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
.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
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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