Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.

Ten eggs are drawn successively, with replacement, from a lot con…

Show that the points whose position vectors are as given below are collinear:
$3\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}},\ \hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$

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Let * be a binary operation on Q - {-1} defined by a * b = a + b + ab for all a, b ∈ Q - {-1}. Then,
Show that '*' is both commutative and associative on Q - {-1}.

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Find the intervals in which the function f given by $f(x) = x ^ { 3 } + \frac { 1 } { x ^ { 3 } } , x \neq 0$ is $x (i)$ increasing. $(ii)$ decreasing.

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Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\text{|x|}\cos\Big(\frac{1}{\text{x}}\Big), & \text{ x}\neq 0\\0 &\text{ x} = 0\end{cases}\text{at x}=0$

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Prove the following:$\tan^{-1}\text{x}+\tan^{-1}\Bigg(\frac{\text{2x}}{\text{1 - x}^{2}}\Bigg)=\tan^{-1}\Bigg(\frac{\text{3x - x}^{2}}{\text{1 - 3x}^{2}}\Bigg)$.

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Prove the following results
$\tan\Big(\cos^{-1}\frac{4}{5}+\tan^{-1}\frac{2}{3}\Big)=\frac{17}{6}$

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If $x=\sqrt{a^{\sin ^{-1} t}}$ and $y=\sqrt{a^{\cos ^{-1} t}}$ then prove that $\frac{d y}{d x}=-\frac{y}{x}$.

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Solve graphically the following linear programming problem:
Maximise $z = 6x + 3y,$
Subject to the constraints:
$4 x+y \geq 80$
$3 x+2 y \leq 150$
$x+5 y \geq 115$
$x > 0, y \geq 0$

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Find the mean and standard deviation of the following probability distributions:

$x_i$	2	3	4
$p_i$	0.2	0.5	0.3

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Prove that the greatest integer function defined by $f\left( x \right) = \left[ x \right],0 < x < 3$ is not differentiable at x = 1 and x = 2.

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