Maths Class X Question Bank

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} $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$\alpha+\beta+\gamma=\dfrac{-b}{a}=\dfrac{-3}{6}=\dfrac{-1}{2}\\ \\ \alpha\beta\gamma=\dfrac{-d}{a}=\dfrac{-1}{6}\\ \\ \alpha\beta+\beta\gamma+\alpha\gamma=\dfrac{c}{a}=\dfrac{-5}{6}\\ \\ \dfrac{1}{\alpha}+\dfrac{1}{\beta}+\dfrac{1}{\gamma}=\dfrac{\beta\gamma+\alpha\gamma+\alpha\beta}{\alpha\beta\gamma}=\dfrac{-5}{6}\times\dfrac{-6}{1}=5$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$\alpha=a-b\\ \\ \beta=a\\ \\ \gamma=a+b$
sum of zeroes $=\dfrac{-b}{a}=\dfrac{-(-3)}{1}=3$
$a-b+a+a+b=3\Rightarrow 3a=3\\ \\ \Rightarrow a=1\\ \\ \alpha\beta\gamma=\dfrac{-c}{a}=\dfrac{-1}{1}\\ \\ (a-b)\ (a)\ (a+b)=-1\Rightarrow 1^2-b^2=-1\\ \\ \Rightarrow b^2=2\Rightarrow b=\pm\sqrt2$
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) $
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$f(x)=x^3-3x^2+x+2\\ \\ q(x)=x-2\\ \\ g(x)=ax^2+bx+c\\ \\ r(x)=-2x+4\\ \\ f(x)=q(x)g(x)+r\\ \\ \Rightarrow x^3-3x^2+x+2=(ax^2+bx+c)(x-2)+(-2x+4)\\ \\ =ax^3+x^2(b-2a)+x(c-2b-2)+(-2c+4)$
Equation
$a=1,b-2a=-3\Rightarrow b=-1\\ \\ c-2b-2=1\Rightarrow c=1\\ \\ g(x)=x^2-x+1$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$\alpha+\beta=\dfrac{-(-2)}{1}=2$
Product $=\alpha\beta=\dfrac{-8}{1}=-8$
$x^2-2(\alpha+\beta)x+4\alpha\beta\\ \\ x^2-2\times2x+4(-8)\\ \\ x^2-4x-32$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$f(x)=x^4-3x^3-x^2+9x-6\\ \\ (x+\sqrt3)\ (x-\sqrt3)=x^2-3$

$x^2-3x+2=(x-2)\ (x-1)=2,1$
The zeroes are $1,2,\sqrt3,-\sqrt3$
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 $
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

No it is not a factor
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} $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$25p^2-15p+2\\ \\ \alpha+\beta=\dfrac{-(-15)}{25}=\dfrac{3}{5}\\ \\ \alpha\beta=\dfrac{2}{25}\\ \\ \dfrac{1}{2\alpha}+\dfrac{1}{2\beta}=\dfrac{1}{2}\times\dfrac{\alpha+\beta}{\alpha\beta}=\dfrac{1}{2}\times\dfrac{3}{\dfrac{5}{\dfrac{2}{25}}}=\dfrac{15}{4}\\ \\ \dfrac{1}{2\alpha}\ \dfrac{1}{2\beta}=\dfrac{1}{4\times\dfrac{2}{25}}=\dfrac{25}{8}\\ \\ P^2-\dfrac{15P}{4}+\dfrac{25}{8}\\ \\ 8P^2-30P+25$
Question 8 of 28

8. Question
1 point(s)
Divide $3x^3-3x+5 $ by $x-1-x^2 $ and verify the division algorithm
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

Verify
$(-x^2+x-1)(-3x-3)+(-3x+2)\\ \\ =3x^3+3-3x+2=3x^3-3x+5$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$\alpha+\beta=\dfrac{1}{21}\\ \\ \alpha\beta=\dfrac{-2}{21}\\ \\ 2\alpha+2\beta=2(\alpha+\beta)=\dfrac{2}{21}\\ \\ 2\alpha\times2\beta=4\left(\dfrac{-2}{21}\right)=\dfrac{-8}{21}\\ \\ x^2-\left(\dfrac{2}{21}\right)x+\left(\dfrac{-8}{21}\right)=0\Rightarrow 21x^2-2x-8=0$
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.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$4\sqrt5x^2+17x+3\sqrt5\\ \\ 4\sqrt5x^2+12x+5x+3\sqrt5\\ \\ (4x+\sqrt5)\ (\sqrt5x+3)\\ \\ x=\dfrac{-\sqrt5}{4}=\alpha\ \text{and }\ \dfrac{-3}{\sqrt5}=\beta\\ \\ \alpha+\beta=\dfrac{-\sqrt5}{4}+\left(\dfrac{-3}{\sqrt5}\right)=\dfrac{-5-12}{4\sqrt{5}}=\dfrac{-17}{4\sqrt5}=\dfrac{-b}{a}\\ \\ \alpha\beta=\left(\dfrac{-\sqrt5}{4}\right)\left(\dfrac{-3}{\sqrt5}\right)=\dfrac{3}{4}=\dfrac{c}{a}$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

