Assignments in Mathematics Class IX (Term I) 2. POLYNOMIALS IMPOrTANT TerMS, DefINITIONS AND reSuLTS l
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polynomials. In general, every real number is a constant polynomial. Clearly, the degree of a nonzero constant polynomial is zero.
An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial. For example, x2 + 5x – 6, x3 – 7x2 + 11, x5 – 3x + 2, x2 + 5 , x4 + 5x3 – 2x2 + 7x – 3, etc. are polynomials. In the polynomial 5x3 – 4x2 + 6x – 3, we say that the coefficients of x3, x2 and x are 5, – 4 and 6 respectively, and we also say that – 3 is the constant term in it.
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A polynomial consisting of one term namely zero only, is called a zero polynomial. The degree of zero polynomial is not defined.
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Let p(x) be a polynomial. If p(α) = 0, then we say that α is a zero of the polynomial p(x). Finding the zeroes of a polynomial p(x) means solving the equation p(x) = 0
In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial. For example, 2x + 3 is a polynomial 3 in x of degree 1, 4x2 – x – 5 is a polynomial in 2 x of degree 2, and 3x4 – 5x2 + 1 is a polynomial in x of degree 4.
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The constant polynomial has no zero.
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Every real number is a zero of the zero polynomial.
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A linear polynomial has one and only one zero.
A polynomial of degree 1 is called a linear polynomial. For example, 3x + 5 is a linear polynomial in x.
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If a polynomial p(x) is divided by d(x) = x – a, then the remainder is given by p(a).
A polynomial of degree 2 is called a quadratic 1 polynomial. For example, x2 + 5x – is a quadratic 2 polynomial in x.
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[degree of p(x) > degree of d(x)]. factor Theorem : Let f(x) be a polynomial of degree n > 1 and let a be any real number. (i) If f(a) = 0, then (x – a) is a factor of f(x). (ii) If (x – a) is a factor of f(x), then f(a) = 0.
A polynomial of degree 3 is called a cubic polynomial. For example, 4x3 – 3x2 + 7x + 1 is a cubic polynomial in x.
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A polynomial of degree 4 is called a biquadratic polynomial. For example, x 4 – 3x 3 + 2x 2 + 5x – 3 is a biquadratic polynomial in x.
Following results are known as identities as they are true for all values of the variables a, b and c. (i) (a + b)2 = a2 + 2ab + b2 (ii) (a – b)2 = a2 – 2ab + b2
A polynomial having one term is called a monomial. Thus, 5x, 7x2, 11x3, 3xy and 2xyz are some examples of monomials in one, two and three variables.
(iii) (a + b) (a – b) = a2 – b2 (iv) (a + b + c)2 = a2 + b2 + c2 + 2ab
A polynomial having two terms is called a binomial. Thus, x + 1, 2x3 + 5, x2 –1, x6 + 1, x + y, x2 + y2 are some examples of binomials in one and two variables.
+ 2bc + 2ca (v) (a + b)3 = a3 + b3 + 3ab (a + b) (vi) (a – b)3 = a3 – b3 – 3ab (a – b)
A polynomial having three terms is called a trinomial. Thus, x2 – 3x + 1, x3 – 7x2 + 11, x + y + z are some examples of trinomials.
(vii) a3 + b3 = (a + b) (a2 – ab + b2) (viii) a3 – b3 = (a – b) (a2 + ab + b2) (ix) a 3 + b 3 + c 3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
A polynomial containing one term only, consisting of a constant is called a constant polynomial. 7 , etc. are all constant For example, 3, – 5, 8
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TIME -1.5 Hr
TEST
Summative aSSeSSment
Multiple ChoiCe Questions
[1 Mark]
a. important Questions 1. (2x + 5) (2x + 7) is equal to : 4x2
(a) (x – 1) (x – 2) (x – 3)
2x2
(a) + 12x + 35 (c) 4x2 + 24x + 35
(b) + 12x + 35 (d) 4x2 + 24x – 35
2. On factorising x2 + 8x + 15, we get : (a) (x + 3) (x – 5) (c) (x + 3) (x + 5)
(b) (x – 3) (x + 5) (d) (x – 3) (x – 5)
3. On dividing x2 – 2x – 15 by (x – 5), the quotient is (x + 3) and remainder is 0. Which of the following statements is true ?
