Calculus Paper-1 [291 marks]

1. Let 8x [5 marks]
f ′ (x) = . Given that f (0) = 5, find f (x).
√ 2x2+1
 Let f (x) = 13 x3 + x2 − 15x + 17.

2a. Find f ′ (x). [2 marks]

 The graph of f has horizontal tangents at the points where x = a and x = b, a < b.

2b. Find the value of a and the value of b . [3 marks]

= ′( )
2c. Sketch the graph of y = f ′ (x). [1 mark]

2d. Hence explain why the graph of f has a local maximum point at x = a. [1 mark]

′′ ( )
2e. Find f ′′ (b). [3 marks]

2f. Hence, use your answer to part (d)(i) to show that the graph of f has a [1 mark]
local minimum point at x = b.

=
2g. The normal to the graph of f at x = a and the tangent to the graph of f [5 marks]
at x = b intersect at the point (p, q) .

Find the value of p and the value of q.

R
 Let f (x) = ln 5x where
kx
x > 0, k ∈ R+ .

3a. Show that f ′ (x) 1−ln 5x [3 marks]

= kx2
.

The graph of f has exactly one maximum point P.

3b. Find the x-coordinate of P. [3 marks]

2 ln 5 −3
 2 ln 5x−3
The second derivative of f is given by f ′′ (x) = kx3
. The graph of f has
exactly one point of inflexion Q.

3c. Show that the x-coordinate of Q is 1 e 32 . [3 marks]

5

3d. The region R is enclosed by the graph of f , the x-axis, and the vertical [7 marks]
lines through the maximum point P and the point of inflexion Q.

Given that the area of R is 3, find the value of k.

sin
 The function f is defined by f (x) = esin x .

4a. Find the first two derivatives of f (x) and hence find the Maclaurin series [8 marks]
for f (x) up to and including the x2 term.

3
4b. Show that the coefficient of x3 in the Maclaurin series for f (x) is zero. [4 marks]

3
4c. Using the Maclaurin series for arctan x and e3x − 1, find the Maclaurin [6 marks]
series for arctan (e3x − 1) up to and including the x3 term.

lim
4d. lim f(x)−1 [3 marks]
Hence, or otherwise, find x→0 .
arctan(e3x−1)

'( )= 3
5. The derivative of a function f is given by f'(x)= 3√x. [6 marks]
Given that f(1)= 3, find the value of f(4).

2
 The following diagram shows the graph of y = 4 − x2 , 0 ≤ x ≤ 2 and rectangle
ORST. The rectangle has a vertex at the origin O, a vertex on the y-axis at the
point R(0, y), a vertex on the x-axis at the point T(x, 0) and a vertex at point
S(x, y) on the graph.

Let P represent the perimeter of rectangle ORST.

6a. Show that P = −2x2 + 2x + 8. [2 marks]

ORST
6b. Find the dimensions of rectangle ORST that has maximum perimeter [6 marks]
and determine the value of the maximum perimeter.

Let A represent the area of rectangle ORST.

6c. Find an expression for A in terms of x. [2 marks]

6d. Find the dimensions of rectangle ORST that has maximum area. [5 marks]

6e. Determine the maximum area of rectangle ORST. [1 mark]

 The following diagram shows a ball attached to the end of a spring, which is
suspended from a ceiling.

The height, h metres, of the ball above the ground at time t seconds after being
released can be modelled by the function h(t)= 0. 4 cos(πt)+1. 8 where t ≥ 0.

7a. Find the height of the ball above the ground when it is released. [2 marks]
7b. Find the minimum height of the ball above the ground. [2 marks]

7c. Show that the ball takes 2 seconds to return to its initial height above [2 marks]
the ground for the first time.
7d. For the first 2 seconds of its motion, determine the amount of time that [5 marks]
the ball is less than 1. 8 + 0. 2√2 metres above the ground.

1
7e. Find the rate of change of the ball’s height above the ground when t 1 [4 marks]
= 3
. Give your answer in the form pπ√q ms−1 where p ∈ Q and q ∈ Z+ .

3
R
 3
A function f is defined by f(x)= x2+2
,x ∈ R.

8a. Sketch the curve y = f(x), clearly indicating any asymptotes with their [4 marks]
equations and stating the coordinates of any points of intersection with the axes.

