If A = 1&1&1 1&1&1 1&1&1 , prove that A'' = 3^ n-1 &3^ n-1 &3^ n-1 3^ n-1 &3^ n-1 &3^ n-1 3^ n-1 &3^ n-1 &3^ n-1 n N.

Given: A= 1&1&1 1&1&1 1&1&1 Let p(n): A^ n = 3^ n-1 &3^ n-1 &3^ n-1 3^ n-1 &3^ n-1 &3^ n-1 3^ n-1 &3^ n-1 &3^ n-1 p(1): A= 3^ 0 &3^ 0 &3^ 0 3^ 0 &3^ 0 &3^ 0 3^ 0 &3^ 0 &3^ 0 1&1&1 1&1&1 1&1&1 p(1) is true for n = 1. Now p(k): A^ k = 3^ k-1 &3^ k-1 &3^ k-1 3^ k-1 &3^ k-1 &3^ k-1 3^ k-1 &3^ k-1 &3^ k-1 ...(ii) Multiplying eq. (ii) by eq. (i) A^ k A= 3^ k-1 &3^ k-1 &3^ k-1 3^ k-1 &3^ k-1 &3^ k-1 3^ k-1 &3^ k-1 &3^ k-1 1&1&1 1&1&1 1&1&1 A^ k+1= 3^ k &3^ k &3^ k 3^ k &3^ k &3^ k 3^ k &3^ k &3^ k =p(k+1) Therefore, p(n) is true for all natural numbers by P.M.I.

If A = 1&1&1 1&1&1 1&1&1 , prove that A'' = 3^ n-1 &3^ n-1 &3^ n-…

Find all points of discontinuity of f, where f is defined by:
$\text f(\text x)=\begin{cases}\left|\text x\right|+3, \text{if x}\leq-3\\-2 \text{x},\text{if}-3 < \text x > 3\\6\text{x}+2,\text{if}\ \text{x}\geq3\end{cases}$

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Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}(3+2\cos\text{x})}\ \text{dx}$

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Evaluvate the following intregals:
$\int\frac{1}{1-\cot\text{x}}\text{ dx}$

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Find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show five.

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Using Lagrange's mean value theorem, prove that
$(\text{b}-\text{a})\sec^2\text{a}<\tan\text{b}-\tan\text{a}<(\text{b}-\text{a})\sec^2\text{b}$
where $0<\text{a}<\text{b}<\frac{\pi}{2}.$

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Find the angle of intersecting of the following curves:
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\text{x}^2+\text{y}^2=\text{ab}$

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Using properties of determinants show that $\begin{vmatrix} 1 & 1 & \text{1 + x} \\ 1 & \text{1 + y} & 1 \\ \text{1 + z} & 1 & 1 \end{vmatrix} = \text{xyz + yz + zx + xy}.$

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Evaluate the following integrals:
$\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\sin^2\text{x dx}$

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Find the equation of the plane which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 and whose x-intercept is twice its z-intercept.
Hence write the vector equation of a plane passing through the point (2, 3, –1) and parallel to the plane obtained above.

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If $\big|\vec{\text{a}}+\vec{\text{b}}\big|=60,\big|\vec{\text{a}}-\vec{\text{b}}\big|=40$ and $\big|\vec{\text{b}}\big|=46,$ find $|\vec{\text{a}}|$

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