(c) (x + 1) (x – 1) (x – 2) (d) (x + 1) (x – 1) (x + 2) 12. Which of the following is a monomial of degree 50 ? (a) 50x
(b) x50 + 50
(c)
(d)
50x
(b) x2 – 2x – 15 is a factor of (x – 5)
(a) monomial of degree 8
(c) (x + 3) is a factor of (x – 5)
(b) monomial of degree 40 (c) binomial of degree 40
(d) (x + 3) is a multiple of (x – 5) 4. The value of the polynomial 3x + x = 0 is : (a) 2 x140
+ with :
(b) 3 x139
+
x138
(c) – 4 + .........
x2
2x2
– 4 at
(d) 4
+ x + 1 is a polynomial
(a) infinite terms
(b) 140 terms
(c) 141 terms
(d) 139 terms
6. The coefficient of x in (x + 5) (x – 7) is : (a) – 12
(b) 2
(c) – 2
(d) 12
7. The remainder when x3 – px2 + 6x – p is divided by x – p is : (a) p
(b) 5p
(c) – 5p
(d) 5p2
8. If (y – p) is a factor of y6 – py5 + y4 – py3 + 3y – p – 2, then the value of p is : (a) 1
(b) 2
(c) 3
(d) – 1
9. On factorising a 3 + 3 3b3 , we get : (a) (b) (c) (d)
(a + 3b)(a
2
(a + 3b)(a2 − (a + 3b)(a2 − ( 3a + b)(a2 − (b) 1
14. If (x – 2) is a factor of the polynomial x4 – 2x3 + ax – 1, then the value of a is : (a) 1
(b) 0
15. If x −
1 x
(a) 11
(c) 2
(d) −
2
= 3 , then x + 2 is : x (b) 75 (c) 10 x0
16. The coefficient of (a) 1
1
1
(b) 5
17. On factorising
x2
in
5x2
1 2
(d) 5
– 7x – 3 is :
(c) 0
(d) – 3
– 3x – 4, we get :
(a) (x – 4) (x + 1) (c) (x + 4) (x – 1)
(b) (x – 4) (x – 1) (d) (x + 4) (x + 1)
18. If p(x) = x + 3, then p(x) + p(–x) is equal to : (a) 3
(b) 2x
(c) 0
(d) 6
2
19. If + kx + 6 = (x + 2) (x + 3) for all x, then the value of k is :
2
20. Which one of the following is a polynomial ?
(a) 1
(a)
2
2
(c) 840
(b) – 1
x2
2 − 2 2 x
(c) x 2 +
10. The value of 73 + 83 – 153 is : (a) 0
(d) binomial of degree 3
x2
) 3ab + 3b ) 3ab − 3b ) 3ab + 3b )
+ 3ab + 3b
50 x 50
13. 8x40 + 3 is a :
(a) x2 – 2x – 15 is a multiple of (x – 5)
5.
(b) (x + 1) (x – 2) (x – 3)
(d) – 2520
11. On factorising x3 – 2x2 – x + 2, we get :
3 3x 2
x
(c) 5
(b)
(d)
(d) 3
2x − 1 x −1 x +1
21. Degree of the polynomial 4x4 + 0x3 + 0x5 + 5x + 7 is : (a) 4 (b) 5 (c) 3 (d) 7
( )
22. If p(x) = x2 – 2 2x + 1, then p 2 2 is equal to : (a) 0 (b) 1 (d) 8 2 + 1
(c) 4 2
23. If x – 1 is a factor of mx 2 − 2 x + 1 , then the value of m is : (a)
2 +1
(b)
2
24. If x – 1 is a factor of value of m is : (a) 0
(b) 1
(c) 1
2x3
+
x2
(d)
2 −1
– 4x + m, then the
(c) 2
(d) –1
25. On factorising x2 + y2 + 2 (xy + yz + zx), we get : (a) (x + y) (x + y + z) (b) (x + y + z)2 (c) (x + y) (x + y + 2z) (d) (x + y) (x + y + z)2 26. The value of a for which (x + a) is a factor of x3 + ax2 – 3x + 16 + a is : (a) – 4
(b) 4
(c) – 2
(d) 2
27. When p(x) is divided by ax – b, then the remainder is :
−b a
(b) p
(a) p(a + b)
b (c) p a 28. (x +
y)3
– (x –
y)3
a
is equal to : (b) 2(y3 + 3x2y)
(c) 2(x3 – 3xy2)
(d) 2(y3 – 3x2y)
+
29. If p(x) = q(x) × g(x) + r(x), r(x) ≠ 0, where p(x), q(x), g(x), and r(x) are polynomials, then : (a) degree of r(x) = degree of g(x) (b) degree of r(x) > degree of g(x) (c) degree of r(x) < degree of g(x) (d) degree of r(x) = 0 30. On factorising x4 + y4 + x2y2, we get :
(a) 2x
(b) 2z
(c) x + z
(d) 2x + 2z
a (a) x + x − b a (c) x − x − b
b + x + 1, we get : b a a b (b) x + x + a b
a
b b
(d) (x + ab) (x – ab)
a
2 is : 35. One of the factors of 2 − 2 2 x y (a) 1 +
