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Integration Examples

Here you will learn some integration examples for better understanding of integration concepts. Example 1 : Evaluate : \(\int\) \(dx\over {3sinx + 4cosx}\) Solution : I = \(\int\) \(dx\over {3sinx + 4cosx}\) = \(\int\) \(dx\over {3[{2tan{x\over 2}\over {1+tan^2{x\over 2}}}] + 4[{1-tan^2{x\over 2}\over {1+tan^2{x\over 2}}}]}\) = \(\int\) \(sec^2{x\over 2}dx\over {4+6tan{x\over 2}-4tan^2{x\over 2}}\) let \(tan{x\over 2}\) = …

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Hyperbola Examples

Here you will learn some hyperbola examples for better understanding of hyperbola concepts. Example 1 : If the foci of a hyperbola are foci of the ellipse \(x^2\over 25\) + \(y^2\over 9\) = 1. If the eccentricity of the hyperbola be 2, then its equation is : Solution : For ellipse e = \(4\over 5\), …

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Function Examples

Here you will learn some function examples for better understanding of function concepts. Example 1 : Find the range of the given function \(log_{\sqrt{2}}(2-log_2(16sin^2x+1))\) Solution : Now 1 \(\le\) \(16sin^2x\) + 1) \(\le\) 17 \(\therefore\)   0 \(\le\) \(log_2(16sin^2x+1)\) \(\le\) \(log_217\) \(\therefore\)   2 – \(log_217\) \(\le\) 2 – \(log_2(16sin^2x+1)\) \(\le\) 2 Now consider 0 …

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Ellipse Examples

Here you will learn some ellipse examples for better understanding of ellipse concepts. Example 1 : Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \(\sqrt{5}\) Solution : Here S = (2, 3) & S’ is (-2, 3) and b = \(\sqrt{5}\) \(\implies\) SS’ …

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Circle Examples

Here you will learn some circle examples for better understanding of circle concepts. Example 1 : Find the equation of the normal to the circle \(x^2 + y^2 – 5x + 2y -48\) = 0 at the point (5,6). Solution : Since the normal to the circle always passes through the centre so the equation …

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Nilpotent Matrix – Definition and Example

Here you will learn what is nilpotent matrix with examples. Let’s begin – Nilpotent Matrix A square matrix of the order ‘n’ is said to be a nilpotent matrix of order m, m \(\in\) N if \(A^m\) = O & \(A^{m-1}\) \(\ne\) O. Example : Show that A = \(\begin{bmatrix} 1 & 1 &  3 …

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