F4 Functions with trig

QUESTION PAPER (QP)

1 The function f : x → 5 sin2 x + 3 cos2 x is defined for the domain 0 ≤ x ≤ π .

(i) Express f(x) in the form a + b sin2 x, stating the values of a and b. [2]

(ii) Hence find the values of x for which f(x) = 7 sin x. [3]

(iii) State the range of f. [2]

9709/01/O/N/04
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2 A function f is defined by f : x → 3 − 2 sin x, for 0 ≤ x ≤ 360 .

(i) Find the range of f. [2]

(ii) Sketch the graph of y = f(x). [2]

A function g is defined by g : x → 3 − 2 sin x, for 0◦ ≤ x ≤ A◦ , where A is a constant.

(iii) State the largest value of A for which g has an inverse. [1]

(iv) When A has this value, obtain an expression, in terms of x, for g−1 (x). [2]
9709/01/M/J/05

3 The function f is defined by f(x) = a + b cos 2x, for 0 ≤ x ≤ π . It is given that f(0) = −1 and f
12 π  = 7.

(i) Find the values of a and b. [3]

(ii) Find the x-coordinates of the points where the curve y = f(x) intersects the x-axis. [3]

(iii) Sketch the graph of y = f(x). [2]

9709/01/M/J/07

4 The function f is such that f(x) = a − b cos x for 0◦ ≤ x ≤ 360◦ , where a and b are positive constants.
The maximum value of f(x) is 10 and the minimum value is −2.

(i) Find the values of a and b. [3]

(ii) Solve the equation f(x) = 0. [3]

(iii) Sketch the graph of y = f(x). [2]

9709/01/O/N/08

5 The function f is defined by f : x → 5 − 3 sin 2x for 0 ≤ x ≤ π .

(i) Find the range of f. [2]

(ii) Sketch the graph of y = f(x). [3]

(iii) State, with a reason, whether f has an inverse. [1]

9709/12/O/N/09
6 The function f is such that f (x) = 2 sin2 x − 3 cos2 x for 0 ≤ x ≤ π .

(i) Express f (x) in the form a + b cos2 x, stating the values of a and b. [2]

(ii) State the greatest and least values of f (x). [2]

(iii) Solve the equation f (x) + 1 = 0. [3]

9709/11/M/J/10

7 The function f : x → 4 − 3 sin x is defined for the domain 0 ≤ x ≤ 2π .

(i) Solve the equation f (x) = 2. [3]

(ii) Sketch the graph of y = f (x). [2]

(iii) Find the set of values of k for which the equation f (x) = k has no solution. [2]

The function g : x → 4 − 3 sin x is defined for the domain 12 π ≤ x ≤ A.

(iv) State the largest value of A for which g has an inverse. [1]

(v) For this value of A, find the value of g−1 (3). [2]

9709/12/M/J/10

8 The function f : x → a + b cos x is defined for 0 ≤ x ≤ 2π . Given that f (0) = 10 and that f 23 π
= 1, find
(i) the values of a and b, [2]
(ii) the range of f, [1]
(iii) the exact value of f 56 π
. [2]
9709/13/M/J/10

9 A function f is defined by f : x → 3 − 2 tan 12 x
for 0 ≤ x < π .

(i) State the range of f. [1]

(ii) State the exact value of f 23 π
. [1]

(iii) Sketch the graph of y = f (x). [2]

(iv) Obtain an expression, in terms of x, for f −1 (x). [3]

9709/11/O/N/10

10 The function f is such that f (x) = 3 − 4 cosk x, for 0 ≤ x ≤ π , where k is a constant.

(i) In the case where k = 2,

(a) find the range of f, [2]
(b) find the exact solutions of the equation f (x) = 1. [3]

(ii) In the case where k = 1,

(a) sketch the graph of y = f (x), [2]
(b) state, with a reason, whether f has an inverse. [1]
9709/12/M/J/11
11 The functions f and g are defined for − 12 π ≤ x ≤ 12 π by

f (x) = 12 x + 16 π ,
g(x) = cos x.

Solve the following equations for − 12 π ≤ x ≤ 12 π .

(i) gf (x) = 1, giving your answer in terms of π . [2]

(ii) fg(x) = 1, giving your answers correct to 2 decimal places. [4]

9709/13/O/N/12

12 A function f is defined by f : x → 3 cos x − 2 for 0 ≤ x ≤ 20.

(i) Solve the equation f x
= 0. [3]

(ii) Find the range of f. [2]

(iii) Sketch the graph of y = f x
. [2]

A function g is defined by g : x → 3 cos x − 2 for 0 ≤ x ≤ k.

