Math Assignment Ch-2 Class X

Mathematics Assignment

Important and useful questions on quadratic polynomial, Extra questions based on quadratic polynomial class 10 for the examination point of view.

ASSIGNMENT | CHAPTER-2 | POLYNOMIALS

Question 1: 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 - 2 $\sqrt{2}$ x - 6 [Ans; 3 $\mathbf{\sqrt{2}}$ , - $\mathbf{\sqrt{2}}$

(c) x² - 8x + 12 [Ans; 6, 2]

(d) 2x² + 5x + 2 [Ans -2, -1/2]

$\mathbf{\: \: \sqrt{3}x^{2}-8x+4\sqrt{3}\: \: \: \left [ Ans:\: \frac{2}{\sqrt{3}},\: 2\sqrt{3} \right ]}$
$\mathbf{\: \: 4\sqrt{3}x^{2}+4\sqrt{3}x-3\sqrt{3}\: \: \: \left [ Ans:\: \frac{-3}{2},\: \frac{1}{2} \right ]}$

(g) x² - 8 [Ans; ± 2 $\mathbf{\sqrt{2}}$ ],

(h) 9y² - 6y + 1 [Ans; 1/3, 1/3],

(i) 3x² - 5x - 2 [Ans; 2, -1/3]

$\mathbf{j)\: \: 6x^{2}+17x+12\: \: \: \left [ Ans:\: \frac{-4}{3},\: \frac{-3}{2} \right ]}$

$\mathbf{k)\: \: 24x^{2}-41x+12\: \: \: \left [ Ans:\: \frac{4}{3},\: \frac{3}{8} \right ]}$

$\mathbf{\textit{l})\: \: 15x^{2}-x-28\: \: \: \left [ Ans:\: \frac{7}{5},\: \frac{-4}{3} \right ]}$

$\mathbf{m)\: \: 5\sqrt{5}x^{2}+20x+3\sqrt{5}\: \: \: \left [ Ans:\: \frac{-3}{\sqrt{5}},\: \frac{-\sqrt{5}}{5} \right ]}$

$\mathbf{n)\: \: 6\sqrt{3}x^{2}-47x+5\sqrt{3}\: \: \: \left [ Ans:\: \frac{5\sqrt{3}}{2},\: \frac{1}{3\sqrt{3}} \right ]}$

Question 2 : Find sum and product of zeroes of

(a) 2x² + 2x + 3 [Ans; -1, 3/2],

(b) x² - 7x - 7 [Ans; 7, -7]

c) $\mathbf{4\sqrt{3}x^{2}+7x-2\sqrt{3}\: \: \: \left [ Ans:\: \frac{-7}{4\sqrt{3}},\: \frac{-1}{2} \right ]}$ Question 3: Find a quadratic polynomial whose sum and product of zeros are

(a) 1/3 and 1/2 [Ans; 6x² - 2x + 3]

(b) $\mathbf{0\: and\: \sqrt{3}\: \: \: \: \: [Ans:\: x^{2}+\sqrt{3}]}$ (c) $\mathbf{-\sqrt{2}\: and\: \sqrt{3}}$ [ Ans; x² + $\sqrt{2}$ x + $\sqrt{3}$ ]

(d) $\mathbf{0\: and\: \frac{-\sqrt{7}}{2}}$ $\mathbf{[Ans:\: 2x^{2}-\sqrt{7}]}$

(e) $\mathbf{\frac{5}{7}\: and\: \frac{10}{21}}$ $\mathbf{[Ans:\: 21x^{2}-15x+10]}$

Question 4: If x + a is the factor of 2x²+ 2ax + 5x + 10, then find a

Answer: a = 2

Question 5: For what value of k, -4 is a zero of the polynomial x²– x - (2k + 2) ?

Answer k = 9

Question 6: Find a if 2x - 3 is a factor of 6x³ - x² - 10x + a.

Answer: a = - 3

Question 7: For what value of k is -3 a zero of x² + 11x + k.

Answer: k = -24

Question 8: If 1 is the zero of ax² - 3(a - 1)x - 1 then find a.

Answer : a = 1

Question 9: Is (x + 2) a factor of P(x) = 2x² + 3x - 2.

