1. What is the value of y as x approaches 5 in the equation y = (x2 + 5x – 50)/(x – 5)
A. 15
B. 0
C. 3
D. 5
2. Evaluate lim (𝑥 − 2)(𝑥2 + 3𝑥 − 2)/(𝑥2 − 4)
𝑥 →2
A.
B.
C.
D.
0
3
2
4
3. Find the lim (𝑥2 − 4)/(𝑥 − 2)
𝑥 →2
A.
B.
C.
D.
–2
4
8
–8
4. The limit of (x2 – a2)/(x – a) as x tends to a is
A. a
B. 0
C. – 2a
D. 2a
5. Find the limit of x2 – x /x as x → 0
A. 1
B. 2
C. – 3
D. – 1
6. Calculate lim 𝐶𝑜𝑠 𝑥/(𝑥 − 2)
𝑥 →0
A.
B.
C.
D.
1/2
– 1/2
1
1/4
7. If y = 13x, find dy/dt
A. 13
B. 13x2
C. 0
D. – 13
8. If y = 4x3 – 12x2 + 3x + 12, find dy/dx
A. 12x2 – 24x + 3
B. 7x4 + 14x3 + 3x2
C. 12x4 – 24x3 + 3x2
D. 7x2 – 14x + 3
9. Find the derivative of f(x) = 2x(x2 + 3)
A. 6(x2 + 1)
B. 2(x2 + 3)
C. 4x2 + 3
D. 6(x2 – 2)
10. Given that y = (x + 1)2, find dy/dx
A. x + 1
B. 2(x + 1)
C. x – 1
D. 2(x – 1)
11. If y = Sin 2x, find dy/dx
A. – 2 Sin x
B. – 2 Cos x
C. 2 Cos 2x
D. Cos 2x
12. Find the derivative of y = e2x + 3 with respect to x
A. – 3e2x + 2
B. 2e2x + 3
C. – 2e2x + 2
D. 6e2x + 4
13. Find the derivative of y = Cos(3x2 – 2x) with respect to x
A. – Sin(6x – 2)
B. – Sin(3x2 – 2x)
C. (6x – 2)Sin(3x2 – 2x)
D. – (6x – 2)Sin(3x2 – 2x)
14. Find dy/dx if y = log 𝑒 (1/𝑥)
A. – x
B. – 1/x
C. 1/x
D. 1/x2
15. If y = 3 Cos(x/3), find dy/dx when x = 3𝜋/2
A. – 1
B. – 3
C. 2
D. 1
16. If y = x Sinx, find dy/dx when x = 𝜋/2
A. – 𝜋/2
B. – 1
C. 1
D. 𝜋/2
17. Differentiate x/Cosx with respect to x
A. 1 + x Secx Tanx
B. 1 + Sec2x
C. Secx + x SecxTanx
D. Cosx + x Tanx
18. Find the gradient of the curve y = 2 √𝑥 – 1/x at the point x = 1
A. 0
B. 1
C. 3
D. 2
19. The slope of the tangent to the curve y = 3x2 – 2x + 5 at the point (1, 6) is
A. 6
B. 5
C. 1
D. 4
20. At what point is the gradient of the curve x2 – 6x + 3 equal to zero?
A. 3
B. – 2
C. – 3
D. 2
21. At what value of x is the function x2 + x + 1 minimum?
A. – 1
B. – 1/2
C. 1/2
D. 1
22. For what value of x will the function f(x) = x3 – 6x2 + 5 have turning point?
A. 0, 2
B. 0, 4
C. 0, - 2
D. 0, - 4
23. Find the value of x for which the function f(x) = 2x3 – x2 – 4x + 4 has a maximum value
A. – 2/3
B. 1
C. 2/3
D. – 1
24. Find the maximum value of the function y = 10 + 4x – x2
A. 10
B. 18
C. 14
D. 24
25. Find the maximum value of the function 2x3 + 3x2 – 72x + 1
A. 209
B. – 134
C. 298
D. 465
26. Find the turning points of the function y = 2x3 – 6x2 – 18x + 3