A/Q $x+2=ax+b$
$\Rightarrow a=1,b=2$
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 $
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$2x+1=0\Rightarrow x=\dfrac{-1}{2}$
The zeroes are $\sqrt3,-\sqrt3,\dfrac{-1}{2}$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$\alpha+\beta=\dfrac{6}{3}=2\\ \\ \alpha\beta=\dfrac{4}{3}\\ \\ \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}+2\left[\dfrac{1}{\alpha}+\dfrac{1}{\beta}\right]+3\alpha\beta\\ \\ =\dfrac{\alpha^2+\beta^2}{\alpha\beta}+2\left[\dfrac{\alpha+\beta}{\alpha\beta}\right]+3\alpha\beta\\ \\ =\dfrac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}+2\left[\dfrac{\alpha+\beta}{\alpha\beta}\right]+3\alpha\beta\\ \\ =\dfrac{2^2-2\times\dfrac{4}{3}+2\left(\dfrac{2\times3}{4}\right]+3\left(\dfrac{4}{3}\right)}{\dfrac{4}{3}}\\ \\ =\dfrac{4}{3}\times\dfrac{3}{4}(1+7)=8$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$x^2-px+q\\ \\ \alpha+\beta=p\\ \\ \alpha\beta=q\\ \\ \dfrac{\alpha^2}{\beta^2}+\dfrac{\beta^2}{\alpha^2}=\dfrac{\alpha^4+\beta^4}{\alpha^2\beta^2}=\dfrac{(\alpha^2+\beta^2)-2\alpha^2\beta^2}{\alpha^2\beta^2}\\ \\ =\dfrac{((\alpha+\beta)^2-2\alpha\beta)^2-2(\alpha\beta)^2}{(\alpha\beta)^2}=\dfrac{(p^2-2q)^2-2q^2}{q^2}\\ \\ \dfrac{p^4}{q^2}-\dfrac{4p^2q}{q^2}+\dfrac{2q^2}{q^2}\\ \\ =\dfrac{p^4}{q^2}-\dfrac{4p^2}{q}+2$ (Proved)
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 $
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$x^2+px+q\\ \\ \alpha+\beta=-p\\ \\ \alpha\beta=q\\ \\ (\alpha+\beta)^2=(-p)^2=p^2\\ \\ (\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=p^2-4q\\ \\ x^2-(a+b)x+ab=0\\ \\ x^2-(p^2+p^2-4q)x+p^2\ (p^2-4q)\\ \\ =x^2-(2p^2-4q)x+p^4-4p^2q$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$x^2-2x+3\\ \\ 2\beta=3\\ \\ \alpha+\beta=2\\ \\ \alpha+2+\beta+2=2+4=6$
Product of roots $=(\alpha+2)\ (\beta+2)$
$=\alpha\beta+2(\alpha+\beta)+4\\ \\ =3+8=11$
$x^2$ – (sum of roots) $x$ + Product of root
$=x^2-6x+11$
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$f(x)=2x^4+x^3-14x^2-19x-6\\ \\ (x+2)(x+1)=0\Rightarrow x^2+3x+2=0$

$2x^2-5x-3=(2x+1)(x-3)\\ \\ x=3,\dfrac{-1}{2}$
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$?
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

So $-17x+17$ must be added
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 $?
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

So $2x-3$ must be subtracted
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.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

The zeroes are $-2,-3,3$
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.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

other two zeroes are -5 and 7
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 $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

$2,-3$ are zeroes.
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.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

Verify
$(3x^2+2x-4)(10x^2-3x)+0\\ \\ =30x^4-9x^3-36x^2+20x^3-6x^2-24x-40x^2+12x+48+0\\ \\ =30x^4+11x^3-82x^2+12x+48$
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}} $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint
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 $
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

Zeroes of polynomial are $\dfrac{1}{2},1,\sqrt3,-\sqrt3$
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) $.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint

So, $114x+77$ should be added
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$.
- Upload your answer to this question.
  
  This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Grading can be reviewed and adjusted.

Grading can be reviewed and adjusted.

Hint