1
(b) 1 −
xy
2 1 +
xy
1
(a) 10 1
(b) 38 1
(c) – 38
(d) 76
1
37. If x 2 + y 2 − z 2 = 0, then the value of (x + y – z)2 is : (a) 2xy
(b) 2yz x3
38. The value of is :
+
(c) 4xz y3
(d) 4xy
+ 9xy – 27, if x = 3 – y
(a) 1
(d) cannot be determined
(d) 3(x – y) (y – z) (z – x)
1
(d) all the above xy 36. If ab = 5 and a – b = 2, then the value of a3 – b3 is equal to : (c)
(c) (1 + x2 + y2) (1 – x2 – y2)
(c) (x – y) (y – z) (z – x)
a
34. On factorising x 2 +
(c) 0
(b) 3xyz
1
m
33. The coefficient of y in (x + y + z)2 is :
(b) (x2 + y2 + xy) (x2 + y2 – xy)
(a) xyz
( x − m)
m
(b) – 1
31. The value of (x – y)3 + (y – z)3 + (z – x)3 is :
1
(d) ( x − m) x −
m
(a) (x2 + y2 + xy)2
(d) (x2 – y2 + xy) (x2 – y2 – xy)
x + 1, we get :
m
(b) x +
1
(c) ( x + m) x +
1
(a) (x + m) (x – m)
b
3x2y)
(a)
2(x3
(d)
32. On factorising x 2 + m +
39. The coefficient of x2 in (3x2 + 2x – 4) (x2 – 3x – 2) is : (a) 2 40. The value of is :
(b) – 16
(x
(c) 16
( x − y )3 + ( y − z )3 + ( z − x)3 2
−
y2
) + (y 3
(a) (x – y) (y – z) (z – x) 1 (b) ( x − y )( y − z )( z − x ) (c)
(d) 8
1
( x + y )( y + z )( z + x )
(d) (x + y) (y + z) (z + x)
2
−
z2
) + (z 3
2
−
x2
)
3
(a) 1
(b) – 1
(c) 0
(d)
1
2 47. One of the zeroes of the polynomial 2x2 + 7x – 4 is :
41. One of the factors of (25x2 – 1) + (1 + 5x)2 is : (a) 5 + x (b) 5 – x 42. The value of (a) 12
2492
–
(c) 5x – 1 2482
(b) 477
(d) 10x
is :
(c) 487
(a) (c)
xy2
+
y2
(b) (2x + 1) (2x + 3) (d) (2x – 1) (2x – 3)
+ 2xy
(b)
x2
+
y2
– xy
45. The coefficient of x in the expansion of (x + 3)3 is :
46. If
x y
(b) 9 y
+
x
(c) 18
(a) – 3
(b) 4
(c) 2
(d) – 2
(b) x3 + x2 + x + 1
(d) 3xy
(a) 1
1
(a) x3 + x2 – x + 1
44. Which of the following is a factor of (x + y)3 – (x3 + y3) ? x2
(c) −
(d) – 2 2 2 48. If x + 1 is a factor of the polynomial 2x2 + kx, then the value of k is : (b)
49. x + 1 is a factor of the polynomial :
(d) 497
43. The factorisation of 4x2 + 8x + 3 is : (a) (x + 1) (x + 3) (c) (2x + 2) (2x + 5)
1
(a) 2
(d) 27
= −1 (x, y ≠ 0), then the value of x3 – y3
(c) x4 + x3 + x2 + 1 (d) x4 + 3x3 + 3x2 + x + 1 1 50. If 49 x 2 − b = 7 x + 7 x − 2 of b is : 1
(a) 0
(b)
(a) 0
(b) abc
(c)
1
, then the value
2
1
(d)
1
4 2 2 3 3 3 51. If a + b + c = 0, then a + b + c is equal to : (c) 3abc
(d) 2abc
is :
B. Questions From CBSE Examination Papers 1. If a polynomial f(x) is divided by x – a, then remainder is : [T-I (2010)] (a) f(0)
(b) f(a)
(c) f(–a)
(d) f(a) – f(0)
(b) 3
(c) –2
(d) 1
3. One of the factors of (x3 – 1) – (x – 1) is : [T-I (2010)] (a) x + 1 (b) x2 – 1
(c) x – 1
(d) x + 4
4. The coefficient of x2 in (2 – 3x2)(x2 – 5) is : [T-I (2010)] (a) –17
(b) –10
(c) –3
5. One of the factors of (x – 1) –
(d) 17 (x2
(a)
– 1 (b) x + 1
– 1) is :
6. The factors of (2a – is :
(c) x – 1 b)3
+ (b –
(a) (2a – b)(b – 2c)(c – a)
2c)3
7. In which of the following (x + 2) is a factor ? [T-I (2010)] (a)
4x3
(c)
4x3
– 13x + 6 + 13x – 25
(b)
x3
(b)
–2x3
+
x2
+x+4
+ x2 – x – 19
8. If P(x) = 7 – 3x + 2x2, then value of P(–2) is : [T-I (2010)] (a) 12 x1/3
(b) 31 y1/3
(c) 21
(d) 22
z1/3
9. If + + = 0, then which one of the following expressions is correct ? [T-I (2010)] (a) x3 + y3 + z3 = 0
[T-I (2010)] x2
(c) 6(2a – b)(b – 2c)(c – a) (d) 2a × b × 2c
