NB = 1 : 2

a Find OM and ON .

b Prove that C, M and N are collinear. 9 Given that vectors p and q are not parallel, find the values of the constants a and b such that

a ap + 3q = 5p + bq b (2p + aq) + (bp − 4q) = 0

c 4aq − p = bp − 2q d (2ap + bq) − (aq − 6p) = 0 10 A C

O B

The diagram shows triangle OAB in which OA = a and OB = b. The point C is the mid-point of OA and the point D is the mid-point of BC.

a Find an expression for OD in terms of a and b.

b Show that if the point E lies on AB then OE can be written in the form a + k(b − a), where k is a constant.

Given also that OD produced meets AB at E,

c find OE ,

d show that AE : EB = 2 : 1

C4 VECTORS Worksheet A continued

D

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VECTORS C4 Worksheet B 1 The points A, B and C have coordinates (6, 1), (2, 3) and (−4, 3) respectively and O is the origin.

Find, in terms of i and j, the vectors

a OA b AB c BC d CA 2 Given that p = i − 3j and q = 4i + 2j, find expressions in terms of i and j for

a 4p b q − p c 2p + 3q d 4p − 2q

3 Given that p = 34

−

and q = 12

, find

a | p | b | 2q | c | p + 2q | d | 3q − 2p | 4 Given that p = 2i + j and q = i − 3j, find, in degrees to 1 decimal place, the angle made with

the vector i by the vector

a p b q c 5p + q d p − 3q 5 Find a unit vector in the direction

a 43

b 724

−

c 11−

d 24

6 Find a vector

a of magnitude 26 in the direction 5i + 12j,

b of magnitude 15 in the direction −6i − 8j,

c of magnitude 5 in the direction 2i − 4j. 7 Given that m = 2i + λj and n = µi − 5j, find the values of λ and µ such that

a m + n = 3i − j b 2m − n = −3i + 8j 8 Given that r = 6i + cj, where c is a positive constant, find the value of c such that

a r is parallel to the vector 2i + j b r is parallel to the vector −9i − 6j

c | r | = 10 d | r | = 53 9 Given that p = i + 3j and q = 4i − 2j,

a find the values of a and b such that ap + bq = −5i + 13j,

b find the value of c such that cp + q is parallel to the vector j,

c find the value of d such that p + dq is parallel to the vector 3i − j.

10 Relative to a fixed origin O, the points A and B have position vectors 36

and 52

−

respectively.

Find

a the vector AB ,

b AB ,

c the position vector of the mid-point of AB,

d the position vector of the point C such that OABC is a parallelogram.

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11 Given the coordinates of the points A and B, find the length AB in each case.

a A (4, 0, 9), B (2, −3, 3) b A (11, −3, 5), B (7, −1, 3) 12 Find the magnitude of each vector.

a 4i + 2j − 4k b i + j + k c −8i − j + 4k d 3i − 5j + k 13 Find

a a unit vector in the direction 5i − 2j + 14k,

b a vector of magnitude 10 in the direction 2i + 11j − 10k,

c a vector of magnitude 20 in the direction −5i − 4j + 2k. 14 Given that r = λi + 12j − 4k, find the two possible values of λ such that | r | = 14.

15 Given that p = 131

−

, q = 42

1

−

and r = 2

53

− −

, find as column vectors,

a p + 2q b p − r c p + q + r d 2p − 3q + r 16 Given that r = −2i + λj + µk, find the values of λ and µ such that

a r is parallel to 4i + 2j − 8k b r is parallel to −5i + 20j − 10k 17 Given that p = i − 2j + 4k, q = −i + 2j + 2k and r = 2i − 4j − 7k,

a find | 2p − q |,

b find the value of k such that p + kq is parallel to r. 18 Relative to a fixed origin O, the points A, B and C have position vectors (−2i + 7j + 4k),

(−4i + j + 8k) and (6i − 5j) respectively.

a Find the position vector of the mid-point of AB.

b Find the position vector of the point D on AC such that AD : DC = 3 : 1 19 Given that r = λi − 2λj + µk, and that r is parallel to the vector (2i − 4j − 3k),

a show that 3λ + 2µ = 0.

Given also that | r | = 292 and that µ > 0,

b find the values of λ and µ.

20 Relative to a fixed origin O, the points A, B and C have position vectors 624

− −

, 12

74

− −

and 618

−

respectively.

a Find the position vector of the point M, the mid-point of BC.

b Show that O, A and M are collinear. 21 The position vector of a model aircraft at time t seconds is (9 − t)i + (1 + 2t)j + (5 − t)k, relative

to a fixed origin O. One unit on each coordinate axis represents 1 metre.

a Find an expression for d 2 in terms of t, where d metres is the distance of the aircraft from O.

b Find the value of t when the aircraft is closest to O and hence, the least distance of the aircraft from O.

C4 VECTORS Worksheet B continued

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VECTORS C4 Worksheet C 1 Sketch each line on a separate diagram given its vector equation.

a r = 2i + sj b r = s(i + j) c r = i + 4j + s(i + 2j) d r = 3j + s(3i − j) e r = −4i + 2j + s(2i − j) f r = (2s + 1)i + (3s − 2)j 2 Write down a vector equation of the straight line

a parallel to the vector (3i − 2j) which passes through the point with position vector (−i + j),

b parallel to the x-axis which passes through the point with coordinates (0, 4),

c parallel to the line r = 2i + t(i + 5j) which passes through the point with coordinates (3, −1). 3 Find a vector equation of the straight line which passes through the points with position vectors

a 10

and 31

b 34

−

and 11−

c 22

−

and 23

−

4 Find the value of the constant c such that line with vector equation r = 3i − j + λ(ci + 2j)

a passes through the point (0, 5),

b is parallel to the line r = −2i + 4j + µ(6i + 3j). 5 Find a vector equation for each line given its cartesian equation.

a x = −1 b y = 2x c y = 3x + 1

d y = 34 x − 2 e y = 5 − 1

2 x f x − 4y + 8 = 0 6 A line has the vector equation r = 2i + j + λ(3i + 2j).

a Write down parametric equations for the line.

b Hence find the cartesian equation of the line in the form ax + by + c = 0, where a, b and c are integers.

7 Find a cartesian equation for each line in the form ax + by + c = 0, where a, b and c are integers.

a r = 3i + λ(i + 2j) b r = i + 4j + λ(3i + j) c r = 2j + λ(4i − j) d r = −2i + j + λ(5i + 2j) e r = 2i − 3j + λ(−3i + 4j) f r = (λ + 3)i + (−2λ − 1)j 8 For each pair of lines, determine with reasons whether they are identical, parallel but not identical

or not parallel.

a r = 12

+ s 31

−

b r = 12−

+ s 14

c r = 25

−

+ s 24

r = 23

−

+ t 62

−

r = 24

−

+ t 41

r = 11−

+ t 36

9 Find the position vector of the point of intersection of each pair of lines.

a r = i + 2j + λi b r = 4i + j + λ(−i + j) c r = j + λ(2i − j) r = 2i + j + µ(3i + j) r = 5i − 2j + µ(2i − 3j) r = 2i + 10j + µ(−i + 3j) d r = −i + 5j + λ(−4i + 6j) e r = −2i + 11j + λ(−3i + 4j) f r = i + 2j + λ(3i + 2j) r = 2i − 2j + µ(−i + 2j) r = −3i − 7j + µ(5i + 3j) r = 3i + 5j + µ(i + 4j)

Solomon Press

10 Write down a vector equation of the straight line

a parallel to the vector (i + 3j − 2k) which passes through the point with position vector (4i + k),

b perpendicular to the xy-plane which passes through the point with coordinates (2, 1, 0),

c parallel to the line r = 3i − j + t(2i − 3j + 5k) which passes through the point with coordinates (−1, 4, 2).

11 The points A and B have position vectors (5i + j − 2k) and (6i − 3j + k) respectively.

a Find AB in terms of i, j and k.

b Write down a vector equation of the straight line l which passes through A and B.

c Show that l passes through the point with coordinates (3, 9, −8). 12 Find a vector equation of the straight line which passes through the points with position vectors

a (i + 3j + 4k) and (5i + 4j + 6k) b (3i − 2k) and (i + 5j + 2k)

c 0 and (6i − j + 2k) d (−i − 2j + 3k) and (4i − 7j + k) 13 Find the value of the constants a and b such that line r = 3i − 5j + k + λ(2i + aj + bk)

a passes through the point (9, −2, −8),

b is parallel to the line r = 4j − 2k + µ(8i − 4j + 2k). 14 Find cartesian equations for each of the following lines.

a r = 230

+ λ352

b r = 41

3

−

+ λ163

c r = 1

52

− −

+ λ421

− −

15 Find a vector equation for each line given its cartesian equations.

a 13

x − = 42

y + = z − 5 b 4x = 1

2y −−

= 73

z + c 54

x +−

= y + 3 = z 16 Show that the lines with vector equations r = 4i + 3k + s(i − 2j + 2k) and r = 7i + 2j − 5k + t(−3i + 2j + k) intersect, and find the coordinates of their point

of intersection. 17 Show that the lines with vector equations r = 2i − j + 4k + λ(i + j + 3k) and

r = i + 4j + 3k + µ(i − 2j + k) are skew. 18 For each pair of lines, find the position vector of their point of intersection or, if they do not

intersect, state whether they are parallel or skew.

