PURE MATHS EXAM BANK QUESTIONS
Differentiation (10 mark questions)
1
1.Given that f(x) = x + .
4x
a) For what range of values of x is f(x) increasing? [5]
b) What are the coordinates of the stationary points? [5]
4
2.Determine the equation of the normal in the form ax +by + c =0 to the curve y= x + at the
x
point P where x = 4. [10]
1
3.Determine the type of stationary points on the curve y = 4x + and the coordinates of these
x
points. [10]
4.A cylindrical can ( with lid) of radius r cm is made from 300 cm3 of thin sheet metal.
2
150−π r
a)Show that its height, h cm, is given by h = .
πr
b) Find r and h so that the can will contain the maximum volume and find this volume. [6]
Differentiation (5 mark questions)
dy
1.i) Given that y = cos 2 x , find . [2]
dx
1 t dy
ii)Given that x = 2 and y = 2 , where t is a parameter, find in terms of t. [3]
1+ t 1+ t dx
2.Find the derivative with respect to x of:
sin x
i)
x
−1 x
ii) sin ( ) [5]
10
1 −1
3.A curve has parametric equations x = = 2 and y = tan t . Determine the gradient of the
1+ t
curve at the point where t = 2. [5]
4. Find the gradient of the tangent at the point (2,3) on the curve 4 x 2 - y 2 = 7 [5]
Vectors (10 mark questions)
()( ) ()
1 1 1
1a)Two lines L1 and L2 have equations r 1 = 6 t −2 and r 2 = s 2 respectively.
1 −3 −1
i)Write down three equations which must be satisfied by the real parameters t and s if the two
lines intersect.
ii) Find the values of s and t satisfying these three equations.
iii)Hence find the coordinates of the point of intersection of the two lines. [5]
() ()
b)The line m has equation r = = 1 +¿ λ a , where λ is a real parameter and a is a
0 5
constant. Find the positive value of a for which the angle between line m
1 5
and L2
(from part a) is 600 .
[5]
() ()
2.The points P and Q have position vectors = 2 and = 3 respectively.
1 3
3 5
i)Find the position vector of the point where the line PQ meets the plane z =0. [4]
ii)Find the equation of the plane through P normal to PQ. [3]
iii)Find O ^
PQ, giving your answer to the nearest 0, 10 . [3]
() ()
0 +¿ t 1
8 −6
3.The point A has coordinates (3, -1, 5) and the line L has equation r =
−1 4
i)Find the coordinates of the point B on L such that AB is perpendicular to
L. [5]
()
II)The plane π has equation r. −1 = 15. Find the coordinates of the point C where L
1
3
intersects . [5]
( ) ()
4.The equation of a plane π is x-6y+ 2z = -5 and line m is r = −3 +¿ t 0 .
3 −2
−13 1
i)Show that m lies in π.
[5]
ii)Given that point B with coordinates (2, 8, 0), find the length of the
perpendicular line from B to m.
[5]
5 mark questions
() ()
1.Given that point P and Q 2 and 3 respectively. Find the values of a, b, c
1 3
3 5
such that the equation of the plane OPQ is r. a = c.
()
1
b
[5]
() ()
2.Show that the cosine of the angle between the line = 5 +¿ λ 0 and the line parallel
1 1
3 1
to the vector
()
3 is .
2
1
−1 √28
[5]
3.a = 12i + 8j + k, b = 3i +2j -5k, c = -3i – 2j + pk. Find the value of p for
which c is parallel to AB.
[5]
()
1
4. Given that r = 4
() ( )
t 2 and r = s 2
( ) . Determine whether the
1 1 1
+¿ 2
+¿
3
2 −3 −4
4
two lines intersect.
[5]
Differential equations (10 mark questions)
1.Under suitable conditions, the rate at which the temperature of a hot
object decreases may be taken to be proportional to the difference between
the temperature of the hot object(T) and the temperature of its surrounding
(T 0) .
i)Express this information as a differential equation connecting T and t,
where T is the temperature of the hot object at time t and T 0 is the
temperature of its surrounding. [2]
ii)Show that the general solution of the differential may be expressed in
the form T = T 0 +Ae−kt .
iii)Given that T = 5T 0 when t = 0 and T = 3T 0 when t = T. Find in terms
of T, the value of t when T = 2 T 0 .
[4]
2.A rectangular tank has a square base with sides 3 m long and height 5
m. The tank is initially full of water. The water leaks out through the base
and sides of the tank at a rate proportional to the total area in contact
with the water. When the depth of the water is 3 m, the level of the water
is falling at a rate of 0.3 m/h. If y denotes the depth of water in the tank
after t, hours.
i)Show that =- ( + 1) .
dy 3 4y
[5]
dx 50 3
ii)Hence solve the differential equation in (i), giving y in terms of t.
