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VERY SHORT ANSWER TYPE QUESTIONS
1. Look at the graph in fig given below. Each is the graph of y = p (x), where p(x) is a polynomial. For each of the graph, find the numbre of zeros of p(x).
2. Consider the cubic polynomial f(x) = x^{3} – 4x. Find from the fig, the number of zeros of the above stated polynomials.
3. Let f(x) = x^{3}
The graph of the polynomial is shown in fig.
(i) Find the number of zeros of polynomial f(x).
(ii) Determine the coordinates of the points, at which the graph intersects the xaxis.
SHORT ANSWER TYPE QUESTIONS
1. Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and their coefficients.
(i) 6x^{2} – x – 1
(ii) 25x (x + 1) + 4
(iii) 4x_{2} + 4x + 1
(iv) 48y^{2} – 13y – 1
(v) 63 – 2x – x^{2}
(vi) 2x^{2} – 5x
(vii)49x^{2} – 81
(viii) 4x^{2} – 4x – 3
2. Find a quadratic polynomial each with the given numbers as the zeros of the polynomials.
(i) 3 +√7,3 – √7
(ii) √3,2√3
(iii) 3/7,2/3
(iv) √3, 3√3
(v) 2 + 3 √2, 2 – 3 √2
(vi) 8/3 , 5/2
3. Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
(i) 4√3,9
(ii) 2√31,3√3
(iii) 0,– 1/4
(iv)
(v) 5/6 ,25/9
(vi)
(vii)
(viii) – 6/5 , 9/25
(ix) √2, –12
4. If α and β are the zeros of the polynomial f(x) = 5x2 + 4x – 9 then evaluate the following :
(i) α – β
(ii) α^{2} + β^{2}
(iii) α^{2} – β^{2}
(iv) α^{3} + β^{3}
(v) α^{3} – β^{3}
(vi) α^{4} – β^{4}
5. If one of the zeros of the quadratic polynomial 2x^{2} + px + 4 is 2, find the other zero. Also find the value of p.
6. If one zero of the polynomial (a^{2} + 9)x2 + 13x + 6a is the reciprocal of the other, find the value of a.
7. If the product of zeros of the polynomial ax^{2} – 6x – 6 is 4, find the value of a.
8. Find the zeros of the quadratic polynomial 5x^{2} – 4 – 8x and verify the relationship between the zeros and
the coefficients of the polynomial.
9. Determine if 3 is a zero of p(x) =
10. If α and β be two zeros of the quadratic polynomial ax2 + bx + c, then evaluate :
(i) a^{2} + b^{2}
(ii) α^{3} + β^{3}
(iii)
(iv)
11. Find the value of k :
(i) I f α and β are the zeros of the polynomial x^{2} – 5x + k where α – β = 1.
(ii) If α and β are the zeros of the polynomial x^{2} – 8x + k such that α^{2} + β^{2} = 40.
(iii) If α and β are the zeros of the polynomial x^{2} – 6x + k such that 3α + 2β = 20.
12. If 2 and 3 are zeros of polynomial 3x^{2} – 2kx + 2m, find the values of k and m.
13. If one zero of polynomial 3x^{2} = 8x + 2k + 1 is seven times the other, then find the zeros and the value of k.
14. If α and β are the zeros of the polynomial 2x^{2} – 4x + 5. Form the polynomial where zeros are :
(i)
(ii)
(iii)
15. If α and β are the zeros of the quadratic polynomial x^{2} – 3x + 2, find a quadratic polynomial whose zeros are :
(i)
(ii)
16. If the sum of the squares of zeros of the polynomial 5x^{2} + 3x + k is – 11/25 , find the value of k.
17. If one zero of the quadratic polynomial 2x^{2} – (3k + 1) x – 9 is negative of the other, find the value of k.
18. If α and β are the two zeros of the quadratic polynomial x^{2} – 2x + 5, find a quadratic polynomial whose zeros are
19. If α and β are the zeros of the quadratic polynomial f(x) = x2 – px + q, prove that
20. Apply the division algorithm to find the quotient q(x) and remainder r(x) on dividing p(x) by g(x) as given below :
(i) p(x) = 3x^{3} + 2x^{2} + x + 1 ; g(x) = x^{3} + 3x + 2
(ii) p(x) = x^{6} + x^{4} – x^{2} – 1 ; g(x) = x^{3} – x^{2} + x – 1
(iii) p(x) = 2x^{5} + 3x^{4} + 4x^{3} + 4x2 + 3x + 2 ; g(x) = x^{3} + x^{2} + x + 1
(iv) p(x) = x^{3} – 3x^{2} – x + 3 ; g(x) = x^{2} – 4x + 3
21. Find the quotient q(x) and remainder r(x) of the following when f(x) is divided by g(x). Verify the division algorithm.
