Questions Bank-Essay Type 2-Polynomials – Class X
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Question 1 of 28
1. Question
1 point(s)If \(\alpha \) and \(\beta,\gamma \) are zeroes of the polynomial \(6x^3+3x^2-5x+1 \), then find the value of \(\alpha^{-1}+\beta^{-1}+\gamma^{-1} \).
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Question 2 of 28
2. Question
1 point(s)If the zeroes of the polynomial \(x^3-3x^2+x+1 \) are \(a-b \) and \(a+b \), find the value of \(a \) and \(b \).
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sum of zeroes
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Question 3 of 28
3. Question
1 point(s)On dividing \( x^3-3x^2+x+2\) by a polynomial \(g(x) \), the quotient and remainder were \(x-2 \) and \(-2x+4 \) respectively. Find \(g(x) \)
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Equation
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Question 4 of 28
4. Question
1 point(s)If \(\alpha,\beta \) are zeroes of the polynomial \(x^2-2x-8 \), then form a quadratic polynomial whose zeroes are \( 2\alpha\) and \(2\beta \).
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Product
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Question 5 of 28
5. Question
1 point(s)Obtain all zeroes of \(f(x)=x^4-3x^3-x^2+9x-6 \) if two of its zeroes are \((-\sqrt3) \) and \(\sqrt3 \).
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The zeroes are
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Question 6 of 28
6. Question
1 point(s)Check whether the polynomial \(g(x)=x^3-3x+1 \) is the factor of polynomial \(px(x)=x^5-4x^3-x^2+3x+1 \)
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No it is not a factor
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Question 7 of 28
7. Question
1 point(s)If \(\alpha,\beta \) are the zeroes of the polynomial \(25p^2-15p+2 \), find a quadratic polynomial whose zeroes are \(\dfrac{1}{2\alpha} \) and \(\dfrac{1}{2\beta} \).
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Question 8 of 28
8. Question
1 point(s)Divide \(3x^3-3x+5 \) by \(x-1-x^2 \) and verify the division algorithm
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Verify
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Question 9 of 28
9. Question
1 point(s)If \(\alpha,\beta \) are the zeroes of the polynomial \(21y^2-y-1 \), find a quadratic polynomial whose zeroes are \( 2\alpha\) and \(2\beta \).
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Question 10 of 28
10. Question
1 point(s)Find the zeroes of \(4\sqrt5x^2+17x+3\sqrt5 \) and verify the relation between the zeroes and coefficients of the polynomial.
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Question 11 of 28
11. Question
1 point(s)If the polynomial \(6x^4+8x^3+17x^2+21x+7 \) is divided by another polynomial \(3x^2+4x+1 \), the remainder comes out to be \((ax+b) \), find \(a \) and \(b \).
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A/Q
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Question 12 of 28
12. Question
1 point(s)Find all the zeroes of the polynomial \(2x^3+x^2-6x-3 \), if two of its zeroes are \(-\sqrt3 \) and \(\sqrt3 \)
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The zeroes are
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Question 13 of 28
13. Question
1 point(s)If \(\alpha \) and \(\beta \) are the zeroes of the quadratic polynomial \(p(s)=3s^2-6s+4 \), find the value of \(\dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}+2\left(\dfrac{1}{\alpha}+\dfrac{1}{\beta}\right)+3\alpha\beta \).
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![\alpha+\beta=\dfrac{6}{3}=2\\ \\ \alpha\beta=\dfrac{4}{3}\\ \\ \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}+2\left[\dfrac{1}{\alpha}+\dfrac{1}{\beta}\right]+3\alpha\beta\\ \\ =\dfrac{\alpha^2+\beta^2}{\alpha\beta}+2\left[\dfrac{\alpha+\beta}{\alpha\beta}\right]+3\alpha\beta\\ \\ =\dfrac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}+2\left[\dfrac{\alpha+\beta}{\alpha\beta}\right]+3\alpha\beta\\ \\ =\dfrac{2^2-2\times\dfrac{4}{3}+2\left(\dfrac{2\times3}{4}\right]+3\left(\dfrac{4}{3}\right)}{\dfrac{4}{3}}\\ \\ =\dfrac{4}{3}\times\dfrac{3}{4}(1+7)=8](https://mathsgenii.com/wp-content/ql-cache/quicklatex.com-e6244190a4b4ea5e790c50e17930e5d6_l3.png "Rendered by QuickLaTeX.com")
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Question 14 of 28
14. Question
1 point(s)If \(\alpha \) and \(\beta \) are the zeroes of the quadratic polynomial \(f(x)=x^2-px+q \), prove that \(\dfrac{\alpha^2}{\beta^2}+\dfrac{\beta^2}{\alpha^2}=\dfrac{p^4}{q^2}-\dfrac{4p^2}{q}+2 \).
