Maths Class X Question Bank

Question 1 of 52

1. Question
1 point(s)
The graph of $y=f(x) $ cuts the x-axis at $(1,0) $ and $\left(\dfrac{-3}{2},0\right) $. Find all the zeroes of $f(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

$y=f(x)=0$
$y=0$ at $1$ and $\dfrac{-3}{2}$
The zeroes of the polynomial are $1$ and $\dfrac{-3}{2}$
Question 2 of 52

2. Question
1 point(s)
Show that $1,-1 $ and $3 $ are the zeroes of the polynomial $x^3-3x^2-x+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

$p(x)=x^3-3x^2-x+3\\ \\ p(1)=1-3-1+3=0\\ \\ p(-1)=(-1)^3-3(-1)^2-(-1)+3\\ \\ =-1-3+1+3=0\\ \\ p(3)=3^3-3(3)^2-3+3\\ \\ =27-27-3+3=0$
Question 3 of 52

3. Question
1 point(s)
Write the polynomial, the product and sum of whose zeroes are $\dfrac{-9}{2} $ and $\dfrac{-3}{2} $ respectively
- 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

$x^2$ – (Sum of zeroes) $x$ + product of zeroes
$x^2-\left(-\dfrac{3}{2}\right)x+\left(-\dfrac{9}{2}\right)\\ \\ x^2+\dfrac{3}{2}x-\dfrac{9}{2}$
Question 4 of 52

4. Question
1 point(s)
If $(x+a) $ is a factor of $2x^2+2ax+5x+10 $, find $ a$.
- 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

$f(-a)=0\\ \\ 2(-a)^2+2(a)\ (-a)+5(-a)+10=0\\ \\ \Rightarrow 2a^2-2a^2-5a+10=0\\ \\ \Rightarrow -5a+10=0\Rightarrow a=2$
Question 5 of 52

5. Question
1 point(s)
If $\alpha $ and $\beta $ are the zeroes of the quadratic polynomial $f(t)=t^2-4t+3 $, find the value of $\alpha^4\beta^4+\alpha^3\beta^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

$\alpha+\beta=4\\ \\ \alpha\beta=3\\ \\ \alpha^4\beta^3+\alpha^3\beta^4=\alpha^3\beta^3(\alpha+\beta)\\ \\ =3^3\times4=108$
Question 6 of 52

6. Question
1 point(s)
Find a quadratic polynomial, the sum and product of whose zeroes are $\sqrt2 $ and $\dfrac{1}{3} $ respectively.
- 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

$x^2-(\alpha+\beta)x+\alpha\beta\\ \\ \alpha+\beta=\sqrt2,\ \alpha\beta=\dfrac{1}{3}\\ \\ \Rightarrow x^2-\sqrt2x+\dfrac{1}{3}$
Question 7 of 52

7. Question
1 point(s)
Divide $6x^3+13x^2+x-2 $ by $2x+1 $, and find the quotient and remainder.
- 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 8 of 52

8. Question
1 point(s)
Divide $x^4-3x^2+4x+5 $ by $x^2-x+1 $, find quotient and remainder.
- 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 9 of 52

9. Question
1 point(s)
$\alpha,\beta $ are the roots of the quadratic polynomial $p(x)=x^2-(k+6)x+2\ (2k-1) $. Find the value of $k $, if $\alpha+\beta=\dfrac{1}{2}\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

$\alpha+\beta=k+6\\ \\ \alpha\beta=2(2k-1)$
A/Q $k+6=\dfrac{1}{2}\times2(2k-1)\\ \\ \Rightarrow k+6=2k-1\\ \\ \Rightarrow k-2k=-1-6\\ \\ \Rightarrow -k=-7\Rightarrow k=7$
Question 10 of 52

10. Question
1 point(s)
$\alpha,\beta $ are the roots of the quadratic polynomial $p(x)=x^2-(k+6)x+2\ (2k-1) $. Find the value of $k $, if $\alpha+\beta=\dfrac{1}{2}\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

$\alpha+\beta=k+6\\ \\ \alpha\beta=2(2k-1)$
A/Q $k+6=\dfrac{1}{2}\times2(2k-1)\\ \\ \Rightarrow k+6=2k-1\\ \\ \Rightarrow k-2k=-1-6\\ \\ \Rightarrow -k=-7\Rightarrow k=7$
Question 11 of 52

