## CLASS: SS2 SUBJECT: Further Mathematics TIME: 2½ HOURS

1. Given that x*y = √xy, evaluate 6* (18*32) (a) 10 (b) 11 (c) 12 (d) 13

2. A binary operation * defined on the set R of real numbers is such M*n = (3m-2n+1)/3, find the identity element (a)1 (b) 2 (c) ½ (d) ¾

3. Given the statements:

P: the subject is difficult

q: I will do my best which of the following is equivalent to “Although the subject is difficult, I will do my best (a) PVq (b) PVq (c) P^(~q)

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4. Given that r = 2i – j, s = 2i + 5j and t = 6i-2j, find the magnitude of (2r+5-1) A. √15 B. 4 C. √14 D. √26.

5. Given that x*y = (x+y)/2, x θy = and (3*b) 48 = 1/3, find b when b >0 A. 8 B. 6 C.5 D.4

6. Given that (AB) ⃗ (■(4@3))and (AC) ⃗ = (■(2@-3)) find |(BC) ⃗ | A. 4√2 B. 6√2 C. 2√10

D. 4√10.

7. Find the angle between (5i+3j) and (3i-5j) (a) 180 (b) 90 (c) 45 (d) 0

8. Function is defined by h: x→2-1/(2x-3), x & 3/2 use this information to answer Question 8 and 9.

8.Find h-1(x) A. (3x-4)/(2x-7) B. (3x-7)/(2x-4) C. (2x-7)/(4x-3) D. (4x-7)/(2x-4)

9. Find h-1(1/2) A. 6 B. 11/6 C. 11/4 D. 5/3

10. A binary operation * is defined on the set of real numbers. R by a*b = a/b +b/a If (√(x+1))^* (√(x-1))=4, find the value of x A. 6 B. 5 C.4 D.3

11. Given that a=i-3j, b=2i+5i, c=3i-j. Calculate |a-b+c| A. √13 B. 3√13 C. 6 D. 9

12. If (ox) ⃗=(■(-7@6)) and (oy) ⃗ = (■(16@-11)) find (xy) ⃗ A. (■(-23@-5)) B. (■(9@-5)) C. (■(9@17)) D. (■(-23@17)).

13. Two functions F and Y are defined on the set of real numbers by f(x) = x2+1 and g(x) = x-2, find fog A. x2+4x-5 B. x2-4x+5 C. x2-1 D. x-1

14. An operation * is defined on the set, R, of real number by p*q = p+q+2pq. If the identity element is zero. Find the value of P for which the operation has no universe A. (-1)/2 B. 0 C. 2/3 D.2

15. Consider the statement:

p: Musa is short

q: Musa is brilliant. Which of the following represent the statement “Musa is short but not brilliant A. Pvq B. P v~q C. P ^~q D. p^q

16. If f(x) = 4/x-1, x≠0, find f-1(7) A. (-3)/7 B. 0 C. 1/2 D. 4

17. What is the angle between a = (3i-4j) and b = (6i-4j) A. 13 B. 87 C. 100 D. 110

18. A binary operation * is defined on the set R, of red numbers by a*b = ab/4.

19. Find the value √2 * √6 A. √3 B. (3√2)/4 C. √3/2 D. √2/2

20. Two functions f and g are defined by f(x) = 3x-1 and g(x) = 2×3, evaluate fg(-2) A. – 49 B. -47 C. -10 D. -9

21. If f (x) = 1/2x, x ≠ 2. Find f-1 ((-1)/2) A. 4 B. 0 C.-2 D.-4’

22. If a = (■(3@2)) and b = (■(-3@5)) find a vector C such that 4a + 3c = b A. (■(3@-1)) B. (■(-5@-1)) C. (■(-5@1)) D. (■(-5@-9))

23. Calculate correct to one decimal place, the angle between 5i+12j and 2i+3j A. 54.8+0 B. 56.30 C. 66.4 D. 76.3

24. Given that p = 4i+3j, find the unit vector in the director of P A. 1/3 (4i+3j) B. 1/3 (3i+4j) C. 1/5 (3i+4j) D. 1/5 (4i+3j).

25. A function F is defined on R, the set real numbers by F(x) = (x+3)/(x-1), x≠2, find F-1 A. (2x+3)/(x-1) B. (x+3)/(x+R) C. (x-1)/(2x+3) D. (x-2)/(x+3).