A. 0, 2
B. – 1, 2
C. 1, 3
D. – 1, 3
27. The turning point of the curve y = 5 – 2x – x2 occurs at
A. – 2
B. – 1
C. 1
D. 2
28. If y = f(x) is an increasing function at a given interval, then
A. dy/dx = 0
B. dy/dx < 0
C. dy/dx > 0
D. dy/dx < 1
29. At stationary point
A. dy/dx = 0
B. dy/dx < 0
C. dy/dx > 0
D. dy/dx = 1
30. The motion of a particle along a straight line is described by the equation x = 4t4 – 3t3.
Find the velocity after 3 seconds
A. 273ms-1
B. 196ms-1
C. 482ms-1
D. 351ms-1
31. Using the question in number 30 above, find the acceleration after 3 seconds
A. 422ms-2
B. 528ms-2
C. 378ms-2
D. 264ms-2
32. If 2x2 – 5y2 = 6xy, find dy/dx
A. (3x + 5y)/(2x – 3y)
B. (2x – 3y)/(3x + 5y)
C. (2x + 3y)/(3x – 5y)
D. 2x/(3x + 5y)
33. Differentiate with respect to x: y2 + x2 – 3xy = 4
A. (3y – 2x)/(2y – 3x)
B. (2y + 3x)/3y – 2x)
C. (2x – y)/(2y + 3x)
D. 3x/(3x + 2y)
34. Find dy/dx if Cosx = Siny
A. – Sinx/Cosy
B. Cosx/Siny
C. – 2Siny/Cosx
D. 2Sinx/Cosx
35. Find the gradient of the curve y2x + 3x2y = 1 at the point (1, 1)
A. 5/3
B. 9/8
C. – 7/5
D. – 3/4
36. The gradient of the curve x2 – 2xy – 2y2 – 2x = 0 at the point (1, - 4) is
A. – 4/3
B. – 1/2
C. – 4/7
D. – 4/9
37. Resolve
2x + 11
into partial fractions
(x – 2)(x + 3)
A. 2x + 11
x–2
x+3
B.
C.
3 1
x–2
x+3
11
+
x–2
D.
3
x–2
1
x+3
+
1
x+3
10x + 1
≡
(x – 2)(x + 1)
A. 7
B. – 3
C. 3
D. 1
38. If
39. Resolve
1
r (r + 2)
A. 1 + 2
r+2
2r
B.
3 . Find the value of k
x+1
into partial fractions
2
- 1
2r + 1
r+2
C.
1
r
D.
1
2r
+
1
2r – 1
-
1
2(r + 2)
40. Given that
A.
B.
C.
D.
k +
x–2
2x + 3
x2 – 5x + 6
≡
K
x–3
+
R Find the value of K
x–2
6
3
7
9
41. Using the question in 40 above, find the value of R
A. 3
B. 7
C. 9
D. 6
42. Express
5 – 12x
6x2 + 5x + 1
A.
13 23
2x – 1
3x + 3
B.
27
3x + 1
-
22
2x + 1
C.
18
3x + 1
+
27
2x + 1
D.
17
+
2x + 1
23
3x + 1
in partial fractions
43. Given that
6x + M
2x + 7x – 15
2
A.
B.
C.
D.
3+x
(x – 1)(x – b)
A. – 2
B. – 1/2
C. 2
D. 3
A.
B.
C.
D.
4
x+5
x+3
express in partial fraction is
x(x2 + 2)
3/2
– 3/2
–2
3
47. Find the value of C in question 45 above
A. 3
B. – 2
C. 3
D. – 3/2
48. Express
3x + 5 in practical fractions
(x + 2)2
+ B
x–2
A.
A
x+2
B.
A - B
x–2
(x + 2)2
D.
2
Find the constant M
2x – 3
express in partial fractions is
46. Find the value of B in question 45 above
A. – 2
B. 3/2
C. – 3/2
D. 3
C.
