ZIMSEC A Level Pure Mathematics Paper 1 June 2010

Question 101

[1 marks]trigonometry

Given that

\cos\theta = -\dfrac{1}{5}

with

-180^\circ < \theta < -90^\circ

, what is the value of

\sin\theta

A $-\dfrac{24}{25}$
B $\dfrac{2\sqrt6}{5}$
C $-\dfrac{2\sqrt5}{5}$
D $-\dfrac{2\sqrt6}{5}$

Question 102

[1 marks]trigonometry

Given that

\cos\theta = -\dfrac{1}{5}

with

-180^\circ < \theta < -90^\circ

(so

\sin\theta = -\dfrac{2\sqrt6}{5}

), find the exact value of

\cot\theta

A $-\dfrac{\sqrt6}{12}$
B $\dfrac{\sqrt5}{10}$
C $2\sqrt6$
D $\dfrac{\sqrt6}{12}$

Question 103

[1 marks]trigonometry

Given

\cos\theta = -\dfrac{1}{5}

, find

\sin^2\theta

using

\sin^2\theta = 1 - \cos^2\theta

Answer this when you sit the paper.

Question 201

[1 marks]differentiation

Differentiate

e^{-2t}\sin t

with respect to

t

A $e^{-2t}\cos t$
B $e^{-2t}(\cos t - 2\sin t)$
C $e^{-2t}(\cos t - \sin t)$
D $e^{-2t}(\cos t + 2\sin t)$

Question 202

[1 marks]differentiation

Differentiate

\sec^2(3t-100)

with respect to

t

A $-6\sec^2(3t-100)\tan(3t-100)$
B $6\sec^2(3t-100)\tan(3t-100)$
C $2\sec^2(3t-100)\tan(3t-100)$
D $6\sec(3t-100)\tan(3t-100)$

Question 203

[2 marks]differentiation

Apply the product rule to

e^{-2t}\sin t

and write the result as an unsimplified sum of two terms, before combining like terms.

Answer this when you sit the paper.

Question 301

[1 marks]inequalities

Solve the inequality

|3x+1|\geq 2|x-2|

A $x\leq -\dfrac{3}{5}$ or $x\geq 5$
B $x\leq -5$ or $x\geq \dfrac{3}{5}$
C $-5\leq x\leq \dfrac{3}{5}$
D $x\leq -1$ or $x\geq 3$

Question 302

[3 marks]inequalities

Square both sides of

|3x+1| \geq 2|x-2|

, simplify to a quadratic

ax^2+bx+c \geq 0

, then factorise the quadratic. State the factorised form.

Answer this when you sit the paper.

Question 401

[1 marks]partial fractions

Express

\dfrac{3x+8}{(2x+1)(x^2+3)}

in partial fractions.

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

Question 402

[2 marks]partial fractions

Substitute

x=-\frac12

into

3x+8 = A(x^2+3) + (Bx+C)(2x+1)

to find

A

Answer this when you sit the paper.

Question 403

[1 marks]partial fractions

Comparing coefficients of

x^2

3x+8 = A(x^2+3) + (Bx+C)(2x+1)

, find

B

Answer this when you sit the paper.

Question 501

[1 marks]exponential and logarithmic functions

A relation

y=ab^x

is linearised and plotted as a straight-line graph passing through the points

(0, 1.1)

and

(1, 2.7)

. What should be plotted on the vertical and horizontal axes respectively to produce this straight line?

A $\ln y$ on the vertical axis and $x$ on the horizontal axis
B $y$ on the vertical axis and $\ln x$ on the horizontal axis
C $\ln y$ on the vertical axis and $\ln x$ on the horizontal axis
D $x$ on the vertical axis and $\ln y$ on the horizontal axis

Question 502

[1 marks]exponential and logarithmic functions

The linearised graph of

y=ab^x

(plotting

\ln y

against

x

) passes through

(0, 1.1)

and

(1, 2.7)

. Calculate the value of

a

, correct to 2 decimal places.

