if the tangent to the curve y x pow 3 ax b at the point 1 5

If the tangent to the curve, $$y = {x^3} + ax - b$$ at the point (1, -5) is perpendicular to the line, $$ - x + y + 4 = 0, $$ then which one of the following points lies on the curve?

A. (-2, 1)

B. (-2, 2)

C. (2, -1)

D. (2, -2)

Answer : (2, -2)

Solution :
$$y = {x^3} + ax - b$$
Since, the point (1, -5) lies on the curve.
$$\eqalign{ & \Rightarrow 1 + a - b = - 5 \cr & \Rightarrow a - b = - 6\,.......\left( 1 \right) \cr & {\text{Now, }}\frac{{dy}}{{dx}} = 3{x^2} + a \Rightarrow {\left( {\frac{{dy}}{{dx}}} \right)_{{\text{at }}x = 1}} = 3 + a \cr} $$
Since, required line is perpendicular to $$y = x - 4,$$ then slope of tangent at the point $$P\left( {1, - 5} \right) = - 1$$
$$\therefore 3 + a = - 1 \Rightarrow a = - 4 \Rightarrow b = 2$$ the equation of the curve is $$y = {x^3} - 4x - 2$$
$$ \Rightarrow \left( {2, - 2} \right)$$ lies on the curve.

Releted MCQ Question on
Calculus >> Application of Derivatives

Releted Question 1

If $$a + b + c = 0,$$ then the quadratic equation $$3a{x^2}+ 2bx + c = 0$$ has

A. at least one root in $$\left[ {0, 1} \right]$$

B. one root in $$\left[ {2, 3} \right]$$ and the other in $$\left[ { - 2, - 1} \right]$$

C. imaginary roots

D. none of these

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Releted Question 2

$$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then

A. the area of $$\Delta ABC$$ is maximum when it is isosceles

B. the area of $$\Delta ABC$$ is minimum when it is isosceles

C. the perimeter of $$\Delta ABC$$ is minimum when it is isosceles

D. none of these

View Answer

Releted Question 3

The normal to the curve $$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point $$'\theta '$$ is such that

A. it makes a constant angle with the $$x - $$axis

B. it passes through the origin

C. it is at a constant distance from the origin

D. none of these

View Answer

Releted Question 4

If $$y = a\ln x + b{x^2} + x$$ has its extremum values at $$x = - 1$$ and $$x = 2,$$ then

A. $$a = 2,b = - 1$$

B. $$a = 2,b = - \frac{1}{2}$$

C. $$a = - 2,b = \frac{1}{2}$$

D. none of these

View Answer

If the tangent to the curve, $$y = {x^3} + ax - b$$ at the point (1, -5) is perpendicular to the line, $$ - x + y + 4 = 0, $$ then which one of the following points lies on the curve?

Releted MCQ Question on
Calculus >> Application of Derivatives

If $$a + b + c = 0,$$ then the quadratic equation $$3a{x^2}+ 2bx + c = 0$$ has

$$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then

The normal to the curve $$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point $$'\theta '$$ is such that

If $$y = a\ln x + b{x^2} + x$$ has its extremum values at $$x = - 1$$ and $$x = 2,$$ then

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Application of Derivatives

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If the tangent to the curve, $$y = {x^3} + ax - b$$ at the point (1, -5) is perpendicular to the line, $$ - x + y + 4 = 0, $$ then which one of the following points lies on the curve?

Releted MCQ Question on Calculus >> Application of Derivatives

If $$a + b + c = 0,$$ then the quadratic equation $$3a{x^2}+ 2bx + c = 0$$ has

$$AB$$ is a diameter of a circle and $$C$$ is any point on the circumference of the circle. Then

The normal to the curve $$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point $$'\theta '$$ is such that

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