11. Question
1 point(s)
What must be added to polynomial $f(x)=x^4+2x^3-2x^2+x-1 $ so that the resulting polynomial is exactly divisible by $x^2+2x-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

$x-2$ must be added
Question 12 of 52

12. Question
1 point(s)
What must be added to polynomial $f(x)=x^4+2x^3-2x^2+x-1 $ so that the resulting polynomial is exactly divisible by $x^2+2x-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

$x-2$ must be added
Question 13 of 52

13. Question
1 point(s)
Find a quadratic polynomial whose zeroes are 2 and –6. Ve rify the relation between the coefficients and zeroes 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

$\alpha+\beta=2+(-6)=-4\\ \\ \alpha\beta=2\times(-6)=-12$
Polynomial $=x^2-(-4)x-12=x^2+4x-12$
Sum of zeroes $=\dfrac{-4}{1}=-4$
Product of zeroes $=\dfrac{-12}{1}=-12$
Question 14 of 52

14. Question
1 point(s)
Find a quadratic polynomial whose zeroes are 2 and –6. Ve rify the relation between the coefficients and zeroes 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

$\alpha+\beta=2+(-6)=-4\\ \\ \alpha\beta=2\times(-6)=-12$
Polynomial $=x^2-(-4)x-12=x^2+4x-12$
Sum of zeroes $=\dfrac{-4}{1}=-4$
Product of zeroes $=\dfrac{-12}{1}=-12$
Question 15 of 52

15. Question
1 point(s)
Can (x + 3) be the remainder on the division of a polynomial p(x) by (2x– 5)? Justify your answer.
- 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

$(x+3)$ cannot be the remainder on the division of a polynomial by $2x-5$ because the degree of remainder should be less than the degree of divisor. Here the degree of remainder is equal to the degree of divisor.
Question 16 of 52

16. Question
1 point(s)
Can (x + 3) be the remainder on the division of a polynomial p(x) by (2x– 5)? Justify your answer.
- 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

$(x+3)$ cannot be the remainder on the division of a polynomial by $2x-5$ because the degree of remainder should be less than the degree of divisor. Here the degree of remainder is equal to the degree of divisor.
Question 17 of 52

17. Question
1 point(s)
Find the zeroes of the polynomial $2x^2-7x+3 $ and hence find the sum of product of its zeroes.
- 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

$2x^2-7x+3\\ \\ 2x^2-6x-x+3\\ \\ (x-3)\ (2x-1)=0\\ \\ x=3,\dfrac{1}{2}$
Sum of zeroes $=3+\dfrac{1}{2}=\dfrac{7}{2}$
Product $=3\times\dfrac{1}{2}=\dfrac{3}{2}$
Question 18 of 52

18. Question
1 point(s)
Find the zeroes of the polynomial $2x^2-7x+3 $ and hence find the sum of product of its zeroes.
- 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

$2x^2-7x+3\\ \\ 2x^2-6x-x+3\\ \\ (x-3)\ (2x-1)=0\\ \\ x=3,\dfrac{1}{2}$
Sum of zeroes $=3+\dfrac{1}{2}=\dfrac{7}{2}$
Product $=3\times\dfrac{1}{2}=\dfrac{3}{2}$
Question 19 of 52

19. Question
1 point(s)
Check whether $x^2+3x+1 $ is a factor of $3x^4+5x^3-7x^2+2x+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

Yes it is a factor
Question 20 of 52

20. Question
1 point(s)
Check whether $x^2+3x+1 $ is a factor of $3x^4+5x^3-7x^2+2x+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

Yes it is a factor
Question 21 of 52

21. Question
1 point(s)
Find the zeroes of the quadratic polynomial $x^2+7x+12 $ and verify the relationship between the zeroes and its coefïcients.
- 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

$x^2+7x+12\\ \\ x^2+4x+3x+12\\ \\ (x+4)\ (x+3)\\ \\ x=-4,-3$
Sum $=-7$ , Product $=12$
Sum of zeroes $=\dfrac{-7}{1}=-7$ , Product $=\dfrac{12}{1}=12$
Question 22 of 52

22. Question
1 point(s)
Find the zeroes of the quadratic polynomial $x^2+7x+12 $ and verify the relationship between the zeroes and its coefïcients.
- 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