A $a\approx1.73$
B $a\approx1.10$
C $a\approx4.95$
D $a\approx3.00$

Question 503

[1 marks]exponential and logarithmic functions

The linearised graph of

y=ab^x

(plotting

\ln y

against

x

) passes through

(0, 1.1)

and

(1, 2.7)

. Calculate the value of

b

, correct to 2 decimal places.

A $b\approx1.60$
B $b\approx3.00$
C $b\approx4.95$
D $b\approx11.64$

Question 504

[2 marks]exponential and logarithmic functions

Taking natural logs of

y=ab^x

gives

\ln y = \ln a + x\ln b

. Which statement correctly identifies the intercept and gradient of this linearised graph (with

\ln y

on the vertical axis and

x

on the horizontal axis)?

A $y$ -intercept $=\ln(ab)$ , gradient $=1$ (constant)
B $y$ -intercept $=\ln b$ , gradient $=\ln a$ (axes swapped)
C $y$ -intercept $=\ln a$ , gradient $=\ln b$
D $y$ -intercept $=a$ , gradient $=b$ (unlogged values)

Question 601

[1 marks]polynomials

Find the value of

a

for which

(x-2)

is a factor of

3x^3+ax^2+x-2

Answer this when you sit the paper.

Question 602

[1 marks]polynomials

With

a=-6

, the cubic

3x^3-6x^2+x-2

factorises as

(x-2)(3x^2+1)

. Which statement correctly explains why

x=2

is the equation's only real root?

AThe quadratic $3x^2+1$ has discriminant $0^2-4(3)(1)=-12$ , which is negative, so it contributes no real roots.
BThe quadratic $3x^2+1$ factorises into two real linear factors, each giving an extra real root.
CDividing by $(x-2)$ leaves a linear remainder, which contributes exactly one further real root.
DThe quadratic $3x^2+1$ has discriminant $0$ , giving one repeated real root equal to $2$ .

Question 603

[1 marks]polynomials

By the factor theorem, write the equation obtained by substituting

x=2

into

3x^3+ax^2+x-2=0

, in terms of

a

, set equal to

0

Answer this when you sit the paper.

Question 604

[2 marks]polynomials

With

a=-6

, divide

3x^3-6x^2+x-2

(x-2)

to find the quadratic factor.

Answer this when you sit the paper.

Question 701

[1 marks]coordinate geometry - circles

Write down the equation of the circle with centre

(-3, 2)

and radius

\sqrt{10}

A $(x-3)^2+(y+2)^2=10$
B $(x+3)^2+(y-2)^2=\sqrt{10}$
C $(x+3)^2+(y-2)^2=10$
D $(x+3)^2+(y-2)^2=100$

Question 702

[1 marks]coordinate geometry - circles

A circle has centre

(-3, 2)

and radius

\sqrt{10}

. The point

A(-2, -1)

lies on this circle. Find the coordinates of

B

, the other end of the diameter through

A

A $B=(-1, 3)$
B $B=(5, -4)$
C $B=(2, 1)$
D $B=(-4, 5)$

Question 703

[1 marks]coordinate geometry - circles

Evaluate

(-2+3)^2+(-1-2)^2

, the left-hand side of the circle equation

(x+3)^2+(y-2)^2=10

at point

A(-2,-1)

Answer this when you sit the paper.

Question 704

[2 marks]coordinate geometry - circles

Calculate the exact length of the diameter

AB

, the distance from

A(-2,-1)

B(-4,5)

Answer this when you sit the paper.