2. The coefficient of x in the product of (x – 1)(1 – 2x) is : [T-I (2010)] (a) –3
(b) 3(2a – b)(b – 2c)(c – a)
(d) x + 4 a)3
+ 8(c – [T-I (2010)]
(b) x + y + z = 3x1/3y1/3z1/3 (c) x + y + z = 3xyz (d) x3 + y3 + z3 = 3xyz 10. (x + 2) is a factor of 2x3 + 5x2 – x – k. The value of k is : [T-I (2010)] (a) 6
(b) –24
(c) –6
(d) 24
1 11. The coefficient of x2 in (3x + x3) x + is : x [T-I (2010)] (a) 3 (b) 1 (c) 4 (d) 2 12. What is the remainder when x3 – 2x2 + x + 1 is divided by (x – 1)? [T-I (2010)] (a) 0 (b) –1 (c) 1 (d) 2 x x2 2 13. If p ( x) = 2 + + x − , then p(–1) is : 2 3 [T-I (2010)] (a)
15 6
(b)
17 6
(c)
1 6
(d)
13 6
14. Zero of the polynomial p(x) where p(x) = ax, a ≠ 0 is : [T-I (2010)] 1 (a) 1 (b) a (c) 0 (d) a 15. Which of the following is a polynomial in x ? [T-I (2010)] (a) x +
1 x
(b) x 2 + x
(d) 3 x + 1 (c) x + 2 x 2 + 1 2 16. The remainder when x + 2x + 1 is divided by (x + 1) is : [T-I (2010)] (a) 4 (b) 0 (c) 1 (d) –2 1 1 2 1 17. Product of x − x + x + 2 is : x x x [T-I (2010)] (a) x 4 −
1 x4
(c) x 4 −
1
(b) x 3 + (d) x 2 +
1
−2
x3 1
+2 x x2 18. Which of the following is a binomial in y ? [T-I (2010)] 1 (a) y 2 + 2 (b) y + + 2 y 4
y + 2y
(c)
(d) y y + 1
19. The remainder obtained when the polynomial p(x) is divided by (b – ax) is : [T-I (2010)] −b a (b) p (a) p a b b (c) p a
−a (d) p b
20. Which of the following is a trinomial in x ? [T-I (2010)] (a)
x3
+1
(c) x x + x + 1
(b)
x3
+
x2
(d) x3 + 2x
+x
21. a2 + b2 + c2 – ab – bc – ca equals : [T-I (2010)] (a) (a + b +
c)2
(b) (a – b – c)2
(c) (a – b + c)2 1 [(a − b)2 + (b − c)2 + (c − a)2 ] 2 22. If x51 + 51 is divided by (x + 1), the remainder is : [T-I (2010)] (d)
(a) 0 23.
(b) 1
(c) 49
(d) 50
2 is a polynomial of degree :
1 2 24. Which of the following is a polynomial in one variable ? [T-I (2010)] (a) 2
(b) 0
(a) 3 – x2 + x
(c) 1
(b)
(d)
3x + 4
1 x 25. The value of p for which x + p is a factor of x2 + px + 3 – p is : [T-I (2010)] (c) x3 + y3 + 7
(d) x +
(a) 1
(c) 3
(b) –1
(d) –3
26. The degree of the polynomial p(x) = 3 is : [T-I (2010)] (a) 3 (b) 1 (c) 0 (d) 2 x y 27. If + = −1,( x, y ≠ 0), the value of x3 – y3 is : y x [T-I (2010)] (a) 1 (b) –1 (c) 1/2 (d) 0 28. (1 + 3x)3 is an example of : [T-I (2010)] (a) monomial (b) binomial (c) trinomial (d) none of these 29. Degree of zero polynomial is : [T-I (2010)] (a) 0 (b) 1 (c) any natural number (d) not defined 30. The coefficient of x2 in (3x2 – 5)(4 + 4x2) is : [T-I (2010)] (a) 12 (b) 5 (c) –8 (d) 8 31. One of the factors of (16y2 – 1) + (1 – 4y)2 is : [T-I (2010)] (a) (4 + y) (b) (4 – y) (c) (4y + 1) (d) 8y 32. If x2 + kx + 6 = (x + 2)(x + 3) for all x, the value of k is : [T-I (2010)] (a) 1 (b) –1 (c) 5 (d) 3 33. Zero of the zero polynomial is : [T-I (2010)] (a) 0 (b) 1 (c) any real number (d) not defined 34. If (x – 1) is a factor of p(x) = x2 + x + k, then value of k is : [T-I (2010)] (a) 3
(b) 2
(c) –2
(d) 1
short Answer type Questions
[2 Marks]
a. important Questions 1. Find the remainder when 4x3 – 3x2 + 2x – 4 is divided by x + 2. 2. Write whether the following statements are true or false : In each case justify your answer. (i)
1 5
1
x 2 + 1 is a polynomial 3
(ii)
6 x+x 2 x
is a polynomial, x ≠ 0.
