higher grade

8/11/2019 higher grade

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1. Explain a scenario where we can apply cosine rule to solve triangle instead of sine rule. Give

examples if there are situations where we can apply only sine rule but not cosine rule. When does the

ambiguity case of solving a triangle occur? Explain it with example situations where the behaviors of

number of solutions vary.

Sol 1-> when to use cosine rule:

i) finding third side of triangle when 2 sides and the angle between them is given.

ii) finding angles of the triangles when all three sides of triangles are known.

When to use sine rule:

i) when the two angles and a side is given ( AAS triangle or ASA triangle)

Ambiguous case while solving triangle:

A

BC

37

118

A

B C

5 8

9

A B

C

C

A

a


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Consider a case when two sides and the angle is given ( SAS triangle)

9.112)1.67180(

1.67

)9215.0(sin

9215.028

39sin*41

sin

sin

28

sin

41

sin

1

R

or

R

R

R

QR

PQR

Here R have 2 possible angles, one is acute and another one is obtuse.

2. Using the parent function, explain how we can graph g(x) = -2cos( x + 2/3) using transformation.

Specify the amplitude, period and phase shift for g(x) = -2cos( x + 2/3). Graph for 2 periods.

Sol2:

given: g(x)=-2 Comparing above equation with : y=a Where |a|=amplitude=|-2|=2

Period =

Phase shift=

where if c > 0 =>left shiftC right shift

Sketching curve using transformation:

1)

sketch y= cos x

Q

P R

41

28

39


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2)sketch y = 2 cos x

Increase the amplitude of the above curve from [1,-1] to [2,-2]

3)

sketch y=-2cos xtake reflection of the above curve about x-axis

4)

sketch y = -2cos(.change period from 6 to 2


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5)

y = -2cos( shift the above curve towards left by 2/3

3. When two sets are said to be one to one correspondence and establish a one to one

correspondence between set of all integers and the set of all positive integers? Let A and B be two

sets, which are in one to one correspondence, can A and B have same number of elements? Justify

your answer.

Sol3->

One to One correspondence: a function f: X -> Y is said to be one to

one correspondence if images of distinct element of X under f are

distinct ie. x1,x2 .Eg. Consider a function from set of positive integer to set of negative

integer

=


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Proof: f(x) is one to one

Consider f(x)= f(y)

or x=y x=y

therefore f: x->y is one to one

4. Derive the formula for Surface Area of Right Circular Cone. How is this

related to volume of cone?

sol 4: formula for Surface Area of Right Circular Cone:

A=area of sector = Total area =area of sector+area of base circle =

O

A B

S s

2

S

r


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Volume of right circular cylinder= =

5. For first find total number of possiblezeros (real and imaginary), Then list out all the zeros, express f as product oflinear factors.

Sol5:

consider Number of possible rational zero are :

Possible factors of 8 are =Possible factors of 1= Possible rational zeros =

f(1)=0.so by factor theorem (x-1) is one factor.

f(-2)=0.so by factor theorem (x+2) is another factor.

Therefore, (x-1)(x+2)g(x) =

g(x)=

g(x)=


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possible rational zeros=

g(1)= 0byfactor theorem (x-1) is one of the possible factor

(x-1)f(x)= f(x)=

f(x)= x^2+4=0

x= therefore possible real or imaginary zeros are 1,1,-2,

6. Find the inverse of the function f(x) = 5sqrt(2x3). State Domain and

range of the inverse.

Sol6.

Finding domain: set of values of x for which y is defined

2x-3 X Domain=[3/2,Finding range: Range is the set of values y can take


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So when x=3/2 .y=0

When x= So range =[0,

Finding inverse:

Express x in terms of y:

Squaring both sides

=2x-3

X=

Interchanging x by y and vice versa we get the inverse of the function

y=

domain of inverse function= range of original function=[0,range of inverse function=domain of original function=[3/2,

7. What is a rational function? What are Asymptotes? Give examples of rational

functions, with vertical asymptotes, horizontal asymptote. State Domain or

range of the function selected as an example, show and explain how domain

and range can be written in inequality notation, interval notation, on a number

line. Explain how would you graph a rational function?. Model a function with

Vertical Asymptote at x=3, and hole at x=5, horizontal asymptote at y= 4.

sol7.

Rational function: a function f(x)=is a rational function if f(x) and g(x) are

polynomial and g(x)


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Asymptotes:distance between the curve and the line approaches zero as they

tend to infinity.

Eg.

Vertical asymptotes: set the denominator equal to zero

x=Domain: set of value other than x=Or

Or

Or

Horizontal asymptotes: if degree of numerator= degree of denominator

Then horizontal asymptotes ,y=

=

Y=1/4

Range: Graph of

i) vertical asymptotes are: x=+3/2 and x=-3/2

ii) horizontal asymptotes : y=1/4

3/2-3/2 0 -


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iii) xintercept : equate => X= () )/2= -0.38So the point of x intercept are (-2.618,0) or (-0.38,0)

iv) y- intercept: put x=0 in y=-1/9

coordinates are (0,0.111)

v)

() ()(

)

So the above quadratic equation has no real roots. so there are no turns

in the graph.

Mark all the asymptotes and x and y intercept on the graph

vi) Nature of derivative in the following intervals:interval f(x) Nature

(- f(-4)= -23/605 f(-4)


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curve is decreasing

(-1.5,-0.38) f(-1)=-13/25 . f(-1)


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horizontal asymptote at y= 4..so numerator and denominator is of same degree

and

8. Find the domain of a composition function (fog)(x) given by and Sol8:

given :

fog(x)=f( ( )

Domain is set of real numbers.

9. How would you determine the volume of the solid generated by rotating the

region bounded by and the x-axis, aboutthe x-axis. State and explain which method can be used, shell, or washer

method or both.

Sol 9. The given problem can be solved using both method:

Method1: washer method:


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= 49.0088Method 2: shell method


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eq1r=y , h=(4-x), a=2 ,b=5

x={

substituting above value of x in equation 1

Volume V= 2.723+36.8614+9.429= 49.01


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10. What is the difference between implicit and explicit derivatives if sqrt (1-

x^2)+ sqrt (1-y^2)=a(x-y) then show that dy/dx= sqrt(1-y^2)/sqrt(1-x^2)

Sol 10:

y = f (x) is an explicit functionof x

f (x, y) = 0 is an implicit functionof x.

If f (x, y) can be and is expressed as y = f (x), it becomes explicit function of x.

Often, it is either not possible or not easy and convenient to convert f (x, y) =0 in

the explicit form y = f(x).

When y = f (x) is given and we differentiate f (x) and find dy/dx, it is calledexplicit

differenciation.

When we differenciate f (x, y) = 0 with respect to x without converting it into the

explicit form y = f (x) either because there is no way to do so or because it is

inconvenient to do so, then the differentiation is called implicit differentiation.

..eq1Let Putting values of x and y we get


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