-
20
12
– 10
– 22
44. If
45. If
≡
A
x
-
B
(x – 2)2
A
+
(x + 2)
B
(x + 2)2
A +
x
5
x–b
-
Bx + C
x2 + 2
4
x–1
Find the value of b
Find the value of A
49. If y = ln 𝑥 . Find dy/dx
A. 1/x
B. x2
C. 1/2x
D. 1/x2
50. If y = 𝑒 𝑥 . Find dy/dx
A. 2ex
B. e-x
C. ex
D. e2x
1. Evaluate the limit of
3x3 + 2x2 + x + 1
as x tends to ∞
x3 + 2x + 5
b. Find dy/dx of the function 2x3 y2 – 3xy2 = 4
2. Find, from first principle, the derivative, with respect to x of the function
y = 2x2 – x + 3
(b). Find the equation of the tangent to the curve x2y + y3x + 3x – 13 = 0 at the point (1, 2)
3.
Differentiate the following with respect to x
(i). 𝑒 𝑥 Cosx
(ii). ln 𝑥 / 1 + Sinx
(iii). ln 𝑠𝑖𝑛𝑥
(b). Find the range of values of x for which x3 – 5 x2 – 2x + 1 is decreasing
2
4.
Resolve
x3 + x2 + 4x
x2 + x – 2
into partial fraction
(b). The motion of a particle starting from O, is describe by the equation S = 2 t3 - 17 t2 + 21t where
3
2
S is the distance in meters, and t the time in seconds. Find the acceleration of the particle when it
is momentarily at rest.
5.
Find the turning points on the curve y = x4 + 5 x3 - 2x2 – 3x + 1 and distinguish between them.
2
3
2
(b). Sketch the curve of y = x - 9
3𝑥+1
1. A function is defined by F(x) = 𝑥 2 −1 . Find F|(2).
11
A. 9
B. -1
C. 4
D. 1
2. The distance s travelled by a car over a time interval, t is given by, s =3t 2 – 4t + 1.
Find the speed of the car after 4 seconds.
A.
B.
−5
m/s
6
−4
m/s
27
C. 20m/s
D.
3
m/s
√2
3. Find the gradient of the curve 5𝑥 2 = 4 + 3𝑦 2 at the point (-2,1)
A. -4
−10
B. 3
C. ¼
D. 4
1
𝑑2 𝑦
4. Given that 𝑦 = 3𝑥 2 + 𝑥 2 , find 𝑑𝑥 2 .
A. 6(1 + 𝑥14 )
2
B. 6𝑥 − 𝑥 3
1
C. 6𝑥 2 + 2𝑥
D. 6𝑥 + 2𝑥 3
𝑑𝐴
5. Given that V = 4π𝑟 3 and A = 4π𝑟 2 , find 𝑑𝑉
A.
B.
C.
D.
4
𝑟
4π𝑟 3
5
2𝑟
2
𝑟
6. Given that f(x) = 8x2+ 4ax. Find the value of a, such that f1(x) =20 at x=-1.
A. 4
B. 8
C. 5
D. 9
7. At what value of x is the function y = x2 -2x – 3 minimum?
A. 7
B. 6
C. 1
D.10
𝑑𝑦
8. If 𝑦 2 + 𝑦𝑥 − 𝑥 = 0, find 𝑑𝑥
1−𝑦
A. 2𝑦
B.
1−2𝑦
𝑥
1−𝑦
C. 𝑥+3𝑦
1−𝑦
D. 𝑥+2𝑦
9. Find what value of x for which f(x) = x3- 6x2 + 5 has turning point?
A.
−3
4
B. 0 or -4
C. -1 or 4
D. 0 or 4
10. A blacksmith produced x articles at a total cost of $(200-48x+3x2). If each article is
sold at
3𝑥
$ 5 . Find the value of x for which the blacksmith makes a maximum profit.
A. 10
B. 6
C. 8
D. 15
11. 𝐼𝑓 𝑓(𝑥) = 𝑚𝑥 2 − 6𝑥 − 3 𝑎𝑛𝑑 𝑓 ′ (1) = 12, 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑚.
A. 9
B. 3
C. -3
D.-4
12. Find the derivative of 𝑦 = (3𝑥 2 + 5)4 with respect to x.
A. 12(3𝑥 2 + 5)3
B. 24(3𝑥 2 + 5)3
C. 12𝑥(3𝑥 2 + 5)2
D. 24𝑥(3𝑥 2 + 5)5
𝑥 2 +4𝑥+3
13. Evaluate lim 2𝑥 2 +5𝑥−3
𝑥→3
A. -0.1
B. 0.8
C. 0.4
D. 1
1−𝑥
14. Evaluate lim 𝑥 2 +𝑥−2
𝑥→1
A. 1
B. ½
C. 0
D.
