iit jee maths

If f'(x)=g'(x) and g'(x)=-f(x) for all x and f(2)=4=f'(2), then f^(4)+g^(4)is
Asked By: ASHIT MAHAJAN 9 Month ago
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1.    A particle moves according to a law of motion

                       s(t) = t- 12t2 + 36t , t ≥ 0

      where, t is measured in seconds and s in meters.

a)    Find the acceleration at time t and after 3 seconds.

b)    Graph the position, velocity and acceleration function for 0≤t≤8

c)    When is the particle speeding up? When is it slowing down?

Asked By: NORNAZIHAH BINTI ZAKARIA 9 Month ago
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SCRA 2013 SET D, QUES. 10

( [x]+[2x]+[3x]+.....[nx] ) /  n2

Asked By: GAUTAM SAGAR 9 Month ago
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SCRA 2013

f(x) = |x2-2|  ; -3<x<3

QUES.-  Consider The Following Statements:

1. The Absolute Maxmimum Value of The Function is 2.

2. The Absolute Minimum Value of  The Fuction is 0.

WHICH OF THE ABOVE STATEMNTS IS/ARE CORRECT?

Asked By: GAUTAM SAGAR 9 Month ago
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Maths >> Calculus >> Integrations IIT JEE
Asked By: AMAN SONI 9 Month ago
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why a funtion is always continue when it is differential but not always differential when it is continue?

Asked By: AKASH YADGIRI 10 Month ago
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d/dx((1+x^2+x^4)/(1+x+x^2))=ax+b

then a=?, b=?

Asked By: RAJIV 10 Month ago
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Answer Strategies and trick (it will help you to solve it by yourself)

1+x2+x4 = (1+x+x2)(1-x+x2)

now solve

HOW TO INTEGRATE  "GREATEST INTEGER FUNCTION"?

Asked By: GAUTAM SAGAR 10 Month agoSolved By: RAMA SUBRAMANIAM
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Maths >> Calculus >> Functions IIT JEE

Find a formula for a function g(x) satisfying the following conditions

a) domain of g is (-∞ , ∞ )     b) range of g is [-2 , 8]     c)  g has a period π   d)  g(2) = 3

Asked By: VAIBHAV GUPTA 10 Month ago
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Answer Strategies and trick (it will help you to solve it by yourself)

g(x) = 3-5sin(2x-4)

 

Maths >> Calculus >> Functions IIT JEE

Let f(x) = x135 + x125 - x115 + x5 +1. If f(x) is divided by x3-x then the remainder is some function of x say g(x). Find the value of g(10)

Asked By: VAIBHAV GUPTA 10 Month ago
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Answer Strategies and trick (it will help you to solve it by yourself)

for getting reminder put x3= x so

x135 + x125 - x115 + x5 +1 will give

 x45 + x41*x2 - x38*x +x*x2 +1

x15 + x13*x2*x2 - x12*x2*x +x*x2 +1

 x17 + x3 +1

x5*x2 + x + 1

x+ x + 1

x2*x + x +1 

x3 + x +1 

x+x+1

2x+1

now put x = 10

Maths >> Calculus >> Functions IIT JEE

Find out for what integral values of n the number 3π is a period of the function :

f(x) = cos nx . sin(5/n)x 

Asked By: VAIBHAV GUPTA 10 Month ago
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Maths >> Calculus >> Functions IIT JEE

Suppose that f is a function such that f(cos x) = cos 17x . Which one of the following functions g has the property that g(sin x) = sin 17x . 

(A) g(x) = f(√1-x2)  (B)  g(x) = f(x - π/4)   (C) g(x) = √1-x2  (D) g(x) = f(x)

Asked By: VAIBHAV GUPTA 10 Month ago
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Asked By: AKHIL CHAND 11 Month ago
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Answer Strategies and trick (it will help you to solve it by yourself)

all are defined 

put π in the first, factorise second and multiply the conjugate of numerator in the third

Asked By: AKHIL CHAND 11 Month ago
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Answer Strategies and trick (it will help you to solve it by yourself)

factorise numerator and denominator

 

How to inegrate ( In t) dt /(t-1)

Asked By: PARTHASARATHY 11 Month ago
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Answer Strategies and trick (it will help you to solve it by yourself)

let lnt = x so dt/t = dx so dt = tdx = exdx

now ∫(ln t)dt/(t-1) = ∫xexdx/(ex-1) = ∫x(ex-1+1)dx/(ex-1)

now solve 

Maths >> Calculus >> Integrations IIT JEE

∫√tanx

Asked By: BONEY HAVELIWALA 1 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

let  √tanx = t

so sec2xdx/2√tanx = dt

so (1+t4)dx/2t = dt

so dx = 2tdt/(1+t4)

now solve

Maths >> Calculus >> Integrations IIT JEE

∫√(x+3)/(x+2)     {x>-2}

Asked By: BONEY HAVELIWALA 1 year ago
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Maths >> Calculus >> Functions AIEEE

if ƒ:R−{0} -> R,  2ƒ(x) − 3ƒ(1⁄x) = x²  then ƒ(3)= ?

