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If x^y = e^(x-y) then prove that (dy)/(d...

If `x^y = e^(x-y)` then prove that `(dy)/(dx)=logx/(1+logx)^2`.

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If e^(x) + e^(y) = e^(x + y) , then prove that (dy)/(dx) = (e^(x)(e^(y) - 1))/(e^(y)(e^(x) - 1)) or (dy)/(dx) + e^(y - x) = 0 .

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ARIHANT PUBLICATION-SAMPLE PAPER 1-SHORT ANSWER TYPE QUESTIONS
  1. If P(A) = 0.4, P(B//A) = 0.3 and P(B^(c)/A^(c)) = 0.2. Find P(B).

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  2. A bag consists of 10 balls each marked with one of the digits 0 to 9 ...

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  3. Using the properties of determinants? show that {:|(y+z,x,y),(z+x,z,...

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  4. Using elementary transformations, find the inverse of [(4,5),(3,4)]

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  5. Find the matrix A such that A[{:(1,2,3),(4,5,6):}]=[{:(-7,-8,-9),(2,4...

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  6. If y=log[x+sqrt(x^(2)+1)], then prove that (x^(2)+1)(d^(2)y)/(dx^(2))+...

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  7. Prove that the function f gives by f(x)=|x-1|x in R is not differentia...

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  8. Using differentials, find approximate value (255)^(1//4)

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  9. The two equal sides of an isosceles triangle with fixed base b are dec...

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  10. If x^y = e^(x-y) then prove that (dy)/(dx)=logx/(1+logx)^2.

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  11. Evaluate int(sqrt(tan x)+sqrt(cot x))dx

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  12. Form the differential equation of the family of circles touching the X...

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  13. Solve each of the following differential equations : x(dy)/(dx)-y= s...

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  14. Using properties of definite integrals, evaluate int(pi//6)^(pi//3)...

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  15. Find the area of the region {(x,y):0 le y le x^(2) +1,0 le y le x +1...

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  16. Find the value of x such that the four points A(3, 2, 1), B(4, x, 5), ...

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  17. Find the equation of the plane which contains the line of intersection...

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  18. Find a unit vector in XY-plane, which makes an angle of 45^(@) with t...

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  19. If a, b and c are vectors such that a .b=a .c, axx b=a xx c, a ne 0 ,...

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  20. Let v=2 hati + hatj-hatk and w=hati+3hatk if u is unit vector then fi...

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