ZIMSEC A Level Pure Mathematics Paper 1 November 2021

Question 101

[1 marks]Exponential equations

Solve the equation

2e^{2x} - 7e^x + 6 = 0

, giving your answers in exact form.

A $x = \ln 2$ or $x = \ln 3$
B $x = \ln\tfrac{1}{2}$ or $x = \ln\tfrac{2}{3}$
C $x = \ln 2$ or $x = \ln\tfrac{3}{2}$
D $x = 2$ or $x = \tfrac{3}{2}$

Question 102

[2 marks]Exponential equations

Let

u=e^x

. Factorise

2u^2-7u+6

Answer this when you sit the paper.

Question 201

[1 marks]Trigonometric identities

Simplify

\dfrac{\sin 2\theta}{1 + \cos 2\theta}

A $2\tan\theta$
B $\cot\theta$
C $\tan\theta$
D $\tan 2\theta$

Question 202

[2 marks]Trigonometric identities

Using double angle formulas, rewrite the numerator

\sin2\theta

and denominator

1+\cos2\theta

\dfrac{\sin2\theta}{1+\cos2\theta}

in terms of

\sin\theta

and

\cos\theta

only. State the simplified denominator, in the form

k\cos^2\theta

. State

k

Answer this when you sit the paper.

Question 301

[1 marks]Differentiation (chain rule)

Given

y = \ln\left[\cos(x^2+1)\right]

, find

\dfrac{dy}{dx}

A $2x\tan(x^2+1)$
B $\dfrac{-2x}{\cos(x^2+1)}$
C $-2x\tan(x^2+1)$
D $-\tan(x^2+1)$

Question 302

[2 marks]Differentiation (chain rule)

Using the chain rule on

y=\ln[\cos(x^2+1)]

with

u=x^2+1

, find

\dfrac{dy}{du}

, in terms of

u

Answer this when you sit the paper.

Question 401

[1 marks]Variation

M

is inversely proportional to the cube root of

(n-1)

, and

M = 5

when

n = 9

. Find the formula connecting

M

and

n

A $M = \dfrac{10}{\sqrt[3]{n-1}}$
B $M = \dfrac{5}{\sqrt[3]{n-1}}$
C $M = 10\sqrt[3]{n-1}$
D $M = \dfrac{40}{\sqrt[3]{n-1}}$

Question 402

[1 marks]Variation

Given

M = \dfrac{10}{\sqrt[3]{n-1}}

, find the exact value of

n

when

M = 25

A $n = \tfrac{2}{5}$
B $n = \tfrac{8}{125}$
C $n = \tfrac{125}{8}$
D $n = \tfrac{133}{125}$

Question 403

[2 marks]Variation

M

is inversely proportional to the cube root of

(n-1)

, so

M=\dfrac{k}{\sqrt[3]{n-1}}

for constant

k

. Using

M=5

when

n=9

, find

\sqrt[3]{n-1}=\sqrt[3]{8}

first, then solve for

k

Answer this when you sit the paper.

Question 404

[1 marks]Variation

Using

M=\dfrac{10}{\sqrt[3]{n-1}}=25

, find

\sqrt[3]{n-1}

(before cubing to find

n

Answer this when you sit the paper.

Question 501

[1 marks]Completing the square

Express

2x^2 + 3x + 1

in the form

a(x+b)^2 + c

A $2\left(x + \tfrac{3}{4}\right)^2 - \tfrac{1}{8}$
B $2\left(x + \tfrac{3}{4}\right)^2 + \tfrac{1}{8}$
C $2\left(x + \tfrac{3}{2}\right)^2 - \tfrac{1}{8}$
D $2\left(x - \tfrac{3}{4}\right)^2 - \tfrac{1}{8}$

Question 502

[1 marks]Completing the square

Solve the equation

2x^2 + 3x + 1 = 0

A $x = -\tfrac{1}{4}$ or $x = -1$
B $x = \tfrac{1}{2}$ or $x = 1$
C $x = -\tfrac{3}{4}$ only
D $x = -\tfrac{1}{2}$ or $x = -1$

Question 503

[2 marks]Completing the square

Completing the square,

2x^2+3x+1=2(x+b)^2+c

. State the values of

b

and

c

, as exact fractions.

