166 - Questions Answers
is there any need to know or learn the proof pf formulas of trigo
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
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 = 2x2 - 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.
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
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
Is area always a vector quantity or it can be a vector quantity?
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)
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 ?
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]
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 ?
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 ?
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
(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