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Differentiation question

2020 · 7 Jan · Shift 1 · Q28
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  5. /2020 · 7 Jan · Shift 1 · Q28

Differentiation question

2020 · 7 Jan · Shift 1 · Q28

JEE MainMathematicsDifferentiationMCQ+4 / −1
Let xk + yk = ak, (a, k > 0 ) and dydx+(yx)13=0{{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0dxdy​+(xy​)31​=0, then k is:
  1. A
    13{1 \over 3}31​
  2. B
    23{2 \over 3}32​
  3. C
    43{4 \over 3}34​
  4. D
    32{3 \over 2}23​
View written solutionFree

Correct answer: B

  1. We are given

xk+yk=ak ,a,k>0x^k + y^k = a^k \, , \quad a,k>0xk+yk=ak,a,k>0

and also

dydx+(yx)1/3=0.\frac{dy}{dx} + \left(\frac{y}{x}\right)^{1/3}=0.dxdy​+(xy​)1/3=0.

We need to find kkk.

  1. Differentiate the relation

xk+yk=akx^k + y^k = a^kxk+yk=ak

with respect to xxx.

Since aka^kak is constant,

ddx(xk)+ddx(yk)=0\frac{d}{dx}(x^k) + \frac{d}{dx}(y^k)=0dxd​(xk)+dxd​(yk)=0

kxk−1+kyk−1dydx=0.k x^{k-1} + k y^{k-1}\frac{dy}{dx}=0.kxk−1+kyk−1dxdy​=0.

Divide by kkk:

xk−1+yk−1dydx=0.x^{k-1} + y^{k-1}\frac{dy}{dx}=0.xk−1+yk−1dxdy​=0.

So,

dydx=−xk−1yk−1=−(xy)k−1.\frac{dy}{dx} = -\frac{x^{k-1}}{y^{k-1}} = -\left(\frac{x}{y}\right)^{k-1}.dxdy​=−yk−1xk−1​=−(yx​)k−1.

  1. From the second given differential equation,

dydx+(yx)1/3=0\frac{dy}{dx} + \left(\frac{y}{x}\right)^{1/3}=0dxdy​+(xy​)1/3=0

we get

dydx=−(yx)1/3.\frac{dy}{dx} = -\left(\frac{y}{x}\right)^{1/3}.dxdy​=−(xy​)1/3.

  1. Equate the two expressions for dydx\dfrac{dy}{dx}dxdy​:

−(xy)k−1=−(yx)1/3.-\left(\frac{x}{y}\right)^{k-1} = -\left(\frac{y}{x}\right)^{1/3}.−(yx​)k−1=−(xy​)1/3.

Hence,

(xy)k−1=(yx)1/3.\left(\frac{x}{y}\right)^{k-1} = \left(\frac{y}{x}\right)^{1/3}.(yx​)k−1=(xy​)1/3.

Now,

(yx)1/3=(xy)−1/3.\left(\frac{y}{x}\right)^{1/3} = \left(\frac{x}{y}\right)^{-1/3}.(xy​)1/3=(yx​)−1/3.

Therefore,

(xy)k−1=(xy)−1/3.\left(\frac{x}{y}\right)^{k-1} = \left(\frac{x}{y}\right)^{-1/3}.(yx​)k−1=(yx​)−1/3.

So the exponents must be equal:

k−1=−13.k-1 = -\frac{1}{3}.k−1=−31​.

Thus,

k=1−13=23.k = 1 - \frac{1}{3} = \frac{2}{3}.k=1−31​=32​.

  1. Check options:
  • A: 13\frac{1}{3}31​
  • B: 23\frac{2}{3}32​
  • C: 43\frac{4}{3}34​
  • D: 32\frac{3}{2}23​

So the correct option is B.

  1. Comparison with stored answer:

Stored correct answer: B

Our derived answer: B

They match.

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