AHB Solutions C2 Jan 2014

8/10/2019 AHB Solutions C2 Jan 2014

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WJEC January 2014 C2 Solutions AHB

2. Find all the values of in the range satisfying

The angles

and are the three angles of a triangle. Given that and that , find the values of

and .

Give each angle correct to the nearest degree.

EITHER OR

, 246.42 , 311.81

Angles cannot be negative

Angles in a triangle add up to :

Substituting into

Let ,


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3. The sum of the third and eighth terms of an arithmetic series is zero. Thesum of the fifth, seventh and tenth terms of the series is 22. Find the firstterm and the common difference of the series.

The first term of another arithmetic series is 9 and the common difference

is 2. The sum of the first terms of this arithmetic series is 3 times thesum of the first terms of the series. Find the value of .

The general formula for a term in an arithmetic series is .

Using this formula, the third term is equal to and theeighth term is equal to .

Fifth term

Substituting into , First term, . Common difference, .

when and .

[ ]

Substituting and into

[ ] [ ] [ ]

or

Formula booklet

page 2

Formula booklet page 2


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4. A geometric series has first term and common ratio . Prove that the sumof the first terms is given by

The fourth term of a geometric series is and the seventh term is 4.

(i) Find the common ratio of the series.

(ii) Find the sum to infinity of the series.

(

(i) The general term for in a geometric series is .

(ii) The sum to infinity of a geometric series is

||.

Substituting and into :

Formulabookletpage 2

Formulabookletpage 2


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5. The diagram shows a sketch of the triangle with The point is on such that

(i) By applying the cosine rule in each of the triangles and , showthat and find a similar expression for (ii) Noting that and are angles on a straight line, use theexpressions derived in part (i) to write down an equation satisfied by .

Hence show that .

Find the area of triangle . Give your answer correct to two decimalplaces. (i) Recall that the cosine rule is .

Using this rule to find ,

QEDTo find


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(ii) Angles on a straight line add up to .

Using

as cannot be negative. QED Recall that the area of a triangle is equal to .Applying this rule to triangle :

From part (i),

(from part (ii))

2dp)

This is k negative argument Let


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6. Find . The diagram below shows a sketch of the curve with equation

.

The shaded region is bounded by the curve, the -axis and the lines. Find the area of the shaded region.

Integrate between and .


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7. Given that , show that

.

Solve the equation

.

Show your working and give your answer correct to three decimal places.

Solve the equation

Let

Taking the th root of both sides,

As and , QED

Taking log of both sides,

3dp)

If ,then


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8. The circle has centre and equation

(i) Write down the coordinates of

.

(ii) The point has coordinates and lies on . Find the equation ofthe tangent to at .

The line has equation . Show that and do not intersect.

(i)

Completing the square:

(ii) The radius of the circle is perpendicular to the tangent at asshown.The gradient of the radius, i.e. the normal to the tangent, at point can be found by:

Substituting , and into :

Equation ofa tangent


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Substituting into (from part :

( )

If the line intersects the circle , then the above quadratic equation will haveat least one solution.

Looking at the discriminant, :

As the discriminant is negative, there are no real solutions to this quadraticequation, so and do not intersect. QED

If the discriminant is negative, then wewould be taking the square root of anegative number in the quadratic formula,which is impossible.

Looking at the graphs of and ,it is also clear that they do notintersect.


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9.

The diagram shows two concentric circles with a common centre . Theradius of the larger circle is 7cm and the radius of the smaller circle is 4cm.The points and lie on the larger circle and and cut the smallercircle at the points and respectively. The area of the shaded region is . Find the perimeter of .

Let angle .Area of a sector

Area of shaded region

radiansLength of cmLength of cmLength of Length of cmPerimeter of cm


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10. The th term of a number sequence is denoted by The th term ofthe sequence satisfies

for all positive integers . Given that ,

evaluate and ,

describe the behaviour of the sequence and hence, without carrying outany further calculation, write down the value of

which means the pattern, will repeat indefinitely.When is a multiple of 3, When is 1 more than a multiple of 3,

When is 2 more than a multiple of 3,

For , is 2 more than a multiple of 3.

where is a positive integer.

,

AHB Solutions C2 Jan 2014

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