LIMITS

Why is the limit as x approaches positive or negative infinity of the function ƒ(x) = (x+1) /(x-2) equal to 1?

f(x)=(x+1)/(x-2) when x→∞

if we put x=∞ then it is ∞/∞ form

or if let x=1/t then t→0

f(1/t)=(1/t+1)/(1/t-2) when t→0

or f(1/t)=(1+t)/(1-2t) when t→0

or t=0, f(1/t)=(1+0)/(1-2×0)=1

hence proved

QUESTIONS ON GRAPH

What is the solution of sinx=3?

No Real solution

Because if we draw the graph of y=sinx and y=3 then it will not intersect each other.

as the maximum value of sinx is 1 when x=90° and y=3 not cut the graph of sinx.

GENERAL QUESTIONS

A cylinder of diameter 14 and height 7 is converted into a cone of radius 6. Now, what could be the percentage height of the new shape?

volume of cylinder=Π×r²×h

=Π×7²×7

volume of cone=Π×R²×H/3=Π×6²×H/3

volume of cylinder and cone will be same

Π×7²×7=Π×6²×H/3

H=28.58

% of height of cone=(28.58-7/7)×100=308.28

If the volume of a cylinder is 300 m3 and the height to diameter ratio is 3:1, what is the height and diameter of the cylinder?

Given h/d=3/1 or h/2r=3/1 or h/r=6 or h=6r

volume of cyinder=Π×r²×h

300=Π×r²×(6r)

300=Π×6×r³

300/6=Π×r³

50/Π=r³

solve for r=2.51

and h=6×2.51=15.06

DOMAIN AND RANGE

What is the range of the real function f defined by f(x) =√(x-1)?     

Given Y=√(x-1)

Range is the value of y for the function to be defined.

convert x is in the form of y

i.e.

y²=x-1

or x=1+y²

hence the x is always defined for all real value of y

so the range of function is y∈R i.e. all real value

What is the domain function f, where f(x) = √(1-x) (x-2)?

Given y=√(1-x) (x-2)

for function to be defined i.e. y to defined domain is value of x for which y is defined

y is defined when value under squre root is greater than and equal to zero

i.e.

(1-x)(x-2)≥0

or (x-1)(x-2)≤0

or 1≤x≤2

What will be the range of the function f(x) =1/ (√ (x^2-1))?

Range is the value of the y for which x is defined

rewritting as y=1/ (√ (x^2-1))

or y²=1/(x^2-1)

or x²-1=1/y²

or x²=1/y²+1

or x=√((1/y²)+1

hence the x is defined for all value of y except when y is zero

range y∈R-(1)

What are the domain and range of the function y=-(x-2)^2 +2?

Domain is the value of x for which function is defined, here for all value of x i.e. all real value of x.

Range is the value of y for which function is defined.

(y-2)=-(x-2)²

or (2-y)=(x-2)²

or (x-2)=√2-y

for this function to defined (2-y)≥0

or (y-2)≤0

or y≤2

What is the range of f(x)=(sinx)sinx+1?

range is the value of y for which function to be defined 

min value of sinx=0 when x=0

hence min value when x=0 is 0⁰+1

which is not defined hence solve by limit

limx→0 (sinx)^sinx 

^{or taking log on both sides}

^{lny=limx→0ln}(sinx)^sinx 

^{or lny=}limx→0sinxln(sinx)

or lny=limx→0ln(sinx)/coescx

solve using L hospital rule

lny=limx→0cosx/(-cosecx.cotx)(sinx)

lny=-1

or y=1/e

hence min value of y is (1/e)Λ(1/e)+1

and max value when x=90°

ie. y(max)=1+1=2

range is ((1/e)Λ(1/e)+1),2)

DEFINITE INTEGRAL

Definite Integration

The definite integral has limit fixed and hence the value obtained of integral is without constant(C).

TIME AND WORK

Time and work

Men days:

It is defined as the men multiplied by the day.

MxD= constant

Note:  If M1 men can do W1 work in D1 days H1 hours per day and M2 men can do W2 work in D2 days working H2 hours per day then

M1xD1xH1/W1=M2xD2xH2/W2

LCM AND HCF

LCM AND HCF

LCM: LCM is the least common multiple of two or more numbers is the smallest number that is divisible by each of these two given numbers.

HCF: HCM is the highest common multiple of two or more numbers is the largest number of the given numbers.

For example:

LCM of (3,6,9) is 18 and HCF of ( 3,6,9) is 3.

AREAS

AREAS OF RECTANGLE

A plan of a small garden is drawn to scale of 1 : 25. A pond covers 24cm2 on the plan. What is the area in m2 of the real pond.”

scale is given in length

so the area of pond will be A=24*(25)^2 cm^2

A=15000cm2

but 1m=100cm

so A=15000/(100*100)=1.5m^2

SIMPLE INTEREST AND COMPOUND INTEREST

SIMPLE INTEREST AND COMPOUND INTEREST

Principal:  The amount of money deposited or borrowed is called Principal.

Interest: The amount earned or paid for the use of money is called interest

SIMPLE INTEREST:  The interest that is earned only on the principal is called simple interst.

SI=(P*R*T)/100

where P=principal
R=rate of interest
T=time in year

Compound interest: The principal which is paid not ony pricipal but also on the interest is called compound interest.

The amount at the end of n years is given by

A=P(1+r/100)ⁿ

P=Principal

r=rate of interest

I=A-P

ADDITION AND SUBTRACTION

Addition and Subtraction

Addition:- Addition can explain by following example

Lets ram has 2 apple and john has 3 Apple

So if ram gives all 2 apple to john then john will have total 2+3=5

i.e. 5 Apple.

 So addition means 2+3=5

Addition method on number line
 lets say we have to add 2 and 3
then draw number line

—1—2—3—4—5—6—7—8-
So after 2 on the no. line we have to add 3, so after 2-3, 3-4 and 4-5 so after 3 interval or addition of 3 on 2 we get 5.

Subtraction: Subtraction is loosing of quantity.

Lets say amy has 5 mangoes and she gives 2 mangoes to john then amy will now remaining 5-2=3 mangoes.

On the number line:

—1—2—3—4—5—6—7—8—9—10-

On the number line if the 5-4 and 4-3 so the total reduction after 2 interval is 3.