4E AM Term 3 Common Test (2011)

8/4/2019 4E AM Term 3 Common Test (2011)

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3

[2 -lJ It is given tha t P = I 2 .

(i) Find the matrix Q such that PQ = .•(7 ) ... 1

(ii) Hence write down the equations of a pair of straight lines which

intersect at the point (3,-1).

2 Th e roots of the quadratic equation 3x2

+4x - 2 = 0 are a and /3.

Form a quadratic equation whose roots are a2

and /32

• [5]

. . tan A -cotB3 (a) Prove the identity == tan A co t B .

. t a n B ~ c o t A · . . . [3 ]

(b) Find all the angles between 0 0and 360 0

which.satisfy the equation

2 cos x = sec x - tan x . [5]

(e) Given that 0:::; x :::; 2, find the values ofx for which

2cosec(x +0.5) = 3 .. [3]

.... .

. . 4 The equation ofthecurveis y III(2x2+.5x )

where x>·O. Find .

"

.... 12] .:. dy(i)

dx'

(ii) the x-coordinate of the point on the curve at which the normal to the. . . . .. "

curve is perpendicular to the line 6y- 2x = 7. [4]

5 (a) T h coefficient of x2

of (2 + x)(l- ~ ) is 32.. Find the value of n. [4]

·39 9

(b). Find, in its simplest form, the coefficient of xl) in the expansion of (x 2- J [3]

3

4E AM Term 3 Common Test 2011



4

6 (a) Find the set of values ofx for which x2+ 2::::: (2x + l)(x 2). [3]

(b) Show that the roots of the equation nr' + (3p +q)x + 3q =

ofp and q.

0 are real for all values [3]

7 (i)

(ii)

x -10Express in partial fractions.

x2 -4

Hence, or otherwise, find the gradientof the curve y.

= \ - 10

x -4 at the point.

[3]

[2]

8 (i)

(ii)

(iii)

Solve [2- ~ x = 4.

Sketch the graph ofY+ fo r - 2 x s; 5.

Hence, find the range of values of x such that 12 xl ::::: 4 for - 2::;; x ::;; 5.

[3]

[2]

[2]

4



5

9 (a) The variables x and yare connected by the equation ay = ebx

2 , where Q and b

are constants. Experimental values ofx and y were obtained. The straight line

graph of lny against x2

was plotted. Given that the straight line passes through

(0, 2) and (2, 8), find the value of Q and of b. [4]

(b)

3

A

Q2x-J

(4,-1)

(i) The givendiagram shows part of the straight line.graph obtained by

[3]plotting against (2x - 3) . Express y in term ofx.x

(ii) The line cuts the horizontal axis at the pointA, where the value on the axis

is Q. Find the value of Q. . [1]

'.· 1 0 A p a r t i . ~ u l a r metallic 'flask consist sof a h ~ m i s p h e r e O f radius rem mounted on a' · .

-cylinder of height h. cm and r ~ d i u s r em.

' . .•. - '.' '. .... .... .'4 '

. [A sphere of radius r has a volume of -Jr r '. 3

and a su;facearea of 41rr2.]

(i) Given that the flask is made of thin aluminium sheets and the surface area of the

flask is 100Jr em", show that the volume, V em", of the flask is given by

V.

5 3= SOJr r - Jr r"6

[4]

Given that r can vary,

(ii) find the stationary value of V (correct to the nearest whole number), [4]

(iii) determine whether this is a maximum or minimum value. [1]

5



6

11 P

Q

4m

R S

The diagram shows it quadrilateral PQRS with PQ = 2 m , QR = 4 m ,

LPSR =90° andLQPS = LQRS = 8° .

(i) . Showthatthe area of PQRS, A m 2, is given by

.[3]A = 5 sin 28- 4 cos 28 + 4.

(ii) Express A in the form R sin(2B - a) + 4, where R is positive and

a is acute. [3]

(iii) State the maximum value of A and find the corresponding value of 8. [3]

(iv) Find the value of 8 for which A = 5. [2]

END OFPAPER

6



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4E AM Term 3 Common Test (2011)