= ( ) =0
 The region R is bounded by the curve y = f(x), the x-axis and the lines x = 0
and x = √6. Let A be the area of R.

8b. √2π [4 marks]

Show that A = 2 .

=
 The line x = k divides R into two regions of equal area.

8c. Find the value of k. [4 marks]

= ( )
 Let m be the gradient of a tangent to the curve y = f(x).

8d. Show that m 6x [2 marks]

=− .
( x2+2 ) 2

√
8e.
Show that the maximum value of m is 27
32
√ 23 . [7 marks]

lim
9. lim [5 marks]
Use l’Hôpital’s rule to determine the value of x→0( ).
2x cos ( x2 )
5tan x

3√ x−5
10a.
The expression can be written as 3 − 5xp . Write down the value [1 mark]
√x
of p.

( )
3 −5
 ( )d x.
10b. 9 3√ x−5 [4 marks]
Hence, find the value of ∫1
√x

R
11. Consider the curve with equation y = (2x − 1)ekx , where x ∈ R and [5 marks]
k ∈ Q.
The tangent to the curve at the point where x = 1 is parallel to the line y = 5ek x.
Find the value of k.

2 R
 A function, f , has its derivative given by f'(x) = 3x2 − 12x + p, where p ∈ R.
The following diagram shows part of the graph of f'.

The graph of f' has an axis of symmetry x = q.

12a. Find the value of q. [2 marks]

The vertex of the graph of f' lies on the x-axis.

12b. Write down the value of the discriminant of f' . [1 mark]

12c. Hence or otherwise, find the value of p. [3 marks]

12d. Find the value of the gradient of the graph of f' at x = 0. [3 marks]
12e. Sketch the graph of f'', the second derivative of f . Indicate clearly the [2 marks]
x-intercept and the y-intercept.

The graph of f has a point of inflexion at x = a.

12f. Write down the value of a . [1 mark]

12g. Find the values of x for which the graph of f is concave-down. Justify [2 marks]
your answer.

13a. Expand and simplify (1 − a)3 in ascending powers of a. [2 marks]

13b. By using a suitable substitution for a , show that [4 marks]

1 − 3 cos 2x + 3 cos2 2x− cos3 2x = 8 sin6 x.

2 3
 Consider f(x)= 4 cos x(1 − 3 cos 2x + 3 cos2 2x − cos3 2x).

13c. Show that ∫ m f(x)d x = 32 sin 7 m, where m is a positive real constant. [4 marks]
0 7

π
13d. It is given that ∫ π2 f(x)d x = 127
, where 0≤m≤ π [5 marks]
m 28 2 . Find the value of
m.
 The following diagram shows part of the graph of a quadratic function f .
The graph of f has its vertex at (3, 4), and it passes through point Q as shown.

14a. Write down the equation of the axis of symmetry. [1 mark]

The function can be written in the form f(x) = a(x − h)2 + k.

14b. Write down the values of h and k. [2 marks]

Q (5, 12)
14c. Point Q has coordinates (5, 12). Find the value of a. [2 marks]

The line L is tangent to the graph of f at Q.

14d. Find the equation of L. [4 marks]

= ( )
 Now consider another function y = g(x). The derivative of g is given by
g'(x) = f(x) − d, where d ∈ R.

14e. Find the values of d for which g is an increasing function. [3 marks]

14f. Find the values of x for which the graph of g is concave-up. [3 marks]

1
Consider the functions f(x)=
x−4
+ 1, for x ≠ 4, and g(x)= x − 3 for x ∈ R.
 The following diagram shows the graphs of f and g.

The graphs of f and g intersect at points A and B. The coordinates of A are

(3, 0).

15a. Find the coordinates of B. [5 marks]

 In the following diagram, the shaded region is enclosed by the graph of f , the
graph of g , the x-axis, and the line x = k, where k ∈ Z.

The area of the shaded region can be written as ln(p) + 8, where p ∈ Z.

15b. Find the value of k and the value of p. [10 marks]

( )
3 −5
 ( )d x.
16. 9 3√ x−5 [5 marks]
Find the value of ∫1
√x

( ) = ex sin ∈R
 The function f is defined by f(x) = ex sin x, where x ∈ R.

