MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

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STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c yo n l i n e . c o m

1

Directions for questions 1 – 6: State true or false

1. If ‘x’ is a variable and ‘a’ is a constant, then x + a is a constant.

2. Every combination of constants and variables is a variable.

3. The coefficient of xo in the polynomial 3x3 – 5x2 + 4x – 8 is ‘4’.

4. The degree of the polynomial 5x2 – 3x3 + 4x5 – 6x + 7 is 11.

5. The degree of the polynomial ax + by + cxy + 7 is 1 (given that a, b, c are constants and ‘x’ and ‘y’ are

variables)

6. x15 – y25 is a polynomial in degree ‘10’.

7. Write the degrees of the following polynomials.

(a) 3x – 5x2 – 7 (b) 25 – 2x2 – 7x3

(c) 1 – x (d) 5

(e) (x2y2)3 + (x2y)–2 + (xy)4

8. If p(x) = 2x2 + x + 3, find the value of p(0), p(1) and p(–3).

9. If p(y) = (y + 3) (y – 1) y, find the values of p(1) and p(y + 1)

10. Given that p(x) = 5x2 + 7x + k. If p(1) = 25 find the value of ‘k’.

11. If p(x) = (x + 2) (x + 3) + 7, find the values of p(–2) and p(–3). Also find p(–3) – p(–2).

12. Find the zeros of the following linear polynomials.

(i) p(x) = lx + m (ii) p(x) = 4x – 3

(iii) p(x) = 3x4

– 5 (iv) p(x) = 8x – 5

13. Check whether the values indicated against each polynomial are its zeros or not

(i) x2 – 1; x = 1, – 1 (ii) 2x + π; x = 2

π

(iii) 7x + 4; x = 4

7− (iv) 4x – 1; x =

4

1

14. Find the zeros of the following polynomials.

(a) p(x) = ax + b (b) p(x) = mx – n

(c) p(x) = 4x + 7 (d) p(y) = y2 – 16

15. Find the value of the polynomial 3x2 + 2x + 8 at x = 0, x = – 1 and x = 2.

16. If zero of the polynomial p(x) = x + k is – 2, find ‘k’.

ASSIGNMENT ON POLYNOMIALS

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

_________________________________________________________________________________

STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c y o n l i n e . c o m

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17. If zero of the polynomial p(y) = 5y – k is 5, find the value of k .

18. Using remainder theorem, find the remainder when 2x2 + x + 3 is divided by (x – 1).

19. Divide 4x3 – 3x2 + 2x + 5 by (x – 2).

20. When p(x) = 5x2 – 7x + 15 is divided by (x + 5), what is the remainder?

21. Find the remainder when 3x2 + 4x – 5 is divided by (3x – 6)

22. The polynomial p(x) = 2x3 + 6x2 + bx + 9 leaves a remainder 15 when divided by (x + 1), find the value of

‘b’.

23. Use the remainder theorem to find the remainder when x3 – 3x2 + 3x – 1 is divided by

(i) x – 1 (ii) x – 2

1 (iii) 2x + 1

24. Find the remainder when x3 + kx2 + 9x + k is divided by (x + k).

25. If f(x) = 5x3 – 2x2 – 2x – 2, find the values of

(i) f(0) (ii) f(–1) (iii) f ⎟⎠⎞

⎜⎝⎛

2

1 (iv) f(1)