a r = 315

+ λ411

−

and r = 324

−

+ µ102

b r = 031

+ λ213

− −

and r = 621

− −

+ µ4

26

−

c r = 824

−

+ λ132

−

and r = 2

28

−

+ µ434

− −

d r = 152

+ λ142

−

and r = 765

− −

+ µ213

−

e r = 41

3

−

+ λ253

−

and r = 32

1

−

+ µ534

− −

f r = 072

−

+ λ64

8

−

and r = 121

11

− −

+ µ523

−

C4 VECTORS Worksheet C continued

Solomon Press

VECTORS C4 Worksheet D 1 Calculate

a (i + 2j).(3i + j) b (4i − j).(3i + 5j) c (i − 2j).(−5i − 2j) 2 Show that the vectors (i + 4j) and (8i − 2j) are perpendicular. 3 Find in each case the value of the constant c for which the vectors u and v are perpendicular.

a u = 31

−

, v = 3c

b u = 21

, v = 3c

c u = 25

−

, v = 4

c −

4 Find, in degrees to 1 decimal place, the angle between the vectors

a (4i − 3j) and (8i + 6j) b (7i + j) and (2i + 6j) c (4i + 2j) and (−5i + 2j) 5 Relative to a fixed origin O, the points A, B and C have position vectors (9i + j), (3i − j)

and (5i − 2j) respectively. Show that ∠ABC = 45°. 6 Calculate

a (i + 2j + 4k).(3i + j + 2k) b (6i − 2j + 2k).(i − 3j − k)

c (−5i + 2k).(i + 4j − 3k) d (3i + 2j − 8k).(−i + 11j − 4k)

e (3i − 7j + k).(9i + 4j − k) f (7i − 3j).(−3j + 6k) 7 Given that p = 2i + j − 3k, q = i + 5j − k and r = 6i − 2j − 3k,

a find the value of p.q,

b find the value of p.r,

c verify that p.(q + r) = p.q + p.r 8 Simplify

a p.(q + r) + p.(q − r) b p.(q + r) + q.(r − p) 9 Show that the vectors (5i − 3j + 2k) and (3i + j − 6k) are perpendicular. 10 Relative to a fixed origin O, the points A, B and C have position vectors (3i + 4j − 6k),

(i + 5j − 2k) and (8i + 3j + 2k) respectively. Show that ∠ABC = 90°. 11 Find in each case the value or values of the constant c for which the vectors u and v are

perpendicular.

a u = (2i + 3j + k), v = (ci − 3j + k) b u = (−5i + 3j + 2k), v = (ci − j + 3ck)

c u = (ci − 2j + 8k), v = (ci + cj − 3k) d u = (3ci + 2j + ck), v = (5i − 4j + 2ck) 12 Find the exact value of the cosine of the angle between the vectors

a 122

−

and 814

−

b 412

−

and 2

36

− −

c 121

−

and 17

2

−

d 53

4

−

and 341

− −

13 Find, in degrees to 1 decimal place, the angle between the vectors

a (3i − 4k) and (7i − 4j + 4k) b (2i − 6j + 3k) and (i − 3j − k)

c (6i − 2j − 9k) and (3i + j + 4k) d (i + 5j − 3k) and (−3i − 4j + 2k)

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14 The points A (7, 2, −2), B (−1, 6, −3) and C (−3, 1, 2) are the vertices of a triangle.

a Find BA and BC in terms of i, j and k.

b Show that ∠ABC = 82.2° to 1 decimal place.

c Find the area of triangle ABC to 3 significant figures. 15 Relative to a fixed origin, the points A, B and C have position vectors (3i − 2j − k),

(4i + 3j − 2k) and (2i − j) respectively.

a Find the exact value of the cosine of angle BAC.

b Hence show that the area of triangle ABC is 3 2 . 16 Find, in degrees to 1 decimal place, the acute angle between each pair of lines.

a r = 131

−

+ λ44

2

−

and r = 52

1

−

+ µ806

−

b r = 03

7

−

+ λ61

18

− −

and r = 463

−

+ µ4123

−

c r = 715

+ λ11

3

−

and r = 2

63

− −

+ µ25

3

−

d r = 239

− −

+ λ46

7

− −

and r = 1112

−

+ µ518

− −

17 Relative to a fixed origin, the points A and B have position vectors (5i + 8j − k) and

(6i + 5j + k) respectively.

a Find a vector equation of the straight line l1 which passes through A and B.

The line l2 has the equation r = 4i − 3j + 5k + µ(−5i + j − 2k).

b Show that lines l1 and l2 intersect and find the position vector of their point of intersection.

c Find, in degrees, the acute angle between lines l1 and l2. 18 Find, in degrees to 1 decimal place, the acute angle between the lines with cartesian equations

23

x − = 2y = 5

6z +−

and 44

x −−

= 17

y + = 34

z −−

. 19 The line l has the equation r = 7i − 2k + λ(2i − j + 2k) and the line m has the equation r = −4i + 7j − 6k + µ(5i − 4j − 2k).

a Find the coordinates of the point A where lines l and m intersect.

b Find, in degrees, the acute angle between lines l and m.

The point B has coordinates (5, 1, −4).

c Show that B lies on the line l.

d Find the distance of B from m. 20 Relative to a fixed origin O, the points A and B have position vectors (9i + 6j) and (11i + 5j + k)

respectively.

a Show that for all values of λ, the point C with position vector (9 + 2λ)i + (6 − λ)j + λk lies on the straight line l which passes through A and B.

b Find the value of λ for which OC is perpendicular to l.

c Hence, find the position vector of the foot of the perpendicular from O to l. 21 Find the coordinates of the point on each line which is closest to the origin.

a r = −4i + 2j + 7k + λ(i + 3j − 4k) b r = 7i + 11j − 9k + λ(6i − 9j + 3k)

C4 VECTORS Worksheet D continued

Solomon Press

VECTORS C4 Worksheet E 1 Relative to a fixed origin, the line l has vector equation

r = i − 4j + pk + λ(2i + qj − 3k),

where λ is a scalar parameter.

Given that l passes through the point with position vector (7i − j − k),

a find the values of the constants p and q, (3)

b find, in degrees, the acute angle l makes with the line with equation

r = 3i + 4j − 3k + µ(−4i + 5j − 2k). (4)

2 The points A and B have position vectors 164

and 506

−

respectively, relative to a

fixed origin.

a Find, in vector form, an equation of the line l which passes through A and B. (2)

The line m has equation

r = 55

3

−

+ t14

2

−

.

Given that lines l and m intersect at the point C,

b find the position vector of C, (5)

c show that C is the mid-point of AB. (2) 3 Relative to a fixed origin, the points P and Q have position vectors (5i − 2j + 2k) and

(3i + j) respectively.

a Find, in vector form, an equation of the line L1 which passes through P and Q. (2)

The line L2 has equation

r = 4i + 6j − k + µ(5i − j + 3k).

b Show that lines L1 and L2 intersect and find the position vector of their point of intersection. (6)

c Find, in degrees to 1 decimal place, the acute angle between lines L1 and L2. (4) 4 Relative to a fixed origin, the lines l1 and l2 have vector equations as follows:

l1 : r = 5i + k + λ(2i − j + 2k),

l2 : r = 7i − 3j + 7k + µ(−i + j − 2k),

where λ and µ are scalar parameters.

a Show that lines l1 and l2 intersect and find the position vector of their point of intersection. (6)

The points A and C lie on l1 and the points B and D lie on l2.

Given that ABCD is a parallelogram and that A has position vector (9i − 2j + 5k),

b find the position vector of C. (3)

Given also that the area of parallelogram ABCD is 54,

c find the distance of the point B from the line l1. (4)

Solomon Press

5 Relative to a fixed origin, the points A and B have position vectors (4i + 2j − 4k) and

(2i − j + 2k) respectively.

a Find, in vector form, an equation of the line l1 which passes through A and B. (2)

The line l2 passes through the point C with position vector (4i − 7j − k) and is parallel to the vector (6j − 2k).

b Write down, in vector form, an equation of the line l2. (1)

c Show that A lies on l2. (2)

d Find, in degrees, the acute angle between lines l1 and l2. (4)


10

− −

and 418

−


fixed origin O.


The line l intersects the y-axis at the point C.

b Find the coordinates of C. (2)

The point D on the line l is such that OD is perpendicular to l.

c Find the coordinates of D. (5)

d Find the area of triangle OCD, giving your answer in the form 5k . (3) 7 Relative to a fixed origin, the line l1 has the equation

r = 162

− −

+ s041

−

.

a Show that the point P with coordinates (1, 6, −5) lies on l1. (1)

The line l2 has the equation

r = 441

− −

+ t32

2

−

,

and intersects l1 at the point Q.

b Find the position vector of Q. (3)

The point R lies on l2 such that PQ = QR.

c Find the two possible position vectors of the point R. (5) 8 Relative to a fixed origin, the points A and B have position vectors (4i + 5j + 6k) and

(4i + 6j + 2k) respectively.

a Find, in vector form, an equation of the line l1 which passes through A and B. (2)

The line l2 has equation

r = i + 5j − 3k + µ(i + j − k).

b Show that l1 and l2 intersect and find the position vector of their point of intersection. (4)

c Find the acute angle between lines l1 and l2. (3)

d Show that the point on l2 closest to A has position vector (−i + 3j − k). (5)

C4 VECTORS Worksheet E continued

Solomon Press

VECTORS C4 Worksheet F


− −

and 034

−


fixed origin.


The line m has equation

r = 65

1

−

+ µ 31

a −

,

where a is a constant.