Asked By: BONEY HAVELIWALA 1 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

2ƒ(x) − 3ƒ(1⁄x) = x²   (1)

so 2ƒ(1/x) − 3ƒ(x) = 1/x²  (2)

multiply (1) to 2 and (2) to 3, then add the two equations, you will get f(x) then calculate f(3)  

Maths >> Calculus >> Functions Others

Sir/Madam,

                    Suddenly a question struck on my mind:   0×∞= 1 or 0?? as 1⁄0=∞.

     

Asked By: BONEY HAVELIWALA 1 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

it will be 0, assume it by considering that ∞ is a big number 

for ex.   0*(1111111111111111111111111) = 0

Maths >> Calculus >> Functions IIT JEE

FIND DOMAIN OF f(x)=log4log3 log2x WHERE 4,3,2 ARE BASES ?

Asked By: SIDDHANT 1 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

f(x)= log4log3log2

so 0 < log3log2x < 

so 1 < log2x < ∞ 

so 2 < x < 

Maths >> Calculus >> Differential Equations Engineering Exam
Differential form of ol conics whose axes coincide with the co-ordinate axes??
Asked By: NEHA GUPTA 1 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

let us consider the case of ellipse with x and y axes as their axes

eq. is x2/a2 + y2/b2= 1

on differentiating we get 2x/a2 + 2yy'/b2 = 0                                y'is first differential

so yy'/x = -b2/a2

again differentiate and get answer. 

You should remember that you should differentiate as many times as the number of constants.

for ex. in the case of parabola only first diffrentiation is sufficient.

now complete it for all conics

Maths >> Calculus >> Differential Equations Engineering Exam
Form DE of all conics whose axes coincide with d co-ordinate axes??
Asked By: NEHA GUPTA 1 year ago
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using the transformation x=r cosθ and y=r sinθ,find the singular solutionof the differential equation x+py=(x-y)(p2+1)½           where p=dy/dx

Asked By: DEBANJAN GHOSHAL 1 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

x=rcosθ and y=rsinθ

so dx=-rsinθdθ  and dy = rcosθdθ

so p = -cotθ

so given eq. will become x-cotθy = (x-y)cosecθ 

on putting values of x and y we get 

0 = r(cosθ-sinθ)/sinθ

so tanθ = 1  so θ = nπ+π/4

so x = rcos(nπ+π/4) and y = rsin(nπ+π/4)

show that the family of parabolas y2=4a(x+a) is self orthogonal. 

Asked By: DEBANJAN GHOSHAL 1 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

y2= 4a(x+a)

so 2yy' = 4a so a = yy'/2

on putting a we get y2= 4yy'/2(x+yy'/2)      

so y2 = yy' (2x+yy')  or y = 2xy' + yy'2    (1)

now on putting -1/y' in the place of y'

we get y2 = -y/y'[2x-y/y']

so -yy'2 = 2xy' - y (2)

similarity of (1) and (2) shows that the given curve is self orthogonal 

show that limit /x/ = 0 as x approaches 0.

Asked By: FAITH MWONGELI 1 year ago
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show that limit /x/ = 0 as x approaches 0.

Asked By: FAITH MWONGELI 1 year ago
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In a class of 78 students 41 are taking French, 22 are taking German. Of the students taking French or German, 9 are taking both courses. How many students are not enrolled in either course?                                                                                                                                                                                                            A. 6
B. 15
C. 24
D. 33
E. 54

Asked By: RAJENDRA J LAVANTRA 1 year ago
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Q1. Let  f be a twice differentiable function on R.Given that f''(x)>0 for all x element of R.Then which one is true and why?

a.f(x)=0 has exactly two solutions on R.

b.f(x) =0 has a positive solution if f(o)=0 and f'(0)=0

c.f(x)=0 has no positive solution if f(o)=0 and f'(o)>0.

