FURTHER MATHEMATICS FIRST TERM EXAMINATION SS 3

FIRST TERM

Examination malpractices may lead to a repeat of the subject or suspensions don’t be involved.

SUBJECT: FURTHER MATHEMATICS         CLASS: SS3            TIME APPROVED: 3HRS

INSTRUCTION: Answer all questions from both parts

PART A – OBJECTIVE:

1. Find the coordinates of the foci of the ellipse 9x²+4y²= 36 (a) (0, √5) and (0, -√5)  (b) (0, √3) and (0, -√3)
   (c) (0, √7)  (0, -√7)             (d)   (0, √4)  (0, -√4)
2. If α and β are roots of an equation such that (α + β)=3 and (αβ)=2, find the equation.
   (A)   x²-3x+2=0 (B) x²-2x+3=0 (C) x²-3x-2=0 (D) x²-2x-3=0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
3. Simplify (1-sinθ)(1+sinθ)? (A) sin²θ (B) sec²θ (C) tan²θ (D) cos²θ
4. The sum of the first three terms of an arithmetic progression (A.P.) is 18. If the first term is 4, find their product. (A) 130 (B) 192 (C)210 (D) 260
5. Two functions f and g are defined on the set R of real numbers by f:x→2x-1, g:x→x²+1. Find the value of f^-1o g(3) (A)12 (B)11 (C)11/2 (D)9/2
6. The gradient of the line passing through the points P(4,5) and Q(x,9) is 9. Find the value of x.

(A)-4 (B)0 (C)4 (D)12

7. Simplify  √3      +      √3

(√3-1)        (√3-1)              (A) ½ (B) √3/2 (C)  3 (D) 2√3

8. Given that log₂ y² =log₅625, find the value of y (A) 16 (B)25 (C)36 (D)4
9. Simplify tan80-tan20         (A)3√2 (B) 2√3 (C) √3 (D) -√3

                     1+ tan80tan20

10. The binary operation * is define 0n the set R, of real numbers x*y=3x+3y-xy for all x,y ϵ R. determine, in terms of x, the identity element of the operation. (A)2x/(x+3) (B) 2x/(x-3) (C) 3x/(x-3) (D)3x/(x+3)
11. Find the fourth term of the binomial expansion of (x-k)⁵ (A) 10x²k³(B) 5x³k² (C)-5x³k² (D) -10x²k³

The tangent to the curve y=4x³+kx²-6x+4 at the point P(1,m) is parallel to the x-axis, where k and m are constants. Use the information above to answer questions 12 and 13.

12. Find the value of k (A) 3 (B) 2 (C) -2 (D) -3
13. Determine the coordinates of P. (A) (1,2) (B) (1,1) (C) (1,-1) (D) (1,-2)
14. Two vectors m and n are defined by m=3i+4j and n=2i-j. find the angle between m and n. (A) 97.9 (B) 79.7 (C) 63.4 (D) 36.4
15. Find the area of the circle whose equation is x² + y² +4x + 8y + 11 = 0 (A) 3П (B) 6П (C) 9П (D) 12П
16. Two bodies of masses 8kg and 5kg travelling in the same direction with speeds of xm/s and 2m/s respectively collide. If after collision, they move together with of 3.85m/s, find correct to the nearest whole number the value of x. (A) 2 (B) 5 (C) 8 (D) 13
17. A particle accelerates at 12ms^-2 and travels a distance of 250m in 6 seconds. Find the initial velocity of the particle  (A) 5.7ms^-1 (B) 6.0ms^-1 (C) 60.0ms^-1 (D) 77.5ms^-1

18. A bag contains 2 red and 4 green sweets of the same size and shape. Two boys pick a sweet each from the box, one after the other, without replacement. What is the probability that at least a sweet with a green wrapper is picked? (A) 1/5 (B) 2/5 (C) 8/15 (D) 14/15
19. If f(x)=mx²-6x-3 and f^-1(1)=12, find the value of the constant, m. (A) 9 (B) 3 (C) -3 (D) -4
20. A body is acted upon by two forces F₁=(5N, 060) and F₂=(10N, 180), find the magnitude of the resultant. (A) 18.75N (B) 15.75N (C) 9.50N (D) 8.66N
21. The equation of a curve is given by y=2x²-5x+k, if the curve has two intercepts on the x-axis, find the value(s) of the constant, k. (A) 8/25 (B) 25/8 (C) <25/8 (D) >8/25
22. Find the value of p which will make x²-x+p a perfect square. (A) -1/2 (B) ½ (C) ¼ (D) 1
23. The polynomial g(x)= 2x³+3x²+qx-1, has the same remainder when divided by (x+2) and (x+1). Find the value of q. (A) -11 (B) -9 (C) -3 (D) -1

Use the chart above to answer questions 25-26

24. Find the upper class boundary of the class that contains the third quartile. (A) 13.0 (B) 16.0 (C) 16.5 (D) 22.5
25. Find the probability that a pupil selected at random had a mark of at least 14. (A) 0.22 (B) 0.41 (C) 0.49 (D) 0.63
26. If T = find T^-1 (a)        (b)        (c)    (d)
27. Given that p = 1+√2 and q=1-√2, evaluate (p²-q²)/2pq .A. 2(2 +√ 2) B. –2(2 +√2) C. 2√2 D.-2√2
28. Solve the equations m² + n² = 29, m + n = 7.
29. (2, 3) and (3, 5) B. (2, 5) and (5, 2) C. (5, 2) and (5, 3) D. (5, 3) and (3, 5)

In a school, 220 students offer Biology or Mathematics or both. 125 offer Biology and 110 Mathematics.