Given that lines l and m intersect,

b find the value of a and the coordinates of the point where l and m intersect. (6) 2 Relative to a fixed origin, the points A, B and C have position vectors (−4i + 2j − k),

(2i + 5j − 7k) and (6i + 4j + k) respectively.

a Show that cos (∠ABC) = 13 . (3)

The point M is the mid-point of AC.

b Find the position vector of M. (2)

c Show that BM is perpendicular to AC. (3)

d Find the size of angle ACB in degrees. (3)

3 Relative to a fixed origin O, the points A and B have position vectors 953

−

and 1173

−

respectively.

a Find, in vector form, an equation of the line L which passes through A and B. (2)

The point C lies on L such that OC is perpendicular to L.

b Find the position vector of C. (5)

c Find, to 3 significant figures, the area of triangle OAC. (3)

d Find the exact ratio of the area of triangle OAB to the area of triangle OAC. (2) 4 Relative to a fixed origin O, the points A and B have position vectors (7i − 5j − k) and

(4i − 5j + 3k) respectively.

a Find cos (∠AOB), giving your answer in the form 6k , where k is an exact fraction. (4)

b Show that AB is perpendicular to OB. (3)

The point C is such that OC = 32 OB .

c Show that AC is perpendicular to OA. (3)

d Find the size of ∠ACO in degrees to 1 decimal place. (3)

Solomon Press

DIFFERENTIATION C4 Worksheet A 1 A curve is given by the parametric equations

x = t 2 + 1, y = 4t

.

a Write down the coordinates of the point on the curve where t = 2.

b Find the value of t at the point on the curve with coordinates ( 54 , −8).

2 A curve is given by the parametric equations

x = 1 + sin t, y = 2 cos t, 0 ≤ t < 2π.

a Write down the coordinates of the point on the curve where t = π2 .

b Find the value of t at the point on the curve with coordinates ( 32 , 3− ).

3 Find a cartesian equation for each curve, given its parametric equations.

a x = 3t, y = t 2 b x = 2t, y = 1t

c x = t 3, y = 2t 2

d x = 1 − t 2, y = 4 − t e x = 2t − 1, y = 22t

f x = 11t −

, y = 12 t−

4 A curve has parametric equations

x = 2t + 1, y = t2.

a Find a cartesian equation for the curve.

b Hence, sketch the curve. 5 Find a cartesian equation for each curve, given its parametric equations.

a x = cos θ, y = sin θ b x = sin θ, y = cos 2θ c x = 3 + 2 cos θ, y = 1 + 2 sin θ

d x = 2 sec θ, y = 4 tan θ e x = sin θ, y = sin2 2θ f x = cos θ, y = tan2 θ 6 A circle has parametric equations

x = 1 + 3 cos θ, y = 4 + 3 sin θ, 0 ≤ θ < 2π.

a Find a cartesian equation for the circle.

b Write down the coordinates of the centre and the radius of the circle.

c Sketch the circle and label the points on the circle where θ takes each of the following values:

0, π4 , π2 , 3π4 , π, 5π

4 , 3π2 , 7π

4 . 7 Write down parametric equations for a circle

a centre (0, 0), radius 5,

b centre (6, −1), radius 2,

c centre (a, b), radius r, where a, b and r are constants and r > 0. 8 For each curve given by parametric equations, find a cartesian equation and hence, sketch the

curve, showing the coordinates of any points where it meets the coordinate axes.

a x = 2t, y = 4t(t − 1) b x = 1 − sin θ, y = 2 − cos θ, 0 ≤ θ < 2π

c x = t − 3, y = 4 − t2 d x = t + 1, y = 2t

Solomon Press

DIFFERENTIATION C4 Worksheet B 1 A curve is given by the parametric equations

x = 2 + t, y = t 2 − 1.

a Write down expressions for ddxt

and ddyt

.

b Hence, show that ddyx

= 2t.

2 Find and simplify an expression for ddyx

in terms of the parameter t in each case.

a x = t 2, y = 3t b x = t 2 − 1, y = 2t 3 + t 2 c x = 2 sin t, y = 6 cos t

d x = 3t − 1, y = 2 − 1t

e x = cos 2t, y = sin t f x = et + 1, y = e2t − 1

g x = sin2 t, y = cos3 t h x = 3 sec t, y = 5 tan t i x = 11t +

, y = 1

tt −

3 Find, in the form y = mx + c, an equation for the tangent to the given curve at the point with the

given value of the parameter t.

a x = t 3, y = 3t 2, t = 1 b x = 1 − t 2, y = 2t − t 2, t = 2

c x = 2 sin t, y = 1 − 4 cos t, t = π3 d x = ln (4 − t), y = t 2 − 5, t = 3 4 Show that the normal to the curve with parametric equations

x = sec θ, y = 2 tan θ, 0 ≤ θ < π2 ,

at the point where θ = π3 , has the equation

3 x + 4y = 10 3 . 5 A curve is given by the parametric equations

x = 1t

, y = 12t +

.

a Show that ddyx

= 2

2t

t +

.

b Find an equation for the normal to the curve at the point where t = 2, giving your answer in the form ax + by + c = 0, where a, b and c are integers.


x = sin 2t, y = sin2 t, 0 ≤ t < π.

a Show that ddyx

= 12 tan 2t.

b Find an equation for the tangent to the curve at the point where t = π6 . 7 A curve has parametric equations

x = 3 cos θ, y = 4 sin θ, 0 ≤ θ < 2π.

a Show that the tangent to the curve at the point (3 cos α, 4 sin α) has the equation

3y sin α + 4x cos α = 12.

b Hence find an equation for the tangent to the curve at the point ( 32− , 2 3 ).

Solomon Press

8 A curve is given by the parametric equations

x = t 2, y = t(t − 2), t ≥ 0.

a Find the coordinates of any points where the curve meets the coordinate axes.

b Find ddyx

in terms of x

i by first finding ddyx

in terms of t,

ii by first finding a cartesian equation for the curve. 9 y O x

The diagram shows the ellipse with parametric equations

x = 1 − 2 cos θ, y = 3 sin θ, 0 ≤ θ < 2π.

a Find ddyx

in terms of θ.

b Find the coordinates of the points where the tangent to the curve is

i parallel to the x-axis,

ii parallel to the y-axis. 10 A curve is given by the parametric equations

x = sin θ, y = sin 2θ, 0 ≤ θ ≤ π2 .

a Find the coordinates of any points where the curve meets the coordinate axes.

b Find an equation for the tangent to the curve that is parallel to the x-axis.

c Find a cartesian equation for the curve in the form y = f(x). 11 A curve has parametric equations

x = sin2 t, y = tan t, − π2 < t < π2 .

a Show that the tangent to the curve at the point where t = π4 passes through the origin.

b Find a cartesian equation for the curve in the form y2 = f(x). 12 A curve is given by the parametric equations

x = t + 1t

, y = t − 1t

, t ≠ 0.

a Find an equation for the tangent to the curve at the point P where t = 3.

b Show that the tangent to the curve at P does not meet the curve again.

c Show that the cartesian equation of the curve can be written in the form

x2 − y2 = k,

where k is a constant to be found.

C4 DIFFERENTIATION Worksheet B continued

Solomon Press

DIFFERENTIATION C4 Worksheet C 1 Differentiate with respect to x

a 4y b y3 c sin 2y d 2

3e y

2 Find ddyx

in terms of x and y in each case.

a x2 + y2 = 2 b 2x − y + y2 = 0 c y4 = x2 − 6x + 2

d x2 + y2 + 3x − 4y = 9 e x2 − 2y2 + x + 3y − 4 = 0 f sin x + cos y = 0

g 2e3x + e−2y + 7 = 0 h tan x + cosec 2y = 1 i ln (x − 2) = ln (2y + 1) 3 Differentiate with respect to x

a xy b x2y3 c sin x tan y d (x − 2y)3

4 Find ddyx

in terms of x and y in each case.

a x2y = 2 b x2 + 3xy − y2 = 0 c 4x2 − 2xy + 3y2 = 8

d cos 2x sec 3y + 1 = 0 e y = (x + y)2 f xey − y = 5

g 2xy2 − x3y = 0 h y2 + x ln y = 3 i x sin y + x2 cos y = 1 5 Find an equation for the tangent to each curve at the given point on the curve.

a x2 + y2 − 3y − 2 = 0, (2, 1) b 2x2 − xy + y2 = 28, (3, 5)

c 4 sin y − sec x = 0, ( π3 , π6 ) d 2 tan x cos y = 1, ( π

4 , π3 ) 6 A curve has the equation x2 + 2y2 − x + 4y = 6.

a Show that ddyx

= 1 24( 1)

xy−

+.

b Find an equation for the normal to the curve at the point (1, −3). 7 A curve has the equation x2 + 4xy − 3y2 = 36.

a Find an equation for the tangent to the curve at the point P (4, 2).

Given that the tangent to the curve at the point Q on the curve is parallel to the tangent at P,

b find the coordinates of Q. 8 A curve has the equation y = ax, where a is a positive constant.

By first taking logarithms, find an expression for ddyx

in terms of a and x.

9 Differentiate with respect to x

a 3x b 62x c 51 − x d 3

2x 10 A biological culture is growing exponentially such that the number of bacteria present, N, at time

t minutes is given by

N = 800 (1.04)t.

Find the rate at which the number of bacteria is increasing when there are 4000 bacteria present.

Solomon Press

DIFFERENTIATION C4 Worksheet D 1 Given that y = x2 + 3x + 5,

and that x = (t − 4)3,

a find expressions for

i ddyx

in terms of x, ii ddxt

in terms of t,

b find the value of ddyt

when

i t = 5, ii x = 8. 2 The variables x and y are related by the equation y = 2 3x x − .