Answer this when you sit the paper.

Question 504

[2 marks]Completing the square

Factorise

2x^2+3x+1

, stating both linear factors.

Answer this when you sit the paper.

Question 601

[1 marks]Circular measure and cosine rule

A

and

B

lie on a circle of centre

O

and radius

r

with angle

AOB = \theta

radians, and

C

is the midpoint of

OA

. Express

BC^2

in terms of

r

and

\theta

A $\tfrac{r^2}{4} - r^2\cos\theta$
B $\tfrac{5r^2}{4} - r^2\cos\theta$
C $\tfrac{5r^2}{4} + r^2\cos\theta$
D $r^2 - \tfrac{r^2}{4}\cos\theta$

Question 602

[1 marks]Circular measure and cosine rule

y^2 = \tfrac{5r^2}{4} - r^2\cos\theta

and

\theta

is small, with

t = r\theta

the arc length, then

y^2 \approx

A $\tfrac{1}{4}r^2 - \tfrac{1}{2}t^2$
B $\tfrac{1}{4}r^2 + \tfrac{1}{2}t^2$
C $\tfrac{5}{4}r^2 + \tfrac{1}{2}t^2$
D $\tfrac{1}{2}r^2 + \tfrac{1}{4}t^2$

Question 603

[2 marks]Circular measure and cosine rule

For small

\theta

\cos\theta\approx1-\dfrac{\theta^2}{2}

. Substitute this into

y^2=\dfrac{5r^2}{4}-r^2\cos\theta

and simplify to the form

y^2\approx\dfrac{r^2}{4}+Kr^2\theta^2

. State

K

Answer this when you sit the paper.

Question 604

[2 marks]Circular measure and cosine rule

Since

\theta=\dfrac{t}{r}

(arc length

t=r\theta

), express

r^2\theta^2

in terms of

t

only.

Answer this when you sit the paper.

Question 701

[1 marks]Proof by induction (matrices)

In a proof by induction that

A^n = \begin{pmatrix} 2n+1 & -n \\ 4n & 1-2n \end{pmatrix}

for

A = \begin{pmatrix} 3 & -1 \\ 4 & -1 \end{pmatrix}

, what is the inductive step?

AShow $A^{k+1} = A^k A$ matches the formula with $n = k+1$ , assuming it holds for $n = k$
BShow that $\det A^n = (\det A)^n$
CShow that $A^n \to 0$ as $n \to \infty$
DVerify the formula holds for $n = 1$ and $n = 2$

Question 702

[2 marks]Proof by induction (matrices)

As part of verifying the base case

n=2

, compute

A^2=AA

for

A=\begin{pmatrix}3&-1\\4&-1\end{pmatrix}

. State the bottom-left entry of

A^2

Answer this when you sit the paper.

Question 703

[2 marks]Proof by induction (matrices)

In the inductive step, assuming

A^k=\begin{pmatrix}2k+1&-k\\4k&1-2k\end{pmatrix}

, compute

A^{k+1}=A^kA

. State the top-left entry of

A^{k+1}

, in terms of

k

Answer this when you sit the paper.

Question 704

[2 marks]Proof by induction (matrices)

Continuing

A^{k+1}=A^kA

with

A^k=\begin{pmatrix}2k+1&-k\\4k&1-2k\end{pmatrix}

and

A=\begin{pmatrix}3&-1\\4&-1\end{pmatrix}

, state the bottom-left entry of

A^{k+1}

, in terms of

k

Answer this when you sit the paper.

Question 801

[1 marks]Exponential equations and modulus inequalities

Solve the equation

\dfrac{2^x + 1}{2^x - 1} = 5

, giving your answer correct to 3 significant figures.