26. Show that (2x – 3) is a factor of 2x3 – 9x2 + 11x – 3.

27. What real number should be added to x4 – x2 – 12 to make it a multiple of (x + 2)?

28. Find the value of ‘k’ for which (x – 1) is a factor of

(i) p(x) = x2 + x + k (ii) p(x) = kx2 – 3x + k (iii) p(x) = 2x2 + kx + 2

29. Find the value of ‘m’ and ‘n’ for which (x – 1) and (x + 2) both are factors of the polynomial

2x3 + mx2 + nx – 10

30. If (x – 3) is a factor of p(x) = x3 – kx2 + (k + 1) x – 12, find the value of ‘k’.

31. Find the value of ‘a’, if (x + 1) is a factor of ax3 – 9x2 + x + 6a

32. Factorize the following polynomials.

(i) x2 – 16x + 63 (ii) 3 – 2x2 + 5x (iii) 15k2 – 4k – 3

(iv) 2x2 + 3 3 x + 3 (v) 15x2 + 3 3 x + 3 (vi) 2(x + y)2 + 9(x + y) (x – y) – 5 (x

– y)2 (vii) 5 – 20x2 (viii) (x + y)3 – x – y (ix) x(x – 1) – y(y – 1)

33. Factorize the following

(i) x10 – y4 (ii) x7 – x3

(iii) a – b – a2 + b (iv) x2 – (y + z)2 (v) 4

y 2

– 4

(vi) a2 + b2 – c2 – d2 + 2ab – 2cd (vii) x2 + y2 + 2xy – 100

34. Factorize the following using factor theorem

(i) x3 – 3x2 – 9x – 5 (ii) 2x3 + x2 – 2x – 1 (iii) 6a2 + a – 1

(iv) 4x3 + 8x2 + x – 3 (v) 2x2 + x – 15

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

_________________________________________________________________________________

STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c yo n l i n e . c o m

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35. Expand

(i) (4x + 3y) (4x + 2y) (ii) (m – 9) (m + 10) (iii) (2k – 5) (2k + 7)

(iv) (x + 2y + 3z)2 (v) 2

z

1

y

1

x

1⎟⎟⎠

⎞⎜⎜⎝

⎛++ (vi) (x – 3)3

(vii) 3

x

3

3

x⎟⎠⎞

⎜⎝⎛ −

36. Evaluate using suitable identities

(i) 993 (ii) (102)3 (iii) (201)3

(iv) (97)3

37. Factorize

(i) x3 – y3 – 1 – 3xy (ii) 8x3 – y3 + 125z3 + 30xyz

(iii) (a + b – c)3 + (a – b + c)3 – 8a3 (iv) 125x3 – 8 + 90xy + 27y3

38. Find the value of

(i) 303 – 203 – 103 (ii) 103 + 203 – 303

(iii) 53 + 33 – 83 (iv) 503 + 303 – 803

39. If a1/3 + b1/3 + c1/3 = 0 show that (a + b + c)3 = 27 abc

40. If a + b = – 5 find the value of a3 + b3 + 125.

41. If a – b = c, show that a3 – b3 – c3 – 3abc = 0

42. If the polynomial p(x) leaves a remainder ‘5’ each when divided by (x – 1) and (x – 2), find p(x).

43. The volume of a cuboid is x4 + 7x2 – 8. Write down the possible expressions for its dimensions.

44. If a + b + c = 3m show that (m – a)3 + (m – b)2 + (m – c)3 – 3 (m – a) (m – b) (m – c) = 0

45. If 2x3 + ax2 – bx – 15 has (2x + 3) as a factor and leaves a remainder – 5, when divided by (x – 1), find the

value of a2 + b2

46. If (x – 1) and (x + 2) both are factors of 2x3 + ax2 + bx – 10, then find the value of (a + b)2

47. State remainder theorem and find the remainder when x3 + 4x2 + 7x – 15 is divided by (5 – 3x)

48. If the zero of the polynomial ax + b is 1, find the value of (a + b + 1)2

49. If the zero of the polynomial ax + bx + a + b is – 3 find the value of (a + b)2.

MATHEMATICS ASSIGNMENT – IX STEPS … A TCY Program

_________________________________________________________________________________

STEPS____________________________________________________________ G e t f r e e n o t e s f o r C l a s s X a n d I X o n

w w w . t c y o n l i n e . c o m

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POLYNOMIALS

ANSWER KEY

1. False 2. False 3. False 4. False 5. False 6. False

7. (a) (2), (b) (3), (c) (1), (d) (0), (e) (12) 8. (3, 6, 18) 9. (0, (y + 4) y(y + 1)