Given that x is increasing at the rate of 0.3 units per second when x = 6, find the rate at which y is increasing at this instant.

3 The radius of a circle is increasing at a constant rate of 0.2 cm s−1.

a Show that the perimeter of the circle is increasing at the rate of 0.4π cm s−1.

b Find the rate at which the area of the circle is increasing when the radius is 10 cm.

c Find the radius of the circle when its area is increasing at the rate of 20 cm2 s−1. 4 The area of a circle is decreasing at a constant rate of 0.5 cm2 s−1.

a Find the rate at which the radius of the circle is decreasing when the radius is 8 cm.

b Find the rate at which the perimeter of the circle is decreasing when the radius is 8 cm. 5 The volume of a cube is increasing at a constant rate of 3.5 cm3 s−1. Find

a the rate at which the length of one side of the cube is increasing when the volume is 200 cm3,

b the volume of the cube when the length of one side is increasing at the rate of 2 mm s−1. 6

h cm 60°

The diagram shows the cross-section of a right-circular paper cone being used as a filter funnel. The volume of liquid in the funnel is V cm3 when the depth of the liquid is h cm.

Given that the angle between the sides of the funnel in the cross-section is 60° as shown,

a show that V = 19 πh3.

Given also that at time t seconds after liquid is put in the funnel

V = 600e−0.0005t,

b show that after two minutes, the depth of liquid in the funnel is approximately 11.7 cm,

c find the rate at which the depth of liquid is decreasing after two minutes.

Solomon Press

DIFFERENTIATION C4 Worksheet E 1 A curve has the equation

3x2 + xy − y2 + 9 = 0.

Find an expression for ddyx

in terms of x and y. (5)


x = a cos θ, y = a(sin θ − θ ), 0 ≤ θ < π,

where a is a positive constant.

a Show that ddyx

= tan 2θ . (5)

b Find, in terms of a, an equation for the tangent to the curve at the point where it crosses the y-axis. (3)

3 y

O x

The diagram shows the curve with parametric equations

x = cos θ, y = 12 sin 2θ, 0 ≤ θ < 2π.

a Find ddyx

in terms of θ. (3)

b Find the two values of θ for which the curve passes through the origin. (2)

c Show that the two tangents to the curve at the origin are perpendicular to each other. (2)

d Find a cartesian equation for the curve. (4) 4 A curve has the equation

x2 − 4xy + y2 = 24.

a Show that ddyx

= 22x y

x y−

−. (4)

b Find an equation for the tangent to the curve at the point P (2, 10). (3)

The tangent to the curve at Q is parallel to the tangent at P.

c Find the coordinates of Q. (4) 5 A curve is given by the parametric equations

x = t 2 + 2, y = t(t − 1).

a Find the coordinates of any points on the curve where the tangent to the curve is parallel to the x-axis. (5)

b Show that the tangent to the curve at the point (3, 2) has the equation

3x − 2y = 5. (5)

Solomon Press

6 Find an equation for the normal to the curve with equation

x3 − 3x + xy − 2y2 + 3 = 0

at the point (1, 1).

Give your answer in the form y = mx + c. (7) 7

h cm

The diagram shows the cross-section of a vase. The volume of water in the vase, V cm3, when the depth of water in the vase is h cm is given by

V = 40π(e0.1h − 1).

The vase is initially empty and water is poured into it at a constant rate of 80 cm3 s−1.

Find the rate at which the depth of water in the vase is increasing

a when h = 4, (5)

b after 5 seconds of pouring water in. (4) 8 A curve is given by the parametric equations

x = 1

tt+

, y = 1

tt−, t ≠ ± 1.

a Show that ddyx

= 21

1tt

+ −

. (4)

b Show that the normal to the curve at the point P, where t = 12 , has the equation

3x + 27y = 28. (4)

The normal to the curve at P meets the curve again at the point Q.

c Find the exact value of the parameter t at Q. (4) 9 A curve has the equation

2x + x2y − y2 = 0.

Find the coordinates of the point on the curve where the tangent is parallel to the x-axis. (8) 10 A curve has parametric equations

x = a sec θ, y = 2a tan θ, − π2 ≤ θ < π2 ,

where a is a positive constant.

a Find ddyx

in terms of θ. (3)

b Show that the normal to the curve at the point where θ = π4 has the equation

x + 2 2 y = 5 2 a. (4)

c Find a cartesian equation for the curve in the form y2 = f(x). (3)

C4 DIFFERENTIATION Worksheet E continued

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DIFFERENTIATION C4 Worksheet F 1 A curve has parametric equations

x = t 2, y = 2t

.

a Find ddyx

in terms of t. (3)

b Find an equation for the normal to the curve at the point where t = 2, giving your answer in the form y = mx + c. (3)

2 A curve has the equation y = 4x.

Show that the tangent to the curve at the point where x = 1 has the equation

y = 4 + 8(x − 1) ln 2. (4) 3 A curve has parametric equations

x = sec θ, y = cos 2θ, 0 ≤ θ < π2 .

a Show that ddyx

= −4 cos3 θ. (4)

b Show that the tangent to the curve at the point where θ = π6 has the equation

3 3 x + 2y = k,

where k is an integer to be found. (4) 4 A curve has the equation

2x2 + 6xy − y2 + 77 = 0

and passes through the point P (2, −5).

a Show that the normal to the curve at P has the equation

x + y + 3 = 0. (6)

b Find the x-coordinate of the point where the normal to the curve at P intersects the curve again. (3)

5 y

O x The diagram shows the curve with parametric equations

x = θ − sin θ, y = cos θ, 0 ≤ θ ≤ 2π.

a Find the exact coordinates of the points where the curve crosses the x-axis. (3)

b Show that ddyx

= −cot 2θ . (5)

c Find the exact coordinates of the point on the curve where the tangent to the curve is parallel to the x-axis. (2)

Solomon Press


x = sin θ, y = sec2 θ, − π2 < θ < π2 .

The point P on the curve has x-coordinate 12 .

a Write down the value of the parameter θ at P. (1)

b Show that the tangent to the curve at P has the equation

16x − 9y + 4 = 0. (6)

c Find a cartesian equation for the curve. (2) 7 A curve has the equation

2 sin x − tan 2y = 0.

a Show that ddyx

= cos x cos2 2y. (4)

b Find an equation for the tangent to the curve at the point ( π3 , π6 ), giving your answer

in the form ax + by + c = 0. (3) 8 y O x

A particle moves on the ellipse shown in the diagram such that at time t its coordinates are given by

x = 4 cos t, y = 3 sin t, t ≥ 0.

a Find ddyx

in terms of t. (3)

b Show that at time t, the tangent to the path of the particle has the equation

3x cos t + 4y sin t = 12. (3)

c Find a cartesian equation for the path of the particle. (3) 9 The curve with parametric equations

x = 1

tt +

, y = 21

tt −

,

passes through the origin, O.

a Show that ddyx

= −221

1tt

+ −

. (4)

b Find an equation for the normal to the curve at O. (2)

c Find the coordinates of the point where the normal to the curve at O meets the curve again. (4)

d Show that the cartesian equation of the curve can be written in the form

y = 22 1

xx −

. (4)

C4 DIFFERENTIATION Worksheet F continued

Solomon Press

INTEGRATION C4 Worksheet A 1 Integrate with respect to x

a ex b 4ex c 1x

d 6x

2 Integrate with respect to t

a 2 + 3et b t + t −1 c t 2 − et d 9 − 2t −1

e 7t

+ t f 14 et − 1

t g 1

3t + 2

1t

h 25t

− 3e7

t

3 Find

a ∫ (5 − 3x

) dx b ∫ (u−1 + u−2) du c ∫ 2e 15

t + dt

d ∫ 3 1yy+ dy e ∫ ( 3

4 et + 3 t ) dt f ∫ (x − 1x

)2 dx

4 The curve y = f(x) passes through the point (1, −3).

Given that f ′(x) = 2(2 1)x

x− , find an expression for f(x).

5 Evaluate

a 1

0∫ (ex + 10) dx b 5

2∫ (t + 1t

) dt c 4

1∫25 x

x− dx

d 1

2

−

−∫6 1

3y

y+ dy e

3

3−∫ (ex − x2) dx f 3

2∫2

24 3 6r r

r− + dr

g ln 4

ln 2∫ (7 − eu) du h 10

6∫1 1 12 2 2(2 9 )r r r− −+ dr i

9

4∫ ( 1x

+ 3ex) dx

6 y

y = 3 + ex

O 2 x The shaded region on the diagram is bounded by the curve y = 3 + ex, the coordinate axes and

the line x = 2. Show that the area of the shaded region is e2 + 5. 7 y y = 2x + 1

x

O 1 4 x

The shaded region on the diagram is bounded by the curve y = 2x + 1x

, the x-axis and the lines

x = 1 and x = 4. Find the area of the shaded region in the form a + b ln 2.

Solomon Press

8 Find the exact area of the region enclosed by the given curve, the x-axis and the given ordinates.

In each case, y > 0 over the interval being considered.

a y = 4x + 2ex, x = 0, x = 1 b y = 1 + 3x

, x = 2, x = 4

c y = 4 − 1x

, x = −3, x = −1 d y = 2 − 12 ex, x = 0, x = ln 2

e y = ex + 5x

, x = 12 , x = 2 f y =

3 2xx− , x = 2, x = 3

9 y y = 9 − 7

x − 2x

O x The diagram shows the curve with equation y = 9 − 7

x − 2x, x > 0.

a Find the coordinates of the points where the curve crosses the x-axis.

b Show that the area of the region bounded by the curve and the x-axis is 1411 − 7 ln 7

2 . 10 a Sketch the curve y = ex − a where a is a constant and a > 1.