Question 801

[1 marks]differential equations

Solve the differential equation

\dfrac{dy}{dx}=xy

, given that

y=1

when

x=0

A $y=e^{x^2}$
B $y=e^{x^2/2}$
C $y=\dfrac{x^2}{2}+1$
D $y=e^{-x^2/2}$

Question 802

[1 marks]differential equations

The solution to

\dfrac{dy}{dx}=xy

with

y=1

x=0

y=e^{x^2/2}

. Using the series expansion for

e^u

, write down the first two terms of the Maclaurin series for

y

A $\dfrac{x^2}{2}$
B $1+x^2$
C $1+x$
D $1+\dfrac{x^2}{2}$

Question 803

[2 marks]differential equations

Separate variables in

\dfrac{dy}{dx}=xy

to write

\int \frac{dy}{y} = \int x\,dx

, integrate, then apply the initial condition

x=0, y=1

to find the constant of integration

C

Answer this when you sit the paper.

Question 804

[1 marks]differential equations

In the Maclaurin series

e^u \approx 1+u+\cdots

, what is

u

in terms of

x

for the solution

y=e^{x^2/2}

Answer this when you sit the paper.

Question 901

[1 marks]parametric differentiation

Given

x=\sin^2t

and

y=\cos2t

, find

\dfrac{dy}{dx}

A $-4$
B $-2$
C $-1$
D $2$

Question 902

[1 marks]parametric differentiation

Given

x=\sin^2t

and

y=\cos2t

, and that

\dfrac{dy}{dx}=-2

for every value of

t

, what shape is the graph of

y

against

x

AA circle, since $x$ and $y$ both come from trigonometric functions of $t$ .
BAn exponential curve, since $y$ decreases as $x$ increases.
CA straight line, since the gradient $dy/dx=-2$ is constant for every value of $t$ .
DA downward parabola, since $y$ is quadratic in $\cos2t$ .

Question 903

[2 marks]parametric differentiation

Differentiate

x=\sin^2 t

to find

\dfrac{dx}{dt}

, giving your answer as a single trig function of

2t

Answer this when you sit the paper.

Question 904

[1 marks]parametric differentiation

Using

\cos 2t = 1-2\sin^2 t

and

x=\sin^2 t

, express

y

directly in terms of

x

Answer this when you sit the paper.

Question 1001

[1 marks]complex numbers

Given

z_1=1-2i

and

z_1z_2=-10i

, find

z_2

in the form

a+ib

A $4-2i$
B $4+2i$
C $-2+4i$
D $-4+2i$

Question 1002

[1 marks]complex numbers

Given

z_2=4-2i

, in which quadrant of the Argand diagram does

z_2

lie?

AThe fourth quadrant, since the real part is positive and the imaginary part is negative.
BThe first quadrant, since both the real and imaginary parts are positive.
CThe third quadrant, since both the real and imaginary parts are negative.
DThe second quadrant, since the real part is negative and the imaginary part is positive.

Question 1003

[1 marks]complex numbers

When rationalising

z_2=\dfrac{-10i}{1-2i}

by multiplying by the conjugate

1+2i

, what does the denominator become?

Answer this when you sit the paper.

Question 1004

[2 marks]complex numbers

Find the modulus

|z_2|

z_2=4-2i

, exact form.

Answer this when you sit the paper.

Question 1101

[1 marks]binomial expansion

Find the term independent of

x

in the expansion of

\left(x^2+\dfrac{3}{x}\right)^6

A $81$
B $135$
C $1215$
D $1620$

Question 1102

[1 marks]binomial expansion

In the series expansion of

(4+x^2)^{1/2}

up to the term in

x^6

, what is the coefficient of

x^4

A $-\dfrac{1}{128}$
B $-\dfrac{1}{64}$
C $-\dfrac{1}{32}$
D $\dfrac{1}{64}$

Question 1103

[2 marks]binomial expansion

Solve

12-3r=0

for

r

, then evaluate the binomial coefficient

\binom{6}{r}

for that value of

r

Answer this when you sit the paper.

Question 1104

[2 marks]binomial expansion

Find the coefficient of

x^6

in the expansion of

(4+x^2)^{1/2}

up to that term.

Answer this when you sit the paper.

