Tangents and Normals

The slope of the tangent to curve  $y=x^3\ –\ x+1$ at the point whose x - coordinate is 2 is 

The slope of the normal to the curve $x=a\cos^3\theta,\ \ y=a\sin^3\theta$ at  $\theta=\frac{\pi}{4}$  is 

The equation of the tangent line to the curve y = x² – 2x +7 which is parallel to the line 2x – y + 9 = 0 is

• $y\ –\ 2x\ –\ 3=0$

• $y\ +2x\ –\ 3=0$

• $y\ –\ 2x\ +\ 3=0$

• $y\ +2x\ +\ 3=0$

The equation of normal to the curve y = tan x at (0, 0) is ________

• $x-y=0$

• $x+y=1$

• $x+y=0$

• $x-y=1$

The point on the curve  $y^2=x$ , where the tangent makes an angle of  $\frac{\pi}{4}$  with x-axis is

•  $\left(\frac{1}{2},\frac{1}{4}\right)$  

•  $\left(\frac{1}{4},\frac{1}{2}\right)$  

•  $\left(4,\ 2\right)$  

•  $\left(1,\ 1\right)$  

More than one correct: The points on the curve $y=x^3$  at which the slope of the tangent is equal to the y - coordinate of the point are

•  $\left(0,\ 0\right)$  

•  $\left(2,\ 8\right)$  

•  $\left(1,\ 1\right)$  

•  $\left(3,\ 27\right)$  

The equation of the normal to the curve $y=x^4–\ 6x^3+13x^2\ –\ 10x+5$  at  $\left(0,\ 5\right)$  is

•  $x\ \ +10y+50=0$  

•  $x\ –\ 10y+50=0$  

•  $x\ –\ 10y-50=0$  

•  $x\ +10y+50=0$  

The equations of all lines having slope 0 which are tangent to the curve $y=\frac{1}{x^2-2x+3}$  is

•  $y=1$  

•  $2y=1$  

•  $2y+1=0$  

•  $y+1=0$