Show on your sketch the coordinates of any points of intersection with the coordinate axes and the equation of any asymptotes.

b Find, in terms of a, the area of the finite region bounded by the curve y = ex − a and the coordinate axes.

c Given that the area of this region is 1 + a, show that a = e2. 11 y P

y = ex

O Q x R The diagram shows the curve with equation y = ex. The point P on the curve has x-coordinate 3,

and the tangent to the curve at P intersects the x-axis at Q and the y-axis at R.

a Find an equation of the tangent to the curve at P.

b Find the coordinates of the points Q and R.

The shaded region is bounded by the curve, the tangent to the curve at P and the y-axis.

c Find the exact area of the shaded region. 12 f(x) ≡ ( 3

x − 4)2, x ∈ , x > 0.

a Find the coordinates of the point where the curve y = f(x) meets the x-axis.

The finite region R is bounded by the curve y = f(x), the line x = 1 and the x-axis.

b Show that the area of R is approximately 0.178

C4 INTEGRATION Worksheet A continued

Solomon Press

INTEGRATION C4 Worksheet B 1 Integrate with respect to x

a (x − 2)7 b (2x + 5)3 c 6(1 + 3x)4 d ( 14 x − 2)5

e (8 − 5x)4 f 21

( 7)x + g 5

8(4 3)x −

h 31

2(5 3 )x−

2 Find

a ∫32(3 )t+ dt b ∫ 4 1x − dx c ∫ 1

2 1y + dy

d ∫ e2x − 3 dx e ∫ 32 7r−

dr f ∫ 3 5 2t − dt

g ∫ 16 y−

dy h ∫ 5e7 − 3t dt i ∫ 43 1u +

du

3 Given f ′(x) and a point on the curve y = f(x), find an expression for f(x) in each case.

a f ′(x) = 8(2x − 3)3, (2, 6) b f ′(x) = 6e2x + 4, (−2, 1)

c f ′(x) = 2 − 84 1x −

, ( 12 , 4) d f ′(x) = 8x − 2

3(3 2)x −

, (−1, 3)

4 Evaluate

a 1

0∫ (3x + 1)2 dx b 2

1∫ (2x − 1)3 dx c 4

2∫ 21

(5 )x− dx

d 1

1−∫ e2x + 2 dx e 6

2∫ 3 2x − dx f 2

1∫4

6 3x − dx

g 1

0∫ 31

7 1x + dx h

1

7

−

−∫1

5 3x + dx i

7

4∫34

2x −

dx


In each case, y > 0 over the interval being considered.

a y = e3 − x, x = 3, x = 4 b y = (3x − 5)3, x = 2, x = 3

c y = 34 2x +

, x = 1, x = 4 d y = 21

(1 2 )x−, x = −2, x = 0

6 y

y = 312

(2 1)x +

O 1 x

The diagram shows part of the curve with equation y = 312

(2 1)x +.

Find the area of the shaded region bounded by the curve, the coordinate axes and the line x = 1.

Solomon Press

INTEGRATION C4 Worksheet C 1 a Express 3 5

( 1)( 3)x

x x+

+ + in partial fractions.

b Hence, find ∫ 3 5( 1)( 3)

xx x

++ +

dx.

2 Show that ∫ 3( 2)( 1)t t− +

dt = ln 21

tt−+

+ c.

3 Integrate with respect to x

a 6 11(2 1)( 3)

xx x

−+ −

b 214

2 8x

x x−

+ − c 6

(2 )(1 )x x+ − d 2

15 14 8

xx x

+− +

4 a Find the values of the constants A, B and C such that

2 6

( 4)( 1)x

x x−

+ − ≡ A +

4B

x + +

1C

x −.

b Hence, find ∫2 6

( 4)( 1)x

x x−

+ − dx.

5 a Express 2

24

( 2)( 1)x x

x x− −

+ + in partial fractions.

b Hence, find ∫2

24

( 2)( 1)x x

x x− −

+ + dx.


a 2

23 5

1xx

−−

b 2(4 13)

(2 ) (3 )x x

x x+

+ − c

2

21

3 10x x

x x− +

− − d

2

22 6 5

(1 2 )x x

x x− +

−

7 Show that 4

3∫3 5

( 1)( 2)x

x x−

− − dx = 2 ln 3 − ln 2.

8 Find the exact value of

a 3

1∫3

( 1)x

x x++

dx b 2

0∫ 23 2

12x

x x−

+ − dx c

2

1∫ 29

2 7 4x x− − dx

d 2

0∫2

22 7 7

2 3x xx x

− +− −

dx e 1

0∫ 25 7

( 1) ( 3)x

x x+

+ + dx f

1

1−∫ 22

8 2x

x x+

− − dx

9 a Express 2 2

1x a−

, where a is a positive constant, in partial fractions.

b Hence, show that ∫ 2 21

x a− dx = 1

2aln x a

x a−+

+ c.

c Find ∫ 2 21

a x− dx.

10 Evaluate

a 1

1−∫ 21

9x − dx b

12

12−∫ 2

41 x−

dx c 1

0∫ 23

2 8x − dx

Solomon Press

INTEGRATION C4 Worksheet D 1 Integrate with respect to x

a 2 cos x b sin 4x c cos 12 x d sin (x + π4 )

e cos (2x − 1) f 3 sin ( π3 − x) g sec x tan x h cosec2 x

i 5 sec2 2x j cosec 14 x cot 1

4 x k 24

sin x l 2

1cos (4 1)x +

2 Evaluate

a π2

0∫ cos x dx b π6

0∫ sin 2x dx c π2

0∫ 2 sec 12 x tan 1

2 x dx

d π3

0∫ cos (2x − π3 ) dx e π3π4∫ sec2 3x dx f

2π3π2∫ cosec x cot x dx

3 a Express tan2 θ in terms of sec θ.

b Show that ∫ tan2 x dx = tan x − x + c.

4 a Use the identity for cos (A + B) to express cos2 A in terms of cos 2A.

b Find ∫ cos2 x dx.

5 Find

a ∫ sin2 x dx b ∫ cot2 2x dx c ∫ sin x cos x dx

d ∫ 2sin

cosxx

dx e ∫ 4 cos2 3x dx f ∫ (1 + sin x)2 dx

g ∫ (sec x − tan x)2 dx h ∫ cosec 2x cot x dx i ∫ cos4 x dx

6 Evaluate

a π2

0∫ 2 cos2 x dx b π4

0∫ cos 2x sin 2x dx c π2π3∫ tan2 1

2 x dx

d π4π6∫ 2

cos2sin 2

xx

dx e π4

0∫ (1 − 2 sin x)2 dx f π3π6∫ sec2 x cosec2 x dx

7 a Use the identities for sin (A + B) and sin (A − B) to show that

sin A cos B ≡ 12 [sin (A + B) + sin (A − B)].

b Find ∫ sin 3x cos x dx.


a 2 sin 5x sin x b cos 2x cos x c 4 sin x cos 4x d cos (x + π6 ) sin x

Solomon Press

INTEGRATION C4 Worksheet E 1 Showing your working in full, use the given substitution to find

a ∫ 2x(x2 − 1)3 dx u = x2 + 1 b ∫ sin4 x cos x dx u = sin x

c ∫ 3x2(2 + x3)2 dx u = 2 + x3 d ∫2

2 exx dx u = x2

e ∫ 2 4( 3)x

x + dx u = x2 + 3 f ∫ sin 2x cos3 2x dx u = cos 2x

g ∫ 23

2x

x − dx u = x2 − 2 h ∫ 21x x− dx u = 1 − x2

i ∫ sec3 x tan x dx u = sec x j ∫ (x + 1)(x2 + 2x)3 dx u = x2 + 2x

2 a Given that u = x2 + 3, find the value of u when

i x = 0 ii x = 1

b Using the substitution u = x2 + 3, show that

1

0∫ 2x(x2 + 3)2 dx = 4

3∫ u2 du.

c Hence, show that

1

0∫ 2x(x2 + 3)2 dx = 1312 .

3 Using the given substitution, evaluate

a 2

1∫ x(x2 − 3)3 dx u = x2 − 3 b π6

0∫ sin3 x cos x dx u = sin x

c 3

0∫ 24

1x

x + dx u = x2 + 1 d

π4π4−∫ tan2 x sec2 x dx u = tan x

e 3

2∫ 2 3

x

x − dx u = x2 − 3 f

1

2

−

−∫ x2(x3 + 2)2 dx u = x3 + 2

g 1

0∫ e2x(1 + e2x)3 dx u = 1 + e2x h 5

3∫ (x − 2)(x2 − 4x)2 dx u = x2 − 4x

4 a Using the substitution u = 4 − x2, show that

2

0∫ x(4 − x2)3 dx = 4

0∫ 12 u3 du.

b Hence, evaluate

2

0∫ x(4 − x2)3 dx.


a 1

0∫22e xx − dx u = 2 − x2 b

π2

0∫sin

1 cosx

x+ dx u = 1 + cos x

Solomon Press

6 a By writing cot x as cos

sinxx

, use the substitution u = sin x to show that

∫ cot x dx = lnsin x + c.

b Show that

∫ tan x dx = lnsec x + c.

c Evaluate

π6

0∫ tan 2x dx.

