Q1- Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
(a) 4x² - 4x + 1 [Ans; 1/2],
(b) x² - 2x - 6
[Ans; 3
, -
],
(c) x² - 8x + 12
[Ans; 6, 2]
(d) 2x² + 5x + 2 [Ans -2, -1/2]
(g) x² - 8 [Ans; ± 2],
(h) 9y² - 6y + 1 [Ans;
1/3, 1/3],
(i) 3x² - 5x - 2 [Ans; 2, -1/3]
Q2- Find sum and product of zeroes of
(a) 2x² + 2x + 3 [Ans; -1, 3/2],
(b) x² - 7x - 7 [Ans; 7, -7]
c)
Q3- Find a quadratic polynomial whose sum and product of zeros are
(a)
(b)
(c) [ Ans; x² +
x +
]
(d)
(e)
Q4- If x + a is the factor of 2x2 + 2ax + 5x + 10, then find a
[Ans a = 2]
Q5) For what value of k, -4 is a zero of the polynomial x2 – x - (2k + 2) ?
[Ans k = 9]
Q6) - Find a if 2x - 3 is a factor of 6x³ - x² - 10x + a.
[Ans; a = - 3]
Q7) For what value of k is -3 a zero of x² + 11x + k.
Q8) If 1 is the zero of ax² - 3(a - 1)x - 1 then find a.
Ans; a = 1
Q9) Is (x + 2) a factor of 2x² + 3x + 1.
Q 10) If α, β are the zeroes of P(x) = ax² + bx + c then find
a)
b)
c)
d)
e)
Q11- If the zeroes of x2 – Kx + 6 are in the ratio 3 : 2, find K
Q12. What is the degree of the polynomial (x + 1)(x2 – x - x4 + 1)
[Ans; k = - 16]
Q14 If α, β are the zeroes of P(y) = 2y² + 7y + 5 then find α + β + αβ.
Q15. A polynomial P(x) is divisible by (x - 4) and 2 is the zero of P(x), then write the corresponding polynomial.
Q16) If α, β are the zeroes of P(x) = x2 - 5x + 4, then find the value of
a) [Ans: 5/4]
b) [Ans: 320]
Q17) If α, β are the zeroes of P(x) = x2 - 1, then find a quadratic polynomial whose zeroes are
Ans[x2 + 4x + 4]
Q18. If α, β are the zeros of the polynomial 25p2 – 15p + 2, then find a quadratic polynomial whose zeroes are
Q19) If α, β are the zeroes of 4x2 + 4x + 1, then form a quadratic polynomial whose zeroes are 2α, 2β
Q20) Find k so that x2 + 2x + k is a factor of 2x4 + x3 – 14x2 + 5x + 6
Ans[k = - 3, - 1]
Q 21) If one zero of (a2 + 9) x2 + 13x + 6a is the reciprocal of the other then find the value of a
Q22) If sum of zeroes of ky2 + 2y – 3k is equal to the twice their product , find the value of k
Q 23) If α, β are the zeroes of P(x) = x2 – x – k such that α – β = 9, find the value of k.
Q 24) If p and q are the zeroes of 2x2 – 7x + 3 then find the value of p2 + q2
Q 25) A quadratic polynomial 2x2 - 3x + 1 has zeroes as α and β
Q 26) If α, β are the zeroes of x2 + 4x + 3, find the polynomial whose zeroes are
Q 27) For what value of p, - 4 is the zero of P(x) = x2 - 2x - (7p + 3)
Q 28) Find a quadratic polynomial whose zeroes are
(a) P(x) = x⁴ + x² + 1; g(x) = x² + x + 1
Ans; q(x) = x² - x + 1
(b) P(y) = 4y⁴ - 10y³ - 10y² + 30y - 15; g(y) = 2y - 5
Ans; q(y) = 2x3 - 5x + 5/2
(c) P(x) = 2x⁴ + 8x³ + 7x² + 4x + 5; g(x) = x + 3
Ans; q(x) = 2x³ + 2x² + x + 1
* Q30) Find a and b so that x⁴ + x³ + 8x² + ax - b is divisible by x² + 1.
[Ans; a = 1, b= - 7]
* Q31- Verify that 1, 1/2, -2 are the zeroes of P(x) = 2x³ + x² - 5x + 2.
