3 Marks

Question 13 Marks

Prove that if a plane has the intercepts $a, b, c$ and is at a distance of $p$ units from the origin, then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{p^2}$.

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Question 23 Marks

Find the mean of the Binomial distribution $B\left(4, \frac{1}{3}\right)$.

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Question 33 Marks

Solve the following linear programming problem graphically.
Minimise $Z=-3 x+4 y$
$\begin{array}{ll}\text { Subject to } & x+2 y \leq 8, \\ & 3 x+2 y \leq 12, \\ & x \geq 0, y \geq 0\end{array}$

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Question 43 Marks

A line makes angles $\alpha, \beta, \gamma$ and $\delta$ with the diagonals of a cube. Prove that $\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma+\cos ^2 \delta=\frac{4}{3}$.

Answer

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Question 53 Marks

If $x=a(\cos t+t \sin t), y=a(\sin t-t \cos t)$ find $\frac{d^2 y}{d x^2}$.

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Question 63 Marks

If $A=\left[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & -1 \\ 1 & 2 &3\end{array}\right]$ then show that matrix $A$ satisfy the equation $
A^3-4 A^2-3 A+11 I=0
$

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Question 73 Marks

Let $f: N \rightarrow R$ be a function defined as $f(x)=4 x^2+12 x+15$. Show that $f: N \rightarrow S$, where S is the range of $f$, is invertible. Find the inverse of $f$.

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Maths 3 Marks

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