3. Write the degree of each of the following polynomials : (i) x5 – x4 + 2x2 – 1
(ii) 6 – x2
(iv) 5 5 4. Find the zeroes of the polynomial p(x) = x2 – 5x + 6. x3 + 2 x + 1 7 2 5. For the polynomial − x − x5 , 5 2 write : (iii) 2x –
6. Give an example of a polynomial which is : (i) monomial of degree 1 (ii) binomial of degree 20 7. Find the value of a, if x + a is a factor of the polynomial x4 – a2x2 + 3x – 6a. 8. Find the value of the polynomial at the indicated value of variable p ( x ) = 3 x 2 − 4 x + 11, at x = 2. 9. Find p(1), p(–2) for the polynomial p(x) = (x + 2) (x – 2). 10. Show that x + 3 is a factor of 69 + 11x – x2 + x3. 11. If (x + 1) is a factor of ax3 + x2 – 2x + 4a – 9, find the value of a. 12. Verify that 1 is not a zero of the polynomial 4y4 – 3y3 + 2y2 – 5y + 1. 13. Factorise : (i) x2 + 9x + 18
(ii) 2x2 – 7x – 15
14. Expand :
(i) the degree of the polynomial
(i) (4a – b + 2c)2
(ii) the coefficient of x3
(ii) (–x + 2y – 3z)2
15. Factorise : a 3 − 2 2b3
(iii) the coefficient of x6 (iv) the constant term
B. Questions From CBSE Examination Papers 1. Evaluate using suitable identity (999)3. [T-I (2010)] 2. Factorise : 3x2 – x – 4.
[T-I (2010)]
3. Using factor theorem, show that (x + 1) is a factor of x19 + 1. [T-I (2010)] 4. Without actually calculating the cubes, find the value of 303 + 203 – 503. [T-I (2010)] 5. Evaluae (104)3 using suitable identity. [T-I (2010)] 6. F i n d t h e v a l u e o f t h e p o l y n o m i a l p ( z ) = 3 z 2 − 4 z + 17 when z = 3. [T-I (2010)] 7. Check whether the polynomial t + 1 is a factor of 4t3 + 4t2 – t – 1. [T-I (2010)] x 1 8. Factorise : x 2 + − . 4 8 [T-I (2010)] 9. Factorise : 27 p3 −
1 9 1 − p2 + p. 216 2 4 [T-I (2010)]
10. If 2x + 3y = 8 and xy = 4, then find the value of 4x2 + 9y2. [T-I (2010)]
11. If x 2 +
1 x2
= 38, then find the value of
1 x − . x
[T-I (2010)]
12. Check whether the polynomial 3x – 1 is a factor of 9x3 – 3x2 + 3x – 1. [T-I (2010)] 1 1 1 13. Find the product of x − , x + , x 2 + 2 x x x 4 1 and x + 4 . [T-I (2010)] x 14. Using factor theorem, show that (2x + 1) is a factor of 2x3 + 3x2 – 11x – 6. [T-I (2010)] 15. Check whether (x + 1) is a factor of x3 + x + x2 + 1. [T-I (2010)] 16. Find the value of a if (x – 1) is a factor of 2 x 2 + ax + 2 . 17. Factorise : 7 2 x 2 − 10 x − 4 2 .
[T-I (2010)] [T-I (2010)]
18. If a + b + c = 7 and ab + bc + ca = 20, find the [T-I (2010)] value of a2 + b2 + c2.
19. If – 1 is a zero of the polynomial p(x) = ax3 – x2 + x + 4, find the value of a : [T-I (2010)]
20. Check whether the polynomial p(s) = 3s3 + s2 – 20s + 12 is a multiple of 3s – 2. [T-I (2010)] 21. Factorise : 125x3 + 27y3.
[T-I (2010)]
short Answer type Questions
[3 Marks]
a. important Questions 1. Check whether p(x) is a multiple of g(x) or not, where p(x) = x3 – x + 1, g(x) = 2 – 3x. 2. Check whether g(x) is a factor of p(x) or not, where : x 1 p(x) = 8x3 – 6x2 – 4x + 3, g(x) = − 3 4 3. Using factor theorem show that x – y is a factor of x (y2 – z2) + y (z2 – x2) + z (x2 – y2). 4. Find the value of a, if x – a is a factor of x3 – ax2 + 2x + a – 1. 5. Find the value of the polynomial 3x3 – 4x2 + 7x – 5, when x = 3 and also when x = – 3.
