Hots Contoh Non Rutin PDF

CONTOH SOALAN HOTS NON RUTIN SEKOLAH MENENGAH SAINS TELUK INTAN, PERAK QUESTION 1 Table 1 shows two functions,

and

.

TABLE 1 A new function,

Express

is defined such that

in terms of

[2 marks]

QUESTION 2 Find all the possible values of

in the equation

[2 marks]

QUESTION 3 A quadratic function is defined as

where ,

and are constants, such that

Determine the quadratic function

,

and

.

.

[2 marks]

QUESTION 4 Calculate the possible values of

and

in the following simultaneous equations.

[2 marks]

QUESTION 5 Find the values of

and in the following simultaneous equations.

[2 marks]

QUESTION 6 Table 2 shows four equations of straight lines.

TABLE 2 The straight lines are drawn on a Cartesian coordinate. The pairs of these lines intersect each other at four distinct points. These points are connected to form a quadrilateral. Find the area, in unit², of the quadrilateral formed.

[2 marks]

QUESTION 7 Table 3 shows the set of distinct prime numbers, where

.

TABLE 3 The set of numbers has a mean of Determine the possible values of

and a standard deviation of

.

and .

[2 marks]

QUESTION 8 Diagram 1 shows a circle with centre .

cm

•

DIAGRAM 1 Line

is the diameter of the circle with the length of

given that the angle

 is rad. 12

Find the length, in cm, of the major arc

cm. Line

is a chord of the circle. It is

, in term of π.

[2 marks]

QUESTION 9 The triangle

has all distinct integer sides. The length of sides

respectively. It is given that Find the length, in cm, of the side

and

are cm and

cm

3 . 2

of triangle

.

[2 marks]

QUESTION 10 A book is first sold with a certain price in the year 2003. In the year 2005, the price of the book increases by from the year 2003. In the year 2007, the price of the book decreases by from the year 2005. In the year 2009, the price of the book increases by from the year 2007. Finally, in the year 2011, the price of the book decreases by from the year 2009. Determine the price index of the book in the year 2011 based on the year 2003.

[2 marks]

QUESTION 11 A polynomial function

is given by

Using differentiation, find the coordinate of all turning point of the curve . Hence, determine whether the turning points are the maximum or minimum points of the curve .

[5 marks]

QUESTION 12 Diagram 2 shows a right cone with vertical height of respectively.

cm and a base diameter of

cm

cm

cm

cm cm

DIAGRAM 2 Water is poured into the cone at the rate of cm³ s¯¹. At a certain time, the vertical height and the surface diameter of the water in the cone reach cm and cm respectively, where and are variables. Using differentiation, find the instantaneous rate of change of the vertical height of water in the cone, in cm s¯¹, when the vertical height of the water is cm.

[5 marks]

JAWAPAN QUESTION 1 By taking a closer look,

Thus,

is actually the inverse function of

.

Hence,

This is also true for any

Simplifying

and

, where

,

Thus,

QUESTION 2 Rearranging the terms in the equation,

Factorising the term

in the equation,

Factorising the equation completely,

is any positive integer.

Thus,

QUESTION 3 Substituting all values of and

,

...(1)

...(2)

...(3)

From equation (1),

Substituting in equations (2) and (3),

...(4) Subtracting equation (1) from equation (2),

From equation (4),

For ,

Thus,

...(5)

QUESTION 4 From the equation,

Since both equations have

, thus,

From these equations, taking a linear equation and a non-linear equation,

Substituting

in another equation,

Substituting

in the equation,

Thus,

QUESTION 5 Treating both equation as quadratic equations,

Comparing both sides of equations,

Solving both equations,

Thus,

QUESTION 6 Sketching the graph,

•

•

•

•

Let the quadrilateral be

.

Point

lies on the -axis. So, point

is

.

For point ,

So, point For point ,

So, point

is

is

.

For point ,

.

So, point

is

.

Using area formula,

Thus, the area of quadrilateral

is

unit².

QUESTION 7 From the standard deviation,

Since and are the only perfect squares whose sum is

, and

, thus

QUESTION 8 Since

and

are the radii of the circle,

So, for minor arc

forms an isosceles triangle. Thus,

,

Thus, for major arc

,

The radius of the circle is cm. Using formula,

Thus, the length of major arc

is

cm.

QUESTION 9 3 , there are two possible values for 2

Since

Using Cosine Rule for both values of

,

cm

cm Since the value of side

.

is integer, thus, the length of side

is

cm.

QUESTION 10 Based on the statement, it can be concluded that the price index, for the specific year based on the base year is given by

where

is the price change from the base year to the specific year. Specific Year 2005 2007 2009 2011

Base Year 2003 2005 2007 2009

Price Index

From the formula,

where ,

and

are the price index, price in the specific year and price in base year.

Thus, by Chain Rule,

So, the price index of the book in the year 2011 based on the year 2003 is

.

QUESTION 11

Using differentiation,

For any turning point, the gradient is always .

Using factorisation,

When



When

,

When

,

1 , 2

Thus, the coordinate of the turning points are  Using differentiation,

1 5 , 2 16

and

.



When



1 2

Thus,  When



1 5 2 16

When

Thus,

1 5 2 16

1 2



1 2

is a maximum point.

,

Thus,



1 , 2

is a minimum point. ,

is a maximum point.

and

are maximum points, while

is a minimum point.

QUESTION 12

From the situation, there are two variables, and . Thus, variable must be eliminated. Taking a closer look, variables and

involved. However, this question deals about

increases in a constant ratio. From the part of the cone, cm

cm cm Thus, using ratio, cm

Using formula of cone,

Substituting in the equation,

Thus, the volume of water in the cone is given by

Using differentiation,

Substituting

When

in the equation,

,

Using chain rule,

From chain rule,

Thus, the rate of change of vertical height of water in the cone is

cm s¯¹.