ZIMSEC A-Level Pure Mathematics Paper 1 November 2021 Question Paper

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Advanced Level

**PURE MATHEMATICS**

**PAPER 1**

NOVEMBER 2021 SESSION

3 hours

Additional materials:

Answer paper

Graph paper

List of Formulae MF7

Scientific calculator [non - programmable]

TIME 3 hours

INSTRUCTIONS TO CANDIDATES

Write your Name, Centre number and Candidate number in the spaces provided on the answer paper/answer booklet.

Answer all questions.

If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given correct to the nearest degree, and in other cases it should be given correct to 2 significant figures.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 120.

The use of a non-programmable scientific calculator is expected, where appropriate.

You are reminded of the need for clear presentation in your answers.

Question 1

Solve the equation,

\(2 e^{2 x}-7 e^{x}+6=0\).

Give the answer in exact form.

Question 2

Prove the identity

$$ \frac{\sin 2 \theta}{1+\cos 2 \theta}=\tan \theta . $$

Question 3

Given that \(y=\ln \left[\cos \left(x^{2}+1\right)\right]\), show that \(\frac{d y}{d x}=-2 x \tan \left(x^{2}+1\right)\).

Question 4

Given that M is inversely proportional to the cube root of \((n-1)\) and that when \(n=9, M=5\),

find

(a)

a formula connecting \(M\) and \(n\)

(b)

the exact value of \(n\) when \(M=25\).

Question 5

(a)

Express \(2 x^{2}+3 x+1\) in the form

$$ a(x+b)^{2}+c \text { where } a, b \text { and } c \text { are constants to be found. } $$

(b)

Hence or otherwise solve the equation \(2 x^{2}+3 x+1=0\).

Question 6

In the diagram, A and B are two points on the circumference of a circle centre O and radius \(r\). Angle \(A O B\) is \(\theta\) radians and \(C\) is the mid-point of \(O A\). The length of \(B C\) is \(y\) and the length of the \(\operatorname{arc} \mathrm{BA}\) is \(t\).

(a)

Express \(y^{2}\) in terms of \(r\) and \(\theta\).

(b)

Hence or otherwise show that if \(\theta\) is small then \(y^{2} \approx \frac{1}{4} r^{2}+\frac{1}{2} t^{2}\).

Question 7

If \(\mathrm{A}=\left(\begin{array}{ll}3 & -1 \\ 4 & -1\end{array}\right)\), prove by induction that \(A^{n}=\left(\begin{array}{cc}2 n+1 & -n \\ 4 n & 1-2 n\end{array}\right)\), where \(n\) is a positive integer.

Question 8

(a)

Solve the equation \(\frac{2^{x}+1}{2^{x}-1}=5\) giving the answer correct to 3 significant figures.

(b)

Solve the inequality \(|x-3|>2|3 x+1|\).

Question 9

Given that \(x\) is so small that \(x^{4}\) and higher powers can be neglected,

(a)

show that

$$ \frac{e^{x}}{1+x} \approx 1+\frac{x^{2}}{2}-\frac{x^{3}}{3} $$

(b)

use the expansion to evaluate

\(\frac{e^{0.01}}{1.01}\) correct to 7 decimal places.

Question 10

(a)

Given that \(f(x)=2-|x|\) and \(g(x)=\frac{1}{3} x+1\), sketch on the same axes the graphs of \(y=f(x)\) and \(y=g(x)\),

(b)

Find the co-ordinates of the points of intersection of \(f(x)\) and \(g(x)\).

Question 11

(a)

Express \(\frac{2 x}{(x+2)(x-2)(x-1)}\), in partial fractions.

(b)

Solve the inequality \(\frac{2 x}{(x+2)(x-2)(x-1)}<0\).

Question 12

If \(p=-4+3 i\) and \(q=-1+\sqrt{3} i\),

(a)

calculate the modulus of,

#### (i)

\(p\),

#### (ii)

\(q\).

(b)

find

#### (i)

the argument of \(q\),

#### (ii)

\(p q^{2}\) in the form \(a+b i\),

#### (iii)

\(\frac{p}{q}\) in the form \(a+b i\).

Question 13

The curve \(y=10-\frac{5}{x}\) and the line \(y+x=6\) intersect at two points P and Q.

Find the

(a)

coordinates of P and Q,

(b)

coordinates of the midpoint of PQ,

(c)

equation of the perpendicular bisector of the line with P and Q as end points.

Question 14

The curve \(y=x^{2}+p x+q\), where p and q are constants, has a turning point at \((-1 ;-5)\)

Find the

(a)

values of \(p\) and \(q\),

(b)

equations of the tangent and the normal in the form \(a y+b x+c=0\), at the point where the curve cuts the \(y\)-axis.

Question 15

(a)

A point ( \(1 ; 2\) ) is mapped onto point ( \(7 ; 2\) ) by a shear parallel to the \(x\)-axis. Find the shear factor.

(b)

A triangle with vertices \(\mathrm{A}(-3 ; 1), \mathrm{B}(3 ; 1)\) and \(\mathrm{C}(3 ; 5)\) is transformed by \(\mathrm{M}=\left(\begin{array}{ll}4 & 1 \\ 2 & 3\end{array}\right)\).

#### (i)

Find the area of triangle \(A B C\).

#### (ii)

Find the co-ordinates of \(\mathrm{A}_{1} \mathrm{~B}_{1}\) and \(\mathrm{C}_{1}\) the images of \(\mathrm{A}, \mathrm{B}\) and C under transformation M.

#### (iii)

Hence or otherwise find the area of the triangle \(A_{1} B_{1} C_{1}\).

Question 16

The gradient function of a curve is directly proportional to \(20-y\). If \(y=0\), \(x=0\) and the gradient is 1

(a)

Show that \(\frac{d y}{d x}=0,05(20-y)\).

(b)

Solve the differential equation giving y in terms of \(x\).

(c)

Find \(y\) when \(x=10\) giving the answer correct to 2 decimal places.

(d)

Briefly describe what happens to \(y\) as \(x\) becomes very large.