$x^2+7x+12\\ \\ x^2+4x+3x+12\\ \\ (x+4)\ (x+3)\\ \\ x=-4,-3$
Sum $=-7$ , Product $=12$
Sum of zeroes $=\dfrac{-7}{1}=-7$ , Product $=\dfrac{12}{1}=12$
Question 23 of 52

23. Question
1 point(s)
Divide $(2x^2+x-20) $ by $(x+3) $ and verify 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 $(x+3)\ (2x-5)+(-5)=2x^2+6x-5x-15-5$
$=2x^2+x-20$
Question 24 of 52

24. Question
1 point(s)
Divide $(2x^2+x-20) $ by $(x+3) $ and verify 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 $(x+3)\ (2x-5)+(-5)=2x^2+6x-5x-15-5$
$=2x^2+x-20$
Question 25 of 52

25. Question
1 point(s)
If $\alpha $ and $\beta $ are the zeroes of $x^2+7x+12 $, then find the value of $\dfrac{1}{\alpha}+\dfrac{1}{\beta}-2\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

$\alpha+\beta=-7,\ \alpha\beta=12\\ \\ \dfrac{1}{\alpha}+\dfrac{1}{\beta}-2\alpha\beta=\dfrac{\alpha+\beta-2\alpha^2\beta^2}{\alpha\beta}\\ \\ =\dfrac{-7-2\times(12)^2}{12}\\ \\ =\dfrac{-7-288}{12}=\dfrac{-295}{12}$
Question 26 of 52

26. Question
1 point(s)
If $\alpha $ and $\beta $ are the zeroes of $x^2+7x+12 $, then find the value of $\dfrac{1}{\alpha}+\dfrac{1}{\beta}-2\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

$\alpha+\beta=-7,\ \alpha\beta=12\\ \\ \dfrac{1}{\alpha}+\dfrac{1}{\beta}-2\alpha\beta=\dfrac{\alpha+\beta-2\alpha^2\beta^2}{\alpha\beta}\\ \\ =\dfrac{-7-2\times(12)^2}{12}\\ \\ =\dfrac{-7-288}{12}=\dfrac{-295}{12}$
Question 27 of 52

27. Question
1 point(s)
If $\alpha $ and $\beta $ are the zeroes of the polynomial $2y^2+7y+5 $, write the value of $\alpha+\beta+\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

$\alpha+\beta=\dfrac{-7}{2}\\ \\ \alpha\beta=\dfrac{5}{2}\\ \\ \alpha+\beta+\alpha\beta=\dfrac{-7}{2}+\dfrac{5}{2}=\dfrac{-7+5}{2}=\dfrac{-2}{2}=-1$
Question 28 of 52

28. Question
1 point(s)
If $\alpha $ and $\beta $ are the zeroes of the polynomial $2y^2+7y+5 $, write the value of $\alpha+\beta+\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

$\alpha+\beta=\dfrac{-7}{2}\\ \\ \alpha\beta=\dfrac{5}{2}\\ \\ \alpha+\beta+\alpha\beta=\dfrac{-7}{2}+\dfrac{5}{2}=\dfrac{-7+5}{2}=\dfrac{-2}{2}=-1$
Question 29 of 52

29. Question
1 point(s)
If the product of zeroes of the polynomial $ax^2-6x-6 $ is $4 $, find the value of $ a$.
- 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

$\alpha\beta=4=\dfrac{-6}{a}\\ \\ =4a=-6\Rightarrow a=\dfrac{-6}{4}=\dfrac{-3}{2}$
Question 30 of 52

30. Question
1 point(s)
If the product of zeroes of the polynomial $ax^2-6x-6 $ is $4 $, find the value of $ a$.
- 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

$\alpha\beta=4=\dfrac{-6}{a}\\ \\ =4a=-6\Rightarrow a=\dfrac{-6}{4}=\dfrac{-3}{2}$
Question 31 of 52

31. Question
1 point(s)
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes 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

quadratic polynomial $=x^2$ – (sum of zeroes) $x$ + Product of zeroes
$=x^2-8x+12\\ \\ =x^2-6x-2x+12\\ \\ x(x-6)-2(x-6)\\ \\ (x-2)\ (x-6)$
zeroes are 2 and 6
Question 32 of 52