4. f(x)=0 has no positive solution if f(0)=0 and f'(0)<0

Asked By: KALPANA SAI 1 year ago
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Maths >> Calculus >> Functions IIT JEE

If f be decreasing continuous function satisfying f(x+y)=f(x)+f(y) for all x,y belongs to R; f'(0)=-1, then

 

   1

   f(x)dx is

0


A. 1                       

B. 1-e

C.2-e

D. none of these

Asked By: MOHIT SAINI 1 year ago
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Maths >> Calculus >> Differential Equations Engineering Exam

 

Solve: dy/dx =[x√(x^2-1) +y]/(√x^2-1)

 

Asked By: SUBHADEEP BASU 1 year ago
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Maths >> Calculus >> Integrations Others

∫√ ex-4 dx

Asked By: NAGAPAVANREDDY 1 year ago
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Maths >> Calculus >> Integrations Others

integral root of ex-4 dx

Asked By: NAGA PAVAN REDDY 1 year ago
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if f is a real valued differentiable function satisfying If(x) - f(y)I ≤ (x-y)2   x,yεR and f(0) =0 then f(1) equals

(a) 1

(b) 2

(c) 0

(d) -1

Asked By: HIMANSHU MITTAL 2 year agoSolved By: VISHAL PHOGAT
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let g(x) be the inverse of the function f(x), and f/(x) = 1/1+x3 then g/(x) equals

(1) 1/1+g3

(2) 1/1+f3

(3) 1+ g3

(4) 1+f3

Asked By: HIMANSHU MITTAL 2 year agoSolved By: VISHAL PHOGAT
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let tr = r/1+r2+ r4  then lim n-->∑ tr

Asked By: HIMANSHU MITTAL 2 year ago
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lim n-->∞ [tanx + 1/2tan(x/2) + 1/22tan(x/22) + ............. + 1/2ntan(x/2n)]

Asked By: HIMANSHU MITTAL 2 year ago
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lim n-->∞ cos(x/2)cos(x/4).......cos(x/2n)

Asked By: HIMANSHU MITTAL 2 year ago
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lt x-->0 [(1+sinx)cosecx- e +sinx/2]/sin2x

Asked By: HIMANSHU MITTAL 2 year ago
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lim X--> 0 (1/x2 - 1/tan2x)

Asked By: HIMANSHU MITTAL 2 year ago
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Maths >> Calculus >> Differentiation Engineering Exam

x(√1+y)+y(√1+x)=0,prove that dy/dx=-1/(1+x)2

Asked By: SANDEEPNALCO 2 year ago
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Maths >> Calculus >> Integrations Olympiad

A bank gives an investor double the initial deposit in 5years, interest being simple interest. Then the rate of Interest is:

(A) 20

(B) 18

Asked By: ARJUN 2 year ago
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Sir , i have solved   limx-->0    (sin-1x  - tan-1x )/x. plz have a look on others.

Asked By: VISHAL PHOGAT 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

limx-->0    (sin-1x  - tan-1x )/x3

use D L Hospital rule or expansion of sin-1x and tan-1x to solve it

no of solutions of  e-x^2/2 - x2 = 0 are

Asked By: VISHAL PHOGAT 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

two intersecting points means two solutions

 if f(X) = z=1n (x- 1/z)(x - 1/ 1+z ) then lim x-->∞ f(0) is ___________

ans. 1

Asked By: VISHAL PHOGAT 2 year ago
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   limx-->0    (sin-1x  - tan-1x )/xis _________  (without using expansion of sin-1x and tan-1x )

 ans. 1/2

 

Asked By: VISHAL PHOGAT 2 year ago
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 lim x-->∞  [x - x2 ln(1+1/x) ] is equal to

ans. 1/2

Asked By: VISHAL PHOGAT 2 year ago
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lim x--->0+   logsinx/2sinx is equal to

a.1   b.o     c. 4 

ans. 1

Asked By: VISHAL PHOGAT 2 year ago
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if lim x-->0 f(x) exists and is finite and non zero and lim x-->∞ [ f(x) + {3f(x) -1/f2(x)} ] =3 then the value of f(x) is equal to

a. 1       b. -1       c. 2

Asked By: VISHAL PHOGAT 2 year ago
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lim x-->0 [ sin2(π/(2-ax)) ][sec(π/(2-bx))]^2

 

 

 

Asked By: VISHAL PHOGAT 2 year ago
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lim x -->  (cos(x+1)½ -cos(x)½ )

Asked By: VISHAL PHOGAT 2 year ago
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Asked By: VISHAL PHOGAT 2 year ago
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lim x»infinity( ((x+1)(x+2)(x+3)(x+4))^¼ - x )

Asked By: VISHAL PHOGAT 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

multiply with the conjugate in numerator again and again you will get 

limx->((x+1)(x+2)(x+3)(x+4) - x4)/((x+1)(x+2)(x+3)(x+4))1/4 + x )*((x+1)(x+2)(x+3)(x+4))1/2 + x2 )

after solving the numerator get a 3 power expression of x then take 3 power of x common, similarly take x and 2 power of x from the first and second denominator terms

Maths >> Calculus >> Functions IIT JEE

 best calculus books for iitjee.

