1. The document is a math exam with 34 multiple choice questions covering topics like polynomials, algebraic fractions, and factorizing expressions. 2. The questions involve tasks like finding expressions when variables are substituted, factorizing polynomials, identifying factors and remainders when polynomials are divided, and determining relationships between variables in expressions. 3. The questions cover a range of skills including simplifying, factorizing, dividing polynomials, and identifying properties of expressions and relationships between variables.

1. Math/CE MC Ex.4
CE MC Ex.4- Unit 4 Formulae, Polynomials and Algebraic Fractions
Section A
1. If f (x) = x2 + 1, then f (x–1) = [CE 87]
A. x2 B. x2 – 1 2
C. x - 2x D. x2 - 2x + 2.
1
2. If f (n) = n(n − 1) , then f (n + 1) – f (n) = [CE 90]
2
A. f (1) B. f (n) C. 1 D. n
ab + 1
3. If x = , then b = [CE 91]
a−b
ax − 1 ax − 1 1 − ax 1 − ax
A. B. C. D.
a+x a−x a+x a−x
1 1
4. If f (x) = x– , then f (x)– f   = [CE 91]
x  x
A. 0 2  1 1 
B. − C. 2 x −  D. 2 − x 
x  x  x 
1
5. If a = 1 – , then b = [CE 92]
1− b
1 1 1 1
A. 1 – B. 1 − C. 1 + D. − 1 +
1− a 1+ a 1− a 1+ a
1 1
6. + = [CE 92]
a b
a+b 1 2 1
A. B. C. D.
ab ab a+b a+b
2x − 1
7. If y = , then x = [CE 94]
x+2
1 + 3y 1 + 2y 1 + 2y 1 − 2y
A. B. C. D.
2 2+ y 2− y 2+ y
8. If f (x) = x2 + 2x, then f (x–1) = [CE 94]
A. x2 B. x2 – 1 2
C. x + 2x – 1 D. x2 + 4x - 1
1

2. Math/CE MC Ex.4
9. Factorize a2 - 2ab + b2 - a + b. [CE 94]
A. (a - b)(a - b -1) C. (a - b)(a + b - 1)
B. (a - b)(a - b +1) D. (a +b)(a - b +1)
x 1
10. If f ( x) = , then f   f (− x) = [CE 95]
1− x x
1 1− x x
A. − B. –1 C. − D.
2 1+ x 1− x2
11. Factorize 2an+1 - 7an - 30an - 1. [CE 95]
A. an(a + 6)(2a - 5) C. an-1(a + 6)(2a - 5)
B. an(a - 6)(2a + 5) D. an-1(a - 6)(2a + 5)
12. If A = 2πr2 + 2πrh, then h = [CE 96]
A. A – r A A A
B. C. –r D. r –
r 2πr 2πr
13. Which of the following expressions has/have b - c as a factor? [CE 96]
I. ab – ac
II. a(b – c) – b + c
III. a(b – c) – b – c
A. I only C. I and III only
B. I and II only D. I, II and III
14. Find the remainder when x 3 − x 2 + 1 is divided by 2 x + 1 . [CE 96]
A. –11 5 7 9
B. C. D.
8 8 8
a+x c
15. If = (c ≠ d ) , then x = [CE 97]
b+ x d
c a a −b b−a ad − bc
A. − B. C. D.
d b c−d c−d c−d
16. If f (x) = 3x2 + bx + 1 and f (x) = f (–x), then f (–3) = [CE 97]
A. –26 B. 0 C. 3 D. 28
y ( z − 3)
17. If x = , then z = [CE 98]
3z
3 −3 3y − 3y
A. B. C. D.
3x − y 3x − y 3x − y 3x − y
2