32. Question
1 point(s)
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes 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

quadratic polynomial $=x^2$ – (sum of zeroes) $x$ + Product of zeroes
$=x^2-8x+12\\ \\ =x^2-6x-2x+12\\ \\ x(x-6)-2(x-6)\\ \\ (x-2)\ (x-6)$
zeroes are 2 and 6
Question 33 of 52

33. Question
1 point(s)
If one zero of the polynomial $(a^2+9)x^2+13x+6a $ is reciprocal of the other, find the value of $ a$.
- 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

Let the zeroes be $\alpha$ and $\dfrac{1}{\alpha}$
$\alpha\times\dfrac{1}{\alpha}=1=\dfrac{6a}{a^2+9}\\ \\ a^2+9=6a\\ \\ \Rightarrow a^2-6a+9=0\\ \\ \Rightarrow (a-3)^2=0\\ \\ \Rightarrow a=3$
Question 34 of 52

34. Question
1 point(s)
If one zero of the polynomial $(a^2+9)x^2+13x+6a $ is reciprocal of the other, find the value of $ a$.
- 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

Let the zeroes be $\alpha$ and $\dfrac{1}{\alpha}$
$\alpha\times\dfrac{1}{\alpha}=1=\dfrac{6a}{a^2+9}\\ \\ a^2+9=6a\\ \\ \Rightarrow a^2-6a+9=0\\ \\ \Rightarrow (a-3)^2=0\\ \\ \Rightarrow a=3$
Question 35 of 52

35. Question
1 point(s)
Find the zeroes of the quadratic polynomial $f(x)=abx^2+(b^2-ac)x-bc $ and verify the relationship between the zeroes and its coefficients.
- 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

$ab x^2+(b^2-ac)x-bc=0\\ \\ abx^2+b^2x-acx-bc=0\\ \\ bx(ax-b)-c\ (ax+b)=0\\ \\ (ax+b)(bx-c)=0\Rightarrow x=\dfrac{-b}{a},\dfrac{c}{b}$
sum $=\dfrac{-b}{a}+\dfrac{c}{b}=\dfrac{-b^2+ca}{ab}$
Product $=\dfrac{-b}{a}\times\dfrac{c}{b}=\dfrac{-c}{a}$
Question 36 of 52

36. Question
1 point(s)
Find the zeroes of the quadratic polynomial $f(x)=abx^2+(b^2-ac)x-bc $ and verify the relationship between the zeroes and its coefficients.
- 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

$ab x^2+(b^2-ac)x-bc=0\\ \\ abx^2+b^2x-acx-bc=0\\ \\ bx(ax-b)-c\ (ax+b)=0\\ \\ (ax+b)(bx-c)=0\Rightarrow x=\dfrac{-b}{a},\dfrac{c}{b}$
sum $=\dfrac{-b}{a}+\dfrac{c}{b}=\dfrac{-b^2+ca}{ab}$
Product $=\dfrac{-b}{a}\times\dfrac{c}{b}=\dfrac{-c}{a}$
Question 37 of 52

37. Question
1 point(s)
Find the zeroes of the quadratic polynomial $p(x)=x^2-(\sqrt3+1)x+\sqrt3 $ and verify the relationship between the zeroes and its coefficients.
- 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

$p(x)=x^2-(\sqrt3+1)x+\sqrt3\\ \\ =x^2-\sqrt3x-x+\sqrt3=0\\ \\ x(x-\sqrt3)-1(x-\sqrt3)=0\\ \\ (x-1)\ (x-\sqrt3)=0\\ \\ x=1,\sqrt3$
sum $=1+\sqrt3$
product $=1\times\sqrt3=\sqrt3$
Question 38 of 52

38. Question
1 point(s)
Find the zeroes of the quadratic polynomial $p(x)=x^2-(\sqrt3+1)x+\sqrt3 $ and verify the relationship between the zeroes and its coefficients.
- 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

$p(x)=x^2-(\sqrt3+1)x+\sqrt3\\ \\ =x^2-\sqrt3x-x+\sqrt3=0\\ \\ x(x-\sqrt3)-1(x-\sqrt3)=0\\ \\ (x-1)\ (x-\sqrt3)=0\\ \\ x=1,\sqrt3$
sum $=1+\sqrt3$
product $=1\times\sqrt3=\sqrt3$
Question 39 of 52

