Find the general solution of \( \tan^2 \theta = 1 \).
Show Hint
Solution and Explanation
Step 1: Solve the equation.
We are given the equation \( \tan^2 \theta = 1 \). Taking the square root of both sides, we get:
\[
\tan \theta = \pm 1
\]
Step 2: Solve for \( \theta \).
The general solution for \( \tan \theta = 1 \) is:
\[
\theta = \frac{\pi}{4} + n\pi \text{where} n \in \mathbb{Z}
\]
The general solution for \( \tan \theta = -1 \) is:
\[
\theta = \frac{3\pi}{4} + n\pi \text{where} n \in \mathbb{Z}
\]
Step 3: Combine the solutions.
Thus, the general solution is:
\[
\theta = \frac{\pi}{4} + n\pi \text{or} \theta = \frac{3\pi}{4} + n\pi \text{where} n \in \mathbb{Z}
\]
Final Answer: \[ \boxed{\theta = \frac{\pi}{4} + n\pi \text{ or } \theta = \frac{3\pi}{4} + n\pi \text{where} n \in \mathbb{Z}} \]
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Questions Asked in Maharashtra Class XII exam
Given below is the Balance Sheet of 'Bhanubai Mahila Seva Kendra' as on 1st April 2019 and Receipts and Payments account for the year ending 31st March 2020:
\[\begin{array}{|l|r|l|r|} \hline \text{Liabilities} & \text{Amount (₹)} & \text{Assets} & \text{Amount (₹)} \\ \hline \text{Capital Fund} & 40{,}000 & \text{Machinery} & 10{,}000 \\ \text{Outstanding Expenses:} & & \text{Furniture} & 20{,}000 \\ \text{Wages} & 8{,}000 & \text{Government Bonds} & 6{,}500 \\ \text{Electricity} & 7{,}000 & \text{Outstanding Subscription} & 8{,}500 \\ \text{Stationery} & 1{,}000 & \text{Cash in Hand} & 1{,}000 \\ & & \text{Cash at Bank} & 10{,}000 \\ \hline \text{Total} & 56{,}000 & \text{Total} & 56{,}000 \\ \hline \end{array}\]
Receipts and Payments Account for the year ended 31st March 2020:
\[\begin{array}{|l|r|l|r|} \hline \text{Receipts} & \text{Amount (₹)} & \text{Payments} & \text{Amount (₹)} \\ \hline \text{To Balance b/d} & & \text{By Electricity Charges} & 25{,}000 \\ \text{Cash in hand} & 1{,}000 & \text{By Wages} & 22{,}000 \\ \text{Cash at bank} & 10{,}000 & \text{By Stationery} & 3{,}000 \\ \text{To Subscription:} & & \text{By Rent and Taxes} & 11{,}800 \\ 2018{-}2019 & 2{,}000 & \text{By Travelling Expenses} & 8{,}000 \\ 2019{-}2020 & 45{,}000 & & \\ 2020{-}2021 & 3{,}000 & \text{By Balance c/d:} & \\ \text{To Entrance fees} & 28{,}000 & \text{Cash in hand} & 4{,}000 \\ \text{To Other receipts} & 5{,}000 & \text{Cash at bank} & 20{,}200 \\ \hline \text{Total} & 94{,}000 & \text{Total} & 94{,}000 \\ \hline \end{array}\]
Additional information:
\[\begin{array}{rl} \bullet & \text{Outstanding wages ₹ 450} \\ \bullet & \text{Entrance fees should be capitalised.} \\ \bullet & \text{Depreciate furniture at 10\% p.a.} \\ \bullet & \text{Subscription for 2019{-}20 was outstanding ₹ 3,000.} \end{array}\]
Prepare:
\[\begin{array}{rl} \bullet & \text{(a) Income and Expenditure account for the year ended 31st March 2020.} \\ \bullet & \text{(b) Balance Sheet as on 31st March 2020.} \end{array}\]
- Maharashtra Class XII - 2025
- Accounting for Non-Profit Organizations
Mahendra, Surendra and Narendra were partners sharing profits and losses in the ratio 5 : 3 : 2 respectively. Their Balance Sheet as on 31st March 2019 was as follows:
\[\begin{array}{|l|r|l|r|} \hline \text{Liabilities} & \text{Amount (₹)} & \text{Assets} & \text{Amount (₹)} \\ \hline \text{Capital Account:} & & \text{Stock} & 17{,}000 \\ \text{Mahendra} & 23{,}000 & \text{Furniture} & 18{,}000 \\ \text{Surendra} & 15{,}000 & \text{Land and Building} & 16{,}000 \\ \text{Narendra} & 12{,}000 & \text{Bank} & 37{,}000 \\ \text{Bills Payable} & 2{,}000 & & \\ \text{Creditors} & 8{,}000 & & \\ \text{Bank Loan} & 12{,}000 & & \\ \text{General Reserve} & 16{,}000 & & \\ \hline \text{Total} & 88{,}000 & \text{Total} & 88{,}000 \\ \hline \end{array}\]
Mr. Narendra died on 30th June 2019 and the following adjustments were agreed as per deed:
\[\begin{array}{rl} \bullet & \text{Stock, furniture, land and building are to be revalued at ₹ 16,700, ₹ 16,200 and ₹ 30,100 respectively.} \\ \bullet & \text{Narendra's share in goodwill is to be valued from the firm's goodwill, which was valued at 3 times the average profit of the last four years. The profits for the last four years were:} \\ & \quad \bullet \ \text{I year – ₹ 30,000} \\ & \quad \bullet \ \text{II year – ₹ 25,000} \\ & \quad \bullet \ \text{III year – ₹ 25,000} \\ & \quad \bullet \ \text{IV year – ₹ 40,000} \\ \bullet & \text{His profit up to the death is to be calculated on the basis of the profit of last year.} \\ \bullet & \text{Narendra was entitled to get a salary of ₹ 1,200 per month.} \\ \bullet & \text{Interest on capital at 10\% p.a. to be allowed.} \\ \bullet & \text{Narendra's drawing up to the date of his death was ₹ 900 per month.} \end{array}\]
Prepare:
\[\begin{array}{rl} \bullet & \text{(A) Narendra's Capital Account showing amount payable to his executor.} \\ \bullet & \text{(B) Give working notes for:} \\ & \quad \bullet \ \text{Share of goodwill due to Narendra} \\ & \quad \bullet \ \text{Share of profit due to Narendra} \end{array}\]
- Maharashtra Class XII - 2025
- Partner's Death and Capital Account
Given below is a Balance Sheet of Aditya, Ajinkya, and Arun who were partners in a firm sharing profits and losses in the ratio 5:3:2. Their Balance Sheet as on 31st March 2020 was as follows:
\[\begin{array}{|l|r|l|r|} \hline \text{Liabilities} & \text{Amount (₹)} & \text{Assets} & \text{Amount (₹)} \\ \hline \text{Creditors} & 10{,}450 & \text{Cash} & 3{,}800 \\ \text{Reserve Fund} & 7{,}500 & \text{Debtors} & 9{,}000 \\ \text{Capital Account:} & & \text{Stock} & 8{,}750 \\ \text{Aditya} & 21{,}000 & \text{Machinery} & 50{,}000 \\ \text{Ajinkya} & 18{,}500 & \text{Furniture} & 2{,}500 \\ \text{Arun} & 16{,}600 & & \\ \hline \text{Total:} & 74{,}050 & \text{Total:} & 74{,}050 \\ \hline \end{array}\]
On 1st April 2020, Arun retired on the following terms:
\[\begin{array}{rl} \bullet & \text{Goodwill of the firm will be raised in the books at ₹ 10,000.} \\ \bullet & \text{Stocks to be reduced by 10\%, Furniture by 5\%, and Machinery by 10\%.} \\ \bullet & \text{A provision of 5\% R.D.D. to be maintained on debtors.} \\ \bullet & \text{₹ 100 to be written off from creditors.} \\ \bullet & \text{All the amount due to Arun will be transferred to his loan account.} \\ \end{array}\]
Prepare:
\[\begin{array}{rl} \bullet & \text{(a) Profit and Loss Adjustment Account} \\ \bullet & \text{(b) Partners' Capital Account} \\ \bullet & \text{(c) Balance Sheet of the new firm} \\ \end{array}\]
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Mr. Deepak and Mr. Abhishek were in partnership sharing profits and losses in the proportion of 3:1 respectively. Their Balance Sheet as on 31st March 2019 stood as follows:
\[\begin{array}{|l|r|l|r|} \hline \text{Liabilities} & \text{Amount (₹)} & \text{Assets} & \text{Amount (₹)} \\ \hline Capital Account: & & Land and Building & 32,000 \\ \hline Mr. Deepak & 1,20,000 & Plant and Machinery & 60,000 \\ \hline Mr. Abhishek & 40,000 & Furniture & 22,000 \\ \hline General Reserve & 16,000 & Stock & 40,000 \\ \hline Sundry Creditors & 80,000 & Sundry Debtors & 64,000 \\ \hline Bank Overdraft & 42,000 & Cash & 80,000 \\ \hline Total: & 2,98,000 & Total: & 2,98,000 \\ \hline \end{array} \]
They admitted Adinath into partnership on 1st April 2019 on the terms that:
\[\begin{array}{rl} \bullet & \text{Adinath shall bring in ₹ 40,000 as his capital for a 1/5 share in future profits and ₹ 20,000 as his share of goodwill.} \\ \bullet & \text{Furniture to be depreciated by 20\%.} \\ \bullet & \text{Stock should be appreciated by 10\%.} \\ \bullet & \text{Building should be appreciated by 5\%.} \\ \bullet & \text{A provision for 5\% R.D.D. to be created on sundry debtors.} \\ \bullet & \text{Capital accounts of all partners be adjusted in their new profit-sharing ratio through cash account.} \\ \end{array}\] Prepare: \[\begin{array}{rl} \bullet & \text{(a) Revaluation Account} \\ \bullet & \text{(b) Partners' Capital Account} \\ \bullet & \text{(c) New Balance Sheet of the firm} \\ \end{array}\]
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- Admission of Partner
- From the following information, find out the current ratio and net profit ratio: (A) From the following information, find out the current ratio: (i) Total assets = ₹22,000
(ii) Fixed assets = ₹10,000
(iii) Capital employed = ₹20,000
(B) Calculate the net profit ratio from the following data: (i) Sales = ₹76,000
(ii) Cost of goods sold = ₹52,000
(iii) Indirect expenses = ₹12,000- Maharashtra Class XII - 2025
- Financial Ratios