Asked By: KRISHN RAJ 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

its I. E. Maron 

 

Maths >> Calculus >> Functions Others

 take 1,1,1,1,1,1 to make sum of the value as 37..

clue: use any operations between the numbers.

Asked By: M.SAKTHIVEL 2 year ago
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lim n--> ∞  4^n/n!

Asked By: HIMANSHU MITTAL 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

limn->∞ 4n/ n!

= limn->∞ [4/1][4/2][4/3][4/4][4/5][4/6][4/7]................................[4/n]

besides first four bracket, all brackets are real numbers less than 1 so their product will be zero for n tends to infinite

 

Maths >> Calculus >> Integrations IIT JEE

a)integrate (1+x)/lnx from limit 0 to 1

Asked By: RAVIKANT 2 year ago
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A tarder buys goods at 19% off the list price he wants to get a profit 20% after allowing a discout of 10%. At what % above the list price should he marks the goods.

Answers - 

1) 4%

2) 6%

3) 8%

4) none

Asked By: UTKARSH BHARDWAJ 2 year agoSolved By: SAGAR
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Maths >> Calculus >> Integrations IIT JEE

                                         ∫dx 


      sin5x   + cos5x

Asked By: NIKUNJ KAUSHIK 2 year ago
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Sir,is the function (2x+5) is differentiable everywhere in its domain set. If yes the what is the case with l2x+5l.
Asked By: GAURAV MAHATE 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

since first one is a straight line so at every point only one tangent is possible so it is differentiable

but second one which is modulus has a sharp turn at x = -5/2 so two tangents at x = -5/2 are possible so is not differentiable.

Maths >> Calculus >> Functions IIT JEE

DOMAIN OF log (X+4) base of log is 2

Asked By: GAURAV SHARMA 2 year ago
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A window frame is shaped like a rectangle with an arch forming the top ( ie; a box with a semicircle on the top)

The vertical straight side is Y, the width at the base and at the widest part of the arch (Diam) is X.

I know that the perimeter is 4.

Find X if the area of the frame is at a maximum?

Asked By: ROBERT DEPANGHER 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

perimeter = 4 so 2Y + X + πX/2 = 4

and area = XY + [π(X/2)2]/2

put Y from 1st equation to 2nd we get A = X[4-X{1+(π/2)}]/2 + πX2/8

now calculate dA/dX and then compare it to zero

X will be 8/(4+π)

y=px+a/p

Asked By: FRANCISBHORGIA 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

this is a special type of differential equation in which p = dy/dx

its solution will be y = cx+(a/c)  here c is a constant

y-xp=x+yp

Asked By: FRANCISBHORGIA 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

on applying p = dy/dx

(ydx-xdy)/dx = (xdx+ydy)/dx

or ydx-xdy = xdx+ydy

or dy/dx = (y-x)/(y+x)

now it is homogeneous

 

 

 

 

Importance High!!!!!!!!!!!!!!!!!

 

Dear Sir / Madam

 

Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is 

(++), (+-), (--) & (-+) at end of the month or year between the below mentioned.

 

Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9

(+) means: an Even number
(-) means: an Odd number

I simply say: for any EO (+-) it exists an OE (-+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83

There is 25 numbers in the EO (+-) group, so 25 in OE (+-).
There is 25 numbers in the EE (++) group, so 25 in OO (--).

Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)

 

So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:

 

There are some results for your review:

MONTH

DATE

DAY

FIRST_SERIES

SECOND_SERIES

FIRST_SERIES

SECOND_SERIES

APR

1

SUN

+

-

-

+

+-

-+

APR

2

MON

+

+

-

+

++

-+

APR

3

TUE

+

-

+

-

+-

+-

APR

4

WED

+

+

+

+

++

++

APR

5

THU

-

+

-

+

-+

-+

APR

6

FRI

+

+

+

-

++

+-

APR

7

SAT

+

-

-

+

+-

-+

APR

8

SUN

-

+

+

-

-+

+-

APR

9

MON

-

+

-

+

-+

-+

APR

10

TUE

+

+

+

+

++

++

APR

11

WED

+

+

-

-

++

--

APR

12

THU

-

+

-

-

-+

--

APR

13

FRI

+

+

+

+

++

++

APR

14

SAT

+

-

+

-

+-

+-

APR

15

SUN

+

-

+

-

+-

+-

APR

16

MON

-

-

+

+

--

++

APR

17

TUE

-

-

+

+

--

++

APR

18

WED

 

