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1
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2.4k
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1
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Prove that $\cos^6\theta + \sin^6\theta =\dfrac 18 (3\cos4\theta+5) $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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2.0k
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IF $\alpha,\beta$ are the roots of the equation $x^2-2\sqrt3 x+4=0$ find the value of $\alpha^3+\beta^3$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
5
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1
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8.6k
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Considering only the principal value, if $ (1+itan \alpha )^{1+itan\beta} $ is real, prove that its value is $ (sec \alpha)^{sec^2 \beta} $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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4.4k
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If $ tan \Big[ log(x+iy) \Big] \;=\; a+ib , $ prove that $ tan[log(x^2+y^2)] \;=\; \dfrac{2a}{1-a^2-b^2} \; \; \; where \; a^2+b^2 \ne 1 $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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If $ \alpha=1+i \;,\; \beta=1-i \; and \; cot \theta \;=\; x+1 \;,\; $ Prove that $ (x+\alpha)^n + (x+\beta)^n\;=\; (\alpha+\beta) cosn\theta \; cosec^n\theta $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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4.6k
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Show that for real values of a and b, $ e^{2aicot^{-1} b } \Bigg[ \dfrac{bi-1}{bi+1} \Bigg]^{-a} \;=\; 1 $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
5
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If $i^{i^{i^{\ldots \infty }}} \;=\; A+iB $ , then then prove that $tan( \dfrac{\pi A}{2} )= \dfrac{B}{2}\; and \; A^2+B^2=e^{-\pi B} $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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If $ y=(x-1)^n $ then prove that $ y+ \dfrac{y_1}{1!} + \dfrac{y_2}{2!} + \dfrac{y_3}{3!}+\ldots+\dfrac{y_n}{n!} \; = \; x^n $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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7.1k
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Expansion Standard Series : Prove that $log(sec x) = \ \dfrac{x^2}{2} + \dfrac{x^4}{12} + \dfrac{x^6}{45}+\ldots $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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Expand $ \dfrac{x}{e^x-1} \; in \; powers \; of \; x. \\ Hence \; Prove \; \dfrac{x}{2} \bigg[ \dfrac{e^x+1}{e^x-1} \bigg] \; = \; 1 + \dfrac{1}{12}x^2-\dfrac{1}{720}x^4+\ldots $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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11k
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Expansion Standard Series : Use Taylor$'$s Theorem and arrange the equation in powers of $ x \; : \; 7 + (x+2) + 3(x+2)^3 + (x+2)^4 - (x+2)^5$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.4k
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Expansion Standard Series : Find the series expansion of log(1+x) power of x. Hence Prove that log x = $ (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3\ldots $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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7.6k
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Expansion Standard Series : Prove that log($ 1-x+x^2 ) \; = \; -x + \frac{x^2}{2} + \frac{2x^3}{3} $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
1
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13k
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1
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Expansion Standard Series : Expand $log(1+x+x^2+x^3) \; in \; powers \; of \; x \; upto \; x^8 $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.3k
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Expansion Standard Series $Show \; that \; log[ 1 \; + \; sin \; x] \; = \; x \; - \; \frac{x^2}{2} \; + \; \frac{x^3}{6} \; + \; \ldots $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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Relation between circular and hyperbolic function : If $tan \frac{x}{2} \; = \; tanh\frac{u}{2}$ , then show that u = $log \; tan \Big( \frac{\pi}{4} + \frac{x}{2}\Big)$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.5k
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Relation between circular and hyperbolic function : If tanh$ \; x = \frac{2}{3}$ , find the value of $x$ and then cosh $2x $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.6k
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Relation between circular and hyperbolic function : Prove that $sec \;h^{-1}(sin\theta) = log(cot(\theta/2))$
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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1.6k
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Relation between circular and hyperbolic function : Prove $ \tfrac{1}{ 1 - \tfrac{1}{1 - \tfrac{1}{cosh^2 x}} } = cosh^2 x $
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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Relation between circular and hyperbolic function : If cosh $x$ = sec$\theta$ , prove that $x$ = log(sec$\theta$ + tan$\theta$ )
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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Relation between circular and hyperbolic function : If $\alpha$ + $\iota\beta$ = tanh($\chi$ + $\iota \frac{\pi}{4}$) , prove that $\alpha^2 + \beta^2$ = 1
engineering mathematics updated 6.0 years ago by teamques10 ★ 70k
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5.0k
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Explain Design for Energy Efficiency principle of Green Chemistry.
advanced engineering chemistry updated 6.0 years ago by ddigantarroy • 0
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4.0k
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Give the conventional and green chemistry route for production of Ibuprofen.
advanced engineering chemistry updated 6.0 years ago by ddigantarroy • 0
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What are Green solvents? Give two industrial applications of Green solvents.
advanced engineering chemistry updated 6.0 years ago by ddigantarroy • 0
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Why is it essential to design safer chemicals and products w.r.t. green chemistry principle? Explain with an example.
advanced engineering chemistry updated 6.0 years ago by ddigantarroy • 0
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