MCQ
Choose the correct answer from the given four options in the following questions : The number of polynomials having zeroes as $-2$ and $5$ is :
- A$1.$
- B$2.$
- C$3.$
- ✓More than $3.$
✓
Answer
Correct option: D.
More than $3.$
Let $p(x) = ax^2+ bx + c$ be the required polynimial whose zeroes are $-2$ and $5$.
$\therefore$ Sum of zeroes $=\frac{-\text{b}}{\text{a}}$
$\Rightarrow\ \frac{-\text{b}}{\text{c}}=-2+5=\frac{3}{1}=\frac{-(-3)}{1}\ .....(\text{i})$
and Procudt of zeroes $=\frac{\text{c}}{\text{a}}$
$\Rightarrow\ \frac{\text{c}}{\text{a}}=-2\times5=\frac{-10}{1}\ .....(\text{ii})$
From Eqs. $(i)$ and $(ii)$
$a = 1, b = -3$ and $c = -10$
$\therefore p(x) = ax^2+ bx + c $
$= 1.x^2- 3x - 10$
$= x^2- 3x - 10$
But we know that. if we multiply/divide any polynomial by any arbitraru constant.
Then, the zeroes of polynomial never change.
$\therefore p(x) = kx^2- 3kx - 10k \ [$where, $k$ is a real number$]$
$\therefore\ \text{p}(\text{x})=\frac{\text{x}^2}{\text{k}}-\frac{3}{\text{k}}\text{x}-\frac{10}{\text{k}}, [$where $k$ is a nonzero real number$].$
$\therefore$ Sum of zeroes $=\frac{-\text{b}}{\text{a}}$
$\Rightarrow\ \frac{-\text{b}}{\text{c}}=-2+5=\frac{3}{1}=\frac{-(-3)}{1}\ .....(\text{i})$
and Procudt of zeroes $=\frac{\text{c}}{\text{a}}$
$\Rightarrow\ \frac{\text{c}}{\text{a}}=-2\times5=\frac{-10}{1}\ .....(\text{ii})$
From Eqs. $(i)$ and $(ii)$
$a = 1, b = -3$ and $c = -10$
$\therefore p(x) = ax^2+ bx + c $
$= 1.x^2- 3x - 10$
$= x^2- 3x - 10$
But we know that. if we multiply/divide any polynomial by any arbitraru constant.
Then, the zeroes of polynomial never change.
$\therefore p(x) = kx^2- 3kx - 10k \ [$where, $k$ is a real number$]$
$\therefore\ \text{p}(\text{x})=\frac{\text{x}^2}{\text{k}}-\frac{3}{\text{k}}\text{x}-\frac{10}{\text{k}}, [$where $k$ is a nonzero real number$].$
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→Here, the maximum frequency is 32 and the corresponding modal class is $30-40$.
$\therefore \quad l=30, h=10, f_0=12, f_1=32, f_2=20($ Step 1$)$
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In which step is there an error in solving?
| Marks | Number of students |
| 0-10 | 6 |
| 10-20 | 10 |
| 20-30 | 12 |
| 30-40 | 32 |
| 40-50 | 20 |
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