6. Find the zeroes of the polynomial p(x) = (x – 2)2 – (x + 2)2. 7. What must be added to x3 – 3x2 – 12x + 19 so that the result is exactly divisible by x2 + x – 6 ? 8. Using suitable identity, evaluate the following : (i) 1033
(ii) 101 × 102
(iii) 9992
9. Factorise : 16x 2 + 4y2 + 9z 2 – 16xy – 12yz + 24xz 10. If x + y + z = 9 and xy + yz + zx = 26, find x2 + y2 + z 2. 11. Find the following product : (2x – y + 3z) (4x2 + y2 + 9z2 + 2xy + 3yz – 6xz).
B. Questions From CBSE Examination Papers 1. Find the value of x3 + y3 – 12xy + 64 when x + y = –4. [T-I (2010)] 2. If x = 2y + 6, then find the value of x3 – 8y3 – 36xy – 216. [T-I (2010)] 3. Factorise : 27(x + y)3 – 8(x – y)3.
[T-I (2010)]
4. Factorise : (x – 2y)3 + (2y – 3z)3 + (3z – x)3. [T-I (2010)] 5. If 2a = 3 + 2b, prove that = 27.
8a3
–
8b3
– 36ab [T-I (2010)]
6. If a – b = 7, a2 + b2 = 85, find a3 – b3.
−1 is a zero of the polynomial p(x) = 27x3 3 – ax2 – x + 3, then find the value of a.
12. If x =
[T-I (2010)] 13. Factorise : 64a3 – 27b3 – 144a2b + 108ab2. [T-I (2010)] c) 2
14. Simplify : (a + b + + (a – b + c) 2 2 + (a + b – c) . [T-I (2010)] 1 1 15. If x + = 9, then find the value of x 3 + 3 . x x
[T-I (2010)] 1 y 7. Expand : (a) + x 3
3
3
1 (b) 4 − . 3x [T-I (2010)]
8. The polynomials kx3 + 3x2 – 8 and 3x3 – 5x + k are divided by x + 2. If the remainder in each case is the same, find the value of k. [T-I (2010)] 9. Find the values of a and b so that the polynomial x3 + 10x2 + ax + b has (x – 1) and (x + 2) as factors. [T-I (2010)] 10. Factorise : 8x3 + y3 + 27z3 – 18xyz. [T-I (2010)] 11. If a2 + b2 + c2 = 90 and a + b + c = 20, then find the value of ab + bc + ca. [T-I (2010)]
[T-I (2010)] 16. Factorise :
4(x2
+
1)2
+
13(x2
+ 1) – 12. [T-I (2010)]
17. Factorise : x 2 +
1
2 + 2 − 2x − . x x
[T-I (2010)]
2
18. Determine whether (3x – 2) is a factor of [T-I (2010)] 3x3 + x2 – 20x + 12 ? 3
3
2 2 19. Simplify : x − y − x + y . 3 3
[T-I (2010)]
20. Factorise : (2x – y – z)3 + (2y – z – x)3 + (2z – x – y)3. [T-I (2010)]
36. Factorise : 2x3 – x2 – 13x – 6.
21. If a + b = 11, a2 + b2 = 61, find a3 + b3. [T-I (2010)]
37. Factorise : a3(b – c)3 + b3(c – a)3 + c3(a – b)3.
22. a 2 + b 2 + c 2 = 30 and a + b + c = 10, then find the value of ab + bc + ca. [T-I (2010)] 23. Using suitable identity evaluate : (42)3 − (18)3 − (24)3 .
[T-I (2010)] 38. If p = 4 – q, prove that p3 + q3 + 12pq = 64.
[T-I (2010)]
[T-I (2010)]
24. Find the values of p and q, if the polynomial x4 + px3 + 2x2 – 3x + q is divisible by the polynomial x2 – 1. [T-I (2010)]
39. Find the value of k so that 2x – 1 be a factor of 8x4 + 4x3 – 16x2 + 10x + k. [T-I (2010)]
25. Simplify (x + y + z)2 – (x + y – z)2.
40. What are the possible expressions for the dimensions of the cuboids whose volume is given below ?
[T-I (2010)]
Volume = 12ky2 + 8ky – 20k.
26. Factorise 9x2 + y2 + z2 – 6xy + 2yz – 6zx. Hence find its value if x = 1, y = 2 and z = –1. 27. Find the value of = 2.
+
b3
+ 6ab – 8 when a + b [T-I (2010)]
28. If x + y + z = 9, then find the value of (3 – + (3 – y)3 + (3 – z)3 – 3(3 – x)(3 – y)(3 – z).
42. Without finding the cubes, find the value of : 3
x)3
3
3
1 1 7 + − . 4 3 12
[T-I (2010)]
[T-I (2010)] 44. Factorise (x – 3y)3 + (3y – 7z)3 + (7z – x)3.