39. Question
1 point(s)
Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and product of its zeroes as 3, –1 and –3 respectively.
- 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

$\alpha+\beta+\gamma=3\\ \\ \alpha\beta+\beta \gamma+\alpha \gamma=-1\\ \\ \alpha\beta \gamma=-3\\ \\ x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta \gamma+\alpha \gamma)x-\alpha\beta \gamma\\ \\ x^3-3x^2-x+3$
Question 40 of 52

40. Question
1 point(s)
Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and product of its zeroes as 3, –1 and –3 respectively.
- 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

$\alpha+\beta+\gamma=3\\ \\ \alpha\beta+\beta \gamma+\alpha \gamma=-1\\ \\ \alpha\beta \gamma=-3\\ \\ x^3-(\alpha+\beta+\gamma)x^2+(\alpha\beta+\beta \gamma+\alpha \gamma)x-\alpha\beta \gamma\\ \\ x^3-3x^2-x+3$
Question 41 of 52

41. Question
1 point(s)
If $\alpha $ and $\beta $ are zeroes of the quadratic polynomial. $f(x)=x^2-1 $, find a quadratic polynomial whose zeroes and $\dfrac{2\alpha}{\beta} $ and $\dfrac{2\beta}{\alpha} $
- 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

$x^2-1=(x+1)\ (x-1)$
1 and -1 are zeroes
$\alpha=1,\beta=-1\\ \\ \dfrac{2\alpha}{\beta}=\dfrac{2(1)}{-1}=-2\\ \\ \dfrac{2\beta}{\alpha}=\dfrac{2(-1)}{1}=-2\\ \\ \dfrac{2\alpha}{\beta}+\dfrac{2\beta}{\alpha}=-2-2=-4\\ \\ \dfrac{2\alpha}{13}\cdot\dfrac{2\beta}{\alpha}=(-2)\times(-2)=4$
Required polynomial $=x^2-(-4)x+4$
$x^2+4x+4$
Question 42 of 52

42. Question
1 point(s)
If $\alpha $ and $\beta $ are zeroes of the quadratic polynomial. $f(x)=x^2-1 $, find a quadratic polynomial whose zeroes and $\dfrac{2\alpha}{\beta} $ and $\dfrac{2\beta}{\alpha} $
- 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

$x^2-1=(x+1)\ (x-1)$
1 and -1 are zeroes
$\alpha=1,\beta=-1\\ \\ \dfrac{2\alpha}{\beta}=\dfrac{2(1)}{-1}=-2\\ \\ \dfrac{2\beta}{\alpha}=\dfrac{2(-1)}{1}=-2\\ \\ \dfrac{2\alpha}{\beta}+\dfrac{2\beta}{\alpha}=-2-2=-4\\ \\ \dfrac{2\alpha}{13}\cdot\dfrac{2\beta}{\alpha}=(-2)\times(-2)=4$
Required polynomial $=x^2-(-4)x+4$
$x^2+4x+4$
Question 43 of 52

43. Question
1 point(s)
If the square of the difference of the zeroes of the quadratic polynomial $f(x)=x^2+px+45 $ is equal to 144, find the value of $p $.
- 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

$\alpha+\beta=-p\\ \\ \alpha\beta=45\\ \\ (\alpha+\beta)^2-4\alpha\beta=144\\ \\ (-p)^2-4\times45=144\\ \\ p^2=324\Rightarrow p=\pm18$
Question 44 of 52

44. Question
1 point(s)
If the square of the difference of the zeroes of the quadratic polynomial $f(x)=x^2+px+45 $ is equal to 144, find the value of $p $.
- 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

$\alpha+\beta=-p\\ \\ \alpha\beta=45\\ \\ (\alpha+\beta)^2-4\alpha\beta=144\\ \\ (-p)^2-4\times45=144\\ \\ p^2=324\Rightarrow p=\pm18$
Question 45 of 52

45. Question
1 point(s)
If the sum of the zeroes of the quadratic polynomial $f(t)=kt^2+2t+3k $ is equal to their product, find the value of $k $.
- 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

$kt^2+2t+3k$
sum of zeroes $=\dfrac{-2}{k}$
Product of zeroes $=\dfrac{3k}{k}=3$
A/Q $\dfrac{-2}{k}=3\Rightarrow k=\dfrac{-2}{3}$
Question 46 of 52

