Higher Maths 2017 Paper 1 — Questions & Video Solutions

1

Question 1

Composite functions

(1, 2)3 Marks

2017 P1 Q1
Functions

\displaystyle f

and

\displaystyle g

are defined on suitable domains by

\displaystyle f(x)=5x

and

\displaystyle g(x)=2 \cos x

(a) Evaluate

\displaystyle f(g(0))

.
(b) Find an expression for

\displaystyle g(f(x))

.

Show answer

(a) 10
(b)

\displaystyle 2 \cos(5x)

2

Question 2

Equation of a tangent to a circle at a point

4 Marks

2017 P1 Q2
The point

\displaystyle P(-2,1)

lies on the circle

\displaystyle x^{2}+y^{2}-8x-6y-15=0

Find the equation of the tangent to the circle at P.

Show answer

\displaystyle y=-3x-5

3

Question 3

Differentiate or evaluate derivative: composite function

2 Marks

2017 P1 Q3
Given

\displaystyle y=(4x-1)^{12}

, find

\displaystyle \frac{dy}{dx}

.

Show answer

\displaystyle 48(4x-1)^{11}

4

Question 4

Discriminant and Quadratics

3 Marks

2017 P1 Q4
Find the value of

\displaystyle k

for which the equation

\displaystyle x^{2}+4x+(k-5)=0

has equal roots.

Show answer

\displaystyle k=9

5

Question 5

Scalar product

(1, 3)4 Marks

2017 P1 Q5

Vectors $\displaystyle \mathbf{u}$ and $\displaystyle \mathbf{v}$ are
$\displaystyle \begin{pmatrix}5\\1\\-1\end{pmatrix}$ and $\displaystyle \begin{pmatrix}3\\-8\\6\end{pmatrix}$
respectively.

(a) Evaluate $\displaystyle \mathbf{u}\cdot\mathbf{v}$ .
(b) Vector $\displaystyle \mathbf{w}$ makes an angle of $\displaystyle \frac{\pi}{3}$ with $\displaystyle \mathbf{u}$ and $\displaystyle |\mathbf{w}|=\sqrt{3}$ .
Calculate $\displaystyle \mathbf{u}\cdot\mathbf{w}$ .

Show answer

(a) 1
(b) 4.5

6

Question 6

Inverse functions

3 Marks

2017 P1 Q6
A function,

\displaystyle h

, is defined by

\displaystyle h(x)=x^{3}+7

where

\displaystyle x\in\mathbb{R}

.
Determine an expression for

\displaystyle h^{-1}(x).

Show answer

\displaystyle h^{-1}(x)=\sqrt[3]{x-7}

7

Question 7

Altitudes, medians, perpendicular bisectors

3 Marks

2017 P1 Q7

\displaystyle A(-3,5)

,

\displaystyle B(7, 9)

and

\displaystyle C(2,11)

are the vertices of a triangle.
Find the equation of the median through C.

Show answer

\displaystyle x=2

8

Question 8

Differentiate or evaluate derivative: polynomial

3 Marks

2017 P1 Q8
Calculate the rate of change of

\displaystyle d(t)=\frac{1}{2t}

,

\displaystyle t\ne0

when

\displaystyle t=5.

Show answer

\displaystyle -\frac{1}{50}

9

Question 9

Limits of recurrence relationsGo backwards to find recurrence relation when terms known

(2, 1, 2)5 Marks

2017 P1 Q9

A sequence is generated by the recurrence relation
$\displaystyle u_{n+1}=mu_{n}+6$
where $\displaystyle m$ is a constant.

(a)  Given $\displaystyle u_{1}=28$ and $\displaystyle u_{2}=13$ , find the value of $\displaystyle m$ .
(b)  (i) Explain why this sequence approaches a limit as $\displaystyle n\rightarrow\infty$ .
      (ii) Calculate this limit.

Show answer

(a)

\displaystyle m=\frac{1}{4}

(b) Limit is 8

10

Question 10

Areas using integration

(5, 4)9 Marks

2017 P1 Q10

Two curves with equations
$\displaystyle y=x^{3}-4x^{2}+3x+1$
and
$\displaystyle y=x^{2}-3x+1$
intersect as shown in the diagram.

(a) Calculate the shaded area.

The line passing through the points of intersection of the curves has equation $\displaystyle y=1-x$ .

(b) Determine the fraction of the shaded area which lies below the line $\displaystyle y=1-x$ .

Show answer

(a)

\displaystyle \frac{8}{3}

(b)

\displaystyle \frac{1}{2}

11

Question 11

Perpendicular and parallel lines

3 Marks

2017 P1 Q11
A and B are the points

\displaystyle (-7, 2)

and

\displaystyle (5, a)

.
AB is parallel to the line with equation

\displaystyle 3y-2x=4

Determine the value of

\displaystyle a

.

Show answer

\displaystyle a=10

12

Question 12

Solving equations containing a logarithm

3 Marks

2017 P1 Q12
Given that

\displaystyle \log_{a}36-\log_{a}4=\frac{1}{2}

find the value of

\displaystyle a

.

Show answer

\displaystyle a=81

13

Question 13

Integrate (definite or indefinite): (px + q)^n

4 Marks

2017 P1 Q13
Find

\displaystyle \int\frac{1}{(5-4x)^{\frac{1}{2}}}dx

,

\displaystyle x \lt \frac{5}{4}.

Show answer

\displaystyle -\frac{1}{2}(5-4x)^{\frac{1}{2}}+c

14

Question 14

Wave function (y = asin x ± bcosx)Identifying/sketching graphs of related functions (trigonometric)

(4, 3)7 Marks

2017 P1 Q14

(a) Express
$\displaystyle \sqrt{3}\sin x^{\circ}-\cos x^{\circ}$
in the form $\displaystyle k\sin(x-a)^{\circ}$ where $\displaystyle k \gt 0$ and $\displaystyle 0 \lt a \lt 360$ .
(b) Hence, or otherwise, sketch the graph with equation
$\displaystyle y=\sqrt{3}\sin x^{\circ}-\cos x^{\circ}$
for $\displaystyle 0\le x\le360$ .

Show answer

(a)

\displaystyle 2\sin(x-30)^{\circ}

(b) Sketch of sine wave with amplitude 2, shifted 30 degrees to the right.

15

Question 15

Identifying/sketching graphs of related functions (non-trigonometric)

(2, 1, 1)4 Marks

2017 P1 Q15

A quadratic function, $\displaystyle f$ , is defined on $\displaystyle \mathbb{R}$ , the set of real numbers.
Diagram 1 shows part of the graph with equation $\displaystyle y=f(x)$ . The turning point is $\displaystyle (2, 3)$ .
Diagram 2 shows part of the graph with equation $\displaystyle y=h(x)$ . The turning point is $\displaystyle (7, 6)$ .

Diagram 1 and 2: Graphs of y=f(x) and y=h(x)

(a)  Given that $\displaystyle h(x)=f(x+a)+b$ . Write down the values of $\displaystyle a$ and $\displaystyle b$ .
(b)  It is known that $\displaystyle \int_{1}^{3}f(x)dx=4$ . Determine the value of $\displaystyle \int_{6}^{8}h(x)dx$ .
(c)  Given $\displaystyle f'(1)=6$ state the value of $\displaystyle h'(8)$ .

Show answer

(a)

\displaystyle a=-5, b=3

(b) 10
(c) -6