CONTOH SOALAN NON RUTIN (HOTS) SEKOLAH MENENGAH SAINS TELUK INTAN, PERAK

Question 1

Given that , find the function .

Note that . By simplifying the function,

By observation,

Thus,

[2 marks]

Question 2 If and are the roots of the equation , form the quadratic equation with the roots and .

From the original equation,

For the new equation,

Hence, the new equation is .

[2 marks]

Question 3 A function is defined by . Point is the point on the curve such that it lies on the axis of symmetry of the curve . Find the coordinates of point .

By sketching the graph,

It is clearly seen that the axis of symmetry is equidistant from the roots of the function. Thus,

Therefore, the equation of the axis of symmetry is

Since point lies on the axis of symmetry, thus

Therefore, the coordinate of point is .

[2 marks]

Axis of symmetry

Question 4

Diagram 1 shows a right-angled triangle .

DIAGRAM 1 It is given that cm, cm and cm. The ratio of the length of side

to the side is34 .

Find the possible value of and .

Based on the ratio,

By Pythagoras theorem,

Solving the simultaneous equation,

or (omitted)

[2 marks]

cm

cm

cm

Question 5

Solve the following equation.

By the Law of Logarithms,

Thus, or

[2 marks]

Question 6

A point, moves such that its distance from the point is units. Another point, moves such that its distance from the point is units. The locus of point intersects the locus of point at two distinct points, and . A straight line is drawn such that it passes through points and . State the equation of the line.

By sketching graph, the locus of is the circle with radius 5 units and center , while the locus of is the circle with radius 5 units and center .

Since the length of radii of both circles is equal and the centers of both circles lie on the same line which is parallel to the -axis, the straight line is the perpendicular bisector of the line connecting centers of both circles. Thus, the equation of the straight line is

[2 marks]

Question 7

The mean and standard deviation of a set of numbers are and respectively. When a number, is added to the set of numbers, the mean remains unchanged and the standard deviation becomes

32p . Find the value of .

From the original set, the standard deviation is given by

Plugging in the value in the new standard deviation,

[2 marks]

Question 8

Diagram 2 shows a circle with radius of cm. The chord divides the circle into two arcs, minor arc and major arc .

DIAGRAM 2

DIAGRAM 2

Given that the length of chord is cm, find the difference between the length, in cm, of minor arc and major arc , in term of .

Let be the center of the circle.

The triangle is equilateral, thus the angle subtended at the center of the circle is or 3 rad.

Let be the circumference, as the minor arc and as the major arc . Thus, The difference between the length of both arcs is given by

[2 marks]

cm

Question 9

The price index for a book in the year based on the year is . In the year , the price of the book increases by from the year . For the year , the price of the book decreases by from the year . Find the price index for the book in the year based on the year .

It is given that the price index for a book in the year based on the year is . Thus,

The price of the book in the year increases by from the year . Thus, the price index for the book in the year based on the year is .

The price of the book in the year decreases by from the year . Thus, the price index for the book in the year based on the year is .

By chain rule, the price index for the book in the year based on the year is

[2 marks]

Question 10 Diagram 3 shows a quadrilateral and rhombus inscribed in a circle with center .

DIAGRAM 3

It is given that and the radius of the circle is cm.

Find the area of the rhombus , in cm.

Since is a cyclic quadrilateral, then . Since is a rhombus, then cm. The area of rhombus is given by

[2 marks]

Section B 10 marks

Question 1

Given that )3(

5)2)(1()(

xxx

xf , find .

[5 marks]

By chain rule,

. [Differentiate by chain rule correctly 1 mark] By product rule,

[Differentiate by product rule correctly 1 mark] By quotient rule,

. [Differentiate by quotient rule correctly 1 mark] Thus,

.. [Insert the value 1 mark] ........... [Correct value 1 mark]

(The marks are still given if the differential functions are not simplified.)

Question 2

Diagram 4 shows a right circular cone.

DIAGRAM 4 It is given that the height of the cone is cm and the diameter of the base is cm, where can take any positive value. Find the maximum volume, in cm, that can be occupied by the cone, in terms of .

[5 marks]

. [Express the volume of the cone in term of 1 mark]

26 dddddV . [Differentiate with respect to 1 mark]

. [Find the values of for the turning points 1 mark]

The maximum volume occurs when . . [Find when the volume is maximum 1 mark]

cm

cm

.......... [Find the maximum volume 1 mark]

END OF MARKING SCHEME