Cambridge International AS and A Level Mathematics Pure Mathematics 1

Answers

P1^

d

d

h

t^ is the rate of change of the height

of the sand with respect to time.

d

d

d

d

V

h

h

× t is the rate of change of the

volume with respect to time.

●^ (Page^ 169)

y = (x^2 − 2)^4

= (x^2 )^4 + 4(x^2 )^3 (−2) + 6(x^2 )^2 (−2)^2

+ 4(x^2 )(−2)^3 + (−2)^4

= x^8 − 8 x^6 + 24 x^4 − 32 x^2 + 16

d

d

y

x^ =^8 x

(^7) − 48 x (^5) + 96 x (^3) − 64 x
= 8 x(x^6 − 6 x^4 + 12 x^2 − 8)
= 8 x(x^2 − 2)(x^4 − 4 x^2 + 4)
= 8 x(x^2 − 2)(x^2 − 2)^2
= 8 x(x^2 − 2)^3
Exercise 5H (Page 171)
  1 (i) 3(x + 2)^2
(ii) 8(2x + 3)^3
(iii) 6 x(x^2 − 5)^2
(iv) 15 x^2 (x^3 + 4)^4
(v) −3(3x + 2)−^2
(vi) (^) (–x 24 –^6 x 3 )
(vii) 3 x(x^2 − 1)
(^12)
(viii)  31 1 1
2
()x+x ()–x 2

(ix) 2 1
3
x()x–
  2 (i) 9(3x − 5)^2
(ii) y = 9 x − 17
3 (i) 8(2x − 1)^3
(ii) (^) ( 21 , 0), minimum
(iii)
  4 (i) 4(2x − 1)(x^2 − x − 2)^3
(ii) (−1, 0), minimum;
(^) (^12 ,^6561256 ), maximum;
(2, 0), minimum
(iii)
  5 4 cm^2 s−^1
  6 −0.015 Ns−^1
  7 10 π m^2 day−^1
(= 0.314 m^2 day−^1 to 3 s.f.)
Chapter  6
●?^ (Page^ 173)
The gradient depends only on the
x co-ordinate. This is the same for
all four curves so at points with the
same x co-ordinate the tangents
are parallel.
Exercise 6A (Page 177)
  1 (i) y = 2 x^3 + 5 x + c
(ii) y = 2 x^3 + 5 x + 2
  2 (i) y = 2 x^2 + 3
(ii) 5
  3 (i) y = 2 x^3 − 6
  4 (ii) t = 4. Only 4 is applicable
here.
  5 (i) y = 5 x + c
(ii) y = 5 x + 3
(iii)
  6 (i) x = 1 (minimum) and
x = −1 (maximum)
(ii) y = x^3 − 3 x + 3
(iii)
  7 (i) y = x^2 − 6 x + 9
(ii) The curve passes through
(1, 4)
  8 (i) y = x^3 − x^2 − x + 1
(ii) (^) ()–,^131275 and (1, 0)
(iii)
  9 (i) y = x^3 − 4 x^2 + 5 x + 3
(ii) max (1, 5), min () 1432 , 2723
(iii) 42327  k  5
(iv) 1  x  123 ; x = (^113)
10 y = 23 x
(^32)

• 2
  11 y = − (^2) x − 3 x + 17
  12 y = 23 x
  (^32)
  − (^1) x + (^513)
  13 y = x^3 + 5 x + 2
  14 (i) y = 2 x x − 9 x + 20
  x (ii) x = 9, minimum
  y
  O –^12
  1
  x
  y
  O –^12
  16
  –1 2
  x
  y
  y = 5x + 3
  O
  3

• –^35

x

y

O





– 



x

y

2 

O x

y

(1, 0)

(0, 1)

(– , 1 )–^1327 –^5