−1
3
15. Given that 𝐴 = 𝑡 3 − 2𝑡 2 + 5𝑡 − 2, find the change in the value of A between t = 1 and
t=3
A. 15
B. 6
C. 20
D. 19
16. Given
2𝑥+1
(𝑥−1)(𝑥+2)
A. A=1, B=2
B. A=-1, B=2
C. A=1, B=-2
D. A=1, B=1
𝐴
𝐵
= 𝑥−1 + 𝑥+2, find the value of A and B.
2
17. If 𝑓(𝑥) = 3𝑥 2 + 𝑥 , 𝑓𝑖𝑛𝑑 𝑓 | (1).
A. 6
B. −6
C. 4
D.−4
𝑑𝑦
18. Find the 𝑑𝑥 𝑖𝑓 9𝑥 2 + 25𝑦 2 = 225
A.
B.
𝑥
𝑦
−9𝑥
25
C. x
D.
𝑦
𝑐
𝑑𝑦
19. Given that the 𝑑𝑥 = 2 , when x=3. Find the value of a, If 𝑦 = 𝑥 2 + 𝑎𝑥 + 3.
A. 4
B. 6
C. 7
D. -4
20. The velocity v of a car after time t is given by v = 2t2 +5t + 2. Find the acceleration
after 6 seconds.
A. 19m/s2
B. 7m/s2
C. 25m/s2
D. 29m/s2
21. Find the equation of the tangent to the curve x2 +3xy = 5 at the point (2,-1).
A. -6y + x -11 = 0
B. 6y + x -12 = 0
C. 6y - x -11 = 0
D. 6y + 2x -11 = 0
𝑑2 𝑦
22. Find
if y = 4x3- 5x2 - 5
𝑑𝑥 2
A. 24X +10
B. 24X – 10
C. 14X +10
D. 24X +5
𝑑𝑦
23. Given that 3x2 + 3y2 = 6, find 𝑑𝑥 at (10, 5).
A. -1
B. 5
C. -2
D. 7
3
24. y = √𝑥 + √𝑥 5 , find the gradient of y.
7
A.
B.
2√𝑥
1
2√𝑥
53
+ 3 √𝑥 2
53
+ 3 √𝑥 2
3
C.. √𝑥 2
D. 2
1
√
73
+ 3 √𝑥 2
𝑥
25. The area of the circular base of a cone is given by A = πr2, Find the derivative of A
with respect to A, when r = 14cm.
A. 48𝜋cm2
B. 18𝜋cm2
C. 28𝜋cm2
D. 28𝜋cm2
26. The distance, S, travelled by a racing car over a period t seconds, is given by s =
4t2+2.
Find the speed of the car in the 4th seconds.
A. 33m|s
B. 25m|s
C.10m|s
D. 13m|s
t3-
1
lim ( 𝑥)
27. Find
𝑥−−→∞
A. 0
B. 1
C. -1
D. 8
28. Given that y = 𝑒 𝑥 sinx, find
A.
B.
C.
D.
𝑥
3𝑒 sinx
𝑒 𝑥 (cosx + sinx)
𝑒 𝑥 𝑐𝑜𝑠x
𝑒 𝑥 sinx
𝑑𝑦
𝑑𝑥
.
29. The gradient of a line (4ax + 2) is -2. Find a.
A. ¾
B. ½
C. – ½
D. 1
𝑑𝑃
30. P = Log (x-5), find 𝑑𝑥
1
A. 𝑥−5
2
B. 4−𝑥
5
C. 𝑥−1
−5
D. 𝑥−1
31. Find the coordinate of the points on the curve y = x3 – 3x2- 9x + 6 where the gradient
is 0.