Answer: p(-2) = 0 ⇒ x + 2 is the factor of p(x)

Question 10: If α, β are the zeroes of P(x) = ax² + bx + c then find

a) $\mathbf{\frac{\alpha }{\beta }+\frac{\beta }{\alpha }}$ $\mathbf{\left [ Ans:\: \frac{b^{2}-2ac}{ac} \right ]}$

b) $\mathbf{\alpha ^{2}+\beta ^{2}}$ $\mathbf{\left [ Ans:\: \frac{b^{2}-2ac}{a^{2}} \right ]}$

c) $\alpha ^{3}+\beta ^{3}$ $\mathbf{\left [ Ans:\: \frac{3abc-b^{3}}{a^{3}} \right ]}$ d) $\frac{1}{\alpha ^{3} }+\frac{1 }{\beta ^{3} }$ $\mathbf{\left [ Ans:\: \frac{3abc-b^{3}}{c^{3}} \right ]}$ e) $\frac{\alpha ^{2}}{\beta }+\frac{\beta ^{2}}{\alpha }$ $\mathbf{\left [ Ans:\: \frac{3abc-b^{3}}{a^{2}c} \right ]}$ Question 11: If the zeroes of x² – Kx + 6 are in the ratio 3 : 2, find K.

Answer: K = 5

Question 12: What is the degree of the polynomial (x + 1)(x²– x - x⁴+ 1)

Answer : Degree of p(x) = 5

Question 13: If α, β are the roots of P(x) = x² - 6x + k then what is the value of k if 3α + 2β = 20. Answer: k = - 16

Question 14: If α, β are the zeroes of P(y) = 2y² + 7y + 5 then find α + β + αβ.

Answer: α + β + αβ = -1

Question 15: A polynomial P(x) is divisible by (x - 4) and 2 is the zero of P(x), then write the corresponding polynomial.

Answer: P(x) = x² - 6x + 8

Question 16: If α, β are the zeroes of P(x) = x²- 5x + 4, then find the value of

a) $\mathbf{\frac{1}{\alpha }+\frac{1}{\beta }}$ [Ans: 5/4]

b) $\alpha ^{4}\beta ^{3}+\alpha ^{3}\beta ^{4}$ [Ans: 320]

Question 17: If α, β are the zeroes of P(x) = x²- 1, then find a quadratic polynomial whose zeroes are $\mathbf{\frac{2\alpha }{\beta }\: and\: \frac{2\beta }{\alpha }}$

Answer: x² + 4x + 4

Question 18: If α, β are the zeros of the polynomial 25p² – 15p + 2, then find a quadratic polynomial whose zeroes are $\mathbf{\frac{1 }{2\alpha }+ \frac{1 }{2\beta }}$

Answe: 15/4

Question 19: If α, β are the zeroes of 4x² + 4x + 1, then form a quadratic polynomial whose zeroes are 2α, 2β

Answer: x² + 2x + 1

Question 20: For what value of p, - 4 is the zero of P(x) = x²- 2x - (7p + 3)

Answer: P = 3
Question 21: If one zero of (a² + 9) x² + 13x + 6a is the reciprocal of the other then find the value of a

Answer: a = 3

Question 22: If sum of zeroes of ky² + 2y – 3k is equal to the twice their product , find the value of k

Answer: k = 1/3

Question 23: If α, β are the zeroes of P(x) = x² – x – k such that α – β = 9, find the value of k.

Answer: k = 20

Question 24: If p and q are the zeroes of 2x² – 7x + 3 then find the value of p² + q²

Answer: 37/4

Question 25: A quadratic polynomial 2x²- 3x + 1 has zeroes as α and β . Now form the quadratic polynomial whose zeroes are 3α and 3β

Answer: P(x) = 2x² – 9x + 9

Question 26: If α, β are the zeroes of x²+ 4x + 3, find the polynomial whose zeroes are $\mathbf{1+\frac{\beta }{\alpha }}$ and $\mathbf{1+\frac{\alpha }{\beta }}$

Answer: P(x) = 3x² – 16x + 16

Question 27: Find a quadratic polynomial whose zeroes are $\mathbf{\frac{3\pm \sqrt{5}}{5}}$

Answer: P(x) = 25x² – 30x + 4

Question 28: If 𝛂 and β are zeroes of the polynomial 5x² + 3x – 7, find the value of $\frac{1}{\alpha}+\frac{1}{\beta}$ .

Ans: 3/7

Question 29: If 𝛂 and β are the zeroes of the polynomial P(x) = kx² – 30x + 45k and 𝛂 + β= 𝛂β, then find the value of k.

Answer : 2/3

Question 30: If 𝛂, β are zeroes of the polynomial P(x) = 5x² – 6x + 1, then find the value of 𝛂 + β + 𝛂β.

