This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Implicit Differentiation”.

1. Find the differentiation of x^{3} + y^{3} – 3xy + y^{2} = 0 ?

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2. x^{3} Sin(y) + Cos(x) y^{3} = 0 , its differentiation is

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3. Find the differentiation of x^{4} + y^{4} = 0

a) – ^{x3}⁄_{y4}

b) – ^{x4}⁄_{y3}

c) – ^{x3}⁄_{y3}

d) ^{x3}⁄_{y3}

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Explanation:

x

^{4}+ y

^{4}= 0

4x^{3} + 4y^{3} ^{dy}⁄_{dx} = 0

^{dy}⁄_{dx} = – ^{x3}⁄_{y3}

^{dy}⁄_{dx} = Sec^{2} (x)Sec(x) e^{x} + Sec^{2} (x)Tan(x) e^{x} + e^{x} Tan(x)Sec(x)

^{dy}⁄_{dx} = Sec^{2} (x) e^{x} [Sec(x)+Tan(x)] + e^{x} Tan(x)Sec(x)

4. Find differentiation of xSin(x) + ayCos(x) + Tan(y) = 0

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5. Find the derivative of Tan(x) = Tan(y)

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6. Implicit functions are those functions

a) Which can be solved for a single variable

b) Which can not be solved for a single variable

c) Which can be eliminated to give zero

d) Which are rational in nature.

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Explanation: Implicit functions are those functions, Which can not be solved for a single variable.

For ex, f(x,y) = x

^{3}+y

^{3}-3xy = 0.

7. Evaluate y^{4}4 + 3xy^{3} + 6x^{2} y^{2} – 7y + 8 = 0.

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8. If Sin(y)=Sin^{(-1)} (y) then

a) (1-y^{2} )(1 – Cos^{2} y) = 1

b) (1-y^{2} )(1 – Sin^{2} y) = 1

c) (1-y^{2} )(1 – Siny)=1

d) (1-y^{2} )(1 – Cosy)=1

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9. If Cos(y)=Cos^{(-1)} (y) then

a) (1 – y^{2} )(1 – Cos^{2} (y))=1

b) (1 – y^{2} )(1 – Cos(y))=1

c) (1 – y^{2} )(1 – Sin^{2} (y))=1

d) (1 – y^{2} )(1 – Sin(y))=1

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Explanation: Cos(y)=Cos

^{(-1)}(y)

Differentiating both sides

-Sin(y) = -1/√(1-y

^{2})

(1 – y

^{2})(1 – Cos

^{2}(y)) = 1.

10. If y^{2} + xy + x^{2} – 2x = 0 then ^{d2y}⁄_{dx2} = ?

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11. If the velocity of car at time t(sec) is directly proportional to the square of its velocity at time (t-1)(sec). Then find the ratio of acceleration at t=10sec to 9sec if proportionality constant is k=10 sec/mt and velocity at t=9sec is 10 mt/sec

a) 100

b) 200

c) 150

d) 250

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Explanation:

Given,v(t)=kv

^{2}(t-1)

Differentiating w.r.t time we get

^{dv(t)}⁄

_{dt}= 2kv(t-1)

^{dv(t – 1)}⁄

_{dt}

a(t) = 2*10*10 a(t-1)

^{a(t)}⁄

_{a(t – 1)}= 200.

12. If z(x,y) = 2Sin(x)+Cos(y)Sin(x) find ^{d2z(xy)}⁄_{dxdy}= ?

a) –Cos(y)Cos(x)

b) -Sin(y)Sin(x)

c) –Sin(y)Cos(x)

d) -Cos(y)Sin(x)

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13. If the car is having a displace from point 1 to point 2 in t sec which is given by equation y(x) = x^{2} + x + 1. Then,

a) Car is moving with constant acceleration

b) Car is moving with constant velocity.

c) Neither acceleration nor velocity is constant.

d) Both aceleration and velocity is contant.

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Explanation: y(x) = x

^{2}+ x + 1

Velocity is , v =

^{dy}⁄

_{dx}= 2x + 1 (not constant)

Acceleration is a =

^{dy}⁄

_{dx}= 2 (constant).

**Sanfoundry Global Education & Learning Series – Engineering Mathematics.**

To practice all areas of Engineering Mathematics, __here is complete set of 1000+ Multiple Choice Questions and Answers__.