166 - Questions Answers

is there any need to know or learn the proof pf formulas of trigo

 

Asked By: AMBUJ SINGLA
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Joshi sir comment

No, For making a good command in trigonometry, only implementation is a need. But for your knowledge you can proof these results

number of real  values of x satisfying 2log2(x+7x+1)+log2(x+1x-1)=1  is

Asked By: SHUBHAM VED
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Joshi sir comment

given eq. can be written as log2 [(x+7)/(x+1)]2 + log2 [(x+1)/(x-1)] = 1

or log2 [(x+7)2(x+1)/(x+1)2(x-1)] = 1

or log2 [(x+7)2/(x2-1)] = 1

or (x+7)2/(x2-1) = 2

or x2 + 14x +49 = 2x- 2

or x2-14x -51 = 0

or x = -3, 17

since at x = -3 first part will become negative inside root so it will not be a root.

so only 1 root

If the letters of the word are arranged as in dictionary, find the rank of the word INDIA.

Asked By: UMESH
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Joshi sir comment

On arranging the letters in ascending order we will get

A D I I N

total no of words = 5*4*3*2*1/2*1 = 60

rank = (60/5)*2 + (24/4)*3 + (6/3)*1 + (2/2)*1 + 1 = 24+18+2+1+1 = 46

here 60/5 means total words/ total letters and 2 is here because in INDIA, I is first letters and this comes at 3rd and 4th place in ascending order arrangement so total letters before it are 2. similarly 24 stands for total words after removing I from the spelling of INDIA and etc 

4[3t+17]=10

Asked By: GLADNESS
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Joshi sir comment

let [ ] is greatest integer function

so 4[3t+17] = 10

or [3t+17] = 2.5

since greatest integer function can not be non integral so no t is possible.

 

now if [ ] is a normal bracket

then 3t+17 = 2.5 or 3t = -14.5 or t =-14.5/3 = -145/30 = -29/6

Maths >> Vector and 3-D >> Plane Others

Is area always a vector quantity or it can be a vector quantity?

Asked By: AMIT DAS
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Joshi sir comment

generally it is scalar but in the measurement of flux through a surface it is considered a vector perpendicular to the surface with same magnitude 

The number of solutions of :  4x²-4x+2 = sin² y   &     x²+y² ≤ 3    are ?    (please solve it graphically i am getting one solution by using graph but the answer is 2)

Asked By: AMIT DAS
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Joshi sir comment

Given eq. can be written as (2x-1)2 + 1 = sin2y

and max. of siny is 1 so only possible value of x = 1/2

and possible values of y = ±nπ/2  where n is odd integer

but by drawing the circle of 2nd eq. we find that only y = ±π/2 are possible values so answer will be 2

What is the range of the function  y = (e^-x)/ (1+[x]) is ?

Asked By: AMIT DAS
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Joshi sir comment

Domain of the given function is (-∞, -1) υ [0, ∞)

Now at x=0, y=1

and at x=∞, y=0

similarly left limiting value of y = -e

and for x=-∞, y=-∞

so range is (-∞, -e] υ (0, 1]

Maths >> Vector and 3-D >> Plane AIEEE

line AB intersects CD internally in the ratio K : 1  where K is undefined , these line may be  ( 1) concurrent  ,   (2)  intersecting ,  (3) skew  <  (4)  all of these .     explain plese   ?

 

 

Asked By: SHUBHAM VED
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Joshi sir comment

I think it will be skew because skew lines do not intersect in real points 

If the lines represented by 2x²-5xy+2y² =0 , be the two sides of a || gm and the line  5x+2y=1 be one of teh diagonals. then  the equation of the other diagonal is ?

Asked By: AMIT DAS
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Joshi sir comment

2x²-5xy+2y² =0

=> (x-2y)(2x-y) = 0 

these 2 lines pass through origin so one of the diagonal will definitely pass through origin. Given diagonal does not contain origin so second diagonal will pass through origin.

For finding any other point in the second diagonal first solve the two lines with given diagonal one by one. After calculating the two vertax use the formula for middle point of given diagonal. This point will be present in the second diagonal also. 

 

 

The AM of two positive numbers a & b exceeds their GM by 3/2 & the GM exceeds the HM by 6/5 such that a+b= α , |a-b|=β , then which of them are correct,

1) α + β² = 96            2) α + β² =74          3)α²+β = 234         4)α²+β = 84 

Asked By: AMIT DAS
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Joshi sir comment

(a+b)/2 = √ab + 3/2                   (1)                        or         a+b = 2√ab + 3

√ab      = 2ab/(a+b)  +  6/5       (2)                        or         5(a+b)√ab  =  10ab + 6(a+b)      

on solving these 2 we get               5[2√ab + 3]√ab = 10ab + 6[2√ab + 3]

                                           or               10ab + 15√ab  =  10ab + 12√ab + 18

                                           or                3√ab = 18     or      ab = 36          and      a+b = 15 = α

 so a-b = √(225 - 144)     =    √81     =     9 = β

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