Level 2 Mathematics and Statistics, 2021 v1

2

9 1 2 6 1

Level 2 Mathematics and Statistics, 2021 v1

91261 Apply algebraic methods in solving problems

Credits: Four

Achievement

Apply algebraic methods in solving problems.

Achievement with Merit

Apply algebraic methods, using relational thinking, in solving problems.

Achievement with Excellence

Apply algebraic methods, using extended abstract thinking, in solving problems.

You should attempt ALL the questions in this booklet.

Make sure that you have Formulae Sheet L2?MATHF.

Show ALL working.

If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.

You are required to show algebraic working in this paper. Guess-and-check methods, and correct answer(s) only, will generally limit grades to Achievement.

YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

TOTAL

2

QUESTION ONE

ASSESSOR'S USE ONLY

(a) Write as a single logarithm log3(x) + log3(6x) - log3(x2 ).

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

(b) Find the value of k if log4 (k) = 3. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

(c)

Find

an

expression for n

in

terms of

m for

the

equation

84m 8m-1

=

64m+2 8n

.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

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6

(e) Julia buys a square box and a rectangular box for packing her clothes. The combined surface area of the two boxes, including their bases, is 106 cm2.

ASSESSOR'S USE ONLY

Find the volume of the square box. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

7 QUESTION THREE

ASSESSOR'S USE ONLY

(a) Find the discriminant of the quadratic equation x2 + 6x = 2.

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

(b) The quadratic equation kx2 ? 11x + r = 0 has the solutions - 2 and 5 . 32

Find the values of k and r. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

(c) Write

y2

y -

49

-

y+2 y2 - 5y -14

as a single fraction in its simplest form.

______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

2

9 1 2 6 2

Level 2 Mathematics and Statistics, 2021 v1

91262 Apply calculus methods in solving problems

Credits: Five

Achievement

Apply calculus methods in solving problems.

Achievement with Merit

Apply calculus methods, using relational thinking, in solving problems.

Achievement with Excellence

Apply calculus methods, using extended abstract thinking, in solving problems.

You should attempt ALL the questions in this booklet. Make sure that you have Formulae Sheet L2?MATHF. Show ALL working. If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question. You must show the use of calculus in answering all questions in this paper. YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

TOTAL

2

QUESTION ONE

(a) A function f is given by f (x) =-x3 + 3x - 7. Find the gradient of the graph of the function at the point where x = 3 .

ASSESSOR'S USE ONLY

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

(b) Find the coordinates of the point(s) on the curve y = x3 + 3x2 - 9x + 2 where the tangent to the curve is parallel to the x -axis. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

(c) Find the value(s) of the constant k for which the graph of the function f (x) =- x3 - 2x2 + kx + 2 3

is always decreasing. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

5 (b) The graph of a function y = g(x) is shown on the axes below.

ASSESSOR'S USE ONLY

It is given that g(2) = 0. Find an expression for g(x). You must use calculus to obtain your answer.

______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

9

(d) The volume, V cm3, of an expanding square pyramid is given by V = 6x3

where x is the side length of the base of the pyramid. Find the side length of the pyramid when the rate of change of the volume of the pyramid with respect to the side length is 36 cm3 / cm.

ASSESSOR'S USE ONLY

______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

(e) The tangent to the curve y = x2 +1 at the point (-2,5) is also a tangent to the curve y = x3 - kx -1, where k is a constant. Use calculus to find the value of k.

______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________

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