(iv) State the maximum value of k for which g has an inverse. [1]

(v) Obtain an expression for g−1 x
. [2]

9709/12/O/N/13
 
13 The function f : x → 6 − 4 cos 12 x is defined for 0 ≤ x ≤ 2.

(i) Find the exact value of x for which f
x = 4. [3]

(ii) State the range of f. [2]

(iii) Sketch the graph of y = f
x. [2]

(iv) Find an expression for f −1
x. [3]

9709/12/O/N/14
 
14 The function f : x → 5 + 3 cos 12 x is defined for 0 ≤ x ≤ 20.

(i) Solve the equation f x
= 7, giving your answer correct to 2 decimal places. [3]

(ii) Sketch the graph of y = f x
. [2]

(iii) Explain why f has an inverse. [1]

(iv) Obtain an expression for f −1 x
. [3]

9709/11/M/J/15
15 The function f is defined by f : x → 4 sin x − 1 for − 12 0 ≤ x ≤ 12 0.

(i) State the range of f. [2]

(ii) Find the coordinates of the points at which the curve y = f x
intersects the coordinate axes. [3]

(iii) Sketch the graph of y = f x
. [2]

(iv) Obtain an expression for f −1 x
, stating both the domain and range of f −1 . [4]

9709/11/M/J/16

16 A function f is defined by f : x → 5 − 2 sin 2x for 0 ≤ x ≤ 0.

(i) Find the range of f. [2]

(ii) Sketch the graph of y = f x
. [2]

(iii) Solve the equation f x
= 6, giving answers in terms of 0. [3]

The function g is defined by g : x → 5 − 2 sin 2x for 0 ≤ x ≤ k, where k is a constant.

(iv) State the largest value of k for which g has an inverse. [1]

(v) For this value of k, find an expression for g−1 x
. [3]

9709/12/O/N/16

 
17 The function f is defined by f x
= 3 tan 12 x − 2, for − 12 0 ≤ x ≤ 12 0.

(i) Solve the equation f x
+ 4 = 0, giving your answer correct to 1 decimal place. [3]

(ii) Find an expression for f −1 x
and find the domain of f −1 . [5]

(iii) Sketch, on the same diagram, the graphs of y = f x
and y = f −1 x
. [3]
9709/12/M/J/17

   
18 (a) The function f, defined by f : x → a + b sin x for x ∈ >, is such that f 16 0 = 4 and f 12 0 = 3.

(i) Find the values of the constants a and b. [3]

(ii) Evaluate ff 0
. [2]
(b) The function g is defined by g : x → c + d sin x for x ∈ >. The range of g is given by −4 ≤ g x
≤ 10.
Find the values of the constants c and d . [3]
9709/12/O/N/17

 
19 The function f is such that f x
= a + b cos x for 0 ≤ x ≤ 20. It is given that f 13 0 = 5 and f 0
= 11.

(i) Find the values of the constants a and b. [3]

(ii) Find the set of values of k for which the equation f x
= k has no solution. [3]
9709/12/M/J/18
20 Functions f and g are defined by
f : x → 2 − 3 cos x for 0 ≤ x ≤ 20,

g : x → 12 x for 0 ≤ x ≤ 20.

(i) Solve the equation fg x
= 1. [3]

(ii) Sketch the graph of y = f x
. [3]

9709/12/O/N/18

21 The function f is defined by f x
= 2 − 3 cos x for 0 ≤ x ≤ 20.

(i) State the range of f. [2]

(ii) Sketch the graph of y = f x
. [2]

The function g is defined by g x
= 2 − 3 cos x for 0 ≤ x ≤ p, where p is a constant.

(iii) State the largest value of p for which g has an inverse. [1]

(iv) For this value of p, find an expression for g−1 x
. [2]

9709/11/M/J/19

22
y

y = f x

x
O 1 0
20

The function f : x → p sin2 2x + q is defined for 0 ≤ x ≤ 0, where p and q are positive constants. The
diagram shows the graph of y = f x
.

(i) In terms of p and q, state the range of f. [2]

(ii) State the number of solutions of the following equations.
(a) f x
= p + q [1]
(b) f x
= q [1]

(c) f x
= 12 p + q [1]

(iii) For the case where p = 3 and q = 2, solve the equation f x
= 4, showing all necessary working.
[5]

9709/13/M/J/19
23 (a) Given that x > 0, find the two smallest values of x, in radians, for which 3 tan 2x + 1
= 1. Show
all necessary working. [4]

(b) The function f : x → 3 cos2 x − 2 sin2 x is defined for 0 ≤ x ≤ 0.

(i) Express f x
in the form a cos2 x + b, where a and b are constants. [1]

(ii) Find the range of f. [2]

9709/12/O/N/19

24 The function f is given by f x
= 4 cos4 x + cos2 x − k for 0 ≤ x ≤ 2π, where k is a constant.

(a) Given that k = 3, find the exact solutions of the equation f x
= 0. [5]

(b) Use the quadratic formula to show that, when k > 5, the equation f x
= 0 has no solutions. [5]

9709/12/M/J/22