If A= [ ccc -1 & 2 & 1 -3 & 2 & -3 ] and B= [ cc 2 & 1 -3 & 2 -1 & 3 ], prove that (A+B^ )^ = A^ +B

& A= ( rrr -1 & 2 & 1 -3 & 2 & -3 ], B= [ rr 2 & 1 -3 & 2 -1 & 3 ] & A ^ T = [ rr -1 & -3 2 & 2 1 & -3 ], B ^ T = ( rrr 2 & -3 & -1 1 & 2 & 3 ) & A + B ^ r = ( rrr -1 & 2 & 1 -3 & 2 & -3 )+ ( rrr 2 & -3 & -1 1 & 2 & 3 ) & = [ rrr -1+2 & 2-3 & 1-1 -3+1 & 2+2 & -3+3 ] & = ( rrr 1 & -1 & 0 -2 & 4 & 0 ) & ( A + B ^ T )^ T = ( rr 1 & -2 -1 & 4 0 & 0 ) & A^T+B= ( rr -1 & -3 2 & 2 1 & -3 )+ ( rr 2 & 1 -3 & 2 -1 & 3 ) & = ( rr -1+2 & -3+1 2-3 & 2+2 1-1 & -3+3 ]= [ rr 1 & -2 -1 & 4 0 & 0 ) & From (1) and (2), (A+B^ )^ =A^ +B

If A= [ ccc -1 & 2 & 1 -3 & 2 & -3 ] and B= [ cc 2 & 1 -3 & 2 -1 …

The probability distribution of a discrete r.v.X is as follows.

x	1	2	3	4	5	6
P(X = x)	k	2k	3k	4k	5k	6k

Complete the following activity.
Solution: Since $\sum p_i=1$
$P(X \geq 3)=\square-\square-\square=\square$

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A company manufactures two types of ladies dresses C and D. The raw material and labour available per day is given in the table.

Resources	Dress C(x)	Dress D(y)	Max. availability
Raw material	5	4	60
Labour	5	3	50

P is the profit, if P = 50x + 100y, solve this LPP to find x and y to get the maximum profit

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If $x =\frac{4 t }{1+ t ^2}, y =3\left(\frac{1- t ^2}{1+ t ^2}\right)$, then show that $\frac{ d y}{ d x}=\frac{-9 x}{4 y}$

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For the differential equation, find the particular solution $\frac{ d y}{ d x}=(4 x + y +1)$, when $y =1, x =0$

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A random variable $X$ has the following probability distribution.

X	$1$	$2$	$3$	$4$	$5$	$6$	$7$
P(x)	$K$	$2K$	$2K$	$3K$	$K^2$	$2K^2$	$7 K^2+K$

Determine (i) $k$, (ii) $P(X < 3)$, (iii) $P(X > 6)$, (iv) $P(0 < X < 3)$.

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The manufacturing company produces x items at the total cost of ₹(180 + 4x). The demand function for this product is P = (240 – x). Find x for which
(i) revenue is increasing
(ii) profit is increasing.

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A chemist has a compound to be made using 3 basic elements X, Y, Z so that it has at least 10 litres of X, 12 litres of Y and 20 litres of Z. He makes this compound by mixing two compounds (I) and (II). Each unit compound (I) had 4 litres of X, 3 litres of Y. Each unit compound (II) had 1 litre of X, 2 litres of Y and 4 litres of Z. The unit costs of compounds (I) and (II) are ₹ 400 and ₹ 600 respectively. Find the number of units of each compound to be produced so as to minimize the cost

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Calculate Walsh’s Price Index Number for the following data.

COMMODITY	BASE YEAR		CURRENT YEAR
COMMODITY	PRICE $p_0$	QUANTITY $q_0$	PRICE $p_1$	QUANTITY $q_0$
I	$8$	$9$	$12$	$25$
II	$10$	$4$	$20$	$16$

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Prove that each of the following statement pattern is a tautology :
(i) (p ∧ q) → q
(ii) (p → q) ↔ (~q → ~p)
(iii) (~p ∧ ~q) → (p → q)
(iv) (~p ∨ ~q) ↔ ~(p ∧ q)

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$\int \frac{1}{\sqrt{x^2-8 x-20}} d x$

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