17a. Find the Maclaurin series for f(x) up to and including the x3 term. [4 marks]

17b. Hence, find an approximate value for ∫ 1 ex2 sin(x2 )d x. [4 marks]

0

( ) = ex cos ∈R
 The function g is defined by g(x) = ex cos x, where x ∈ R.

17c. Show that g(x) satisfies the equation g '' (x) = 2(g'(x) − g(x)). [4 marks]

17d. Hence, deduce that g ( 4 ) (x)= 2(g '''(x)−g ''(x)). [1 mark]

( )
17e. Using the result from part (c), find the Maclaurin series for g(x) up to [5 marks]
and including the x4 term.

lim
17f. lim ex cos x−1−x [3 marks]
Hence, or otherwise, determine the value of x→0 .
x3

The continuous random variable X has probability density function

k
, 0≤x≤1
f(x)={ √ 4−3x2
0, otherwise.

18a. Find the value of k. [4 marks]

18b. Find E(X). [4 marks]

1
R
 1
A function f is defined by f(x)= x2−2x−3
, where x ∈ R, x ≠ −1, x ≠ 3.

19a. Sketch the curve y = f(x), clearly indicating any asymptotes with their [6 marks]
equations. State the coordinates of any local maximum or minimum points and
any points of intersection with the coordinate axes.

1
R
 1
A function g is defined by g(x)= x2−2x−3
, where x ∈ R, x > 3.

The inverse of g is g −1 .

19b. √ 4 x 2+ x [6 marks]
Show that g −1 (x)= 1 + x
.

19c. State the domain of g −1 . [1 mark]

( )= arctan x ∈R
 A function h is defined by h(x)= arctan x2 , where x ∈ R.

π
19d. Given that (h ∘ g)(a)=
4
, find the value of a . [7 marks]
q
Give your answer in the form p + 2 √ r, where p, q, r ∈ Z+ .

( )= √1 − 2 −1 ≤ ≤1
 A function f is defined by f(x)= x√1 − x2 where −1 ≤ x ≤ 1.
The graph of y = f(x) is shown below.

20a. Show that f is an odd function. [2 marks]

≤ ≤ , ∈R
20b. The range off is a ≤ y ≤ b, where a, b ∈ R. [6 marks]
Find the value of a and the value of b .

= sec
21. By using the substitution u = sec x or otherwise, find an expression for [6 marks]
π
3
∫
0 secn x tan x d x in terms of n, where n is a non-zero real number.
22. A continuous random variable X has the probability density function [6 marks]

⎧
⎪
2
⎪ ( b−a ) ( c−a )
(x − a), a ≤ x ≤ c
f(x)=⎨ 2
⎪
(b − x), c<x≤b.
⎩
⎪
( b− a ) ( b− c )
0, otherwise
The following diagram shows the graph of y = f(x) for a ≤ x ≤ b.

a+b
Given that c≥ 2 , find an expression for the median of X in terms of a, b and c.
23. Given that d y = cos(x − π4 ) and y = 2 when x = 3π
, find y in terms of [4 marks]
dx 4
x.

The function f is defined for all x ∈ R. The line with equation y = 6x − 1 is the
tangent to the graph of f at x = 4.

24a. Write down the value of f'(4). [1 mark]

24b. Find f(4). [1 mark]

R 2
 The function g is defined for all x ∈ R where g(x)= x2 − 3x and h(x)= f(g(x)).

24c. Find h(4). [2 marks]

24d. Hence find the equation of the tangent to the graph of h at x = 4. [3 marks]

−1
 A particle P moves along the x-axis. The velocity of P is v m s−1 at time t
seconds, where v(t) = 4 + 4t − 3t2 for 0 ≤ t ≤ 3. When t = 0, P is at the origin
O.

25a. Find the value of t when P reaches its maximum velocity. [2 marks]

88
25b. Show that the distance of P from 88 [5 marks]
O at this time is 27
metres.
25c. Sketch a graph of v against t, clearly showing any points of [4 marks]
intersection with the axes.
 25d. Find the total distance travelled by P. [5 marks]

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