A $x = 1.50$
B $x = 0.405$
C $x = 0.585$
D $x = 1.58$

Question 802

[1 marks]Exponential equations and modulus inequalities

Solve the inequality

|x-3| > 2|3x+1|

A $-1 < x < \tfrac{1}{7}$
B $x < -1$ or $x > \tfrac{1}{7}$
C $-\tfrac{1}{7} < x < 1$
D $-\tfrac{1}{3} < x < 3$

Question 803

[2 marks]Exponential equations and modulus inequalities

Let

u=2^x

. Solve

\dfrac{u+1}{u-1}=5

for

u

Answer this when you sit the paper.

Question 804

[2 marks]Exponential equations and modulus inequalities

Squaring both sides of

|x-3|>2|3x+1|

and simplifying gives an inequality of the form

7x^2+bx+c<0

. State

b

and

c

Answer this when you sit the paper.

Question 805

[2 marks]Exponential equations and modulus inequalities

Factorise

7x^2+6x-1

, stating both linear factors.

Answer this when you sit the paper.

Question 901

[1 marks]Series expansions

Given that

x

is small enough for

x^4

and higher powers to be neglected, expand

\dfrac{e^x}{1+x}

A $1 + \tfrac{x^2}{2} + \tfrac{x^3}{3}$
B $1 + \tfrac{x^2}{2} - \tfrac{x^3}{3}$
C $1 + x + \tfrac{x^2}{2} - \tfrac{x^3}{3}$
D $1 - \tfrac{x^2}{2} + \tfrac{x^3}{3}$

Question 902

[1 marks]Series expansions

Use the expansion

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

to evaluate

\dfrac{e^{0.01}}{1.01}

correct to 7 decimal places.

A $0.9999503$
B $1.0000497$
C $1.0000500$
D $1.0100000$

Question 903

[2 marks]Series expansions

State the Maclaurin expansion of

e^x

up to the term in

x^3

Answer this when you sit the paper.

Question 904

[2 marks]Series expansions

State the binomial expansion of

(1+x)^{-1}

up to the term in

x^3

Answer this when you sit the paper.

Question 905

[2 marks]Series expansions

Multiplying the series for

e^x

and

(1+x)^{-1}

, state the coefficient of

x^1

in the product

e^x(1+x)^{-1}

(before collecting further terms).

Answer this when you sit the paper.

Question 1001

[1 marks]Modulus functions and graphs

Find the coordinates of the points of intersection of

f(x) = 2 - |x|

and

g(x) = \tfrac{1}{3}x + 1

A $\left(-\tfrac{3}{2}, \tfrac{1}{2}\right)$ and $\left(\tfrac{3}{4}, \tfrac{5}{4}\right)$
B $\left(-\tfrac{3}{4}, \tfrac{3}{4}\right)$ and $\left(\tfrac{3}{2}, \tfrac{3}{2}\right)$
C $\left(\tfrac{3}{2}, \tfrac{1}{2}\right)$ and $\left(-\tfrac{3}{4}, \tfrac{5}{4}\right)$
D $(-3, 0)$ and $(2, 0)$

Question 1002

[1 marks]Modulus functions and graphs

State the

y

-intercept of

f(x)=2-|x|

Answer this when you sit the paper.

Question 1003

[1 marks]Modulus functions and graphs

State the

y

-intercept of

g(x)=\tfrac13x+1

Answer this when you sit the paper.

Question 1004

[2 marks]Modulus functions and graphs

Find both

x

-intercepts of

f(x)=2-|x|

Answer this when you sit the paper.

Question 1005

[1 marks]Modulus functions and graphs

Find the

x

-intercept of

g(x)=\tfrac13x+1

Answer this when you sit the paper.

Question 1006

[2 marks]Modulus functions and graphs

For

x\ge0

, solve

2-x=\tfrac13x+1

to find the

x

-coordinate of the intersection point in this region (before finding

y

Answer this when you sit the paper.