26. Two statement are represented by P and q as follows

p: He is brilliant

q: He is regular in classwhich of the following statement he is regular in class but dull? A. qv~p B. q^~p C. ~q^~p D. ~qv~p

27. A binary operation * is defined on the set R, of real number by a*b = a2+b+ab. Find the value of x for which 5*x=37 A. 37 B. 2 C. -2 D. 7

28. Given that (AC) ⃗= 2i+5j, find (BC) ⃗ A. -7i-8j B. -3j+2j C. 3i-2j D. 3i+8j

29. If P = (■(2@-2)) and q = (■(3@4)) find |q-1/p p| A. 2√2 B. √3 C. 5 D. √2 9

30. Find the value of constant K for which a=4j-kj and b = 3i + 8j are perpendicular A. 2/3 B. 3/2 C. 2 D. 3.

31. If (OA) ⃗ = 3I+4j and (OB) ⃗ = 5i-6j where O is the origin and m is the mid-point of (AB) ⃗, find (OM) ⃗ A. -2i-10j B. -2i+2j C. 4i-j D. 4i+j

32. Find the direction consine of the vector 4i-3j A. 9/10, 27/10 B. √17/17, – √17/17 C. 4/5, – 3/5 D. (3√10)/10, √10/10

33. The function F and g are defined on the set, R, of real numbers by f(x)=x2-x-6, g(x) = x-1. Find fg(3) A. -8 B. -6 C. -4 D. -3.

34. Find the unit vector in the director of -5i + 12j A. 1/13(-5i-12j) B. 1/13(5i-12j) C. 1/13 1-5i + 12j) D. 1/13 (5i+12j).

35. Find, correct to two decimal places the acute angle between P = (■(12@4)) , q = (■(12@5)) A. 23.52 B. 24.50 C. 29.52 D. 29.82

36. Find the value of K if P= (2k+1)i + 3j, q = -5i + (k-4)j and p.q = 11 A. 3 B. 4 C. -4 D. -3

37. A binary operation is defined on the set of real numbers, R, by a*b = (a+b)/√ab, where a ≠ 0, b ≠ 0 evaluate -3*-1 A. (-4√3)/3 B. (-4√2)/3 C. (-3)/4 D. 3/4

38. Calculate the positive value of X if 5i + 5xj and qi-xj are perpendicular A. -3 B. -9 C. 3 D. 5

39. If a = 2i-3j+4k and b = i+2j-2k. Find a x b A. 2i+8j-7k B. 02i+8j+7k C. -2i-8j-7k D. 8i+2j-7k.

40. Calculate the area of the parallelogram spanned by vectors a=i+j-2k and b=2i-3j+4k A. √29 B. √69 C. √93 D. √125.

41. Given the vector p = i+2j+3k and q = 3i+5j+3k, find the unit vector in the direction of p-q A. 1/√10 (-2i-3j) B. 1/√13 (-2i-3j) C. 1/√13 (2i+3j) D. 1/√13(2i-3j).

42. Find the inverse of ‘a’ under the binary operation * defined by; a*b = a+b+ab, where a and b are real numbers A. a^(-1)/a B. (a-1)/a C. a/(a-1) D. (-a)/(a+1)

PART B

Three vectors a,b and c are mutually perpendicular, |b| = 1 and |c| = √2. If the vectors p = 2a+3b+4c, q=a+2b-3c are perpendicular.

Find |a|

**Determine the magnitude of q**

b. Show that the vectors ∪=2i+3j-6k, V = 6i+2j+3k and w=3i-6j-2k are mutually perpendicular, and find the unit vectors, U, V and W in the direction of U, V and W respectively.

2. Given that p = (■(5@3)), q = (■(-1@2)) r = (■(17@5)) and r = α P+βq, where α and β are scalars express q in terms of r and p

b. A binary operation is defined by a*b = ab. If a*2 = 2-a, find the possible value of a.

3. If f(x) = 2x/(x-4), x≠4 and g(x) = ax2+bx. Given that g(3) = 9 and that gf(2) = 14, find the value of a and b.

b. A binary operation * is defined by a*b = a+b+1 for real numbers a and b. find the inverse of the real number 7 under the operation *, if the identity element is -1

4. Given that a=3i+2j, b=4i+5j and c= xi + uj, when x and μ and scalars, find x and μ are scalars, find x and μ if 2a+3b = c

b. Given that a = I + j – k, b = 2i + 3j + 4k and c = 5i-4j-3k find

i. 2a + 3b – 5c

ii. a + b – c

|2a+3b-5c|

A unit vector in the direction of a + b – c

5. Given that F (x) = 2x – 3

g (x) = ax-5 and fg (2) = 3 find the value of a

b. A binary operation * is defined on the set R of real number by a*b = a+b- 2ab, where a,b ∈ R.

i. Find the identity element of under the operation *

ii. Determine the inverse under 4

6. The adjacent sides of a parallelogram and PQ = 4i + 3j + K, PR = 5i + 2j + 3k. Find the area of the parallogram

b. The vertices of a triangle have position vectors 3i-2j+7k, 2i+4j+k and 5i+3j-2k. find the area of the triangle.

7. If a = 3i-2j +k

B = 2i+ 4j – 3k

Find a x b (b) |a x b|

b. Two binary operation are defined as m*n = mn – n-1 and M (x) = mn+n = Mn-n-1 numbers m, n find the value of 3 (x) (4*5)

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