A. (3,-21)(-1,11)
B. (3,21)(1,11)
C. (3,-1)(-1,1)
D. (5,-2)(1,4)
𝑑𝑠
32. The total surface area of cylinder is given by S = 2𝜋𝑟 2 + 2𝜋𝑟ℎ, find 𝑑𝑟
A. 4𝜋𝑟 + 2𝜋ℎ
B. 4𝜋𝑟 + 𝜋ℎ
C. 3𝜋𝑟 + 2𝜋ℎ
D. 4𝜋𝑟 + 3𝜋ℎ
33. Find the x-coordinate of the point on the curve y = 2x3+ x2- 2x + 1 where the curve is
parallel to the line y = 2x.
2
A. 3 or -1
B.
2
3
or 1
2
C. - 3 or -1
2
D. 3 or 2
34. What the nature of the stationary point of y = 1 – 2x -2x2 have and what is the value
of y at that point?.
A. minimum (- ½ ,1½ )
B. maximum (- ½ ,1½ )
C. minimum (½ , 1½ )
D. maximum (- ½ , -1½ )
35. Let s be the distance covered in time t. Which of the following equation is true of the
acceleration of a moving object?
A. a =
𝑑2 𝑠
𝑑𝑡
𝑑2 𝑠
B. a = 𝑑𝑡 2
𝑑2 𝑡
C. a = 𝑑𝑠2
𝑑2 𝑡
D. a = 𝑑𝑠2
36. If
10𝑥+1
(𝑥−2)(𝑥+1)
𝐾
=
𝑥−2
+
3
, then K has the vaule?
𝑥+1
A. 4
B. 8
C. 7
D. 2
37.Which of the forms below is true of the fraction
𝐴
𝐵
𝐴
(𝑥+2)2
𝐵
𝐴
(𝑥−2)2
𝐵
𝐴
(𝑥+2)2
𝐵
A. 𝑥− 2 +
B. 𝑥 + 2 +
C. 𝑥 + 2 +
D. 𝑥− 2 +
(𝑥−1)2
5−12𝑥
38. Resolve into6𝑥 2 +5𝑥+1 its component form.
27
22
25
2𝑥+1
22
27
2𝑥+1
20
27
2𝑥+1
12
A. 3𝑥+1 −
B. 3𝑥+1 +
C. 3𝑥+1 +
D. 3𝑥+1 −
2𝑥+1
39. Resolving
1
A. 2𝑟 1
1
𝑟(𝑟+2)
1
3(𝑟+2)
1
B. 2𝑟 -5(𝑟+2)
1
1
1
1
C. 2𝑟 -7(𝑟+2)
D. 2𝑟 -2(𝑟+2)
into partial fractionbgives?
2𝑥 2 −𝑥+2
(𝑥+2)2
40. The radius of a circle is increasing at the rate of 0.4cms-1. Find the rate of increase
in cm2s-1 of the area when the radius is 5cm.
A. 2𝜋
B. 5𝜋
C. 12𝜋
D. 4𝜋
𝑥2
𝑑𝑦
1. (a) A curve has the equation y = 𝑥 +1 . Find 𝑑𝑥 where x = 2
(b) Find the coordinate of the stationary point of the curve
(c) Find the equation of the tangent to the curve at the point x = - 2
2. (a) A cuboid has a total surface area of 120cm2. Their base measure x cm by 2x cm
and its height is h cm. Show that the height h in terms of x is, h =
(b) Show that the volume of the cuboid is V = 40x (c) Show that V has a stationary value when h =
60−2𝑥 2
3𝑥
4𝑥 3
3
4𝑥
3
3. (a) A function is given by f(x) = 5x2 – 3x + 7. Find the derivative of f(x) using first
principle.
(b) Sketch the curve y = (x – 1)(x – 2)(x + 3)
3𝑥+5
(c) A function g(x) = 2𝑥−3. Find the domain and range of g -1(x)
4. (a) Resolve into partial fraction
(b) Given the rational function
2−4𝑥
𝑥(𝑥 2 −𝑥−2)
2√2− √5
3√2+2√5
= 𝑝 + 𝑞√10
where p and q are constants.
Find the value of p – q.
(c) A particle travels in a straight line so that, t seconds after passing through a fixed
𝑡
point O, its speed v m/s, is given by v = 8cos(2). Find the acceleration of the particle
when t = 1 seconds.