30. Find the value of x3 + y3 + 9xy – 27, if x + y = 3. [T-I (2010)]
[T-I (2010)]
a)3
31. If a + b + c = 6, then find the value of (2 – + (2 – b)3 + (2 – c)3 – 3(2 – a)(2 – b)(2 – c).
45. Factorise : 2 2 a3 + 8b3 − 27c3 + 18 2 abc. [T-I (2010)]
[T-I (2010)] 46. Factorise : x6 – y6.
32. If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c. [T-I (2010)]
[T-I (2010)]
1 47. If both (x – 2) and x − are factors of 2 2 px + 5x + r, show that p = r. [T-I (2010)]
1 1 3 = 7, then find the value of x + 3 . x x
48. Find the value of a if (x + a) is a factor of x4 – a2x2 + 3x – a. [T-I (2010)]
[T-I (2010)] 34. If x −
[T-I (2010)]
43. Simplify : (a + b + c)2 – (a – b – c)2.
29. If x – 3 is a factor of x2 – kx + 12, then find the value of k. Also, find the other factor for this value of k. [T-I (2010)]
33. If x +
[T-I (2010)]
41. If the polynomial P(x) = x4 – 2x3 + 3x2 – ax + 8 is divided by (x – 2), it leaves a remainder 10. Find the value of a : [T-I (2010)]
[T-I (2010)] a3
[T-I (2010)]
1 1 3 = 3, then find the value of x − 3 . x x
49. Factorise by splitting the middle term : 9(x – 2y)2 – 4(x – 2y) – 13.
[T-I (2010)]
[T-I (2010)]
50. Find the remainder obtained on dividing 1 2 x 4 − 3 x 3 − 5 x 2 + x + 1 by x − . 2 [T-I (2010)]
35. If ax3 + bx2 + x – 6 has (x + 2) as a factor and leaves a remainder 4 when divided by x – 2, find the values of a and b. [T-I (2010)]
8
long Answer type Questions
[4 Marks]
a. important Questions 1. If (x + 2) is a factor of factorise it.
x3
+
13x2
4. Without actual division prove that (x – 2) is a factor of the polynomial 3x3 – 13x2 + 8x + 12. Also, factorise it completely.
+ 32x + 20, then
5. If a, b, c are all non-zero and a + b + c = 0, prove a 2 b2 c2 + + = 3. that bc ac ab 6. Prove that (a + b + c) 3 – a 3 – b 3 – c 3 = 3 (a + b) (b + c) (c + a).
2. If the polynomials ax3 + 4x2 + 3x – 4 and x3 – 4x + a leave the same remainder when divided by x – 3, find the value of a. 3. The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1, leaves the remainder 19. Find the value of a. Also, find the remainder when p(x) is divided by x + 2.
7. If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 – 3abc = – 25.
B. Questions From CBSE Examination Papers 1 ( x + y + z) 2 [( x − y)2 + ( y − z )2 + ( z − x )2 ]. [T-I (2010)]
12. Find the value of (x – a)3 + (x – b)3 + (x – c)3 – 3(x – a)(x – b)(x – c), if a + b + c = 3x. [T-I (2010)]
1. Verify : x 3 + y3 + z 3 − 3 xy =
13. Simplify by factorisation method :
2 2 3 2 2 3 2 2 3 2. Simplify : (a − b ) + (b − c ) + (c − a ) . 3 3 3 ( a − b) + (b − c ) + ( c − a)
9 − 2 3x − x
3 − x2 – + bx + 3 leaves a remainder 14. If p(x) = –19 when divided by (x + 2) and a remainder 17 when divided by (x – 2), prove that a + b = 6.
[T-I (2010)]
x3
3. Prove that : 2x3 + 2y3 + 2z3 – 6xyz = (x + y + z) [(x – y)2 + (y – z)2 + (z – x)2]. Hence evaluate 2(7)3 + 2(9)3 + 2(13)3 – 6(7) (9) (13).
