Monday, November 8, 2021

Equation of line

 Find the equation of line having intercept are 3 and 2

General equation of line having intercept a and b is 

X/a + Y/b=1

X/3+Y/2=1

The angle on a straight line is called 

Angle on line with x axis is called slope of line.







Thursday, December 24, 2020

The equation of a line whose slope 5 and x-intercept 2 is

 The equation of a line whose slope 5 and x-intercept 2 is

Slope intercept form

y=mx+c

given slope of line i.e. m=5 
then 
y=5x+c
cut the line at x-axis at (2,0)
i.e. 
0=5×2+c
c=-10
i.e. 
y=5x-10

Friday, May 8, 2020

TRIGONOMETRY

If tan(x+y)=tanx+tany , what is tanxtany ?
tan(x+y)=(tanx+tany)/(1-tanxtany)
but given tan(x+y)=tanx+tany
so tan(x+y)=tan(x+y)/(1-tanxtany)
or (1-tanxtany)=tan(x+y)/tan(x+y)
or (1-tanxtany)=1
or -tanxtany=0
or tanxtany=0

MENSURATION

If four interior angles of a quadrilateral are (x+15), (x+10), (2x-45) and (x-30), what is the value of x?
sum of interior angle of quadrilateral is 360
so, (x+15)+(x+10)+(2x-45)+(x-30)=360
5x+25-75=360
5x=360+50
5x=410
x=82

Thursday, May 7, 2020

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


Wednesday, May 6, 2020

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

Tuesday, May 5, 2020

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 ?
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).

Thursday, April 16, 2020

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




Equation of line

 Find the equation of line having intercept are 3 and 2 General equation of line having intercept a and b is  X/a + Y/b=1 X/3+Y/2=1 The angl...