Question 1101

[1 marks]Partial fractions and rational inequalities

Express

\dfrac{2x}{(x+2)(x-2)(x-1)}

in partial fractions.

A $-\dfrac{2}{3(x+2)} + \dfrac{1}{x-2} - \dfrac{1}{3(x-1)}$
B $\dfrac{1}{3(x+2)} + \dfrac{1}{x-2} + \dfrac{2}{3(x-1)}$
C $-\dfrac{1}{3(x+2)} + \dfrac{1}{x-2} - \dfrac{2}{3(x-1)}$
D $-\dfrac{1}{3(x+2)} - \dfrac{1}{x-2} + \dfrac{2}{3(x-1)}$

Question 1102

[1 marks]Partial fractions and rational inequalities

Solve the inequality

\dfrac{2x}{(x+2)(x-2)(x-1)} < 0

A $x < -2$ or $0 < x < 1$
B $-2 < x < 0$ or $1 < x < 2$
C $0 < x < 1$ or $x > 2$
D $-2 < x < 2$

Question 1103

[2 marks]Partial fractions and rational inequalities

Using the cover-up rule, evaluate

A=\dfrac{2x}{(x-2)(x-1)}

x=-2

, the coefficient of

\dfrac{1}{x+2}

in the partial fraction decomposition of

\dfrac{2x}{(x+2)(x-2)(x-1)}

Answer this when you sit the paper.

Question 1104

[2 marks]Partial fractions and rational inequalities

Using the cover-up rule, evaluate

B=\dfrac{2x}{(x+2)(x-1)}

x=2

, the coefficient of

\dfrac{1}{x-2}

Answer this when you sit the paper.

Question 1105

[2 marks]Partial fractions and rational inequalities

Using the cover-up rule, evaluate

C=\dfrac{2x}{(x+2)(x-2)}

x=1

, the coefficient of

\dfrac{1}{x-1}

Answer this when you sit the paper.

Question 1201

[1 marks]Complex numbers

Given

p = -4 + 3i

and

q = -1 + \sqrt{3}i

, calculate

|p|

and

|q|

A $|p| = 5$ , $|q| = \sqrt{2}$
B $|p| = 25$ , $|q| = 4$
C $|p| = 7$ , $|q| = 2$
D $|p| = 5$ , $|q| = 2$

Question 1202

[1 marks]Complex numbers

Find the argument of

q = -1 + \sqrt{3}i

A $\tfrac{\pi}{3}$
B $\tfrac{2\pi}{3}$
C $-\tfrac{2\pi}{3}$
D $\tfrac{5\pi}{6}$

Question 1203

[1 marks]Complex numbers

Given

p = -4 + 3i

and

q = -1 + \sqrt{3}i

, find

pq^2

in the form

a + bi

A $(8 - 6\sqrt{3}) + (8\sqrt{3} + 6)i$
B $(8 + 6\sqrt{3}) - (8\sqrt{3} - 6)i$
C $(-2 - 2\sqrt{3})i$
D $(8 + 6\sqrt{3}) + (8\sqrt{3} - 6)i$

Question 1204

[2 marks]Complex numbers

Find

q^2

for

q=-1+\sqrt3i

, in the form

a+bi

Answer this when you sit the paper.

Question 1205

[2 marks]Complex numbers

To find

\dfrac{p}{q}

for

p=-4+3i

q=-1+\sqrt3i

, multiply numerator and denominator by the conjugate

-1-\sqrt3i

. Find the denominator

(-1+\sqrt3i)(-1-\sqrt3i)

Answer this when you sit the paper.

Question 1206

[2 marks]Complex numbers

Find the numerator

(-4+3i)(-1-\sqrt3i)

, in the form

a+bi

(in terms of

\sqrt3

Answer this when you sit the paper.