* Q33- Find all the zeroes of 2x³ + x² - 5x + 2 if one of zero is 1/2.
Ans; 1, - 2
* Q34- If -1 is one of the zero of P(x) = 3x³ - 5x² - 11x - 3. Find other zeroes and verify the relation between zeroes and coefficients.
[Ans; All zeroes are -1, 3, -1/3]
* Q35) Obtain all zeroes of f(x) = x3 + 13x2 + 32x + 20, if one zero is -2
[Ans: All zeroes are -10, -1, - 2]
* Q36) Find all the zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5 if its two zeroes are
[Ans: - 1, - 1]
* Q37) Find all zeroes of 2x⁴ - 3x³ - 3x² + 6x - 2 if its two zeroes are
[Ans: 1]
* Q38) Obtain all zeroes of f(x) = 2x4 - 2x3 - 7x2 + 3x + 6, if its two zeroes are
* Q39) Obtain all zeroes of p(x) = 2x4 + x3 – 14x2 – 19x – 6, if two of zeroes are -2 and -1
[Ans: -1/2, 3, -2, -1]
* Q40) Find all zeroes of P(x) = x⁴ + x³ - 23x² - 3x + 60 if its 2 zeroes are
* Q41) Find all zeroes of P(x) = x⁴ + x³ - 34x² - 4x + 120 if its two zeroes are ± 2,
* Q42) P(x) = x⁴ - 5x + 6, g(x) = 2 - x² find q(x) and r(x) if P(x) is divided by g(x).
[Ans: q(x) = - x² - 1, r(x) = - 5x + 10]
* Q43- On dividing 10x⁴ - 6x³ - 40x² + 41x - 5 by g(x) the quotient is 5x - 3 and remainder is 2x + 4, Find g(x).
[Ans: 2x³ - 8x + 3]
* Q44- If one zero of the P(x) = 5x² + 13x + k is the reciprocal of the other then the value of k is?
[Ans; 5]
* Q45- What must be subtracted from 8x⁴ + 14x³ - 2x² + 7x - 8 so that the resulting polynomial is divisible by 4x² + 3x - 2.
[Ans; 14x - 10]
* Q46) What must be subtracted from z⁵ - 9z + 6 so that it is exactly divisible by z² + 3.
[ Ans; 6]
* Q47) What must be added to the polynomial p(x) = x4 + 2x3 - 13x2 - 12x + 21 so that the resulting polynomial is exactly divisible by x2 - 4x + 3
[Ans: - 2x + 3]
* Q48- What must be added to x⁴ + 2x³ - 2x² - x - 2 so that the resulting polynomial is divisible by x² + 2x - 3.
[Ans; 3x - 1]
* Q49) If the polynomial 6x4 + 8x3 + 17x2 + 21x + 7 is divided by another polynomial 3x2 + 4x + 1, the remainder comes out to be ax + b, find a and b
Ans [a = 1 & b = 2]
* Q50) Using division algorithm show that 6y5 + 15y4 + 16y3 + 4y2 + 10y – 35 is divisible by 3y2 + 5
* Q51) On dividing the polynomial 9x4 – 4x2 + 5 by 3x2 + x - 1
* Q 52) Polynomial x4 + 7x3 + 7x2 + px + q is exactly divisible by x2 + 7x + 12, then find the value of p and q .
* Q 54) If x3 + 8x2 + kx + 18 is divisible by x2 + 6x + 9 then find the value of k.
* Q 55) If 3x4 – 9x3 + x2 + 15x + k is divisible by 3x2 - 5
* Q 56) On dividing x3 - 8x2 + 20x - 10 by g(x) the quotient and remainder were x - 4 and 6 respectively . Find g(x)
* Q 57) Find all zeroes of x4 - 17x2 - 36x - 20, if two of its zeroes are 5 and -2
* Q 59) What must be added or subtracted to 8x4 + 14x3 - 2x2 + 8x - 12 so that 4x2 + 3x - 2 is a factor of p(x).
* Q 60) If x4 + 2x3 + 8x2 + 12x + 18 is divided by another polynomial x2 + 5, the remainder is px + q . Find the value of p and q .
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