7 By recognising a function and its derivative, or by using a suitable substitution, integrate with

respect to x

a 3x2(x3 − 2)3 b esin x cos x c 2 1x

x +

d (2x + 3)(x2 + 3x)2 e 2 4x x + f cot3 x cosec2 x

g e1 e

x

x+ h cos2

3 sin 2x

x+ i

3

4 2( 2)x

x −

j 3(ln )x

x k

312 2 2(1 )x x+ l

25

x

x−

8 Evaluate

a π2

0∫ sin x (1 + cos x)2 dx b 0

1−∫2

2e

2 e

x

x− dx

c π4π6∫ cot x cosec4 x dx d

4

2∫ 21

2 8x

x x+

+ + dx

9 Using the substitution u = x + 1, show that

∫ x(x + 1)3 dx = 120 (4x − 1)(x + 1)4 + c.

10 Using the given substitution, find

a ∫ x(2x − 1)4 dx u = 2x − 1 b ∫ 1x x− dx u2 = 1 − x

c ∫ 322

1

(1 )x− dx x = sin u d ∫ 1

1x − dx x = u2

e ∫ (x + 1)(2x + 3)3 dx u = 2x + 3 f ∫2

2xx −

dx u2 = x − 2


a 12

0∫ 2

1

1 x− dx x = sin u b

2

0∫ x(2 − x)3 dx u = 2 − x

c 1

0∫24 x− dx x = 2 sin u d

3

0∫2

2 9x

x + dx x = 3 tan u

C4 INTEGRATION Worksheet E continued

Solomon Press

INTEGRATION C4 Worksheet F 1 Using integration by parts, show that

∫ x cos x dx = x sin x + cos x + c. 2 Use integration by parts to find

a ∫ xex dx b ∫ 4x sin x dx c ∫ x cos 2x dx

d ∫ 1x x + dx e ∫ 3e xx dx f ∫ x sec2 x dx

3 Using

i integration by parts, ii the substitution u = 2x + 1,

find ∫ x(2x + 1)3 dx, and show that your answers are equivalent. 4 Show that

2

0∫ xe−x dx = 1 − 3e−2.

5 Evaluate

a π6

0∫ x cos x dx b 1

0∫ xe2x dx c π4

0∫ x sin 3x dx

6 Using integration by parts twice in each case, show that

a ∫ x2ex dx = ex(x2 − 2x + 2) + c,

b ∫ ex sin x dx = 12 ex(sin x − cos x) + c.

7 Find

a ∫ x2 sin x dx b ∫ x2e3x dx c ∫ e−x cos 2x dx 8 a Write down the derivative of ln x with respect to x.

b Use integration by parts to find

∫ ln x dx. 9 Find

a ∫ ln 2x dx b ∫ 3x ln x dx c ∫ (ln x)2 dx 10 Evaluate

a 0

1−∫ (x + 2)ex dx b 2

1∫ x2 ln x dx c 13

1

∫ 2xe3x − 1 dx

d 3

0∫ ln (2x + 3) dx e π2

0∫ x2 cos x dx f π4

0∫ e3x sin 2x dx

Solomon Press

INTEGRATION C4 Worksheet G 1 y

O 2 x The diagram shows part of the curve with parametric equations

x = 2t − 4, y = 1t

.

The shaded region is bounded by the curve, the coordinate axes and the line x = 2.

a Find the value of the parameter t when x = 0 and when x = 2.

b Show that the area of the shaded region is given by 3

2∫2t

dt.

c Hence, find the area of the shaded region.

d Verify your answer to part c by first finding a cartesian equation for the curve. 2 y

A

B

O x

The diagram shows the ellipse with parametric equations

x = 4 cos θ, y = 2 sin θ, 0 ≤ θ < 2π,

which meets the positive coordinate axes at the points A and B.

a Find the value of the parameter θ at the points A and B.

b Show that the area of the shaded region bounded by the curve and the positive coordinate axes is given by

π2

0∫ 8 sin2 θ dθ.

c Hence, show that the area of the region enclosed by the ellipse is 8π. 3 y

O x

The diagram shows the curve with parametric equations

x = 2 sin t, y = 5 sin 2t, 0 ≤ t < π.

a Show that the area of the region enclosed by the curve is given by π2

0∫ 20 sin 2t cos t dt.

b Evaluate this integral.

Solomon Press

INTEGRATION C4 Worksheet H 1 Using an appropriate method, integrate with respect to x

a (2x − 3)4 b cosec2 12 x c 2e4x − 1 d 2( 1)

( 1)x

x x−+

e 23

cos 2x f x(x2 + 3)3 g sec4 x tan x h 7 2x+

i xe3x j 22

2 3x

x x+

− − k 3

14( 1)x +

l tan2 3x

m 4 cos2 (2x + 1) n 23

1xx−

o x sin 2x p 42

xx

++

2 Evaluate

a 2

1∫ 6e2x − 3 dx b π3

0∫ tan x dx c 2

2−∫2

3x − dx

d 3

2∫ 26

4 3x

x x+

+ − dx e

2

1∫ (1 − 2x)3 dx f π3

0∫ sin2 x sin 2x dx


a 32

0∫ 2

1

9 x− dx x = 3 sin u b

1

0∫ x(1 − 3x)3 dx u = 1 − 3x

c 2 3

2∫ 21

4 x+ dx x = 2 tan u d

0

1−∫2 1x x + dx u2 = x + 1


a 25 3x−

b (x + 1)2 2ex x+ c 1

2 1x

x−+

d sin 3x cos 2x

e 3x(x − 1)4 f 2

23 6 2

3 2x xx x

+ ++ +

g 3

52 1x −

h cos2 3sin

xx+

i 2

3 1

x

x − j (2 − cot x)2 k 2

6 5( 1)(2 1)

xx x

−− −

l x2e−x

5 Evaluate

a 4

2∫1

3 4x − dx b

π4π6∫ cosec2 x cot2 x dx c

1

0∫2

27

(2 ) (3 )x

x x−

− − dx

d π2

0∫ x cos 12 x dx e

5

1∫1

4 5x + dx f

π6π6−∫ 2 cos x cos 3x dx

g 2

0∫22 1x x + dx h

1

0∫2 1

2xx

+−

dx i 1

0∫ (x − 2)(x + 1)3 dx


a y = 2 3( 2)x

x +, x = 1, x = 2 b y = ln x, x = 2, x = 4

7 Given that

6

3∫2ax bx+ dx = 18 + 5 ln 2,

find the values of the rational constants a and b.

Solomon Press

8 y y = 6 − 2ex

O P x The diagram shows the curve with equation y = 6 − 2ex.

a Find the coordinates of the point P where the curve crosses the x-axis.

b Show that the area of the region enclosed by the curve and the coordinate axes is 6 ln 3 − 4. 9 Using the substitution u = cot x, show that

π4π6∫ cot2 x cosec4 x dx = 2

15 ( 21 3 − 4).

10 y O x The diagram shows the curve with parametric equations

x = t + 1, y = 4 − t 2.

a Show that the area of the region bounded by the curve and the x-axis is given by

2

2−∫ (4 − t2) dt.

b Hence, find the area of this region. 11 a Given that k is a constant, show that

ddx

(x2 sin 2x + 2kx cos 2x − k sin 2x) = 2x2 cos 2x + (2 − 4k)x sin 2x.

b Using your answer to part a with a suitable value of k, or otherwise, find

∫ x2 cos 2x dx. 12 y y = 2

ln xx

O 2 x

The shaded region in the diagram is bounded by the curve with equation y = 2ln xx

, the x-axis and

the line x = 2. Use integration by parts to show that the area of the shaded region is 12 (1 − ln 2).

13 f(x) ≡ 3 216

3 11 8 4x

x x x+

+ + −

a Factorise completely 3x3 + 11x2 + 8x − 4.

b Express f(x) in partial fractions.

c Show that 0

1−∫ f(x) dx = −(1 + 3 ln 2).

C4 INTEGRATION Worksheet H continued

Solomon Press

INTEGRATION C4 Worksheet I 1 y

y = 2x

O 12 2 x

The shaded region in the diagram is bounded by the curve y = 2x

, the x-axis and the lines x = 12

and x = 2. Show that when the shaded region is rotated through 360° about the x-axis, the volume of the solid formed is 6π.

2 y

y = x2 + 3 O 2 x The shaded region in the diagram, bounded by the curve y = x2 + 3, the coordinate axes and the

line x = 2, is rotated through 2π radians about the x-axis.

Show that the volume of the solid formed is approximately 127. 3 The region enclosed by the given curve, the x-axis and the given ordinates is rotated through 360°

about the x-axis. Find the exact volume of the solid formed in each case.

a y = 22ex, x = 0, x = 1 b y = 2

3x

, x = −2, x = −1

c y = 1 + 1x

, x = 3, x = 9 d y = 23 1xx+ , x = 1, x = 2

e y = 12x +

, x = 2, x = 6 f y = e1 − x, x = −1, x = 1

4 y

y = 42x +

R O 2 x

The diagram shows part of the curve with equation y = 42x +

.

The shaded region, R, is bounded by the curve, the coordinate axes and the line x = 2.

a Find the area of R, giving your answer in the form k ln 2.

The region R is rotated through 2π radians about the x-axis.

b Show that the volume of the solid formed is 4π.

Solomon Press

5 y y =

122x +

12x−

O 1 3 x

The diagram shows the curve with equation y = 122x +

12x− .

The shaded region bounded by the curve, the x-axis and the lines x = 1 and x = 3 is rotated through 2π radians about the x-axis. Find the volume of the solid generated, giving your answer in the form π(a + ln b) where a and b are integers.

6 a Sketch the curve y = 3x − x2, showing the coordinates of any points where the curve

intersects the coordinate axes.

