Revision Test
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.
- Differentiate the following with respect to the
variable:
(a)y= 5 + 2
√
x^3 −
1
x^2
(b)s=4e^2 θsin3θ
(c)y=
3ln5t
cos2t
(d)x=
2
√
(t^2 − 3 t+ 5 )
(13)
- Iff(x)= 2. 5 x^2 − 6 x+2 find the co-ordinates at
the point at which the gradient is−1. (5) - The displacementscm of the end of a stiff spring
at timetseconds is given by:
s=ae−ktsin2πft. Determine the velocity and
acceleration of the end of the spring after
2seconds ifa=3,k= 0 .75 andf=20. (10) - Find the co-ordinates of the turning points on
the curvey= 3 x^3 + 6 x^2 + 3 x−1 and distinguish
between them. (7) - The heat capacityCof a gas varies with absolute
temperatureθas 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)
- Determine for the curvey= 2 x^2 − 3 xat the point
(2, 2): (a) the equation of the tangent (b) the
equation of the normal. (6) - A rectangular block of metal with a square cross-
section has a total surface area of 250cm^2 .Find
the maximum volume of the block of metal. (7)
8. A cycloid has parametric equations given by:
x=5(θ−sinθ)andy= 5 ( 1 −cosθ). Evaluate
(a)
dy
dx
(b)
d^2 y
dx^2
when θ= 1 .5radians.Give
answers correct to 3 decimal places. (8)
- Determine the equation of (a) the tangent, and (b)
the normal, drawn to an ellipsex=4cosθ,
y=sinθatθ=
π
3
.(8)
- Determine expressions for
dz
dy
for each of the
following functions:
(a)z= 5 y^2 cosx (b)z=x^2 + 4 xy−y^2 .(5)
- Ifx^2 +y^2 + 6 x+ 8 y+ 1 =0, find
dy
dx
in terms ofx
andy.(3)
- Determine the gradient of the tangents drawn to
the hyperbolax^2 −y^2 =8atx=3. (3) - Use logarithmic differentiation to differentiate
y=
(x+ 1 )^2
√
(x− 2 )
( 2 x− 1 )^3
√
(x− 3 )^4
with respect tox.(6)
- Differentiatey=
3eθsin2θ
√
θ^5
and hence evaluate
dy
dθ
, correct to 2 decimal places, whenθ=
π
3
.
(9)
- Evaluate
d
dt
[√t
( 2 t+ 1 )
]
whent=2, correct to 4
significant figures. (5)