RI Differentiation C6B Add Prac (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _______________________________________________________________ Additional Practice Questions for Chapter 6B: Applications of Differentiation Page 1 of 7 Additional Practice Questions for Chapter 6B: Applications of Differentiation 1 A curve has equation f( )yx where 32 33f( ) 7 22xx x x   . (i) Show that the gradient of the curve is always positive. Hence explain why the equation f( ) 0x  has only one real root and find this root. [3] (ii) Find the x-coordinates of the points where the tangent to the curve is parallel to the line 23yx . [3] [(ii) 0.145x or 1.15x ] 2 The equation of a closed curve is 22(2 ) 3 ( )2 7xy x y   . (i) Show, by differentiation, that the gradient at the point (x, y) on the curve may be expressed in the form d4 d7 yy x xy x   . (ii) Find the equations of the tangents to the curve that are parallel to (a) the x-axis, (b) the y-axis. [(ii)(a) y = 2 (b) 7x ] 3 The diagram shows the curve C with parametric equations x = 2a cot t , y = 2a sin2 t, where 0 t  and a is a positive constant. (i) Show that 3d 2sin cosd y ttx  . [1] (ii) Hence find, in terms of a, the equation of the tangent l to C which is parallel to the x-axis. [2] (iii) A point P on C has parameter 3t  . Given that the tangent to C at P meets l at the point Q and the point R is the foot of the perpendicular from P to l, find the exact area of triangle PQR in terms of a. [5] [(ii) 2ya (iii) 2 33 a ] x y (0,2a) y = 0

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 6B: Applications of Differentiation Page 2 of 7 4 [It is given that a sphere of radius r has surface area 24 r and volume 34 .3 r ] A model of a concert hall is made up of three parts.  The roof is modelled by the curved surface area of a hemisphere of radius r cm.  The walls are modelled by the curved surface of a cylinder of radius r cm and height h cm.  The floor is modelled by a circular disc of radius r cm. The three parts are joined together as show n in the diagram. The model is made of material of negligible thickness. (i) It is given that the volume of the model is a fixed value k cm3, and the external surface area is a minimum. Use differ entiation to find the values of r and h in terms of k. Simplify your answers. [7] (ii) It is given instead that the volume of the model is 200 cm 3 and its external surface area is 180 cm2. Show that there are two possible values of r. Given also that r < h, find the value of r and the value of h. [5] [(i) 3 3 5 khr  (ii) 3.04r  , 4.88h ]

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 6B: Applications of Differentiation Page 3 of 7 5 Fig. 1 shows a piece of card, ABC, in the form of an equilateral triangle of side a. A kite shape is cut from each corner, to give the shape shown in Fig. 2. The remaining card shown in Fig. 2 is folded along the dotted lines, to form the open triangular prism of height x shown in Fig. 3. (i) Show that the volume V of the prism is given by 21 3( 2 3) .4Vx a x [3] (ii) Use differentiation to find, in terms of a, the maximum value of V, proving that it is a maximum. [6] [(ii) 3 54 a ] 6 NJC Prelim 9740/2014/02/Q1 A solid has height 3 r cm and a base that is made up of a circular sector with radius r cm. The sector subtends an angle of θ radians, where 02 , at the centre of the circle as shown in the diagram. Given that the total surface area of the container is 10 cm2, find the exact values of r and θ such that the volume of the container is a maximum. [8] [ 5 3r , 3  ] B Fig. 2 x Fig.1 A C Fig. 3 a x x x x x x r 3r

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 6B: Applications of Differentiation Page 4 of 7 7 [It is given that a sphere of radius r has volume 34 3 r .] A container is made up of 2 parts.  A cylindrical body, opened at one end, of external radius r cm (where r > 1), external height h cm and thickness 1 cm throughout.  A cover consisting of a solid cylindrical disc with radius r cm and height 1 cm and a hemispherical handle with radius one-third of the radius of the disc. The internal volume of the cylindrical body is 30 cm 3 when full. The material used to make the body and the cover of the container costs $0.50 per cm3. (i) Find an expression for h in terms of r. [2] (ii) Show that the cost, $ C, to make the container is given by  2 32 2 11 5 1581 1 rCr r r     . [3] (iii) Given that 1rr is the value of r which gives the stationary value of C, show that 1r satisfies the equation  2 3 30 2027 1 rr r r     . Hence find the values of r and h which give the minimum value of C. [6] [(i)  2 30 1 1 h r   (iii) r = 2.66 cm, h = 4.48 cm] Cross-section of body Front view of container Cover 1 cm 1 cm 1 cm 1 cm r cm Disc h cm r cm Handle Body cm

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________________ Additional Practice Questions for Chapter 6B: Applications of Differentiation Page 5 of 7 8 A company requires a box made of cardboard of negligible thickness to hold 300 cm3 of powder when full. The top and the base of the box are made up of six identical isosceles triangles. The two identical sides of the isosceles triangle are of length ax cm, where a is a constant and 1 2a , and the remaining side is of length x cm. The height of the box is y cm (see diagram). (i) Use differentiation to find, in terms of a, the value of x which gives a minimum surface area of the box. [7] (ii) Show that, in this case, 213 21 ya xa   . Hence find the range of y x . [3] [(i)  1 3200 32 1x a   (ii) 0 < y x <3] 9 (a) A rocket, R, rises vertically from a point A on level ground. It is observed from another point B on the ground where B is 10 km from A. When the angle of elevation has the value 4  radians, this angle is increasing at the rate of 0.005 radians per second. Find, in m s1, the velocity of the rocket at that instant. (b) JPJC Promo 9758/2021/Q8(b) The height of an upright cone is twice the radius, r, of its circular base. It is known that the volume of the cone is increasing at the rate of 15 cm 3 min 1 when the radius is 3 cm. Find the rate of increase of the base area of the cone at this instant. [4] [(a) 100 m s1 (b) 215 cm min  ] ax ax x x x ax ax

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