29. How many offer both Biology and Mathematics?A.110 B. 75 C. 80 D. 125
30. How many offer Mathematics only? (A) 110 (B) 75 (C) 80 (D) 125
31. How many offer Biology only? (A) 110 (B) 75 (C) 80 (D) 125
32. Given that (1 + 3x)⁴ = 1 + Px +Qx²… find 3P+2 144           B. 14        C. 441 D. 444
33. Find the coefficient of x⁵ in the expansion of A. 1512 504     C. -15 D. 35.
34. Find the coefficient of a³b⁵ in the expansion of (5a + ½b)⁸ 175/2         B. 875/2    C. 875/4. D. 375/2
35. Given that (1 + 5x)⁴ = 1 + Px + Qx² + Rx³ + … Find the value of P – 2Q + 3R.
36. Evaluate Sin 15⁰ A ¼(√6 -√2)     √3/2   C.  ¼(√2 – √6) D. ¼(√2 + √6)
37. Simplify          7 + √2     B.  7 + 7√2        C. 1 – 7√2     D.    1 + √2
38. Without using table, find the value of 2sin15⁰cos15⁰   3/2        B. 1/2     C.1/√2
39. The common ratio in the G.P; log 3, log 9, log 81… is. (a) 1 (b) 3 (c) 6 (d) 2.
40. What is log₁₀ (6.95 x 10^-3)? (a) 3.8420 (b) 3.9777 (c) 3.9777 (d) 3.8420.
41. One of the roots of the equation 6x² = 5-7x is (a) (b)  (c)  (d) .
42. A student found the approximate value of 0.02548 correct to two places of decimal instead of two significant figures. Find the percentage error. (a) 20% (b) 0% (c) 13% (d) 16%.
43. Which of the equations has its roots as 4 and 5? (a) x² + x – 20 = 0 (b) x² + x + 20 = 0 (c) x² – x +20 (d) x² + 4x – 20 = 0.
44. Given that = 2, find the value of . (a)  (b)  (c)  (d) .
45. Solve the equation 5x² – 4x – 1 = 0. (a) -1, (b) -1, (c) ,1  (d) -1, 5.
46. Find (x – y) if 4x – 3y = 7 and 3x – 2y = 5. (a) 4 (b) 3 (c) 2 (d) -3.
47. If k + – = 0, Find k. (a) -2 (b) 1 (c) -1 (d) 2.
48. If y = express x in terms of y, a and b. (a) x = (b) x =    (c) x =   (d) x = .
49. If x² + 15x + 50 = ax² + 6x + c = 0; which of the following statements is not true? (a) x = -5 (b) x = 10 (c) bc = 750 (d) x + 10 = 0.
50. Find the value of sin45° – cos30°. A. (2√2-√3)/2 B. (√3-2√2)/2 C. (√2-√3)/3 D. (√3-√2)/3

THEORY

Answer eight (8) questions in this part. No 1 is compulsory and any other five

1. a) The table below gives the marks obtained by members of a class in mathematics and physics examination

1. Construct a rank table for the table above and calculate the coefficient of rank correlation.
2. Comment on your result.

2)    find the conic section centre, vertices, foci and eccentricity of the equation 4x² +y²-8x+4y-8=0, Hence graph the equation.

1. find the centre and radius of (x+2)² + (y-3)²=4 Hence, sketch the graph of the circle.

3    a) Resolve into partial fractions        (i)                       (ii)

1. b) The life span of inverter batteries purchased domestic use could be assumed to be assumed to be normally distributed about a mean of 1750 hours, with a standard deviation of 70 hours. Calculate the probability that the batteries will drain between 1600 and 1900 hours.

4. a) Given vectors p=3i+4j and q= -8i – 15j, find
5. i) the magnitude of vectors p and q
6. ii) the unit vectors in the direction of p and q

iii) the angle between the vectors

1. b) Use determinants method to the simultaneous equations:

2x – y + z = 0

x + 3y + 2z = 16

3x – y + z = 3

5  a) A missile is projected at angle, θ, with a speed of 300ms^-1 such that the magnitude of its range is twice
that of its height. Calculate (i) the range (ii) height (iii) total time of flight.

1. b) P =             Find P^-1?

6    (a)   Solve the inequality (i) (x²+3) / (2x-8) ≥ 0 (ii) 2x² + x ≥ x² – 4x – 6

(b)   Solve the graph of   (x +1)^{2   –} (y-2)²    =    1
^{25                   16}
      and hence, find (i) the vertices (ii) the focus and (iii) the latus rectum  (iv) Minor and Major axes
(v)  eccentricity

7. (a) Points (2,1)  and (6,7) are opposite vertices of a square which is inscribed in a circle. Find the :
   (i)  Centre Of the Circle                (ii)  Equation Of the Circle

(b)       Evaluate

8. (a) Differentiate   2x+3    with respect to x.
   4x²-8

(b)        (i) Sketch the curve of y = 9x-x³, hence find the point of inflexion.
(ii) Calculate the total area bounded by the x-axis and the curve y = 9x-x³

9. (a) Using the trapezium rule with ordinates at x = 1,2,3,4 and 5, calculate, correct to two decimal places,
   the value of   x + ) dx

(b) Find the distance  between the points  (2,  5)  and   (5,  9)

10. (a)  The first, second and fifth terms of a linear sequence  (A.P) are three consecutive  terms of an
    exponential sequence (G.P).  If the first term of the linear sequence is 7, find its common difference.

(b) Given that  N= 15,  P= ¼,              find  (i) p(X=2)                          (ii) p(X < 3)