Question

Q.3) Laplace transform of t2 sin(2t).

a) [(12S^2-16)/((S^2+4)^4 )]

b) [(3S^2-4)/((S^2+4)^3 )]]

c) [(12S^2-16)/((S^2+4)^6 )]

d) [(12S^2-16)/((S^2+4)^3 )]​

Answer 1

Answer:

₹ .

Q. 4. Dipali and Rajshri are partners in a fi

rm sharing profits and losses in the ratio

of 3 : 2. They decided to dissolve their firm on 31st March, 2018, when their balance

sheet was as under:

Liabilities

Assets

Capital Accounts:

Freehold Property

16,000

Dipali

17,500

Investments

4,000

Rajshri

10,000 27,500 Sundry Debtors

2,000

Sundry Creditors

2,000 Stock

3,000

Profit & Loss A/c

1,500 Cash at Bank

6,000

31,000

31,000

Dipali took over the investments at an agreed value of 3,800, other assets were

realised as follows: Freehold property 18,000; Sundry Debtors 1,800 and Stock

2,800.

Creditors of the firm agreed to accept 5% less. Expenses of realisation of assets

amounted to 400. There was a type-writer in the firm bought out of the firm's money

but the same has not been shown in the above balance sheet. The type-writer is now

sold for 10,000.

Close the firm's books of accounts by preparing a realisation account, partners'

capital accounts and bank account.

Answer 2

Answer:

d)  $[\frac{12s^{2}-16}{(s^{2}+4)^{3}} ]$

Step-by-step explanation:

Given:- The given expression is t² sin( 2t ).

To find:- Laplace transformation of the above expression.

Solution:-

The Laplace transform of a function is represented by L{f( t )} or F(s). Laplace transform helps us solve the differential equations, where it reduces the differential equations into an algebraic problem. It is the integral transform of any given derivative function with real variable t to convert into a complex function with variable s.

We know that,   $L( t^{n} f(t)) = (-1 )^{n} \frac{d^{n}F_{s} }{d_{s^{n}}}$   ------- ( 1 )

In this problem , the given expression is t² sin( 2t ), here n = 2

If f( t ) = sin2t, then F( s ) = $\frac{2}{s^{2}+4}$

Now putting all the given values in equation ( 1 ),

$L( t^{2} sin(2t)) = (-1 )^{2}( \frac{d^{2}(\frac{2}{s^{2}+4}) }{d_{s^{2}}})$

                   = $\frac{d^{2}}{d_{s^{2}}} (\frac{2}{s^{2}+4} )$     [ ∵ ( -1 )² = + 1 ]

                   = $\frac{d}{ds} (\frac{(s^{2}+4).0-2(2s)}{(s^{2}+4)^{2}} )$

                   = $-4[\frac{(s^{2}+4)^{2}-2s(s^{2}+4)2s}{(s^{2}+4)^{4}} ]$

                   = $-4[\frac{(s^{2}+4)-4s^{2}}{(s^{2}+4)^{3}} ]$

                   = $-4[\frac{4 - 3s^{2}}{(s^{2}+4)^{3}} ]$

                   =  $[\frac{12s^{2}-16}{(s^{2}+4)^{3}} ]$

Hence, the Laplace transformation of t²sin( 2t ) = [(12s² - 16) / (s² + 4)³]

Therefore, the correct option is d)  [(12s² - 16) / (s² + 4)³].

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