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Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen

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2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Limits and Derivatives

Kindly consider the following

$(3 x + 5) (1 + t a n x)$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} L e t y = (3 x + 5) (1 + t a n x) \\ \frac{d y}{d x} = \frac{d}{d x} (3 x + 5) (1 + t a n x) \\ = (3 x + 5) \frac{d}{d x} (1 + t a n x) + (1 + t a n x) \frac{d}{d x} (3 x + 5) [\begin{array}{l} ? \frac{d}{d x} [f (x) . g (x)] \\ = f (x) . g' (x) + g (x) . f' (x) \end{array}] \\ = (3 x + 5) s e c^{2} x + (1 + t a n x) (3) \\ = 3 x s e c^{2} x + 5 s e c^{2} x + 3 + 3 t a n x \end{array}$

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2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

Kindly consider the following
${(x + \frac{1}{x})}^{3}$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} L e t y = {(x + \frac{1}{x})}^{3} \\ \frac{d y}{d x} = \frac{d}{d x} {(x + \frac{1}{x})}^{3} \\ = \frac{d}{d x} (x^{3} + \frac{1}{x^{3}} + 3 x + \frac{3}{x}) \\ = \frac{d}{d x} (x^{3} + x^{- 3} + 3 x + 3 . x^{- 1}) = 3 x^{2} - 3 x^{- 4} + 3 - 3 . x^{- 2} \\ = 3 x^{2} - \frac{3}{x^{4}} + 3 - \frac{3}{x^{2}} \end{array}$

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New answer posted

2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

Differentiate each of the functions w.r.t. $x$ in Exercises 29 to 42.

$\frac{x^{4} + x^{3} + x^{2} + x + 1}{x}$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} L e t y = (\frac{x^{4} + x^{3} + x^{2} + x + 1}{x}) \\ ∴ \frac{d y}{d x} = \frac{d}{d x} (\frac{x^{4} + x^{3} + x^{2} + x + 1}{x}) \\ = \frac{d}{d x} (x^{3} + x^{2} + x + \frac{1}{x}) = 3 x^{2} + 2 x + 1 - \frac{1}{x^{2}} \end{array}$

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New answer posted

2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

If $l i m_{x \to 1} \frac{x^{4} - 1}{x - 1} = l i m_{x \to k} \frac{x^{3} - k^{3}}{x^{2} - k^{2}}$ , then find the value of $k$ .

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \underset{x \to 1}{l i m} \frac{x^{4} - 1}{x - 1} = \underset{x \to k}{l i m} \frac{x^{3} - k^{3}}{x^{2} - k^{2}} \\ \Rightarrow 4 {(1)}^{4 - 1} = \underset{x \to k}{l i m} \frac{(x - k) (x^{2} + k^{2} + k x)}{(x - k) (x + k)} \\ \Rightarrow 4 = \underset{x \to k}{l i m} \frac{x^{2} + k^{2} + k x}{x + k} \Rightarrow 4 = \frac{k^{2} + k^{2} + k^{2}}{2 k} \\ \Rightarrow 4 = \frac{3 k^{2}}{2 k} \Rightarrow 4 = \frac{3}{2} k \Rightarrow k = \frac{8}{3} . \end{array}$

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2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

If $l i m_{x \to 1} \frac{x^{4} - 1}{x - 1} = l i m_{x \to k} \frac{x^{3} - k^{3}}{x^{2} - k^{2}}$ , then find the value of $k$ .

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New answer posted

2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

Kindly consider the following
$l i m_{x \to 0} \frac{s i n x - 2 s i n 3 x + s i n 5 x}{x}$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \underset{x \to 0}{l i m} \frac{s i n x - 2 s i n 3 x + s i n 5 x}{x} \\ = \underset{x \to 0}{l i m} \frac{s i n x}{x} - \frac{2 s i n 3 x}{x} + \frac{s i n 5 x}{x} = \underset{x \to 0}{l i m} \frac{s i n x}{x} - \underset{3 x \to 0}{l i m} 2 (\frac{s i n 3 x}{3 x}) * 3 + \underset{5 x \to 0}{l i m} (\frac{s i n 5 x}{5 x}) * 5 \\ = 1 - 6 + 5 = 0 . \end{array}$

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2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

Kindly consider the following

$l i m_{x \to 0} \frac{\sqrt[]{2} - \sqrt[]{1 + c o s x}}{s i n^{2} x}$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \underset{x \to 0}{l i m} \frac{\sqrt[]{2} - \sqrt[]{1 + c o s x}}{s i n^{2} x} \\ = \underset{x \to 0}{l i m} \frac{\sqrt[]{2} - \sqrt[]{1 + c o s x}}{s i n^{2} x} * \frac{\sqrt[]{2} + \sqrt[]{1 + c o s x}}{\sqrt[]{2} + \sqrt[]{1 + c o s x}} \\ = \underset{x \to 0}{l i m} \frac{2 - (1 + c o s x)}{s i n^{2} x [\sqrt[]{2} + \sqrt[]{1 + c o s x}]} = \underset{x \to 0}{l i m} \frac{1 - c o s x}{s i n^{2} x [\sqrt[]{2} + \sqrt[]{1 + c o s x}]} \\ = \underset{x \to 0}{l i m} \frac{2 s i n^{2} x / 2}{{(2 s i n x / 2 c o s x / 2)}^{2}} * \frac{1}{[\sqrt[]{2} + \sqrt[]{1 + c o s x}]} \\ = \underset{x \to 0}{l i m} \frac{2 s i n^{2} x / 2}{4 s i n^{2} x / 2 c o s^{2} x / 2} * \frac{1}{[\sqrt[]{2} + \sqrt[]{1 + c o s x}]} \\ = \underset{x \to 0}{l i m} \frac{2}{4 c o s^{2} x / 2} * \frac{1}{[\sqrt[]{2} + \sqrt[]{1 + c o s x}]} \\ T a k i n g, w e h a v e \\ = \frac{2}{4 c o s^{2} 0} * \frac{1}{[\sqrt[]{2} + \sqrt[]{2}]} = \frac{1}{2} * \frac{1}{2 \sqrt[]{2}} = \frac{1}{4 \sqrt[]{2}} \end{array}$