(i) f(x) = x^{6} + 5x^{3} + 7x + 3 ; g(x) = x^{2} + 2
(ii) f(x) = x^{4} + 2x^{2} + 1 ; g(x) = x^{3} + 1
(iii) f(x) = 4x^{4} – 7x^{2} + 18x – 1 ; g(x) = 2x + 1
(iv) f(x) = 5x^{3} – 70x^{2 }+ 153x – 342 ; g(x) = x^{2} – 10x + 6
22. Check whether g(y) is a factor of f(y) by applying the division algorithm :
(i) f(y) = 2y^{4} + 3y^{3} – 2y^{2} – 9y – 12, g(y) = y^{2} – 3
(ii) f(y) = 3y^{4} + 5y^{3} – 7y^{2} + 2y + 2, g(y) = y^{2} + 3y + 1
(iii) f(y) = y^{5} – 4y^{3} + y^{2} + 3y + 1, g(y) = y^{3} – 3y + 1
23. (a) If 1 is the zero of f(x) = k^{2}x^{2} – 3kx + 3k – 1 then find the value(s) of k.
(b) If 1 and – 2 are the zeros of f(x) = x^{3} + 10x^{2} + ax + b, then find the values of a and b.
(c) Find p and q such that 3 and – 1 are the zeros of f(x) = x4 + px^{3} + qx^{2} + 12x – 9.
(d) If 3 is the zero of f(x) = x4 – x^{3} – 8x^{2} + kx + 12, then find the value of k.
Also show that – 2 is the zero of x^{3} – 2x + 4
24. (a) Find all the zeros of 3x^{3} + 16x^{2} + 23x + 6 if two of its zeros are – 3 and – 2.
(b) Determine all the zeros of 4x^{3} + 12x^{2} – x – 3 if two of its zeros are –1/2 and 1/2.
(c) Determine all the zeros of x^{3} + 5x^{2} – 2x – 10 if two of its zeros are √2 and √2
(d) Determine all the zeros of 4x^{3} + 12x^{2} – x – 3 if one of its zeros is 5/2
(e) Determine all the zeros of 4x^{3} + 5x^{2} – 180x – 225 if one of its zeros is –5/4 .
25. (a) Find all the zeros of 3x^{4} – 10x^{3} + 5x^{2} + 10x – 8 if three of its zeros are 1, 2 and – 1.
(b) Obtain all the zeros of 2x^{4} + 5x^{3} – 8x^{2} – 17x – 6 if three of its zeros are –1, –3, 2.
(c) Determine all the zeros of x^{4} – x^{3} – 8x^{2} + 2x + 12 if two of its zeros are √2 and √2.
26. (a) Obtain all other zeros of the polynomial 2x^{3} – 4x – x^{2} + 2 if two of its zeros are √2 and – √2 .
(b) Find all the zeros of 2x^{4} – 9x^{3} + 5x^{2} + 3x – 1, if two of its zeros are 2+√3 and 2√3.
(c) Find all the zeros of the polynomial x^{4} + x^{3} – 34x^{2} – 4x + 120, if two of its zeros are √2 and – √2
(d) Find all the zeros of the polynomial 2x^{4} + 7x^{3} – 19x^{2} – 14x + 30, if two of its zeros are √2 and – √2.
27. (a) On dividing f(x) = 3x^{3} + x^{2} + 2x + 5 by a polynomial g(x) = x^{2} + 2x + 1, the remainder r(x) = 9x + 10.
Find the quotient polynomial q(x).
(b) On dividing f(x) by a polynomial x – 1 – x2, the quotient q(x) and remainder r(x) are (x – 2) and 3 respectively. Find f(x).
(c) On dividing x^{5} – 4x^{3} + x^{2} + 3x + 1 by polynomial g(x), the quotient and remainder are (x^{2} – 1) and 2 respectively. Find g(x).
(d) On dividing f(x) = 2x^{5} + 3x^{4} + 4x^{3} + 4x^{2} + 3x + 2 by a polynomial g(x), where g(x) = x^{3} + x^{2} + x + 1, the quotient obtained as 2x^{2} + x + 1. Find the remainder r(x).