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(Proved) -
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Question 15 of 28
15. Question
1 point(s)If \(\alpha \) and \(\beta \) are the zeroes of the quadratic polynomial \(f(x)=x^2+px+q \), form a polynomial whose zeroes are \((\alpha+\beta)^2 \) and \((\alpha-\beta)^2 \)
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Question 16 of 28
16. Question
1 point(s)If \(\alpha \) and \(\beta \) are the zeroes of the polynomial \(f(x)=x^2-2x+3 \), find a polynomial whose zeroes are \(\alpha+2 \) and \(\alpha+\beta \).
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Product of roots
– (sum of roots) 
+ Product of root 
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Question 17 of 28
17. Question
1 point(s)Obtain all the zeroes of the polynomial \(f(x) = 2x^4+x^3-14x-19x-6 \), if two of its zeroes are \( -2\) and \(-1 \).
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Question 18 of 28
18. Question
1 point(s)What must be added to \(6x^5+5x^4+11x^3-3x^2+x+5 \) so that it may be exactly divisible by \( 3x^2-2x+4\)?
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So
must be added -
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Question 19 of 28
19. Question
1 point(s)What must be subtracted from the polynomial \(f(x)= x^4+2x^3-13x^2-12x+21 \) so that the resulting polynomial is exactly divisible by \(g(x) = x^2-4x+3 \)?
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So
must be subtracted -
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Question 20 of 28
20. Question
1 point(s)Find the other zeroes of the polynomial \(2x^4-3x^3-3x^2+6x-2 \), if \(-\sqrt2 \) and \( \sqrt2\) are the zeroes of the given polynomial.
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Question 21 of 28
21. Question
1 point(s)If the remainder on division of \(x^3+2x^2+kx+3 \) by \(x-3 \) is \(21 \), find the quotient and the value of \( k\). Hence, find the zeroes of the cubic polynomial \(x^3+2x^2+kx-18 \).
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The zeroes are
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Question 22 of 28
22. Question
1 point(s)If two zeroes of \( p(x) = x^4-6x^3-26x^2+138x–35 \) are \(2\pm\sqrt3 \), find the other zereos.
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other two zeroes are -5 and 7
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Question 23 of 28
23. Question
1 point(s)Find other zeroes of the polynomial \( x^4+x^3-9x^2-3x+18 \), if it is given that two of its zeroes are \(\sqrt3 \) and \(-\sqrt3 \).
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are zeroes. -
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Question 24 of 28
24. Question
1 point(s)Divide \(30x^4+11x^3-82x^2-12x+48 \) by \( (3x^2+2x-4)\) and verify the result by division algorithm.
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Verify
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Question 25 of 28
25. Question
1 point(s)Find all the zeros of the polynomial \(2x^4-10x^3+5x^2+15x-12 \), if it is given that two of its zeroes are \(\sqrt{\dfrac{3}{2}} \) and \(-\sqrt{\dfrac{3}{2}} \).
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Question 26 of 28
26. Question
1 point(s)Find all the zeroes of the polynomial \(2x^4-3x^2-5x^2+9x-3 \), it being given that two of its zeroes are \(\sqrt3 \) and \(-\sqrt3 \)
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Zeroes of polynomial are
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Question 27 of 28
27. Question
1 point(s)What must be added to the polynomial \(p(x)=5x^4+6x^3-13x^2-44x+7 \) so that the resulting polynomial is exactly divisible by the polynomial \(Q(x)=x^2+4x+3 \) and the degree of the polynomial to be added must be less than degree of the polynomial \(Q(x) \).
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So,
should be added -
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Question 28 of 28
28. Question
1 point(s)If the polynomial \(6x^4+8x^3-5x^2+ax+b \) is exactly divisible by the polynomial \(2x^2-5 \), then find the value of \(a \) and \( b\).
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