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 \(tan{x\over 2}\) = …
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 = \(4\over 5\), …
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 …
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’ …
Here you will learn some differentiability examples for better understanding of differentiability concepts. Example 1 : If f(x + y) = f(x) + f(y) – 2xy – 1 for all x and y. If f'(0) exists and f'(0) = -sin\(\alpha\), then find f{f'(0)}. Solution : f'(x) = \(\displaystyle{\lim_{h \to 0}}\) \(f(x+h) – f(x)\over h\) = …
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 …
Here you will learn what is involutory matrix with examples. Let’s begin – Involutory Matrix If \(A^2\) = I . the matrix A is said to be an involutory matrix, i.e. the square roots of the identity matrix (I) is involutory matrix. Note : The determinant value of this matrix (A) is 1 or -1. …
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 …
Here you will learn what is periodic matrix with examples. Let’s begin – Periodic Matrix A square matrix which satisfies the relation \(A^{k+1}\) = A for some positive integer k, is called a periodic matrix. The period of the matrix is the least value of k for which \(A^{k+1}\) = A holds true. Note that …
Here you will learn what is idempotent matrix with examples. Let’s begin – Idempotent Matrix A square matrix is idempotent matrix provided \(A^2\) = A. For this matrix note the following : (i) \(A^n\) = A \(\forall\) n \(\ge\) 2, n \(\in\) N. (ii) The determinant value of this matrix is either 1 or 0. …
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