Differentiation - Past Edexcel Exam Questions
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![Differentiation - Past Edecel Eam Questions 1. (a) Given that y = 5 3 + 7 + 3, find i. ii. dy d d 2 y d 2. Question 2ai, 2aii - January 2005 [3] [1] 2. The curve C has equation y = 4 2 + 5, 0.](/docs-images/54/35194502/images/1-0.png "The point P on C has -coordinate 1. (a) Show that the value of dy at P is 3. d [5] (b) Find an equation of the tangent to C at P. [3] This tangent meets the -ais at the point (k, 0).")
1 Differentiation - Past Edecel Eam Questions 1. (a) Given that y = , find i. ii. dy d d 2 y d 2. Question 2ai, 2aii - January 2005 [3] [1] 2. The curve C has equation y = , 0. The point P on C has -coordinate 1. (a) Show that the value of dy at P is 3. d [5] (b) Find an equation of the tangent to C at P. [3] This tangent meets the -ais at the point (k, 0). (c) Find the value of k. [2] Question 7 - January Given that y = 6 4 2, 0, (a) find dy d, [2] Question 2a - May The curve C has equation y = The point P has coordinates (3, 0). (a) Show that P lies on C. [1] (b) Find the equation of the tangent to C at P, giving your answer in the form y = m + c, where m and c are constants. [5] Another point Q also lies on C. The tangent to C at Q is parallel to the tangent to C at P.
2 (c) Find the coordinates of Q. [5] Question 10 - May Given that y = , 0, (a) find dy d, [2] Question 4 - January Figure 1: y 4 C P 1 Q Figure 1 shows part of the curve C with equation y = ( 1) ( 2 4 ). The curve cuts the -ais at the points P, (1,0) and Q as shown in Figure 1. (a) Write down the -coordinate of P and the -coordinate of Q. [2] (b) Show that dy d = [3] (c) Show that y = + 7 is an equation of the tangent to C at the point ( 1, 6). [2] The tangent to C at the point R is parallel to the tangent at the point ( 1, 6). (d) Find the eact coordinates of R. [5]
3 Question 9 - January Differentiate with respect to (a) 4 + 6, [3] (b) (+4)2. [4] Question 5 - May Given that y = , > 0, find dy d. [4]. Question 1 - January The curve C has equation y = , > 0. (a) Find an epression for dy d. [3] (b) Show that the point P (4, 8) lies on C. [1] (c) Show that an equation of the normal to C at the point P is 3y = The normal to C at P cuts the -ais at the point Q. [4] (d) Find the length P Q, giving your answer in a simplified surd form. [3] Question 8 - January Given that y = , > 0, find (a) dy d, [2] (b) d2 y d 2. [2]
4 Question 3a,b - May The curve C has equation y = 2 ( 6) + 4, > 0. The points P and Q lie on C and have -coordinates 1 and 2 respectively. (a) Show that the length of P Q is 170. [4] (b) Show that the tangents to C at P and Q are parallel. [5] (c) Find an equation for the normal to C at P, giving your answer in the form a + by + c = 0, where a, b and c are integers. [4] Question 10 - May (a) Write 2 +3 in the form 2 p + 3 q, where p and q are constants. [2] Given that y = , > 0, (b) find dy, simplifying the coefficient of each term. [4] d Question 5 - January f() = 3 + 3, > 0. (a) Differentiate to find f (). [2] Given that f () = 15, (b) find the value of. [3] Question 4 - June The curve C has equation y = k , where k is a constant.
