If A= [ cc 2 & -4 3 & -2 0 & 1 ], B= [ ccc 1 & -1 & 2 -2 & 1 & 0 ] then showthat (A B)^ =B^ A^ .

& AB = [ cc 2 & -4 3 & -2 0 & 1 ] [ ccc 1 & -1 & 2 -2 & 1 & 0 ] & = [ ccc 2+8 & -2-4 & 4+0 3+4 & -3-2 & 6-0 0-2 & 0+1 & 0+0 ] & = [ ccc 10 & -6 & 4 7 & -5 & 6 -2 & 1 & 0 ] & ( AB )^ = [ ccc 10 & 7 & -2 -6 & -5 & 1 4 & 6 & 0 ] & Now,AT = [ ccc 2 & 3 & 0 -4 & -2 & 1 ] and B^ = [ cc 1 & -2 -1 & 1 2 & 0 ] & B ^ A ^ = [ cc 1 & -2 -1 & 1 2 & 0 ] [ ccc 2 & 3 & 0 -4 & -2 & 1 ] & = [ ccc 2+8 & 3+4 & 0-2 -2-4 & -3-2 & 0+1 4-0 & 6-0 & 0+0 ] & B ^ A ^ = [ ccc 10 & 7 & -2 -6 & -5 & 1 4 & 6 & 0 ] From (i) and (ii), we get (A B)^ =B^ A^ .

If A= [ cc 2 & -4 3 & -2 0 & 1 ], B= [ ccc 1 & -1 & 2 -2 & 1 & 0 …

Calculate the regression equations of X on Y and Y on X from the following date:

X	10	12	13	17	18
Y	5	6	7	9	13

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Assuming the first statement as p and second as q, write the following statements in symbolic form:
(i) The Sun has set and Moon has risen.
(ii) Mona likes Mathematics and Physics.
(iii) 3 is a prime number if 3 is a perfect square number.
(iv) Kavita is brilliant and brave.
(v) If Kiran drives a car, then Sameer will walk.
(vi) The necessary condition for the existence of a tangent to the curve of the function is continuity.
(vii) To be brave is necessary and sufficient condition to climb Mount Everest.

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In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hours, find the number of times the bacteria are increased in 12 hours.

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Solve y $\log y \frac{d x}{d y}+ x -\log y =0$

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The following table gives the aptitude test scores and productivity indices of 10 works selected at workers selected randomly.

Aptitude score (X)	60	62	65	70	72	48	53	73	65	82
Productivity Index (Y)	68	60	62	80	85	40	52	62	60	81

Obtain the two regression equation and estimate
(i) The productivity index of a worker whose test score is 95.
(ii) The test score when productivity index is 75.

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If $a x^2+2 h x y+b y^2=0$, then show that $\frac{d^2 y}{d x^2}=0$.

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Write the negations of the following statements:
(i) 7 is a prime number and the Taj Mahal is in Agra.
(ii) 10 > 5 and 3 < 8.
(iii) I will have tea or coffee.
(iv) ∀n ∈ N, n + 3 > 9.
(v) ∃x ∈ A, such that x + 5 < 11.

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If the two regression lines for a bivariate data are $2 x=y+15(x$ on $y)$ and $4 y-3 x+25(y$ on $x$ ) find
(i) $\bar{x}$
(ii) $\bar{y}$
(iii) $b_{y x}$
(iv) $b_{x y}$
(v) $r$ [Given $\sqrt{ } 0.375=0.61$ ]

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Solve the following equations by method of inversion : $x+y-z=2, x-2 y+z=3$ and $2 x-y-3 z=-1$

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Which of the following sentences are statements? In case of a statement, write down the truth value:
(i) The Himalayas is the highest mountain range.
(ii) $(x – 2)(x – 3) = x^2 – 5x + 6$ for all $x \in R.$
(iii) What are the causes of rural unemployment?
(iv) $0! = 1.$
(v) The quadratic equation $ax^2 + bx + c = 0 (a \neq 0)$ always has two real roots.

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