3. Math/CE MC Ex.4
18. If ( x + 3) 2 − ( x + 1)( x − 3) ≡ P( x + 1) + Q , find P and Q. [CE 98]
A. P = 2, Q = 4 C. P = 4, Q = 8
B. P = 2, Q = 10 D. P = 8, Q = 4
19. Let f ( x) = 2 x 3 − x 2 − 7 x + 6 . It is known that f (−2) = 0 and f (1) = 0 . [CE 98]
f (x) can be factorized as
A. ( x + 1)( x + 2)(2 x − 3) C. ( x − 1)( x + 2)(2 x + 3)
B. ( x + 1)( x − 2)(2 x + 3) D. ( x − 1)( x + 2)(2 x − 3)
1+ b
20. If a = , then b = [CE 99]
1− b
a −1 a −1 a +1 a −1
A. B. C. D.
2 2a a −1 a +1
21. If (3x − 1)( x − a ) ≡ 3x 2 + bx − 2 , then [CE 99]
A. a = 2, b = –1 C. a = –2, b = 5
B. a = 2, b = –7 D. a = –2, b = –5
22. Let f ( x) = x 3 − 2 x 2 − 5 x + 6 . It is known that f (1) = 0 . [CE 00]
f (x) can be factorized as
A. ( x + 1) 2 ( x + 6) C. ( x − 1)( x − 2)( x + 3)
B. ( x − 1)( x + 1)( x + 6) D. ( x − 1)( x + 2)( x − 3)
23. If 3x 2 + ax + 7 ≡ 3( x − 2) 2 + b , then [CE 00]
A. a = –12, b = –5 C. a = –4, b = 3
B. a = –12, b = 7 D. a = 0, b = –5
24. Let f ( x) = (2 x − 1)( x + 1) + 2 x + 1 . [CE 01]
Find the remainder when f (x) is divided by 2x + 1.
A. –1 1 C. 0 D. 1
B. −
2
25. If ( x + 1) 2 + P( x + 1) ≡ x 2 + Q , then [CE 02]
A. P = –2, Q = –1 C. P = 2, Q = –1
B. P = –2, Q = 1 D. P = 2, Q = 1
3

4. Math/CE MC Ex.4
1
26. If f (x) = 2x2 + kx–1 and f (−2) = f   , then k = [CE 03]
 2
17
A. − . C. 3.
3
31
B. –5. D. .
5
27. If f (x) = x3 + 2x2 + k, where k is a constant. If f (–1) = 0, find the remainder when f (x) is divided
by x–1. [CE 03]
A. –1. B. 0. C. 2. D. 6.
b −1
28. If a = , then b = [CE 03]
b−2
2a − 1 2a − 1 1 1
A. . B. . C. . D. .
a −1 a +1 a −1 a +1
y − 2x
29. If x = , then y = [CE 04]
2y
2x 2x 1 − 2x 2x − 1
A. . B. . C. . D. .
1 − 2x 2x − 1 2x 2x
30. If f (x) = x2 – x + 1, then f (x + 1) – f (x) = [CE 04]
A. 0. B. 2. C. 2x. D. 4x.
31. If a(2x – x2) + b(2x2 – x) ≡ –5x2 + 4x, then a = [CE 04]
A. –1. B. 1. C. –2. D. 2.
32. If a = 1 – 2b, then b = [CE 04]
a −1 a +1 −1− a 1− a
A. . B. . C. . D. .
2 2 2 2
33. If f (x) = 2x2 – 3x + 4, then f(1) – f(–1) = [CE 05]
A. –6. B. –2. C. 2. D. 6.
34. (2x – 3)(x2 + 3x – 2) ≡ [CE 05]
3 2 3 2
A. 2x + 3x + 5x – 6. C. 2x + 3x – 13x – 6.
B. 2x3 + 3x2 + 5x + 6. D. 2x3 + 3x2 – 13x + 6.
4

5. Math/CE MC Ex.4
35. If x2 + 2ax + 8 ≡ (x + a)2 + b, then b = [CE 05]
2
A. 8. C. a – 8.
B. a2 + 8. D. 8 – a2.
Section B
x− y
1−
x+ y
36. = [CE 90]
x+ y
1−
x− y
y−x x−y x D. x + y
A. B. C.
x+y x+y y
37. Let f ( x) = 3 x 3 − 4 x + k . If f ( x) is divisible by x – k, [CE 90]
find the remainder when f ( x) is divisible by x + k.
A. 2k C. 0
B. k D. –k
1 1
+
x3 y3
38. = [CE 91]
1 1
+
x y
1 1 1 2 1
A. 2
+ 2 C. 2
− + 2
x y x xy y
1 1 1 1 1 1
B. 2
+ + 2 D. 2
− + 2
x xy y x xy y
39. If a polynomial f (x) is divisible by x – 1, then f ( x − 1) is divisible by [CE 92]
A. x – 2 C. x – 1
B. x + 2 D. x + 1
40. P(x) is a polynomial. When P(x) is divided by (5 x − 2) , the remainder is R. [CE 94]
If P (x) is divided by (2 − 5 x) , then the remainder is
A. R B. –R 2 2
C. R D.
5 5
5