5 (a) Find dy d. [2] The point A with -coordinate 1 lies on C. The tangent to C at A is parallel 2 to the line with equation 2y = 0. Find (b) the value of k, [4] (c) the value of the y-coordinate of A. [2] Question 9 - June Given that can be written in the form 2 p q, (a) write down the value of p and the value of q. [2] Given that y = , > 0, (b) find dy, simplifying the coefficient of each term. [4] d Question 6 - January The curve C has equation y = 9 4 8, > 0. The point P on C has -coordinate equal to 2. (a) Show that the equation of the tangent to C at the point P is y = 1 2. [6] (b) Find an equation of the normal to C at the point P. [3] The tangent at P meets the -ais at A and the normal at P meets the -ais at B. (c) Find the area of the triangle AP B. [4] Question 11 - January Given that y = , 0, find
6 (a) dy d, [3] Question 3a - June f() = (3 4 ) 2, > 0 (a) Show that f() = A B, where A and B are constants to be found.. [3] (b) Find f (). [3] (c) Evaluate f (9). [2] Question 9 - June The curve C has equation The point P has coordinates (2,7). y = , > 0 (a) Show that P lies on C. [1] (b) Find the equation of the tangent to C at P, giving your answer in the form y = m + c, where m and c are constants. [5] The point Q also lies on C. Given that the tangent to C at Q is perpendicular to the tangent to C at P, (c) show that the -coordinate of Q is 1 3 (2 + 6). [5] Question 11 - June Given that y = , find dy d. [3]. Question 1 - January 2010
7 21. The curve C has equation y = ( + 3)( 8), > 0. (a) Find dy d in its simplest form. [4] (b) Find an equation of the tangent to C at the point where = 2. [4] Question 6 - January Given that y = , > 0, find dy d. [6]. Question 7 - May The curve C has equation y = , > 0. (a) Find dy d. [4] (b) Show that the point P (4, 8) lies on C. [2] (c) Find an equation of the normal to C at the point P, giving your answer in the form a + by + c = 0, where a, b and c are integers. [6] Question 11 - January Given that y = , 0, find, in their simplest form, (a) dy d, [3] Question 2a - May 2011
8 25. The curve C has equation y = ( + 1)( + 3) 2. (a) Sketch C, showing the coordinates of the points at which C meets the aes. [4] (b) Show that dy d = [3] The point A, with -coordinate -5, lies on C. (c) Find the equation of the tangent to C at A, giving your answer in the form y = m + c, where m and c are constants. [4] Another point B also lies on C. The tangents to C at A and B are parallel. (d) Find the -coordinate of B. [3] Question 10 - June Given that y = , find in their simplest form (a) dy d, [3] Question 1a - January The curve C 1 has equation y = 2 ( + 2) (a) Find dy d. [2] (b) Sketch C 1, showing the coordinates of the points where C 1 meets the -ais. [3] The curve C 2 has equation y = ( k) 2 ( k + 2) where k is a constant and k > 2. (c) Find the gradient of C 1 at each point where C 1 meets the -ais. [2] (d) Sketch C 2, showing the coordinates of the points where C 2 meets the and y aes.. [3]
9 Question 8 - January y = (a) Find dy, giving each term in its simplest form. [4] d (b) Find d2 y d 2. [2] Question 4 - May The curve C has equation y = , 0 (a) Find dy, giving each term in its simplest form. [3] d The point P on C has -coordinate equal to 1 4. (b) Find the equation of the tangent to C at the point P, giving your answer in the form y = a + b, where a and b are constants. [4] The tangent to C at the point Q is parallel to the line with equation 2 3y + 18 = 0. (c) Find the coordinates of Q. [5] Question 11 - January (a) Show that f () = (3 2 ) 2 2, 0 f () = A + B 2, where A and B are constants to be found. [3] (b) Find f (). [2]
10 Question 19a,b - May Differentiate with respect to, giving each answer in its simplest form. (a) (1 2) 2 (b) [3] [4] Question 7 - May 2014
11 Solutions 1. (a) i (a) - ii. 30 (b) y = (c) k = (a) (a) - (b) y = (c) ( ) 5, (a) (a) -2,2 (b) - (c) - (d) ( 5, ) (a) (b) (a) (b) - (c) - (d) (a) (b) (a) - (b) - (c) 13y 14 = 0
12 12. (a) (b) (a) (b) = (a) 3k (b) k = 2 (c) (a) p = 3, q = 1 2 (b) (a) - (b) y = (c) (a) (a) A = 16, B = 24 (b) (c) (a) - (b) y = (c) (a) (b) y = (a) (b) - (c) 2 7y 64 =
13 25. (a). y (0, 9) y = ( + 1)( + 3) 2 ( 3, 0) ( 1, 0) (b) - (c) y = (d) = (a) (b). y y = 2 ( + 2) ( 2, 0) (0, 0) (c) 4,0
14 (d). y y = ( k) 2 ( k + 2) (k 2, 0) (k, 0) (0, 2k 2 k 3 ) 28. (a) (b) (a) (b) y = (c) (9, 1) 30. (a) A = 6, B = 1 (b) (a) (b)
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