2 Marks Questions

Question 12 Marks

Write the value of $\Big(\frac{\text{dy}}{\text{dx}}\Big),$ if the normal to the curve y = f(x) at (x, y) is parallel to y-axis.

Answer

The slope of the y-axis is $\infty.$Also, the tangent at a point (x, y) on the curve y = f(x) is parallel to the y-axis.
$\therefore$ Slope of the normal = Slope of the y-axis = $\infty$
$\Rightarrow\frac{\text{dy}}{\text{dx}}$ Slope of the tangent $=\frac{1}{\text{slope of the normal}}=\frac{-1}{\infty}=0$

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Question 22 Marks

If the tangent to a curve at a point (x, y) is equally inclined to the co-ordinates axes then write the value of $\frac{\text{dy}}{\text{dx}}.$

Answer

Since. the tangent to a curve y = f(x) at (x, y) is equally inclined to the co-ordiante axes.
$\therefore\theta=45^\circ\text{or }\theta-45^\circ$
$\therefore\text{m}=\tan45^\circ\text{or }\text{m}=-\tan45^\circ=\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=1\text{ or }\frac{\text{dy}}{\text{dx}}=-1$

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Question 32 Marks

Write the equation of the tangent drawn to the curve $\text{y}=\sin\text{x}$ at the point (0, 0).

Answer

We have,$\text{y}=\sin\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\cos\text{x}$
Slope at (0, 0) $=\text{m}=\Big[\frac{\text{dy}}{\text{dx}}\Big]_{\text{x}=0}=\cos0=1$
So, the equation of the tangent at (0, 0) is given by,
y = mx
putting m = 1, we get
The equation of the tangent is y = x.

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Question 42 Marks

Write the slop of the normal to the curve $\text{y}=\frac{1}{\text{x}}$ at the point $\Big(3,\frac{1}{3}\Big).$

Answer

$\text{y}=\frac{1}{\text{x}}$
On differentiating both sides w.r.t. x, we get
$\frac{\text{dy}}{\text{dx}}=\frac{-1}{\text{x}^2}$
Now,
Slope of the tangent $=\Big(\frac{\text{dy}}{\text{dx}}\Big)_{\big(3,\frac{1}{3}\big)}=\frac{-1}{9}$
Slope of the normal $=\frac{-1}{\text{slope of tangent}}=\frac{-1}{\big(\frac{-1}{9}\big)}=9$

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Question 52 Marks

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of $\frac{\text{dy}}{\text{dx}}.$

Answer

The slope of the x-axis is $\infty.$
Also, the tangent at a point (x, y) on the curve y = f(x) is parallel to the x-axis.
$\therefore$ Slope of the tangent $\Big(\frac{\text{dy}}{\text{dx}}\Big)$ = Slope of the x-axis = $\infty.$
$\frac{\text{dx}}{\text{dy}}=\frac{1}{\Big(\frac{\text{dy}}{\text{dx}}\Big)}=\frac{1}{\infty}=0$

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Question 62 Marks

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of $\Big(\frac{\text{dy}}{\text{dx}}\Big).$

Answer

The slope of the x-axis is 0.
Also, the tangent at a point (x, y) on the curve y = f(x) is parallel to the x-axis.
$\therefore$ Slope of the tangent $\Big(\frac{\text{dy}}{\text{dx}}\Big)$ = Slope of the x-axis = 0

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