6. Math/CE MC Ex.4
 y  x
 − 11 −
 
 x  y

41. Simplify . [CE 95]
x y
−
y x
x− y x− y x+ y x+ y
A. B. – C. D. –
x+ y x+ y x− y x− y
2 a b
42. If ≡ + , find a and b. [CE 96]
x −1 x +1 x −1
2
A. a = 2, b = 1 C. a = 1, b = –1
B. a = 1, b = 2 D. a = –1, b = 1
43. m and n are multiples of 3 and 4 respectively. [CE 96]
Which of the following must be true?
I. mn is a multiple of 12.
II. The H.C.F. of m and n is even.
III. The L.C.M. of m and n is even.
A. I only C. I and III only
B. I and II only D. II and III only
44. If 2 x 2 + x + m is divisible by x – 2, then it is also divisible by [CE 97]
A. x + 3 C. 2x + 3
B. 2x – 3 D. 2x + 5
45. It is given that F ( x) = x 3 − 4 x 2 + ax + b . F(x) is divisible by x – 1. [CE 99]
When it is divided by x + 1, the remainder is 12. Find a and b.
A. a = 5, b = 10 C. a = –4, b = 7
B. a = 1, b = 2 D. a = –7, b = 10
46. Let f ( x) = x 3 + 2 x 2 + ax + b . If f (x) is divisible by x + 1 and x – 2, [CE 01]
f (x) can be factorized as
A. ( x − 1)( x + 1)( x − 2) C. ( x − 3)( x + 1)( x − 2)
B. ( x + 1) ( x − 2)
2
D. ( x + 3)( x + 1)( x − 2)
2x
47. 1 − = [CE 02]
1
x−
x
x−3 x2 − 3 x2 + 1 x2 + 1
A. B. 2 C. 2 D. − 2
x −1 x −1 x −1 x −1
6

7. Math/CE MC Ex.4
48. The remainder when x 2 + ax + b is divided by x + 2 is –4. [CE 02]
The remainder when ax 2 + bx + 1 is divided by x – 2 is 9. The value of a is
A. –3 B. –1 C. 1 D. 3
27
49. x 3 − = [CE 03]
x3
 3  9   3  9 
A.  x +  x 2 − 6 + . C.  x −  x 2 + 6 + .
 x  x2   x  x2 
 3  9   3  9 
B.  x +  x 2 − 3 + . D.  x −  x 2 + 3 + .
 x  x2   x  x2 
3 2
−
x y
50. = [CE 04]
4x 9 y
−
y x
1 −1
A. . C. .
2x − 3y 2x − 3y
1 −1
B. . D. .
2x + 3y 2x + 3y
51. If f (x) = x3 – 7x + 6 is divisible by x2 – 3x + k, then k = [CE 04]
A. – 2. C. –3.
B. 2 D. 3.
52. Let k be a positive integer. When x2k + 1 + kx + k is divided by x + 1, the remainder is [CE 05]
A. –1. C. 2k – 1.
B. 1. D. 2k + 1.
Answers:
1. D 12. C 23. A 34. D 45. D
2. D 13. B 24. A 35. D 46. D
3. A 14. B 25. A 36. A 47. D
4. C 15. D 26. C 37. A 48. D
5. A 16. D 27. C 38. D 49. D
6. A 17. D 28. A 39. A 50. D
7. C 18. D 29. A 40. A 51. B
8. B 19. D 30. C 41. A 52. A
9. A 20. D 31. B 42. D
10. D 21. C 32. D 43. C
11. D 22. D 33. A 44. D
7 ~ End of Unit 4 MC ~