Revision Test 9
This Revision Test covers the material contained in Chapters 27 to 31. The marks for each question are shown in
brackets at the end of each question.
1. Differentiate the following with respect to the
variable:
(a) y = 5 +2
√
x 3 −
1
x 2 (b) s = 4e
2θ sin 3θ
(c) y =
3 ln5t
cos 2t
(d) x =
2
(t 2 − 3t + 5)
(13)
2. If f (x) = 2.5x 2 − 6x + 2 find the co-ordinates at
the point at which the gradient is −1.
(5)
3. The displacement s cm of the end of a stiff spring
at time t seconds is given by:
s = ae −kt sin 2π f t. Determine the velocity and
acceleration of the end of the spring after
2 seconds if a = 3, k = 0.75 and f = 20.
(10)
4. Find the co-ordinates of the turning points on
the curve y = 3x 3 + 6x 2 + 3x − 1 and distinguish
between them.
(7)
5. The heat capacity C of a gas varies with absolute
temperature θ as shown:
C = 26.50 + 7.20 × 10
−3
θ − 1.20 × 10
−6
θ
2
Determine the maximum value of C and the
temperature at which it occurs.
(5)
6. Determine for the curve y = 2x 2 − 3x at the point
(2, 2): (a) the equation of the tangent (b) the
equation of the normal.
(6)
7. A rectangular block of metal with a square crosssection has a total surface area of 250 cm 2 . Find
the maximum volume of the block of metal. (7)
8. A cycloid has parametric equations given by:
x = 5(θ − sin θ) and y = 5(1 − cos θ). Evaluate
(a)
d y
dx
(b)
d 2 y
dx 2 when θ = 1.5 radians. Give
answers correct to 3 decimal places.
(8)
9. Determine the equation of (a) the tangent, and (b)
the normal, drawn to an ellipse x = 4 cos θ,
y = sin θ at θ =
π
3
.
( 8 )
10. Determine expressions for
dz
dy
for each of the
following functions:
(a) z =5y 2 cos x (b) z = x 2 + 4x y − y 2 .
(5)
11. If x 2 + y 2 + 6x + 8y + 1 = 0, find
dy
dx
in terms of x
and y.
( 3 )
12. Determine the gradient of the tangents drawn to
the hyperbola x 2 − y 2 = 8 at x = 3.
(3)
13. Use logarithmic differentiation to differentiate
y =
(x + 1) 2 √
(x − 2)
(2x − 1)
3
(x − 3) 4
with respect to x.
(6)
14. Differentiate y =
3e θ sin 2θ
√
θ 5
and hence evaluate
dy
dθ
, correct to 2 decimal places, when θ =
π
3
.
(9)
15. Evaluate
d
dt
t
√
(2t + 1)
when t = 2, correct to 4
significant figures.
(5)
This Revision Test covers the material contained in Chapters 27 to 31. The marks for each question are shown in
brackets at the end of each question.
1. Differentiate the following with respect to the
variable:
(a) y = 5 +2
√
x 3 −
1
x 2 (b) s = 4e
2θ sin 3θ
(c) y =
3 ln5t
cos 2t
(d) x =
2
(t 2 − 3t + 5)
(13)
2. If f (x) = 2.5x 2 − 6x + 2 find the co-ordinates at
the point at which the gradient is −1.
(5)
3. The displacement s cm of the end of a stiff spring
at time t seconds is given by:
s = ae −kt sin 2π f t. Determine the velocity and
acceleration of the end of the spring after
2 seconds if a = 3, k = 0.75 and f = 20.
(10)
4. Find the co-ordinates of the turning points on
the curve y = 3x 3 + 6x 2 + 3x − 1 and distinguish
between them.
(7)
5. The heat capacity C of a gas varies with absolute
temperature θ as shown:
C = 26.50 + 7.20 × 10
−3
θ − 1.20 × 10
−6
θ
2
Determine the maximum value of C and the
temperature at which it occurs.
(5)
6. Determine for the curve y = 2x 2 − 3x at the point
(2, 2): (a) the equation of the tangent (b) the
equation of the normal.
(6)
7. A rectangular block of metal with a square crosssection has a total surface area of 250 cm 2 . Find
the maximum volume of the block of metal. (7)
8. A cycloid has parametric equations given by:
x = 5(θ − sin θ) and y = 5(1 − cos θ). Evaluate
(a)
d y
dx
(b)
d 2 y
dx 2 when θ = 1.5 radians. Give
answers correct to 3 decimal places.
(8)
9. Determine the equation of (a) the tangent, and (b)
the normal, drawn to an ellipse x = 4 cos θ,
y = sin θ at θ =
π
3
.
( 8 )
10. Determine expressions for
dz
dy
for each of the
following functions:
(a) z =5y 2 cos x (b) z = x 2 + 4x y − y 2 .
(5)
11. If x 2 + y 2 + 6x + 8y + 1 = 0, find
dy
dx
in terms of x
and y.
( 3 )
12. Determine the gradient of the tangents drawn to
the hyperbola x 2 − y 2 = 8 at x = 3.
(3)
13. Use logarithmic differentiation to differentiate
y =
(x + 1) 2 √
(x − 2)
(2x − 1)
3
(x − 3) 4
with respect to x.
(6)
14. Differentiate y =
3e θ sin 2θ
√
θ 5
and hence evaluate
dy
dθ
, correct to 2 decimal places, when θ =
π
3
.
(9)
15. Evaluate
d
dt
t
√
(2t + 1)
when t = 2, correct to 4
significant figures.
(5)