Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) 1a. [2 marks] Let . Find 1b. [5 marks] Hence find the values of θ for which . 2a. [2 marks] Let Find . 2b. [7 marks] Find . 3a. [5 marks] Let The graph of . has a local maximum at A. Find the coordinates of A. 3b. [5 marks] Show that there is exactly one point of inflexion, B, on the graph of . 3c. [3 marks] The coordinates of B can be expressed in the form B where a, b a and the value of b. 3d. [4 marks] Sketch the graph of showing clearly the position of the points A and B. 4. [6 marks] Consider the curve . Find the x-coordinates of the points on the curve where the gradient is zero. Mobile: 9892590312 Email: mohitbvyas@hotmail.com, . Find the value of Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) 5a. [2 marks] Consider the functions defined for , given by and . Find . 5b. [1 mark] Find . 6a. [2 marks] Consider the function Determine whether . is an odd or even function, justifying your answer. 6b. [3 marks] Hence or otherwise, find an expression for the derivative of with respect to . 6c. [8 marks] Show that, for , the equation of the tangent to the curve at is . 7. [8 marks] The folium of Descartes is a curve defined by the equation following diagram. Mobile: 9892590312 Email: mohitbvyas@hotmail.com, , shown in the Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the -axis. 8a. [4 marks] A window is made in the shape of a rectangle with a semicircle of radius the diagram. The perimeter of the window is a constant P metres. Mobile: 9892590312 Email: mohitbvyas@hotmail.com, metres on top, as shown in Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Find the area of the window in terms of P and . 8b. [5 marks] Find the width of the window in terms of P when the area is a maximum, justifying that this is a maximum. 8c. [2 marks] Show that in this case the height of the rectangle is equal to the radius of the semicircle. 9a. [2 marks] Consider the function Showing any and defined by where is a positive constant. intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes. ; 9b. [4 marks] Showing any and intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes. ; 9c. [2 marks] Showing any and intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes. . 9d. [4 marks] The function is defined by By finding explain why for . is an increasing function. 10a. [2 marks] Let . Find an expression for . Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) 10b. [2 marks] Show that . 10c. [2 marks] Consider the function defined by Show that the function . has a local maximum value when . 10d. [2 marks] Find the -coordinate of the point of inflexion of the graph of . 10e. [3 marks] Sketch the graph of , clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts. 10f. [3 marks] The curvature at any point on a graph is defined as Find the value of the curvature of the graph of . at the local maximum point. 10g. [2 marks] Find the value for and comment on its meaning with respect to the shape of the graph. 11a. [5 marks] A curve has equation Find an expression for . in terms of and . 11b. [4 marks] Find the equations of the tangents to this curve at the points where the curve intersects the line . 12. [7 marks] Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Find the -coordinates of all the points on the curve at which the tangent to the curve is parallel to the tangent at . 13. [7 marks] A curve is given by the equation . . Find the coordinates of all the points on the curve for which 14. [6 marks] The function is defined as where Hayley conjectures that . . Show that Hayley’s conjecture is correct. 15a. [2 marks] Consider the curve Find . . 15b. [4 marks] Determine the equation of the normal to the curve at the point where in the form . 16a. [4 marks] A curve is defined by . Show that there is no point where the tangent to the curve is horizontal. 16b. [4 marks] Find the coordinates of the points where the tangent to the curve is vertical. 17. [1 mark] Show that . 18a. [2 marks] Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Let Find . . 18b. [5 marks] Find the coordinates of any local maximum and minimum points on the graph of . Justify whether any such point is a maximum or a minimum. 18c. [5 marks] Find the coordinates of any points of inflexion on the graph of . Justify whether any such point is a point of inflexion. 19a. [2 marks] Consider the functions and . , stating its domain. Find an expression for 19b. [2 marks] Hence show that . 19c. [6 marks] Let , find an exact value for at the point on the graph of , expressing your answer in the form . 