 

 

 

 

 

APR

19

 THU

 

 

 

 

 

 

APR

20

 FRI

 

 

 

 

 

 

APR

21

 SAT

 

 

 

 

 

 

APR

22

 

 

 

 

 

 

 

APR

23

 

 

 

 

 

 

 

 

  (Complete sheets for the year 2011 and 2012 are attached with this mail)

 

You can see that there are two times falls in a day of pair make by even (+) and odd (-) signs, i.e. (++), (+-), (--) & (-+) as above said.

The total of pair combination signs become approx equal to each other at the end of the month or year (every year- so, I am sending you attachment to go through the year’s result).

By this analysis:

1-     I want to know that what is the relation between current falling signs and past fell signs.

2-     How can I come to know, what type of combination of sign would be any particular date or day.

3-     As you can seen on 22nd and 23rd April’2012 I don’t know what combination of sign is.

4-     There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.

 

Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.

 

 I would be very grateful to you till entire life.

 

Regards

 

RAJEEV SHRIVASTAVA

 

 

 

Importance High!!!!!!!!!!!!!!!!!

 

Dear Sir / Madam

 

Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is 

(++), (+-), (--) & (-+) at end of the month or year between the below mentioned.

 

Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9

(+) means: an Even number
(-) means: an Odd number

I simply say: for any EO (+-) it exists an OE (-+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83

There is 25 numbers in the EO (+-) group, so 25 in OE (+-).
There is 25 numbers in the EE (++) group, so 25 in OO (--).

Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)

 

So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:

 

There are some results for your review:

MONTH

DATE

DAY

FIRST_SERIES

SECOND_SERIES

FIRST_SERIES

SECOND_SERIES

APR

1

SUN

+

-

-

+

+-

-+

APR

2

MON

+

+

-

+

++

-+

APR

3

TUE

+

-

+

-

+-

+-

APR

4

WED

+

+

+

+

++

++

APR

5

THU

-

+

-

+

-+

-+

APR

6

FRI

+

+

+

-

++

+-

APR

7

SAT

+

-

-

+

+-

-+

APR

8

SUN

-

+

+

-

-+

+-

APR

9

MON

-

+

-

+

-+

-+

APR

10

TUE

+

+

+

+

++

++

APR

11

WED

+

+

-

-

++

--

APR

12

THU

-

+

-

-

-+

--

APR

13

FRI

+

+

+

+

++

++

APR

14

SAT

+

-

+

-

+-

+-

APR

15

SUN

+

-

+

-

+-

+-

APR

16

MON

-

-

+

+

--

++

APR

17

TUE

-

-

+

+

--

++

APR

18

WED

 

 

 

 

 

 

APR

19

 THU

 

 

 

 

 

 

APR

20

 FRI

 

 

 

 

 

 

APR

21

 SAT

 

 

 

 

 

 

APR

22

 

 

 

 

 

 

 

APR

23

 

 

 

 

 

 

 

 

  (Complete sheets for the year 2011 and 2012 are attached with this mail)

 

You can see that there are two times falls in a day of pair make by even (+) and odd (-) signs, i.e. (++), (+-), (--) & (-+) as above said.

The total of pair combination signs become approx equal to each other at the end of the month or year (every year- so, I am sending you attachment to go through the year’s result).

By this analysis:

1-     I want to know that what is the relation between current falling signs and past fell signs.

2-     How can I come to know, what type of combination of sign would be any particular date or day.

3-     As you can seen on 22nd and 23rd April’2012 I don’t know what combination of sign is.

4-     There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.

 

Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.

 

 I would be very grateful to you till entire life.