The region bounded by the curve and the x-axis is rotated through 360° about the x-axis.

b Show that the volume of the solid generated is 8110 π.

7 y x − 3 = 0

y = 3 − 1x

O P x

The diagram shows the curve with equation y = 3 − 1x

, x > 0.

a Find the coordinates of the point P where the curve crosses the x-axis.

The shaded region is bounded by the curve, the straight line x − 3 = 0 and the x-axis.

b Find the area of the shaded region.

c Find the volume of the solid formed when the shaded region is rotated completely about the x-axis, giving your answer in the form π(a + b ln 3) where a and b are rational.

8 y

y = x − 1x

O 3 x

The diagram shows the curve y = x − 1x

, x ≠ 0.

a Find the coordinates of the points where the curve crosses the x-axis.

The shaded region is bounded by the curve, the x-axis and the line x = 3.

b Show that the area of the shaded region is 4 − ln 3.

The shaded region is rotated through 360° about the x-axis.

c Find the volume of the solid generated as an exact multiple of π.

C4 INTEGRATION Worksheet I continued

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INTEGRATION C4 Worksheet J 1 y

y = x2 + 1 A

O B x The diagram shows the curve y = x2 + 1 which passes through the point A (1, 2).

a Find an equation of the normal to the curve at the point A.

The normal to the curve at A meets the x-axis at the point B as shown.

b Find the coordinates of B.

The shaded region bounded by the curve, the coordinate axes and the line AB is rotated through 2π radians about the x-axis.

c Show that the volume of the solid formed is 365 π.

2 y y = 4x + 9

x

O 1 e x

The shaded region in the diagram is bounded by the curve with equation y = 4x + 9x

,

the x-axis and the lines x = 1 and x = e.

a Find the area of the shaded region, giving your answer in terms of e.

b Find, to 3 significant figures, the volume of the solid formed when the shaded region is rotated completely about the x-axis.

3 The region enclosed by the given curve, the x-axis and the given ordinates is rotated through

2π radians about the x-axis. Find the exact volume of the solid formed in each case.

a y = cosec x, x = π6 , x = π3 b y = 32

xx

++

, x = 1, x = 4

c y = 1 + cos 2x, x = 0, x = π4 d y = 12x e2 − x, x = 1, x = 2

4 y

y = 12e xx −

O 1 x

The shaded region in the diagram, bounded by the curve y = 12e xx − , the x-axis and the line x = 1,

is rotated through 360° about the x-axis.

Show that the volume of the solid formed is π(2 − 5e−1).

Solomon Press

5 y y = 2 sin x + cos x

O π2 x

The diagram shows part of the curve with equation y = 2 sin x + cos x.

The shaded region is bounded by the curve in the interval 0 ≤ x < π2 , the positive coordinate

axes and the line x = π2 .

a Find the area of the shaded region.

b Show that the volume of the solid formed when the shaded region is rotated through 2π radians about the x-axis is 1

4 π(5π + 8). 6 y

O 1 x

The diagram shows part of the curve with parametric equations

x = tan θ, y = sin 2θ, 0 ≤ θ < π2 .

The shaded region is bounded by the curve, the x-axis and the line x = 1.

a Write down the value of the parameter θ at the points where x = 0 and where x = 1.

The shaded region is rotated through 2π radians about the x-axis.

b Show that the volume of the solid formed is given by

4ππ4

0∫ sin2 θ dθ.

c Evaluate this integral. 7 y

O x The diagram shows part of the curve with parametric equations

x = t 2 − 1, y = t(t + 1), t ≥ 0.

a Find the value of the parameter t at the points where the curve meets the coordinate axes.

The shaded region bounded by the curve and the coordinate axes is rotated through 2π radians about the x-axis.

b Find the volume of the solid formed, giving your answer in terms of π.

C4 INTEGRATION Worksheet J continued

Solomon Press

INTEGRATION C4 Worksheet K 1 Find the general solution of each differential equation.

a ddyx

= (x + 2)3 b ddyx

= 4 cos 2x c ddxt

= 3e2t + 2

d (2 − x) ddyx

= 1 e ddNt

= 2 1t t + f ddyx

= xex

2 Find the particular solution of each differential equation.

a ddyx

= e−x, y = 3 when x = 0 b ddyt

= tan3 t sec2 t, y = 1 when t = π3

c (x2 − 3) ddux

= 4x, u = 5 when x = 2 d ddyx

= 3 cos2 x, y = π when x = π2

3 a Express 2

86

xx x

−− −


b Given that

(x2 − x − 6) ddyx

= x − 8,

and that y = ln 9 when x = 1, show that when x = 2, the value of y is ln 32. 4 Find the general solution of each differential equation.

a ddyx

= 2y + 3 b ddyx

= sin2 2y c ddyx

= xy

d (x + 1) ddyx

= y e ddyx

= 2 2xy− f d

dyx

= 2 cos x cos2 y

g x ddyx

= ey − 3 h y ddyx

= xy2 + 3x i ddyx

= xy sin x

j ddyx

= e2x − y k (y − 3) ddyx

= xy(y − 1) l ddyx

= y2 ln x

5 Find the particular solution of each differential equation.

a ddyx

= 2xy

, y = 3 when x = 4 b ddyx

= (y + 1)3, y = 0 when x = 2

c (tan2 x) ddyx

= y, y = 1 when x = π2 d ddyx

= 21

yx

+−

, y = 6 when x = 3

e ddyx

= x2 tan y, y = π6 when x = 0 f ddyx

= 3

yx +

, y = 16 when x = 1

g ex ddyx

= x cosec y, y = π when x = −1 h ddyx

= 21 cos2 sin

yx y+ , y = π3 when x = 1

Solomon Press

INTEGRATION C4 Worksheet L 1 a Express 4

(1 )(2 )xx x

++ −


b Given that y = 2 when x = 3, solve the differential equation

ddyx

= ( 4)(1 )(2 )

y xx x

++ −

.

2 Given that y = 0 when x = 0, solve the differential equation

ddyx

= ex + y cos x.

3 Given that ddyx

is inversely proportional to x and that y = 4 and ddyx

= 53 when x = 3, find an

expression for y in terms of x. 4 A quantity has the value N at time t hours and is increasing at a rate proportional to N.

a Write down a differential equation relating N and t.

b By solving your differential equation, show that

N = Aekt,

where A and k are constants and k is positive.

Given that when t = 0, N = 40 and that when t = 5, N = 60,

c find the values of A and k,

d find the value of N when t = 12. 5 A cube is increasing in size and has volume V cm3 and surface area A cm2 at time t seconds.

a Show that

ddVA

= k A ,

where k is a positive constant.

Given that the rate at which the volume of the cube is increasing is proportional to its surface area

and that when t = 10, A = 100 and ddAt

= 5,

b show that A = 1

16 (t + 30)2. 6 At time t = 0, a piece of radioactive material has mass 24 g. Its mass after t days is m grams and

is decreasing at a rate proportional to m.

a By forming and solving a suitable differential equation, show that

m = 24e−kt,


After 20 days, the mass of the material is found to be 22.6 g.

b Find the value of k.

c Find the rate at which the mass is decreasing after 20 days.

d Find how long it takes for the mass of the material to be halved.

Solomon Press

7 A quantity has the value P at time t seconds and is decreasing at a rate proportional to P .


P = (a − bt)2,

where a and b are constants.

Given that when t = 0, P = 400,

b find the value of a.

Given also that when t = 30, P = 100,

c find the value of P when t = 50. 8

h cm

The diagram shows a container in the shape of a right-circular cone. A quantity of water is poured into the container but this then leaks out from a small hole at its vertex.

In a model of the situation it is assumed that the rate at which the volume of water in the container, V cm3, decreases is proportional to V. Given that the depth of the water is h cm at time t minutes,

a show that

ddht

= −kh,


Given also that h = 12 when t = 0 and that h = 10 when t = 20,

b show that h = 12e−kt,

and find the value of k,

c find the value of t when h = 6.

9 a Express 1(1 )(1 )x x+ −


In an industrial process, the mass of a chemical, m kg, produced after t hours is modelled by the differential equation

ddmt

= ke−t(1 + m)(1 − m),


Given that when t = 0, m = 0 and that the initial rate at which the chemical is produced is 0.5 kg per hour,

b find the value of k,

c show that, for 0 ≤ m < 1, ln 11

mm

+ −

= 1 − e−t.

d find the time taken to produce 0.1 kg of the chemical,

e show that however long the process is allowed to run, the maximum amount of the chemical that will be produced is about 462 g.

C4 INTEGRATION Worksheet L continued

Solomon Press

INTEGRATION C4 Worksheet M 1 Use the trapezium rule with n intervals of equal width to estimate the value of each integral.

a 5

1∫ x ln (x + 1) dx n = 2 b π2π6∫ cot x dx n = 2

c 2

2−∫2

10ex

dx n = 4 d 1

0∫ arccos (x2 − 1) dx n = 4

e 0.5

0∫ sec2 (2x − 1) dx n = 5 f 6

0∫ x3e−x dx n = 6 2 y y = 2 − cosec x O x

The diagram shows the curve with equation y = 2 − cosec x, 0 < x < π.

a Find the exact x-coordinates of the points where the curve crosses the x-axis.

b Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve and the x-axis.