15. The volume of a cube is given by the polynomial p(x) = x3 – 6x2 + 12x – 8. Find the possible expressions for the sides of the cube. Verify the truth of your answer when the length of cube is 3 cm. [T-I (2010)]
4. Using factor theorem show that x2 + 5x + 6 is factor of x4 + 5x3 + 9x2 + 15x + 18. [T-I (2010)] 5. Prove that ( x + y + z ) × [( x − y)2 + ( y − z)2 ] = 2( x 3 + y3 + z 3 − 3 xyz )
16. Using factor theorem, factorise the polynomial : x4 + 3x3 + 2x2 – 3x – 3. [T-I (2010)]
[T-I (2010)] 6. The polynomials p(x) = ax3 + 4x2 + 3x – 4 and q(x) = x3 – 4x + a leave the same remainder when divided by x – 3. Find the remainder when p(x) is divided by (x – 2). [T-I (2010)]
17. Factorise a7 + ab6. x4 + 2x3 – 7x2 – 8x + 12.
8. Simplify by factorisation method : 6 − 2 2x − x2 [T-I (2010)] . 2 − x2 9. Show that (x – 1) is a factor of P(x) = 3x3 – x2 – 3x + 1 and hence factorise P(x). [T-I (2010)]
20. If x and y be two positive real numbers such that 8x3 + 27y3 = 730 and 2x2y + 3xy2 = 15, then evaluate 2x + 3y. [T-I (2010)] 21. Factorise : (x2 – 2x)2 – 2(x2 – 2x) – 3.
10. The polynomials x3 + 2x2 – 5ax – 8 and x3 + ax2 – 12x – 6 when divided by (x – 2) and (x – 3) leave remaindens p and q respectively. If q – p = 10, find the value of a. [T-I (2010)] 11. Prove that (x + = 8y3.
– (x –
–
[T-I (2010)]
19. Without actual division, show that the polynomial 2x4 – 5x3 + 2x2 – x + 2 is exactly divisible by x2 – 3x + 2. [T-I (2010)]
7. If both (x + 2) and (2x + 1) are factors of + 2x + b, prove that a – b = 0. [T-I (2010)]
6y(x2
[T-I (2010)]
18. Using factor theorem, factorise the polynomial.
ax2
y)3
ax2
[T-I (2010)]
T-I (2010)]
y)3
[T-I (2010)]
2
22. If x 2 +
1
x2 1 (i) x − x
y 2)
– [T-I (2010)]
9
[T-I (2010)] = 51, find (ii) x 3 −
1 x3
.
[T-I (2010)]
23. Find the values of m and n so that the polynomial f(x) = x3 – 6x2 + mx – n is exactly divisible by (x – 1) as well as (x – 2). [T-I (2010)] 24. Factorise : x8 – y8.
[T-I (2010)] x4
25. Without actual division prove that + 2x3 – 2x2 2 + 2x – 3 is exactly divisible by x + 2x – 3. [T-I (2010)] 26. Factorise : a12x4 – a4x12.
[T-I (2010)]
27. Without actual division, prove that the polynomial 2x4 – 5x3 + 2x2 – x + 2 is exactly divisible by x2 – 3x + 2. [T-I (2010)] 28. Factorise :
(x2
–
3x)2
–
8(x2
– 3x) – 20. [T-I (2010)]
29. The polynomial p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by (x + 1), leaves the remainder 19. Find the value of a. Also, find the remainder, when p(x) is divided by x + 2. [T-I (2010)] 30. Find the values of a and b so that (x + 1) and (x – 1) are factors of x4 + ax3 – 3x2 + 2x + b. [T-I (2010)] 31. Multiply 9x2 + 25y2 + 15xy + 12x – 20y + 16 by 3x – 5y – 4 using suitable identity. [T-I (2010)] 32. If x2 – 3x + 2 is a factor of x4 – ax2 + b then find a and b. [T-I (2010)] 33. Without actual division show that x4 + 2x3 – 2x2 + 2x – 3 is exactly divisible by x2 + 2x – 3. [T-I (2010)]
27a2 9a + . 3 4b 16b2 64b [T-I (2010)] 35. Find the values of a and b so that (x + 1) and (x – 2) are factors of (x3 + ax2 + 2x + b). [T-I (2010)]
34. Factorise : 27a3 +
1
+
36. W i t h o u t a c t u a l d i v i s i o n , p r o v e t h a t (2 x 4 − 6 x 3 + 3 x 2 + 3 x − 2) is exactly divisible by [T-I (2010)] ( x 2 − 3 x + 2). 37. Simplify : (5a + 3b)3 – (5a – 3b)3. [T-I (2010)] 38. Find the value of a if (x – a) is a factor of x5 – a2x3 + 2x + a + 3, hence factorise x2 – 2ax – 3. [T-I (2010)] 39. The polynomial ax3 + 3x2 – 3 and 2x3 – 5x + a when divided by x – 4 leave the same remainder in each case. Find the value of a. [T-I (2010)] 40. Factorise : 3u3 – 4u2 – 12u + 16. [T-I (2010)] 1 1 41. If x + = 5 , then evaluate x 6 + 6 . x x [T-I (2010)] 42. Without actual division, prove that 2x4 – 8x.3 + 3x2 + 12x – 9 is exactly divisible by x2 – 4x + 3. [T-I (2010)] 43. If f(x) = x4 – 2x3 + 3x2 – ax + b is divided by (x – 1) and (x + 1), it leaves the remainders 5 and 19 respectively. Find a and b. [T-I (2010)]