 

Regards

 

RAJEEV SHRIVASTAVA

 
Asked By: RAJEEV SHRIVASTAVA 2 year ago
is this question helpfull: 8 1 read solutions ( 1 ) | submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

If you are making the sequence on the basis of dates then you should take signs accordingly then you can make this sequence for many years

for example if you take 23 june then first sign series will be +-, + for even number and - for odd number , now as conversion rule as you gave in your explanation 23 will be converted to 32 so second series will be -+

similarly you can make it for any date 

for example 29 december first series will be +- and its conversion is 38 so next series will be -+

if you are thinking something else then explain your question correctly 

how to find differential eqn. of all conics whose axes coincide with coordinate axes??   tell me the eqn. of tht. conic

Asked By: SHALINI SINGH 2 year ago
is this question helpfull: 7 2 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

Parabola   eq.  y2 = 4ax ,

on differentiating this eq. with respect to x we get 2y(dy/dx) = 4a                        

on taking this value of 4a in eq. y2= 4ax we will get the differential eq. of parabola.

similarly for ellipse    x2/a2  +   y2/b2  =   1

on differentiaing this eq. twice we will get two additional eq. 

by solving these eq. get an eq. free from a and b, this will be the differential eq. of the ellipse.

similarly for hyperbola

 

 

 

Maths >> Calculus >> Integrations IIT JEE

0π [cot-1x]dx

Asked By: NIKHIL VARSHNEY 2 year ago
is this question helpfull: 2 0 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

i think the question will be 0π [cotx]dx because limits are in angular terms.

by graph given below we get answer as -π/2

here in the graph same coloured shaded region are dx*(-1) always, because for every similar pair [+k]+[-k] = -1 , here k is a real number, so complete area = -1(π/2)

 lim x tends to 0 x tan( cos-1x)

Asked By: SHUBHAM VED 2 year agoSolved By: NIKHIL VARSHNEY
is this question helpfull: 2 0 read solutions ( 1 ) | submit your answer
Maths >> Calculus >> Integrations Others
  • what is the integration of cosx 2
Asked By: VIKRANT KUMAR 2 year ago
is this question helpfull: 3 0 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

cosx2 = (cos2)x

so ∫ (cos2)x dx = (cos2) x / logecos2

Solution by manish sir

 

cosx2 = (cos2)x

so ∫ (cos2)x dx = (cos2) x / logecos2 + C            formula ∫axdx = ax/logea

 

lim ∑ nr=1 1/n er/n n tends to infinity.

Asked By: NIKHIL VARSHNEY 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

limn->∞ r=1er/n/n

This question is related to definite integration, consider 1 and divide it into n parts, upto nth part total value = r/n is equivalent to x and 1/n is dx

so question will be

01 exdx 

now solve it

ABC is an isosceles triangle inscribed in a circle of radius r. If  AB=AC and h is the altitude from A to BC. and P and φ denote the perimeter and area of the triangle respectively, then  lim n−» 0   φ/p³ is equal to??

Asked By: AMIT DAS 2 year ago
is this question helpfull: 7 0 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

 

first draw a cirle and triange inside it , consider angle B and angle C asα so angle A = 180-2α, let centre of the circle is O and D is the point in the line BC where altitude meets in the line BC, given that altitude = h and radius = r

so AB = AC = h cosecα and BC = 2 h cotα so 

p = 2 h (cosecα + cotα)

and φ = 1/2  2 h cotα h = h2 cotαa

and in triangle OBD angle OBD = 2α - 90    so cos(2α - 90) = h cotα /r   implies that h = 2 r sin2α 

now i think limit will be based on h not n

 lim h−» 0   φ/p³  = 1/128r

for getting solution put p and φ first then put h in terms of r, you will get an equation based on α and r,

convert limh->0  to  limα->0     because when h will be 0, α will also be 0 

 

If f(x) is a monotonically increasing function " x Î R, f "(x) > 0 and f -1(x) exists, then prove that å{f -1(xi)/3} < f -1({x1+x2+x3}/3), i=1,2,3

Asked By: KAMAL 2 year ago
is this question helpfull: 15 0 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

f(x) is monotonically increasing so f '(x)>0 and f '' (x) >0 implies that increment of function increases rapidly with increase in x 

These informations provide the following informations about the nature of inverse of f(x) 

1) f -1 (x) will also be an increasing function but its rate of increase decreases with increasing x

2) for x< x2 < x3 ,  å{f -1(xi)/3} < f -1({x1+x2+x3}/3), 

The same result for f(x) will be 

  å{f (xi)/3} > f ({x1+x2+x3}/3), 

Maths >> Calculus >> Integrations IIT JEE

Evaluate: 0ò11/{ (5+2x-2x2)(1+e(2-4x)) } dx

Asked By: KAMAL 2 year ago
is this question helpfull: 7 0 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

let I = 0ò11/{ (5+2x-2x2)(1+e(2-4x)) } dx

on using the property 0a f(x) dx = 0af(a-x)dx

we get 2I = 0ò11/(5+2x-2x2) dx

now solve this 

Maths >> Calculus >> Functions IIT JEE

Let f(x) be a real valued function not identically equal to zero such that f(x+yn)=f(x)+(f(y))n; y is real, n is odd and n >1 and f'(0) ³ 0. Find out the value of f '(10) and f(5).

