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Hello Anu, Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes. As per question, 1)In deck of cards there are 52 cards. There are 13 cards each of spades,hearts, diamond and clubs.Out of which there is one ace card from each spades,hearts, diamond and clubsStandard card decks usually have Hearts and Diamond in Red and Clubs and Spades in Black. 1)There are 4 ace cards out of 52 cards.This is our favorable event. p(ace)=no.of times a favorable event occurs/ Total no.of possible events. P(ace)=4/52=1/13 2)There are 26 red cards and rest are black. As per the definition P(not a red card)=no.of black cards/total no.of cards =26/52=1/2
Power
In geometry two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure. It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. If The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent. That is: If ∠BAC is equal in measure to ∠B′A′C′, and ∠ABC is equal in measure to ∠A′B′C′, then this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar. If the triangles are similar all the corresponding sides have lengths in the same ratio: AB / A′B′ = BC / B′C′ = AC / A′C′ . This is equivalent to saying that one triangle (or its mirror image) is an enlargement of the other. Now coming to the question. △COA~△DOB AAA axiom Hence AC/BD=OC/OD------1 Area(△COA)=1/2*base * height=1/2*OC*AC=60 On solving AC=24 cm From 1st equation, 24/BD=5/8 BD=192/5 cm AreaDOB=1/2*DO*BD=1/2*8*192/5=153.6 sq.cm
Let the length of straight wire be L units.
One half is bent in the shape of square i.e. perimeter of square = L/2 units.
Let each side of square be A units.
Then as per question,
4A=L/2
=>A=L/8
Area of square=A²=L²/64 sq.units;
Now,the other half of wire is bent into sector,
Length of sector,S =L/2
Let R be radius of the sector
Converting 120 degrees into radians=(π/360)*120=π/3 radians
Therefore ,Length of sector,S=R*π/3 units
=>R=3S/π,
=>R=3L/2π
Area of sector=1/2 R²Φ=1/2* length of arc *radius;
where Φ is in radians
=>Area of sector=1/2 R²Φ=1/2*(9L²/4π²)*π/3 sq.unit
Area of sector=3L²/8π sq.unit
Area of square:Area of sector=L²/64:3L²/8π=11/84 sq.units
Theorem:Angles in the same arc of a circle are equal.
In the above diagram,
∠PTR=∠STQ;
∠PRT=∠SQT(ByTheorem)
∠TPR=∠TSQ(ByTheorem)
In ∆STQ According to Pythagoras theorem, (ST)^2 = (SQ)^2-(TQ)^2 Or ST =√36-25 = √11
By AAA similarity theorem,
∆PTR~∆STQ
Hence,
TR/TQ =PT/ST=4/√11
TR=20/√11=6.03
Option c
Let (x1,y1) be the one of the vertices of above Isosceles triangle.
As per question,
√{(x1-8)^2 + (y1-3)^2}=√{(x1-14)^2 + (y1-3^2}
=>√{(x1-8)^2={(x1-14)^2
=>x1=11;
Putting value of x1=11 in below eqn
√{(x1-8)^2 + (y1-3)^2}=5
on solving y1=-1 or 7
Hence there can be 2 possible coordinates for vetices of this triangle (11,-1) or(11,7)
As per question,
Distance between two possible vertices=√{(11-11)^2 + (-1-7)^2 =8
Option A is the correct answer
Let PR be the Pole.
As per the question,
∟DQR=∟PQD=30°
We have to find PD:DR
In r PQR
tan 60°= PR/QR=---------eqn1
In r QDR
tan 30°=DR/QR ------eqn2
From eqn2
QR= DR
From eqn1
(PD+DR)/QR=,
=>(PD+DR/) DR= DR,
=>PD=2DR, or PD/DR=1/2
As per question,
The given cylinder is hollow ,
Let r and h be radius of base and height of cylinder respectively.
Surface Area of hollow cylinder
=2π*r*h
=>2π*r*h=2500
finding r from above equation,
we get r=250/3π;
Now ,Volume of hollow cylinder =π*r² *h
=>π*r² *h=(π*r*h) *r
=>1250*(250/3π)
on solving we get volume of hollow cylinder=33144 cm3..which when rounded gives 33150.06 cm3
Option B is correct
In tri ABC,
height of actual tree=AC+BC=a+b
AB=c=6cm
tan60=b/c;
b=6√3 cm
cos60=c/a;
a=2c=12cm;
now,height of tree=12+6√3 cm=22.39cm
None of the options are correct
Consider Image uploaded for the full explanation
In triangleTQR,
tan 60=QR/QT,
QR/QT=√3,
QT=10√3
tan 45=PQ/QT
=>PQ=QT
Height of Tower=PQ+QR=PR
=>30+10√3
=>47.32
Option C is correct
In Triangle PSR,
sinx=PR/PS=PR/L
=>PR=LSinx;
In Triangle QTR,
Siny=QR/RT=(QP+PR)/L
LSiny=a+LSinx;
=>L=a/(Siny-Sinx)
Rate of change of total revenue==Marginal value=d(total revenue)/dx=d(R(x))/dx=6x+36
when x=5,marginal value=6x5+36=Rs.66
you can see that
when x=5,total revenue=R(x)=3x²+36x+5=Rs.260
Two vectors are perpendicular then their dot product is 0.
i.e. if a nd b are vectors and they are perpinducular to each other ,Ω=90 then a.b=a . b cos(Ω)=0
As per question
ar = iˆ – jˆ + 7kˆ and b r = 5iˆ – jˆ + λkˆ
ar +br=6iˆ-2jˆ +(7+λ)kˆ-------vector1
ar-br=-4iˆ+(7-λ)kˆ -----vector2
Now As per question
(ar +br) . (ar-br)=0
=> λ²=25
=>λ=±5
Data (or graph) on sales of the branches missing. The problem cannot be answered
Let x ,y,z be the age of husband,wife,child three years ago.
As per question,
(x+y+z)/3=27--1
x+y+z=81;
((y-2) +(z-2))/2=20
y+z=44--2
putting eqn 2in eqn 1
we get x=37;
Hence present age of husband=x+3=37+3=40
option(B) is the correct answer
area of court=3.78*5.25=1109;
let a be the side of square tiles and let there be n square tiles that are required to pave;
as per question,
n x a² =1109;
2 variables and only one equation.
option D is correct
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