20a. [4 marks] Consider the function defined by Determine the values of for which . is a decreasing function. 20b. [3 marks] There is a point of inflexion, Find the coordinates of , on the curve . . 21a. [4 marks] Mobile: 9892590312 Email: mohitbvyas@hotmail.com, where Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) In triangle , and Show that length . . 21b. [4 marks] Given that has a minimum value, determine the value of for which this occurs. 22a. [4 marks] The function is defined as (i) Find (ii) State the domain of . . . 22b. [5 marks] The function is defined as The graph of . and the graph of intersect at the point . . Find the coordinates of 22c. [3 marks] intersects the The graph of Show that the equation of the tangent -axis at the point . to the graph of at the point is . 23. [6 marks] A tranquilizer is injected into a muscle from which it enters the bloodstream. The concentration in , of tranquilizer in the bloodstream can be modelled by the function where is the number of minutes after the injection. Find the maximum concentration of tranquilizer in the bloodstream. 24a. [2 marks] Consider two functions and and their derivatives values for the two functions and their derivatives at and , and . The following table shows the . Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Given that and , find ; 24b. [4 marks] . 25a. [2 marks] Consider the function . The sketch below shows the graph of Show that and its tangent at a point A. . 25b. [3 marks] Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Find the coordinates of B, at which the curve reaches its maximum value. 25c. [5 marks] Find the coordinates of C, the point of inflexion on the curve. 25d. [4 marks] The graph of crosses the -axis at the point A. Find the equation of the tangent to the graph of at the point A. 26. [9 marks] . A curve has equation (a) Find in terms of x and y. (b) Find the gradient of the curve at the point where and . 27a. [2 marks] Consider the following functions: , Sketch the graph of , . 27b. [2 marks] Find an expression for the composite function and state its domain. 27c. [7 marks] Given that (i) find (ii) show that , in simplified form; for . 27d. [3 marks] Nigel states that is an odd function and Tom argues that is an even function. Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) (i) State who is correct and justify your answer. (ii) Hence find the value of for . 28. [6 marks] Paint is poured into a tray where it forms a circular pool with a uniform thickness of 0.5 cm. If the paint is poured at a constant rate of , find the rate of increase of the radius of the circle when the radius is 20 cm. 29. [9 marks] The curve C has equation . Determine the coordinates of the four points on C at which the normal passes through the point (1, 0) . 30. [6 marks] Let . Using implicit differentiation, show that . 31a. [8 marks] At 12:00 a boat is 20 km due south of a freighter. The boat is travelling due east at the freighter is travelling due south at , and . Determine the time at which the two ships are closest to one another, and justify your answer. 31b. [3 marks] If the visibility at sea is 9 km, determine whether or not the captains of the two ships can ever see each other’s ship. 32a. [3 marks] The diagram below shows a circular lake with centre O, diameter AB and radius 2 km. Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Jorg needs to get from A to B as quickly as possible. He considers rowing to point P and then walking to point B. He can row at and walk at . Let radians, and t be the time in hours taken by Jorg to travel from A to B. Show that . 32b. [2 marks] Find the value of for which . 32c. [3 marks] What route should Jorg take to travel from A to B in the least amount of time? Give reasons for your answer. 33a. [5 marks] Consider the function (i) (ii) (iii) , Solve the equation Hence show the graph of . . has a local maximum. Write down the range of the function . 33b. [5 marks] Show that there is a point of inflexion on the graph and determine its coordinates. 33c. [3 marks] Mobile: 9892590312 Email: mohitbvyas@hotmail.com, Mohit Vyas Mathematics Higher Level HL Diff. Non GDC April19 Group & Personal Tutoring For IB , IGCSE & SAT (All Subjects) Sketch the graph of , indicating clearly the asymptote, x-intercept and the local maximum. 33d. [6 marks] Now consider the functions and (i) Sketch the graph of (ii) Write down the range of (iii) Find the values of , where . . such that . Mobile: 9892590312 Email: mohitbvyas@hotmail.com, .
