A Level Questions And Answers Pure Mathematics Pdf
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0 | S o l u t i o n b y F a j a r S . H a r s a
CAMBRIDGE INTERNATIONAL EXAMINATION
QUESTIONS (ADVANCE SUBSIDIARY AND ADVANCE
LEVEL : PURE MATHEMATICS ) 2018
(Solution by Fajar S. Harsa)
PART 1 (PROBLEM OF THE MONTH)
Question 1 – 6
1 |Solution by Fajar S. Harsa
1. The coefficient of in the expansion of
is 330. Find the value of
the constant
Solution
With binomial newton, we get :
Then,
2 |Solution by Fajar S. Harsa
2. The equation of a curve is , where
is a constant
(i) Find set of values of for which the whole of the curve lies above the
x-axis
(ii) Find the value of for which the line is a tangent to the curve
Solution
(i) If a curve lies above the
x-axis
, then the equation of a curve must have
discriminant less than 0 or has no real x solution.
So, The discriminant of the equation with
;
The curve lies above the
x-axis ,
when
(ii) Slope of tangent line to the curve at point , is
derivative of or . Suppose the line which has
slope is a tangent to the curve at point , , so ;
Because point , lies on line and also curve ,
so :
The value of for which the line is a tangent to the curve is 5
3 |Solution by Fajar S. Harsa
3. A company producing salt from sea water changed to a new process. The amount of
salt obtained each week increased by 2% of the amount obtained in preceding week.
It is given that in the first week after the change the company obtained 8000 kg of
salt.
(i) Find the amount of salt obtained in the 12th week after the change.
(ii) Find the total amount of salt obtained in the first 12 weeks after the change
Solution
Suppose , amount of salt in -th week
(i) Week 1 :
Week 2 :
Week 3 :
Week 4 :
By seeing the pattern , we conclude :
So, in 12th week, amount of salt is
(ii) Total amount of salt obtained in the first 12 weeks after the change (S)
So, the total amount of salt obtained in the first 12 weeks after the change is
4 |Solution by Fajar S. Harsa
4. The function is such that . It is given that
and .
(i) Find the values of the constant
(ii) Find the set of values of
k
for which the equation has no solution
Solution
(i) , then :
By eliminating (1) and (2) , we get
(ii) If , then and we get
.
must be less than 1 or greater than 1 in order to get the value of which
the equation has no solution .
5 |Solution by Fajar S. Harsa
5.
The diagram shows a three-dimensional shape. The base OAB is a horizontal triangle
in which angle AOB is 90o. The side OBCD is a rectangle and the side OAD lies in a
vertical plane. Unit vectors I and j are parallel to OA and OB respectively and the unit
vector k is vertical. The position vectors of A, B, and D are given by
.
(i) Express each the vectors
in terms of and
(ii) Use a scalar product to find angle CAD
Solution
(i)
6 |Solution by Fajar S. Harsa
(ii)
Suppose , the scalar product :
7 |Solution by Fajar S. Harsa
6.
The diagram shows points A and B on a circle with centre O and radius r. The tangents
to the circle at A and B meet at T. The shaded region is bounded by the minor arc AB
and the lines AT and BT. Angle AOB is radians
(i) In the case where the area of the sector AOB is the same as the area of the
shaded region, show that than
(ii) In the case where and the length of the minor arc AB is 19.2 cm, find
the area of the blue region
Solution
(i)
8 |Solution by Fajar S. Harsa
(Proved)
(ii) The minor arc AB is 19.2 cm and , then :
cm2
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A Level Questions And Answers Pure Mathematics Pdf
Source: https://www.researchgate.net/publication/331155025_CAMBRIDGE_INTERNATIONAL_EXAMINATION_QUESTIONS_ADVANCE_SUBSIDIARY_AND_ADVANCE_LEVEL_PURE_MATHEMATICS_2018_PART_1_PROBLEM_OF_THE_MONTH
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