Solve Q 11, 12 explaining detailly each step 10. The parameric equations of a curve are given by x -2. y 21-1, Show that anequation of the tangent to the curve at the point with parameters t is ty =x +t +2-t. Verifythat the point A (-1, 1) lies on the curve and that the tangent at this point passes throughB(5, 7). Show that the line 3y x + 8 is also a tangent to the curve and find the coordinatesof the point of contact of this line with the curve.11. Find an equation of the tangent to the curve y = In(4 + x*) at the point where x = 1.12. i) Find the maximum value of the function f defined by f(x) =d'yii) Given that y = e'sinbx, where b is a constant, show that: + 2 + (1+b')y=dx213. Show that the tangent at the point P with parameters t, on the curve whose parametricequations are x = ct, y3Dwhere c is a constant, has equation x+t'y= 2ct. This tangentmeets the x-axis at a point Q and the line through P parallel to the x-axis cuts the y-axis at apoint R. Show that the area of the triangle QOR, where 0 is the origin, is a constant.14. The graph of the function f, where f(x)=ax+bhas stationary values at the point (-1. 1).Find the values of a and b. show that the stationary value is a maximum. Find the value of xfor which the function has a maximum value. Sketch the curve y= (x).15. Find the equations of the tangent to the curve y+3xy+4x 8 at the point x= 116. Given that parametric equations of a curve are x =y = obtain a Cartesian equation of%3Dt+1the curve. Hence, or otherwise, find an equation of the normal to the curve at the pointwhere t= 2.17. Find yat the point (-2, 2) given that 2x-5y2 3xydx18. The parametric equations of a curve are given by x= at". y= 2at, where a is a constant and ta parameter. Show that an equation of tangent to the curve at the point with parameter t isty =x + at23. i) Given that: y = cos1("x) show thatdyVb-ax%3Ddxii) Prove that the curve: y = x"(1 – x)",where m, nE z*, has a stationary pointwhen:mm+nShow that when: m =n=2, the stationary point is a minimum point.56

Question

Solve Q 11, 12 explaining detailly each step 10. The parameric equations of a curve are given by x -2. y 21-1, Show that anequation of the tangent to the curve at the point with parameters t is ty =x +t +2-t. Verifythat the point A (-1, 1) lies on the curve and that the tangent at this point passes throughB(5, 7). Show that the line 3y x + 8 is also a tangent to the curve and find the coordinatesof the point of contact of this line with the curve.11. Find an equation of the tangent to the curve y = In(4 + x*) at the point where x = 1.12. i) Find the maximum value of the function f defined by f(x) =d'yii) Given that y = e'sinbx, where b is a constant, show that: + 2 + (1+b')y=dx213. Show that the tangent at the point P with parameters t, on the curve whose parametricequations are x = ct, y3Dwhere c is a constant, has equation x+t'y= 2ct. This tangentmeets the x-axis at a point Q and the line through P parallel to the x-axis cuts the y-axis at apoint R. Show that the area of the triangle QOR, where 0 is the origin, is a constant.14. The graph of the function f, where f(x)=ax+bhas stationary values at the point (-1. 1).Find the values of a and b. show that the stationary value is a maximum. Find the value of xfor which the function has a maximum value. Sketch the curve y= (x).15. Find the equations of the tangent to the curve y+3xy+4x 8 at the point x= 116. Given that parametric equations of a curve are x =y = obtain a Cartesian equation of%3Dt+1the curve. Hence, or otherwise, find an equation of the normal to the curve at the pointwhere t= 2.17. Find yat the point (-2, 2) given that 2x-5y2 3xydx18. The parametric equations of a curve are given by x= at&#34;. y= 2at, where a is a constant and ta parameter. Show that an equation of tangent to the curve at the point with parameter t isty =x + at23. i) Given that: y = cos1(&#34;x) show thatdyVb-ax%3Ddxii) Prove that the curve: y = x&#34;(1 – x)&#34;,where m, nE z*, has a stationary pointwhen:mm+nShow that when: m =n=2, the stationary point is a minimum point.56

Accepted Answer

Answered: Image /qna-images/answer/f35f588d-6f00-413a-a0a7-7ac9d3f12904.jpg

11. Find an equation of the tangent to the curve y = In(4 + x") at the point %3D 12. i) Find the maximum value of the function f defined by f(x) = 1+x2 ii) Given that y=e'sinbx, where b is a constant, show that: 24+(1+b)y- dx2 dx