CBSE mathematics

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If ab denotes the larger of 'a' and 'b' and if a o b = (ab) + 3, then write the value of (5) o (10), where and o are binary operations.

Solution

(5) o (10) = (5*10) + 3

= 10 + 3 = 13

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If the matrix $A = [\begin{matrix} 0 & a & - 3 \\ 2 & 0 & - 1 \\ b & 1 & 0 \end{matrix}]$ is skew symmertic, find the values of 'a' and 'b'.

Solution

Given, A is skew-symmetric,

∴ $(A^{T} = - A) [\begin{matrix} 0 & 2 & b \\ a & 0 & 1 \\ - 3 & - 1 & 0 \end{matrix}] [\begin{matrix} 0 & - a & 3 \\ - 2 & 0 & 1 \\ - b & - 1 & 0 \end{matrix}] Hence, a_{12} = - a_{21} \Rightarrow a = - 2 and a_{31} = - a_{13} \Rightarrow b = 3$

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Find the value of $\tan^{- 1} \sqrt{3} - c o t^{- 1} (- \sqrt{3})$

Solution

$\tan^{- 1} (\sqrt{3}) - \cot^{- 1} (- \sqrt{3}) = y as \cot^{- 1} (- x) = π - \cot^{- 1} x ∴ y = \tan^{- 1} (\sqrt{3}) - (π - \cot^{- 1} (\sqrt{3})) = \frac{π}{3} - (π - \frac{π}{6}) = \frac{π}{3} - π + \frac{π}{6} = \frac{π}{2} - π = - \frac{π}{2}$

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The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ - 0.02 x² + 30 x +5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of total cost at any level of output.

Solution

Given,

C(x) = 0.05x³ - 0.02 x² + 30 x +5000

Marginal cost (C_M),

$\frac{d}{dx} (C (x)) = 0.005 x 3 x^{2} - 0.02 x 2 x + 30 ∵ x = 3 C_{M} = 0.005 x 3 x 9 - 0.02 x 2 x 3 + 30 = 0.135 - 0.12 = 30.015$

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Differentiate $\tan^{- 1} (\frac{1 + \cos x}{\sin x})$ with respect x.

Solution

${Let f(x) = tan}^{- 1} (1 + \frac{\cos x}{\sin x}) ∴ f' (x) = \frac{1}{1 + {(\frac{1 + \cos x}{\sin x})}^{2}} x \frac{d}{dx} (\frac{1 + \cos x}{\sin x}) = \frac{\sin^{2} x}{\sin^{2} x + (1 + \cos x)^{2}} x \frac{\sin x (- sinx) - (1 + \cos x) cosx}{\sin^{2} x} = \frac{\sin^{2} x}{\sin^{2} x + 1 + \cos^{2} x + 2 \cos x} x \frac{(- \sin^{2} x - \cos x - \cos^{2} x)}{\sin^{2} x} = - \frac{\cos x - 1}{1 + 1 + 2 \cos x} = \frac{- (1 + \cos x)}{2 (1 + \cos x)} = \frac{- 1}{2}$

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CBSE mathematics

If ab denotes the larger of 'a' and 'b' and if a o b = (ab) + 3, then write the value of (5) o (10), where and o are binary operations.

If the matrix $A = [\begin{matrix} 0 & a & - 3 \\ 2 & 0 & - 1 \\ b & 1 & 0 \end{matrix}]$ is skew symmertic, find the values of 'a' and 'b'.

Find the value of $\tan^{- 1} \sqrt{3} - c o t^{- 1} (- \sqrt{3})$

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ - 0.02 x² + 30 x +5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of total cost at any level of output.

Differentiate $\tan^{- 1} (\frac{1 + \cos x}{\sin x})$ with respect x.

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CBSE mathematics

If a*b denotes the larger of 'a' and 'b' and if a o b = (a*b) + 3, then write the value of (5) o (10), where and o are binary operations.

If the matrix A = 0a-320-1b10 is skew symmertic, find the values of 'a' and 'b'.

Find the value of tan-1 3 - cot-1 (-3)

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x3 - 0.02 x2 + 30 x +5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of total cost at any level of output.

Differentiate tan-1 1 + cos xsin x with respect x.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

If ab denotes the larger of 'a' and 'b' and if a o b = (ab) + 3, then write the value of (5) o (10), where and o are binary operations.

If the matrix $A = [\begin{matrix} 0 & a & - 3 \\ 2 & 0 & - 1 \\ b & 1 & 0 \end{matrix}]$ is skew symmertic, find the values of 'a' and 'b'.

Find the value of $\tan^{- 1} \sqrt{3} - c o t^{- 1} (- \sqrt{3})$

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ - 0.02 x² + 30 x +5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of total cost at any level of output.

Differentiate $\tan^{- 1} (\frac{1 + \cos x}{\sin x})$ with respect x.