Asked By: KAMAL 2 year ago
is this question helpfull: 12 0 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

take x = 0 and y = 0 and n = 3

we get f(0) = f(0) + f(0)n or f(0) = 0

now take x = 0,  y = 1 and n = 3

so f(1) = f(0) + f(1)3

so get f(1) = 1, other values are not excetable because f(x) be a real valued function not identically equal to zero and f'(0) ³ 0

similarly get the other values 

you will get f(5) = 5

so f(x) = x generally then find f (x) , i think it will be 1 

Maths >> Calculus >> Integrations IIT JEE

Evaluate 0òx [x] dx .

Asked By: KAMAL 2 year ago
is this question helpfull: 6 0 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

 

 

integer nearst to x and less than x will be [x]

so 0òx [x] dx = 0ò1 0 dx + 1ò2 1 dx + 2ò3 2 dx + ..................... + [x]-1[x] [x]-1 dx + [x][x] dx

now solve it

 

A function f : R® R satisfies f(x+y) = f(x) + f(y) for all x,y Î R and is continuous throughout the domain. If I1 + I2 + I3 + I+ I5 = 450, where In = n.0òn f(x) dx. Find f(x).

Asked By: KAMAL 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

according to the given condition f(nx) = n f(x)

0òn f(x) dx

consider x = ny so this integration will become 0ò1 f(ny) ndy = n0ò1 f(y) dy = n2 0ò1 f(x) dx

now by using these conditions I1 = 13 0ò1 f(x) dx similarly others 

put these values and get the answer

Show that 0òp q3 ln sin q dq = 3p/2 0òp q2 ln [Ö2 sin q] dq.

Asked By: KAMAL 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

 let I = 0òp q ln sin q dq

on aplying property of definite integral

I =  0òp (π-q) ln sin q dq

so 2 I =  0òp π ln sin q dq 

or I = π/2  0ò ln sin q dq


or I = 2π/2  0òp/2 ln sin q dq    this is due to property

similarly solve  0òp q2 ln sin q dq  and   then 0òp q3 ln sin q d


finally solve the right hand side of the equation to prove LHS = RHS

 

 

f(x+y) = f(x) + f(y) + 2xy - 1 " x,y. f is differentiable and f'(0) = cos a. Prove that f(x) > 0 " x Î R.

Asked By: KAMAL 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

f(x+y) = f(x) + f(y) + 2xy - 1 

so f(0) = 1      obtain this by putting x = 0 and y = 0

now f (x+y) = f ' (x)  + 2y         on differentiating with respect to x

take x = 0, we get f ' (y) = f '(0) + 2y

or f ' (x) = cosα + 2x

now integrate this equation within the limits 0 to x

we get f(x) = x2 + cosα x + 1

its descreminant is negative and coefficient of x2 is positive so f(x) > 0

Find area of region bounded by the curve y=[sinx+cosx] between x=0 to x=2p.

Asked By: KAMAL 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

 

[sinx + cosx] = [√2 sin{x+(π/4)}]    here [ ] is greatest integer function

its value in different interval are

1       for 0 to π/2

0       for π/2 to 3π/4

-1        for 3π/4 to π

-2     for π to 3π/2

-1     for 3π/2 to 7π/4

0      for 7π/4 to 2π

now solve

Solve: [3Öxy + 4y - 7Öy]dx + [4x - 7Öxy + 5Öx]dy = 0.

Asked By: KAMAL 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

 

[3Öxy + 4y - 7Öy]dx + [4x - 7Öxy + 5Öx]dy = 0

=> dy/dx = [3Öxy + 4y - 7Öy] / [-4x + 7Öxy - 5Öx]

=> dy/dx = √y/√x [(3√x + 4√y - 7) / (-4√x + 7√y - 5)]

=> (dy/√y)/(dx/√x) = [(3√x + 4√y - 7) / (-4√x + 7√y - 5)]

now take √x = X + h and √y = Y + k

you will get a homogeneous equation dY/dX = (3X+4Y+3h+4k-7)/(7Y-4X+7k-4h-5)

take 3h+4k-7 = 0 and 7k-4h-5 = 0 and solve the equation

Solution by manish sir

 

[3Öxy + 4y - 7Öy]dx + [4x - 7Öxy + 5Öx]dy = 0

=> dy/dx = [3Öxy + 4y - 7Öy] / [-4x + 7Öxy - 5Öx]