PREVIOUS YEARS BOARD (CBSE) QUESTIONS
QUESTIONS CARRYING 1 MARK
1. Write the zeros of the polynomial x^{2} + 2x + 1.
[Delhi2008]
2. Write the zeros of the polynomial, x^{2} – x – 6.
[Delhi2008]
3. Write a quadratic polynomial, the sum and product of whose zeros are 3 and – 2 respectively.
[Delhi2008]
4. Write the number of zeros of the polynomial y = f(x) whose graph is given in figure.
[AI2008]
5. If (x + a) is a factor of 2x2 + 2ax + 5x + 10, find a [Foreign2008]
6. For what value of k, (–4) is a zero of the polynomial x^{2} – x – (2k + 2)?
[Delhi2009]
7. For what value of p, (–4) is a zero of the polynomial x_{2} – 2x – (7p + 3)?
[Delhi2009]
8. If 1 is a zero of the polynomial p(x) = ax^{2} – 3(a – 1) x – 1, then find the value of a.
[AI2009]
9. Write the polynomial, the product and sum of whose zeros – 29 and – 23 respectively
[Foreign2009]
10. Write the polynomial, the product and sum of whose zeros are –153 and – 35 respectively.
[Foreign2009]
ANSWER KEY
VERY SHORT ANSWER TYPE QUESTIONS
1. (i) One zero, (ii) Two zero, (iii) Three zeros, (iv) One zero, (v) One zero, (vi) Four zeros
2. Three zeros 3. (i) One zero, (ii) (0, 0)
SHORT ANSWER TYPE QUESTIONS
(v) 7, –9, (vi) 0, 5/2
2. (i) x^{2} – 6x + 2 , (ii) x^{2} – 12, (iii) 21x^{2} + 23x + 6, (iv) x^{2} – 4√3x + 9, (v) x^{2} – 4x – 14, (vi) 6x^{2} – 31x + 40
3. (i) x^{2} – 4 √3x + 9, (ii) x^{2} – 2√3 – 1x + 3 – √3, (iii) 4x^{2} – 1, (iv) 3x^{2} + 10√3x + 21, (v) 18x^{2} – 15x + 50,
(vi) 3x^{2} + 2 √5x – 5, (vii) 4x^{2} + 4√3x + 1, (viii) 25x^{2} + 30x + 9, (ix) x^{2} – √2x –12
4. (i) 14/5 (ii) 106/25 (iii) 56/25 (iv) 604/125 (v) 854/125 (vi) 5936/125
5. p = – 6, other zero = 1
6. a = 3
7. a = 3/2
8. 2 and 2/5
9. yes
10.
11. (i) 6 (ii) 12 (iii) – 16
12. k = 15/2 , m = 9
13.
14. (i) 1/5 (5x^{2} – 4x + 2) (ii) 1/25 (25x^{2} + 4x + 4) (iii) 1/5 (5x^{2} – 8x + 8)
15. (i) 20x^{2} – 9x + 1 (ii) 3x^{2} – x
16. 2
17. – 1/3
18. 5x^{2} – 12x + 4
20. (i) q(x) = 3, r = 2x^{2} – 8x – 5, (ii) q(x) = x^{3} + x2 + x + 1, r(x) = 0, (iii) q(x) = 2x^{2} + x + 1, (x) = x + 1, (iv) q(x) = x + 1, r(x) = 0
21. (i) q(x) = x^{4} – 2x^{2} + 5x + 4, r(x) = – (3x + 5), (ii) q(x) = x, r(x) = 2x^{2} – x + 1, (iii) q(x) = 2x^{3} – x^{2} – 3x + 11/2 , r(x) = –13/2 , (iv) q(x) = 5x – 20, r(x) = – 127x – 22
22. (i) g(y) is a factor of f(y), (ii) g(x) is a factor of f(x), (iii) g(t) is not a factor of f(t)
23. (a) k = ± 1, (b) a = 7, b = – 18, (c) p = – 8, q = 12, (d) k = 2
24.
25. (c) √2,√2,3,2
26. (a) 1/2
27. (a) q(x) = 3x – 5, (b) f(x) = – x^{3} + 3x^{2} – 3x + 5, (c) g(x) = x^{3} – 3x + 1, (d) r(x) = x + 1,
PREVIOUS YEARS BOARD (CBSE) QUESTIONS
Answer Key
1. x = – 1
2. 3, – 2
3. x^{2} – 3x – 2
4. 3
5. 2
6. 9
7. 3
8. a = 1
9. 2x^{2} + 3x – 9
10. 5x^{2} + 3x – 13
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