[5]
3 (a) Find the general solution of the differential equation =( 1− y ¿ ¿2 ,
dy
expressing y in terms x.
dx
[5]
(b) Solve the differential equation = 2xy and sketch one of the
dy
solution curves other than y = 0 .
dx
[5]
4. The positive quantities x and y are related by the differential equation
=
dy
dx √y
x
i)Find the general solution of this differential equation , expressing y in
terms of x. [5]
ii)The solution curve passing through the point (1,4) is denoted by C.
Show that the gradient at all points of C is greater than 1.
[5]
Differential equations (5 mark questions)
1.Solve the differential equation = 1-y, given that y¿ 1 and that y =0
dy
when x = 0 [5]
dx
2.Find the particular solution of the differential equation y = x 3, where y
dy
= 4 when x = 2.
dx
[5]
3.If = , determine the particular solution when r = 2 at t = 1.
dr 1
[5]
dt r
4.Given that cosy = , y = when x = 1. Find the particular solution of
dy 1 π
the differential equation.
dx x 2
[5]
Integration (10 mark questions)
1.(a) By means of the substitution u = 1 + √ x , or otherwise, find dx.
1
∫ 1+√ x
[5]
(b) ∫ ( x +1)(x +2) dx.
x
[5]
2.
i)Show that ∫ x e 2 x dx = 4 (3e 4+ 1).
2
1
[5]
0
ii) Find the exact value of ∫ x 2 e2 x dx.
2
[5]
0
3. Given the curves y = x 2 and y = 2- x 2. Calculate
(a)The area bounded by the two curves.
[5]
(b) The volume of the solid generated when the bounded region is rotated
through 3600 about the x- axis.
[5]
4.For a curve y f(x) , = 6x-2. Given that y =11 and = 10 when x = 2.
2
d y dy
Determine the equation of the curve.
dx
2
dx
[10]
Integration (5 mark questions)
1.∫ x e dx
4x
[5]
2. Using the substitution u2 = x- 2, or otherwise , find ∫ x √ x −2 dx.
[5]
3.Use the trapezium rule with intervals of width 0.5 to find an
approximation for
dx
2.5
∫ 1+1Inx
[5]
1
4.The region bounded by the curve y = x 2 + 1, the x-axis, the y- axis and the
line x = 2 is rotated completely about the x-axis. Find the volume of the
solid formed. [5]
Logic mathematics(10 mark questions)
1.Solve the equation x ¿(5 ¿ x) = 2¿ (2¿ x) , given that a¿ b = a+ b – 2; a,b ∈
N. [10]
2.Examine the validity of the following verbal argument:
Civil servants are not wealthy or philanthropists.
Civil servants are wealthy or not generous.
Civil servants are generous.
∴ Civil servants are philantropists.
[10]
3.Show that in the algebra of the divisors of 110 under the binary
operations of gcd(greatest common divisor) and lcm( least common
multiple)
i.gcd is distributive over lcm
ii. ¿ = a lcm b where = a = .
¿ ¿ ¿ 110
a
iii.x gcd (x lcm y) = x, by considering suitable elements.
[10]
4. Given that A - B = A ∩ B , where A and B are sets, prove that
¿
(A – B) ∪ (B- A) = (A∪B) – (A∩ B).
[10]
5 mark questions
1.Show that a ∨ (a ∧ b) = a ∨ b is a Boolean Algebra.
[5]
¿ ¿
2.Prove that A∪ A = A in the Algebra of sets.
[5]
3.Investigate the validity of the following argument:
p⟹q
q
∴p
[5]
4. The binary operation ¿ on the set R of real numbers except -1 is defined
by
a¿ b = a + b + ab. Determine whether the operation is associative.
[5]
Group systems (10 mark questions)
1.Prove that numbers of the form 2k , k ϵ Z form a group under
multiplication. [10]
2.Prove that the set M of 2x2 matrices of the form (0k 01) form a group
under multiplication.
[10]
3.Given that C = {1 ; 3 ; 5 ; 9 ; 11; 13 } modulo 14 under X . Show that C form a
group structure.
[10]
4.Given that D = } under ¿. Show
{f 1 : x → x ; f 1 ; f 3 : x →1− x ; f 1 ;f x−1 ; f x
that composition of functions under ¿ form a group.
2: x→ 4:x → 5: x→ 6: x→
x 1− x x x−1
[10]
5 mark questions
1.Let B = {1 ; 2 ; 3 ; 4 ; 5 ; 6 } modulo 7 under X. Construct a Cayle table for the
set. [5]
2.Prove that in a group , the identity element is unique.
[5]
3. B = {1 ; 2 ; 3 ; 4 ; 5 ; 6 } modulo 7 under X . Is the group Abelian?
[5]
4. B = {2 ; 4 ; 6 ; 8 } modulo 10 under X. Is the group Cyclic?
[5]