=> dy/dx = √y/√x [(3√x + 4√y - 7) / (-4√x + 7√y - 5)]

=> (dy/√y)/(dx/√x) = [(3√x + 4√y - 7) / (-4√x + 7√y - 5)]

now take √x = X + h and √y = Y + k

you will get a homogeneous equation dY/dX = (3X+4Y+3h+4k-7)/(7Y-4X+7k-4h-5)

take 3h+4k-7 = 0 and 7k-4h-5 = 0 and solve the equation

dY/dX = (3X+4Y)/(7Y-4X)

take Y = uX then dY/dX = u + Xdu/dX

after solving put the values of X and Y in terms of x and y, 

 

Let g(x) be a continuous function such that    0ò1 g(t) dt = 2.   Let f(x) = 1/2 0òx (x-t)2 g(t) dt then find f '(x) and hence evaluate f "(x).

Asked By: KAMAL 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

 

f(x) = 1/2 0òx (x-t)2 g(t) dt

or f(x) = 1/2 0òx2 g(t) dt + 1/2 0òt2 g(t) dt - 0òx t g(t) dt 

or f(x) = 1/2 x2 0ò g(t) dt + 1/2 0òt2 g(t) dt - x 0ò t g(t) dt

now use newton leibniz formula for middle function and product rule for first and last functions for finding f' (x)

similarly get f'' (x)

 

 

 

limit  n tends to ∞

then

(x^n)/(n!) equals

Asked By: AMIT DAS 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

let y = lim n->∞ (xn/n!)

or y = x lim n->∞ (1/n) lim n->∞ xn-1/(n-1)!

or y = 0

 

limit n tends to ∞

x{ [tan‾¹ (x+1/x+4)] - (π/4)}

Asked By: AMIT DAS 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

 

I think n is printed by mistake, so considering it as x

the question will be limit x tends to ∞

x{ [tan-1 (x+1/x+4)] - (π/4)}

consider [tan-1 (x+1/x+4)] - (π/4) = θ

so [tan-1 (x+1/x+4)] = (π/4)+θ

so (x+1/x+4) = tan(π/4+θ)

or x = (-3-5tanθ)/2tanθ

so converted question will be limθ->0  (-3-5tanθ)θ/2tanθ

solve

Solution by manish sir

 

I think n is printed by mistake, so considering it as x

the question will be limit x tends to ∞

x{ [tan-1 (x+1/x+4)] - (π/4)}

consider [tan-1 (x+1/x+4)] - (π/4) = θ

so [tan-1 (x+1/x+4)] = (π/4)+θ

so (x+1/x+4) = tan(π/4+θ)

or x = (-3-5tanθ)/2tanθ

so converted question will be limθ->0  (-3-5tanθ)θ/2tanθ

now we know that limθ->0 tanθ/θ = limθ->0 θ/tanθ = 1

so next line will be limθ->0 (-3-5tanθ)/2 = -3/2 

limit n tends  to ∞

then

[³√(n²-n³) + n ] equals

Asked By: AMIT DAS 2 year ago
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Answer Strategies and trick (it will help you to solve it by yourself)

 

We have to calculate limit n -> ∞ [{³√(n²-n³) }+ n ]

Use the formula a3+b3 = (a+b)(a2+b2-ab) in the format 

(a+b) = (a3+b3)/(a2+b2-ab)

here a = ³√(n²-n³)  and b = n

on solving we get 1/(1+1+1) = 1/3

 

if f(X)=xlxl then find f-1(x) can you please explain me the meaning and use of sgn

 

Asked By: NIKHIL VARSHNEY 2 year ago
is this question helpfull: 10 1 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

f(x) = x|x|  => f(x) = -x2  for negative real values of x  and f(x) = x2 for positive real values of x

so f-1(x) = -√|x|  for negative real values and = +√|x| for positive real values of x

which of the following is differntiable at x=0 ?

cos(lxl)+lxl

sin(lxl)-lxl

Asked By: NIKHIL VARSHNEY 2 year ago
is this question helpfull: 10 2 submit your answer
Answer Strategies and trick (it will help you to solve it by yourself)

sin(|x|)+|x| is differentiable at x = 0

LHD of first = {cos|0+h|+|0+h|-cos|0|-|0|}/0+h-0  = (cosh+h-1)/h = (1-2sin2h+h-1)/h = 1 (on taking limits)

now check RHD, its value will be -1

similarly in second both values are 0 so it is differentiable

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