Question 1201

[1 marks]integration by substitution

Using the substitution

x=a\sin\theta

, what does

\sqrt{a^2-x^2}

simplify to?

A $a\sin\theta$
B $\cos\theta$
C $a^2\cos\theta$
D $a\cos\theta$

Question 1202

[1 marks]integration by substitution

Use the substitution

x=a\sin\theta

to evaluate

\displaystyle\int_0^a\sqrt{a^2-x^2}\,dx

A $\pi a^2$
B $\dfrac{\pi a^2}{4}$
C $a^2$
D $\dfrac{\pi a^2}{2}$

Question 1203

[2 marks]integration by substitution

For

\int_0^a\sqrt{a^2-x^2}\,dx

with substitution

x=a\sin\theta

, which statement correctly describes the effect on the differential and the limits of integration?

A $dx=a\cos\theta\,d\theta$ ; limits become $\theta=0$ to $\theta=\pi/2$
B $dx=a\,d\theta$ ; limits become $\theta=0$ to $\theta=\pi/2$
C $dx=a\cos\theta\,d\theta$ ; limits become $\theta=0$ to $\theta=\pi$
D $dx=\sin\theta\,d\theta$ ; limits become $\theta=0$ to $\theta=\pi/2$

Question 1204

[2 marks]integration by substitution

Evaluate

\int_0^{\pi/2}\cos^2\theta\,d\theta

Answer this when you sit the paper.

Question 1301

[1 marks]circles and trigonometry

A circle has centre

O

. Tangents

AP

and

BP

are drawn from an external point

P

, with

AP=BP=20

cm and chord

AB=12

cm. Let

M

be the midpoint of

AB

(so

AM=6

cm) and let

r

be the radius. Using right angles

OA\perp AP

and

AM\perp OP

, together with

OP=OM+MP

, which equation correctly determines

r

A $\sqrt{r^2-36}+\sqrt{364}=\sqrt{r^2+400}$
B $\sqrt{r^2+36}+\sqrt{364}=\sqrt{r^2-400}$
C $\sqrt{r^2-36}+\sqrt{364}=r+400$
D $\sqrt{r^2-36}-\sqrt{364}=\sqrt{r^2+400}$

Question 1302

[1 marks]circles and trigonometry

Using

AP=BP=20

cm, chord

AB=12

cm, and the relation

\sqrt{r^2-36}+\sqrt{364}=\sqrt{r^2+400}

, calculate the radius

r

of the circle.

A $39.56$ cm
B $8.49$ cm
C $6.29$ cm
D $1.89$ cm

Question 1303

[1 marks]circles and trigonometry

A circle of radius

r\approx6.29

cm (

r^2\approx39.56

^2

) has a chord

AB=12

cm. The angle

AOB\approx2.532

radians is the obtuse angle between the two radii measured on the side of the chord nearer the external point

P

(where the two tangents meet). Calculate the area of the shaded segment

AB

, which is the region cut off by the chord on the opposite side from

P

(the larger region, containing the centre

O

A $38.8$ cm $^2$
B $124.3$ cm $^2$
C $85.5$ cm $^2$
D $50.1$ cm $^2$

Question 1304

[1 marks]circles and trigonometry

Using

AM=6

cm and

AP=20

cm (the tangent length), find

PM=\sqrt{AP^2-AM^2}

Answer this when you sit the paper.

Question 1305

[2 marks]circles and trigonometry

Using

r\approx6.29

cm, calculate

OM=\sqrt{r^2-36}

Answer this when you sit the paper.

Question 1306

[2 marks]circles and trigonometry

Calculate the area of triangle

OAB = \frac12 \times AB \times OM

, using

AB=12

cm and

OM\approx1.89

cm.

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2010

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 203

Question 301

Question 302

Question 401

Question 402

Question 403

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 901

Question 902

Question 903

Question 904

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1103

Question 1104

Question 1201

Question 1202

Question 1203

Question 1204

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1306

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