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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
6 Express in partial fractions
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.
9 Express 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)
Solomon Press
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
2 , in ascending powers of x up to and including the term in x3.
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
−− −
in partial fractions.
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+
+ −
Solomon Press
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
6 , in ascending powers of x up to and including the term in x3.
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+ −.
a Express f(x) in partial fractions. (3)
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.
a Express f(x) in partial fractions. (3)
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)
Solomon Press
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)
6 The points A and B have position vectors 51
10
− −
and 418
−
respectively, relative to a
fixed origin O.
a Find, in vector form, an equation of the line l which passes through A and B. (2)
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
1 The points A and B have position vectors 215
− −
and 034
−
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 = 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.
6 A curve has parametric equations
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)
2 A curve has parametric equations
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
Solomon Press
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
6 A curve has parametric equations
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
5 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 = 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.
6 Integrate with respect to x
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.
8 Integrate with respect to x
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.
5 Using the given substitution, evaluate
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
11 Using the given substitution, evaluate
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
3 Using the given substitution, evaluate
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
4 Integrate with respect to x
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
6 Find the exact area of the region enclosed by the given curve, the x-axis and the given ordinates.
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.
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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
−− −
in partial fractions.
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
++ −
in partial fractions.
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,
where k is a positive constant.
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 .
a By forming and solving a suitable differential equation, show that
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,
where k is a positive constant.
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 partial fractions.
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),
where k is a positive constant.
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
−− +
.
a Express f(x) in partial fractions. (4)
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 +
.
The shaded region is bounded by the curve, the coordinate axes and the line x = 4.
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−
+ −.
a Find the values of the constants A, B and C such that
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.
The shaded region is bounded by the curve, the coordinate axes and the line x = 3.
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+
− +.
a Express f(x) in partial fractions. (3)
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.
a By forming and solving a suitable differential equation, show that
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−
+ +.
a Find the values of the constants A, B and C such that
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