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2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

Kindly consider the following

$l i m_{x \to \frac{π}{6}} \frac{c o t^{3} x - 3}{c o s e c x - 2}$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \underset{x \to \frac{π}{6}}{l i m} \frac{c o t^{2} x - 3}{c o s e c x - 2} \\ = \underset{x \to \frac{π}{6}}{l i m} \frac{c o s e c^{2} x - 1 - 3}{c o s e c x - 2} = \underset{x \to \frac{π}{6}}{l i m} \frac{c o s e c^{2} x - 4}{c o s e c x - 2} \\ = \underset{x \to \frac{π}{6}}{l i m} \frac{(c o s e c x - 2) (c o s e c x + 2)}{(c o s e c x - 2)} = \underset{x \to \frac{π}{6}}{l i m} (c o s e c x + 2) \\ T a k i n g, w e h a v e \\ = c o s e c \frac{π}{6} + 2 = 2 + 2 = 4 \end{array}$

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New answer posted

2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

Kindly consider the following
$l i m_{x \to a} \frac{s i n x - s i n a}{\sqrt[]{x} - \sqrt[]{a}}$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \underset{x \to a}{l i m} \frac{s i n x - s i n a}{\sqrt[]{x} - \sqrt[]{a}} \\ = \underset{x \to a}{l i m} \frac{s i n x - s i n a}{\sqrt[]{x} - \sqrt[]{a}} * \frac{\sqrt[]{x} + \sqrt[]{a}}{\sqrt[]{x} + \sqrt[]{a}} = \underset{x \to a}{l i m} \frac{(s i n x - s i n a) (\sqrt[]{x} + \sqrt[]{a})}{x - a} \\ = \underset{x \to a}{l i m} \frac{(2 c o s \frac{x + a}{2} . s i n \frac{x - a}{2}) (\sqrt[]{x} + \sqrt[]{a})}{x - a} [? s i n A - s i n B = 2 c o s \frac{A + B}{2} . s i n \frac{A - B}{2}] \\ = \underset{\frac{x - a}{2} \to 0}{l i m} (2 c o s \frac{x + a}{2} . \frac{s i n \frac{x - a}{2}}{2 * \frac{x - a}{2}}) (\sqrt[]{x} + \sqrt[]{a}) \\ = \underset{x \to a}{l i m} c o s (\frac{x + a}{2}) (\sqrt[]{x} + \sqrt[]{a}) [? \underset{\frac{x - a}{2} \to 0}{l i m} \frac{s i n \frac{x - a}{2}}{\frac{x - a}{2}} = 1] \\ T a k i n g, w e h a v e \\ = c o s (\frac{a + a}{2}) (\sqrt[]{a} + \sqrt[]{a}) = c o s a * 2 \sqrt[]{a} = 2 \sqrt[]{a} . c o s a \end{array}$

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New answer posted

2 months ago

Maths NCERT Exemplar Solutions Class 11th Chapter Thirteen Exemplar Solution

Kindly consider the following
$\underset{x \to 0}{l i m} \frac{s i n 2 x + 3 x}{2 x + t a n 3 x}$

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alok kumar singh

Contributor-Level 10

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \underset{x \to 0}{l i m} \frac{s i n 2 x + 3 x}{2 x + t a n 3 x} \\ = \underset{x \to 0}{l i m} \frac{(\frac{s i n 2 x + 3 x}{2 x} * 2 x)}{(\frac{2 x + t a n 3 x}{3 x} * 3 x)} = \underset{x \to 0}{l i m} \frac{(\frac{s i n 2 x}{2 x} + \frac{3 x}{2 x}) * 2 x}{(\frac{2 x}{3 x} + \frac{t a n 3 x}{3 x}) * 3 x} \\ = \frac{(\underset{2 x \to 0}{l i m} \frac{s i n 2 x}{2 x} + \frac{3}{2})}{[\frac{2}{3} + \underset{3 x \to 0}{l i m} \frac{t a n 3 x}{3 x}]} * \frac{2}{3} [? \underset{x \to 0}{l i m} \frac{s i n x}{x} = 1] \\ = (\frac{1 + \frac{3}{2}}{\frac{2}{3} + 1}) * \frac{2}{3} [? \underset{x \to 0}{l i m} \frac{t a n x}{x} = 1] \\ = \frac{5 / 2}{5 / 3} * \frac{2}{3} = \frac{3}{2} * \frac{2}{3} = 1 \end{array}$

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