Question 1207

[2 marks]Complex numbers

Hence state

\dfrac{p}{q}

for

p=-4+3i

q=-1+\sqrt3i

, correct to 3 decimal places, in the form

a+bi

Answer this when you sit the paper.

Question 1208

[1 marks]Complex numbers

State the real part of

pq^2

, in the form

8+k\sqrt3

. State

k

Answer this when you sit the paper.

Question 1301

[1 marks]Coordinate geometry

The curve

y = 10 - \dfrac{5}{x}

meets the line

y + x = 6

P

and

Q

. Find their coordinates.

A $P(-1, 7)$ and $Q(5, 1)$
B $P(-5, 11)$ and $Q(1, 5)$
C $P(5, 1)$ and $Q(-1, 7)$
D $P(-5, 1)$ and $Q(1, 11)$

Question 1302

[1 marks]Coordinate geometry

Find the equation of the perpendicular bisector of the segment joining

P(-5, 11)

and

Q(1, 5)

A $y - x - 10 = 0$
B $y + x - 10 = 0$
C $y + x - 6 = 0$
D $y - x + 10 = 0$

Question 1303

[2 marks]Coordinate geometry

Substituting

y=6-x

into

y=10-\dfrac{5}{x}

and simplifying (multiply through by

x

) gives a quadratic in the form

x^2+bx+c=0

. State

b

and

c

Answer this when you sit the paper.

Question 1304

[1 marks]Coordinate geometry

Factorise

x^2+4x-5

Answer this when you sit the paper.

Question 1305

[2 marks]Coordinate geometry

Find the coordinates of the midpoint of

PQ

, where

P(-5,11)

and

Q(1,5)

Answer this when you sit the paper.

Question 1306

[1 marks]Coordinate geometry

Find the gradient of the line segment

PQ

, where

P(-5,11)

and

Q(1,5)

Answer this when you sit the paper.

Question 1307

[2 marks]Coordinate geometry

State the gradient of the perpendicular bisector of

PQ

(perpendicular to a line of gradient

-1

Answer this when you sit the paper.

Question 1401

[1 marks]Differentiation, tangents and normals

The curve

y = x^2 + px + q

has a turning point at

(-1, -5)

. Find

p

and

q

A $p = -2$ , $q = -4$
B $p = 2$ , $q = 4$
C $p = 2$ , $q = -4$
D $p = -2$ , $q = 4$

Question 1402

[1 marks]Differentiation, tangents and normals

The curve

y = x^2 + 2x - 4

cuts the

y

-axis at

(0, -4)

. Find the equations of the tangent and normal there, in the form

ay + bx + c = 0

Atangent $y - 2x - 4 = 0$ ; normal $2y + x - 8 = 0$
Btangent $2y - x + 8 = 0$ ; normal $y - 2x + 4 = 0$
Ctangent $y - 2x + 4 = 0$ ; normal $2y + x + 8 = 0$
Dtangent $y + 2x + 4 = 0$ ; normal $2y - x + 8 = 0$

Question 1403

[2 marks]Differentiation, tangents and normals

Using

\dfrac{dy}{dx}=2x+p=0

at the turning point

x=-1

y=x^2+px+q

, find

p

Answer this when you sit the paper.

Question 1404

[2 marks]Differentiation, tangents and normals

Substituting the turning point

(-1,-5)

into

y=x^2+2x+q

, find

q

Answer this when you sit the paper.

Question 1405

[2 marks]Differentiation, tangents and normals

For the curve

y=x^2+2x-4

, find

\dfrac{dy}{dx}

x=0

(the gradient of the tangent at the

y

-axis crossing).

Answer this when you sit the paper.

Question 1406

[2 marks]Differentiation, tangents and normals

State the gradient of the normal to

y=x^2+2x-4

x=0

(perpendicular to the tangent of gradient 2).

Answer this when you sit the paper.

Question 1501

[1 marks]Matrix transformations

A shear parallel to the

x

-axis maps

(1, 2)

onto

(7, 2)

. Find the shear factor.