10. –12 11. (7, 7, 0) 12. (i) ⎟⎠⎞

⎜⎝⎛ −l

m (ii) ⎟

⎠⎞

⎜⎝⎛

4

3 (iii) ⎟

⎠⎞

⎜⎝⎛

4

15 (iv) ⎟

⎠⎞

⎜⎝⎛

8

5

13. (i) yes (ii) No (iii) No (iv) Yes 14. (a) ⎟⎠⎞

⎜⎝⎛ −

a

b (b) ⎟

⎠⎞

⎜⎝⎛

m

n (c) ⎟

⎠⎞

⎜⎝⎛ −

4

7 (d) (± 4)

15. (8, 9, 24) 16. 2 17. 5 18. 6 20. 175 21. 15

22. –2 23. (i) (0), (ii) ⎟⎠⎞

⎜⎝⎛ −

8

1 (iii) ⎟

⎠⎞

⎜⎝⎛ −

25

24. –8k

25. (i) –2 (ii) –7 (iii) ⎟⎠⎞

⎜⎝⎛ −

817

(iv) –3 27. 0

28. (i) (–2) (ii) ⎟⎠⎞

⎜⎝⎛

23

(iii) (–2 – 2 ) 29. ⎟⎠⎞

⎜⎝⎛ −

==25

n,221

m 30. 3 31. 2

32. (i) (x – 9) (x – 7) (ii) –(2x + 1) (x – 3) (iii) (5k – 3) (3k + 1)

(iv) (2x + 3 ) (x + 3 ) (v) 3(5x2 + 3 x + 3) (vi) 2(3x2 – 4y2 + 7xy)

(vii) 5(1 + 2x) (1 – 2x) (viii) (x + y) (x2 + y2 + 2xy – 1) (ix) (x – y) (x + y – 1)

33. (i) (x5 – y2) (x5 + y2) (ii) x3(x – 1) (x + 1) (x2 + 1) (iii) (a – b) (1 – a – b)

(iv) (x – y – z) (x + y + z) (v) ⎟⎠⎞

⎜⎝⎛ − 2

2y

⎟⎠⎞

⎜⎝⎛ + 2

2y

(vi) (a + b + c + d) (a + b – c – d) (vii) (x + y + 10) (x + y –10)

34. (i) (x + 1)2 (x – 5) (ii) (2x + 1) (x + 1) (x – 1) (iii) (3a – 1) (2a + 1)

(iv) (2x – 1) (2x + 3) (x + 1) (v) (2x – 5) (x + 3)

35. (i) 16x2 + 20xy + 6y2 (ii) m2 + m – 90 (iii) 4k2 + 4k – 35

(iv) x2 + 4y2 + 9z2 + 4xy + 12yz + 6xz (v) zx

2

y

2

xy

2

z

1

y

1

x

12222+++++

(vi) x3 – 9x2 + 27x – 27 (vii) 27

x 3

– 3x

27– x +

x

9

36. (i) 970299 (ii) 1061208 (iii) 8120601 (iv) 912673

37. (i) (x – y – 1) (x2 + y2 – xy + y – x + 1) (ii) (2x – y + 5z) (4x2 + y2 + 25z2 – 2xy – 5yz + 100xz)

(iii) 2a(b2 + c2 – a2 – 2bc) (iv) (5x + 3y – 2) (25x2 + 9y2 + 4 – 15xy + 6y + 10x)

38. (i) 18000 (ii) –18000 (iii) –360 (iv) –360000

40. 0 42. (x2 – 3x +7) 43. (x–1, x + 1, x2 + 8) 45. 82

46. 81 47. 27335

48. 1 49. 0