3 f(x) ≡ π6 + arcsin ( 1

2 x), x ∈ , −2 ≤ x ≤ 2.

a Use the trapezium rule with three strips to estimate the value of the integral I = 2

1−∫ f(x) dx.

b Use the trapezium rule with six strips to find an improved estimate for I. 4 y y = ln x

O 5 x

The shaded region in the diagram is bounded by the curve y = ln x, the x-axis and the line x = 5.

a Estimate the area of the shaded region to 3 decimal places using the trapezium rule with

i 2 strips ii 4 strips iii 8 strips

b By considering your answers to part a, suggest a more accurate value for the area of the shaded region correct to 3 decimal places.

c Use integration to find the true value of the area correct to 3 decimal places. 5 y y = ex − x −4 O x The shaded region in the diagram is bounded by the curve y = ex − x, the coordinate axes and the

line x = −4. Use the trapezium rule with five equally-spaced ordinates to estimate the volume of the solid formed when the shaded region is rotated completely about the x-axis.

Solomon Press

INTEGRATION C4 Worksheet N 1 Show that

7

2∫8

4 3x − dx = ln 25. (4)

2 Given that y = π4 when x = 1, solve the differential equation

ddyx

= x sec y cosec3 y. (7)

3 a Use the trapezium rule with three intervals of equal width to find an approximate

value for the integral

1.5

0∫2 1ex − dx. (4)

b Use the trapezium rule with six intervals of equal width to find an improved approximation for the above integral. (2)

4 f(x) ≡ 2

3(2 )(1 2 ) (1 )

xx x

−− +

.


b Show that

2

1∫ f(x) dx = 1 − ln 2. (6)

5 The rate of growth in the number of yeast cells, N, present in a culture after t hours is

proportional to N.

a By forming and solving a differential equation, show that

N = Aekt,

where A and k are positive constants. (4)

Initially there are 200 yeast cells in the culture and after 2 hours there are 3000 yeast cells in the culture. Find, to the nearest minute, after how long

b there are 10 000 yeast cells in the culture, (5)

c the number of yeast cells is increasing at the rate of 5 per second. (4) 6 y

y = 12 1x +

O 4 x

The diagram shows part of the curve with equation y = 12 1x +

.


a Find the area of the shaded region. (4)

The shaded region is rotated through four right angles about the x-axis.

b Find the volume of the solid formed, giving your answer in the form π ln k. (5)

Solomon Press

7 Using the substitution u2 = x + 3, show that

1

0∫ 3x x + dx = k( 3 3 − 4),

where k is a rational number to be found. (7) 8 a Use the identities for sin (A + B) and sin (A − B) to prove that

2 sin A cos B ≡ sin (A + B) + sin (A − B). (2)

y O x

The diagram shows the curve given by the parametric equations

x = 2 sin 2t, y = sin 4t, 0 ≤ t < π.

b Show that the total area enclosed by the two loops of the curve is given by

π4

0∫ 16 sin 4t cos 2t dt. (4)

c Evaluate this integral. (5)

9 f(x) ≡ 2 22

( 2)( 4)x

x x−

+ −.


f(x) ≡ A + 2

Bx +

+ 4

Cx −

. (3)

The finite region R is bounded by the curve y = f(x), the coordinate axes and the line x = 2.

b Find the area of R, giving your answer in the form p + ln q, where p and q are integers. (5) 10 a Find ∫ sin2 x dx. (4)

b Use integration by parts to show that

∫ x sin2 x dx = 18 (2x2 − 2x sin 2x − cos 2x) + c,

where c is an arbitrary constant. (4)

y

y = 12 sinx x

R O x

The diagram shows the curve with equation y = 12 sinx x , 0 ≤ x ≤ π.

The finite region R, bounded by the curve and the x-axis, is rotated through 2π radians about the x-axis.

c Find the volume of the solid formed, giving your answer in terms of π. (3)

C4 INTEGRATION Worksheet N continued

Solomon Press

INTEGRATION C4 Worksheet O 1 a Express 2

13 2x x− +

in partial fractions. (3)

b Show that

4

3∫ 213 2x x− +

dx = ln ab

,

where a and b are integers to be found. (5) 2 Evaluate

π6

0∫ cos x cos 3x dx. (6)

3 a Find the quotient and remainder obtained in dividing (x2 + x − 1) by (x − 1). (3)

b Hence, show that

∫2 1

1x x

x+ −

− dx = 1

2 x2 + 2x + lnx − 1 + c,

where c is an arbitrary constant. (2) 4 y

y = 2 − 1x

O 1 4 x The diagram shows the curve with equation y = 2 − 1

x.

The shaded region bounded by the curve, the x-axis and the lines x = 1 and x = 4 is rotated through 360° about the x-axis to form the solid S.

a Show that the volume of S is 2π(2 + ln 2). (6)

S is used to model the shape of a container with 1 unit on each axis representing 10 cm.

b Find the volume of the container correct to 3 significant figures. (2) 5 a Use integration by parts to find ∫ x ln x dx. (4)

b Given that y = 4 when x = 2, solve the differential equation

ddyx

= xy ln x, x > 0, y > 0,

and hence, find the exact value of y when x = 1. (5)

6 a Evaluate π3

0∫ sin x sec2 x dx. (4)

b Using the substitution u = cos θ, or otherwise, show that

π4

0∫ 4sin

cosθθ

dθ = a + 2b ,

where a and b are rational. (6)

Solomon Press

7 y O 3 x

The diagram shows part of the curve with parametric equations

x = 2t + 1, y = 12 t−

, t ≠ 2.


a Find the value of the parameter t at the points where x = 0 and where x = 3. (2)

b Show that the area of the shaded region is 2 ln 52 . (5)

c Find the exact volume of the solid formed when the shaded region is rotated completely about the x-axis. (5)

8 a Using integration by parts, find

∫ 6x cos 3x dx. (5)

b Use the substitution x = 2 sin u to show that

3

0∫ 2

1

4 x− dx = π3 . (5)

9 In an experiment to investigate the formation of ice on a body of water, a thin circular

disc of ice is placed on the surface of a tank of water and the surrounding air temperature is kept constant at −5°C.

In a model of the situation, it is assumed that the disc of ice remains circular and that its area, A cm2 after t minutes, increases at a rate proportional to its perimeter.

a Show that

ddAt

= k A ,

where k is a positive constant. (3)

b Show that the general solution of this differential equation is

A = (pt + q)2,

where p and q are constants. (4)

Given that when t = 0, A = 25 and that when t = 20, A = 40,

c find how long it takes for the area to increase to 50 cm2. (5) 10 f(x) ≡ 5 1

(1 )(1 2 )x

x x+

− +.


b Find 12

0∫ f(x) dx, giving your answer in the form k ln 2. (4)

c Find the series expansion of f(x) in ascending powers of x up to and including the term in x3, for | x | < 1

2 . (6)

C4 INTEGRATION Worksheet O continued

Solomon Press

INTEGRATION C4 Worksheet P 1 y

3

y = 1x

O 3 x

The diagram shows the curve with equation y = 1x

, x > 0.

The shaded region is bounded by the curve, the lines x = 3 and y = 3 and the coordinate axes.

a Show that the area of the shaded region is 1 + ln 9. (5)

b Find the volume of the solid generated when the shaded region is rotated through 360° about the x-axis, giving your answer in terms of π. (5)

2 Given that

I = 4

0∫ x sec ( 13 x) dx,

a find estimates for the value of I to 3 significant figures using the trapezium rule with

i 2 strips,

ii 4 strips,

iii 8 strips. (6)

b Making your reasoning clear, suggest a value for I correct to 3 significant figures. (2) 3 The temperature in a room is 10°C. A heater is used to raise the temperature in the room

to 25°C and then turned off. The amount by which the temperature in the room exceeds 10°C is θ °C, at time t minutes after the heater is turned off.

It is assumed that the rate at which θ decreases is proportional to θ.


θ = 15e−kt,

where k is a positive constant. (6)

Given that after half an hour the temperature in the room is 20°C,

b find the value of k. (3)

The heater is set to turn on again if the temperature in the room falls to 15°C.

c Find how long it takes before the heater is turned on. (3) 4 a Find the values of the constants p, q and r such that

sin4 x ≡ p + q cos 2x + r cos 4x. (4)

b Hence, evaluate

π2

0∫ sin4 x dx,

giving your answer in terms of π. (4)

Solomon Press

5 a Find the general solution of the differential equation

ddyx

= xy3. (4)

b Given also that y = 12 when x = 1, find the particular solution of the differential

equation, giving your answer in the form y2 = f(x). (3) 6 a Show that, using the substitution x = eu,

∫ 22 ln x

x+ dx = ∫ (2 + u)e−u du. (3)

b Hence, or otherwise, evaluate

1

e∫ 2

2 ln xx

+ dx. (6)

7 y O x The diagram shows the curve with parametric equations

x = cos 2t, y = tan t, 0 ≤ t < π2 .

The shaded region is bounded by the curve and the coordinate axes.

a Show that the area of the shaded region is given by

π4

0∫ 4 sin2 t dt. (4)

b Hence find the area of the shaded region, giving your answer in terms of π. (4)

c Write down expressions in terms of cos 2A for

i sin2 A,

ii cos2 A,

and hence find a cartesian equation for the curve in the form y2 = f(x). (4)

8 f(x) ≡ 2

26 2

( 1) ( 3)x

x x−

+ +.


f(x) ≡ 2( 1)A

x + +

1B

x + +

3C

x +. (4)

The curve y = f(x) crosses the y-axis at the point P.

b Show that the tangent to the curve at P has the equation

14x + 3y = 6. (5)

c Evaluate

1

0∫ f(x) dx,

giving your answer in the form a + b ln 2 + c ln 3 where a, b and c are integers. (5)

C4 INTEGRATION Worksheet P continued

math c4 practice

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