A $3$
B $7$
C $\tfrac{1}{3}$
D $6$

Question 1502

[1 marks]Matrix transformations

Triangle

ABC

has vertices

A(-3, 1)

B(3, 1)

and

C(3, 5)

. Find its area.

A $12$ units $^2$
B $24$ units $^2$
C $6$ units $^2$
D $10$ units $^2$

Question 1503

[1 marks]Matrix transformations

Triangle

ABC

of area 12 units

^2

is transformed by

M = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}

. Find the area of the image triangle.

A $12$ units $^2$
B $120$ units $^2$
C $144$ units $^2$
D $10$ units $^2$

Question 1504

[2 marks]Matrix transformations

Find

A_1

, the image of

A(-3,1)

under

M=\begin{pmatrix}4&1\\2&3\end{pmatrix}

, in the form

(x,y)

Answer this when you sit the paper.

Question 1505

[2 marks]Matrix transformations

Find

B_1

, the image of

B(3,1)

under

M=\begin{pmatrix}4&1\\2&3\end{pmatrix}

, in the form

(x,y)

Answer this when you sit the paper.

Question 1506

[2 marks]Matrix transformations

Find

C_1

, the image of

C(3,5)

under

M=\begin{pmatrix}4&1\\2&3\end{pmatrix}

, in the form

(x,y)

Answer this when you sit the paper.

Question 1507

[2 marks]Matrix transformations

Write the shear matrix parallel to the

x

-axis with shear factor

k=3

, in the form

\begin{pmatrix}1&k\\0&1\end{pmatrix}

. State its determinant (confirming shears preserve area).

Answer this when you sit the paper.

Question 1601

[1 marks]Differential equations

The gradient function of a curve is directly proportional to

20 - y

. Given that the gradient is 1 when

y = 0

, find the differential equation.

A $\dfrac{dy}{dx} = 20(20 - y)$
B $\dfrac{dy}{dx} = 0.05y$
C $\dfrac{dy}{dx} = 20 - y$
D $\dfrac{dy}{dx} = 0.05(20 - y)$

Question 1602

[1 marks]Differential equations

Solve

\dfrac{dy}{dx} = 0.05(20 - y)

given that

y = 0

when

x = 0

A $y = 20e^{-0.05x}$
B $y = 20 + 20e^{-0.05x}$
C $y = 20 - 20e^{0.05x}$
D $y = 20 - 20e^{-0.05x}$

Question 1603

[1 marks]Differential equations

For

y = 20 - 20e^{-0.05x}

, describe what happens to

y

x

becomes very large.

A $y$ approaches 0
B $y$ oscillates about 20
C $y$ increases without limit
D $y$ approaches 20 from below

Question 1604

[2 marks]Differential equations

Separating variables in

\dfrac{dy}{dx}=0{,}05(20-y)

gives

\displaystyle\int\dfrac{1}{20-y}\,dy=\int0{,}05\,dx

. Evaluate the left-hand integral (ignore the constant of integration), in terms of

y

Answer this when you sit the paper.

Question 1605

[2 marks]Differential equations

Using

y=0

when

x=0

-\ln(20-y)=0{,}05x+C

, find the constant

C

Answer this when you sit the paper.

Question 1606

[2 marks]Differential equations

Using

y=20-20e^{-0.05x}

, find

y

when

x=5

, correct to 2 decimal places.

Answer this when you sit the paper.

Question 1607

[3 marks]Differential equations

Using

y=20-20e^{-0.05x}

, find

y

when

x=10

, correct to 2 decimal places.

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2021

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 301

Question 302

Question 401

Question 402

Question 403

Question 404

Question 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 604

Question 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 804

Question 805

Question 901

Question 902

Question 903

Question 904

Question 905

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1006

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

Question 1207

Question 1208

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1306

Question 1307

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

Question 1506

Question 1507

Question 1601

Question 1602

Question 1603

Question 1604

Question 1605

Question 1606