46. Question
1 point(s)
If the sum of the zeroes of the quadratic polynomial $f(t)=kt^2+2t+3k $ is equal to their product, find the value of $k $.
- 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

$kt^2+2t+3k$
sum of zeroes $=\dfrac{-2}{k}$
Product of zeroes $=\dfrac{3k}{k}=3$
A/Q $\dfrac{-2}{k}=3\Rightarrow k=\dfrac{-2}{3}$
Question 47 of 52

47. Question
1 point(s)
Find the zeroes of the quadratic polynomial $x^2+\dfrac{7}{2}x+\dfrac{3}{4} $, and verify relationship between the zeroes and the coefficients.
- 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

$x^2+\dfrac{7}{2}x+\dfrac{3}{4}=0\\ \\ x^2+\dfrac{7}{2}x=\dfrac{-3}{4}\\ \\ x^2+\dfrac{7}{2}x+\left(\dfrac{7}{4}\right)^2=\dfrac{-3}{4}+\left(\dfrac{7}{4}\right)^2\\ \\ \left(x+\dfrac{7}{4}\right)^2=\dfrac{-12+49}{16}=\dfrac{37}{16}$
$x=\dfrac{-7}{4}\pm\dfrac{\sqrt37}{4}$
root are $\left(\dfrac{-7+\sqrt37}{4}\right)$ and $\left(\dfrac{-7-\sqrt37}{4}\right)$
sum of roots $=\dfrac{-7+\sqrt37+\ -7-\sqrt37}{4}=\dfrac{-7}{2}=\dfrac{-b}{a}$
Product $=\dfrac{3}{4}=\dfrac{c}{a}$
Question 48 of 52

48. Question
1 point(s)
Find the zeroes of the quadratic polynomial $x^2+\dfrac{7}{2}x+\dfrac{3}{4} $, and verify relationship between the zeroes and the coefficients.
- 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

$x^2+\dfrac{7}{2}x+\dfrac{3}{4}=0\\ \\ x^2+\dfrac{7}{2}x=\dfrac{-3}{4}\\ \\ x^2+\dfrac{7}{2}x+\left(\dfrac{7}{4}\right)^2=\dfrac{-3}{4}+\left(\dfrac{7}{4}\right)^2\\ \\ \left(x+\dfrac{7}{4}\right)^2=\dfrac{-12+49}{16}=\dfrac{37}{16}$
$x=\dfrac{-7}{4}\pm\dfrac{\sqrt37}{4}$
root are $\left(\dfrac{-7+\sqrt37}{4}\right)$ and $\left(\dfrac{-7-\sqrt37}{4}\right)$
sum of roots $=\dfrac{-7+\sqrt37+\ -7-\sqrt37}{4}=\dfrac{-7}{2}=\dfrac{-b}{a}$
Product $=\dfrac{3}{4}=\dfrac{c}{a}$
Question 49 of 52

49. Question
1 point(s)
If $\alpha $ and $\beta $ are zeroes of the quadratic polynomial $x^2-6x+a $; find the value of $a $ if $3\alpha+2\beta=20 $.
- 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

$\alpha+\beta=6$ ….(1)
$\alpha\beta=k$ …(2)
on solving we get $\alpha=8,\beta=-2$
$k=8\times-2=-16$
Question 50 of 52

50. Question
1 point(s)
If $\alpha $ and $\beta $ are zeroes of the quadratic polynomial $x^2-6x+a $; find the value of $a $ if $3\alpha+2\beta=20 $.
- 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

$\alpha+\beta=6$ ….(1)
$\alpha\beta=k$ …(2)
on solving we get $\alpha=8,\beta=-2$
$k=8\times-2=-16$
Question 51 of 52

51. Question
1 point(s)
Divide $(6+19x+x^2-6x^2) $ by $(2+5x-3x^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 $(-3x^2+5x+2)\ (2x+3)+0=-6x^3+10x^2+4x-9x^2+15x+6\\ \\ =-6x^3+x^2+19x+6$
Question 52 of 52

52. Question
1 point(s)
Divide $(6+19x+x^2-6x^2) $ by $(2+5x-3x^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 $(-3x^2+5x+2)\ (2x+3)+0=-6x^3+10x^2+4x-9x^2+15x+6\\ \\ =-6x^3+x^2+19x+6$