Answer: 7/5

Question 31: If 𝛂 and β are the zeroes of the polynomial x² + x – 2, then find the value of

$\mathbf{\large \mathbf{\frac{\alpha}{\beta}+\frac{\beta}{\alpha}}}$

Answer: - 5/2

Question 32: If the sum of zeroes of the polynomial P(x) = 2x² – k √2x + 1 is √2 then find the value of k.

Answer: 2

Question 33: The zeroes of the polynomial x² + px + q are twice the zeroes of the polynomial 4x² – 5x – 6, then find the value of p.

Answer : - 5/2

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QUESTIONS DELETED FROM CBSE SYLLABUS

Q1) Find k so that x² + 2x + k is a factor of 2x⁴ + x³ – 14x² + 5x + 6
Ans[k = - 3, - 1]

Q2- Find quotient and remainder on dividing P(x) by g(x)

(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

Q3) Find a and b so that x⁴ + x³ + 8x² + ax - b is divisible by x² + 1.

[Ans; a = 1, b= - 7]

Q4- Verify that 1, 1/2, -2 are the zeroes of P(x) = 2x³ + x² - 5x + 2.

Q5- Find all zeroes of 2x³ + x² - 6x - 3 if its 2 zeroes are ± $\sqrt{3}$

Q6- Find all the zeroes of 2x³ + x² - 5x + 2 if one of zero is 1/2.

Ans; 1, - 2

Q7- 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]

Q8) Obtain all zeroes of f(x) = x³+ 13x²+ 32x + 20, if one zero is -2

[Ans: All zeroes are -10, -1, - 2]

Q9) Find all the zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5 if its two zeroes are $\pm \sqrt{\frac{5}{3}}$

[Ans: - 1, - 1]

Q10) Find all zeroes of 2x⁴ - 3x³ - 3x² + 6x - 2 if its two zeroes are $\pm \sqrt{2}$

[Ans: 1]

Q11) Obtain all zeroes of f(x) = 2x⁴- 2x³- 7x²+ 3x + 6, if its two zeroes are $\pm \sqrt{\frac{3}{2}}$

[Ans: 2, -1]

Q12) 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]

Q13) Find all zeroes of P(x) = x⁴ + x³ - 23x² - 3x + 60 if its 2 zeroes are $\pm \sqrt{3}$

Q14) Find all zeroes of P(x) = x⁴ + x³ - 34x² - 4x + 120 if its two zeroes are ± 2,

[Ans; 5, - 6]

Q15) 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]

Q16- 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]

Q17- If one zero of the P(x) = 5x² + 13x + k is the reciprocal of the other then the value of k is?

[Ans; 5]

Q18- What must be subtracted from 8x⁴ + 14x³ - 2x² + 7x - 8 so that the resulting polynomial is divisible by 4x² + 3x - 2.

[Ans; 14x - 10]

Q19) What must be subtracted from z⁵ - 9z + 6 so that it is exactly divisible by z² + 3.

[ Ans; 6]

Q20) 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]

Q21- What must be added to x⁴ + 2x³ - 2x² - x - 2 so that the resulting polynomial is divisible by x² + 2x - 3.

[Ans; 3x - 1]

Q22) 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]

Q23) Using division algorithm show that 6y⁵ + 15y⁴ + 16y³ + 4y² + 10y – 35 is divisible by 3y² + 5

Q24) On dividing the polynomial 9x⁴ – 4x² + 5 by 3x² + x - 1 remainder is ax - b. Find a and b

Q 25) Polynomial x⁴ + 7x³ + 7x² + px + q is exactly divisible by x² + 7x + 12, then find the value of p and q .

Q 26) If x - $\sqrt{5}$ is the factor of x³- 3 $\sqrt{5}$ x² - 5x + 15 $\sqrt{5}$ then find all zeroes of this polynomial.
Q 27) If x³ + 8x² + kx + 18 is divisible by x² + 6x + 9 then find the value of k.
Q 28) If 3x⁴ – 9x³ + x² + 15x + k is divisible by 3x² - 5 , find the value of k and other two zeroes of the polynomial .
Q 29 On dividing x³- 8x²+ 20x - 10 by g(x) the quotient and remainder were x - 4 and 6 respectively . Find g(x)
Q 30) Find all zeroes of x⁴- 17x²- 36x - 20, if two of its zeroes are 5 and -2
Q 31) Find all zeroes of x⁴ – 3 x³ + 3x² + 3 x - 4 if two of its zeroes are $\sqrt{2}$ and 2 $\sqrt{2}$
Q 32) What must be added or subtracted to 8x⁴+ 14x³- 2x²+ 8x - 12 so that 4x²+ 3x - 2 is a factor of p(x).
Q 33) 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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