SEE Assignment 1 - Summit Coming Sunday to Your Sir

FOCUS EDGE EDUCATION NETWORK (FEEN)

Assignment 1: SEE Mathematics Practice

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NOTICE TO STUDENTS: Everyone must bring their practice copy and show this assignment to Sabal Sir. Strict action will be taken if not completed!

SECTION 1: COMPOUND INTEREST (मिश्रित ब्याज)

1. A man borrowed Rs 5,00,000 from a bank for 2 years at 12% per annum. If the interest is compounded half-yearly, find the total compound interest he has to pay.

एक मानिसले बैंकबाट 2 वर्षका लागि प्रतिवर्ष 12% ब्याजदरमा रु 5,00,000 ऋण लिए। यदि ब्याज अर्ध-वार्षिक रूपमा गणना गरिन्छ भने, उनले तिर्नुपर्ने कुल मिश्रित ब्याज पत्ता लगाउनुहोस्।

Hint: Use the formula: CI = P [ ( 1 + R / 200 ) ^ 2T - 1 ]

2. The difference between yearly compound interest and simple interest on a certain sum for 2 years at 10% per annum is Rs 500. Find the principal.

10% प्रतिवर्ष ब्याजदरमा 2 वर्षको वार्षिक मिश्रित ब्याज र साधारण ब्याज बीचको अन्तर रु 500 छ। सावाँ पत्ता लगाउनुहोस्।

Hint: Use the relation: CI - SI = P ( R / 100 ) ^ 2 = 500

3. At what rate of compound interest per year will a sum of Rs 1,00,000 amount to Rs 1,21,000 in 2 years?

कुन वार्षिक मिश्रित ब्याजदरमा रु 1,00,000 को 2 वर्षमा मिश्रधन रु 1,21,000 हुन्छ?

Hint: Solve for R in the formula: CA = P ( 1 + R / 100 ) ^ T

4. A person deposited Rs 2,00,000 in a bank for 2 years at 10% p.a. compounded half-yearly. After 1 year, the bank changed its policy to compound the interest quarterly. Find the difference in interest between the two years after a 5% tax deduction.

एक व्यक्तिले रु 2,00,000 बैंकमा 2 वर्षका लागि 10% वार्षिक अर्ध-वार्षिक ब्याजदरमा जम्मा गरे। १ वर्षपछि बैंकले ब्याज त्रैमासिक रूपमा गणना गर्ने नीति लियो। 5% कर कट्टा गरेपछि दुई वर्षको ब्याज बीचको अन्तर पत्ता लगाउनुहोस्।

Hint: Calculate net interest for Year 1 (half-yearly) and Year 2 (quarterly) separately, then subtract tax.

5. Find the compound interest of Rs 20,000 for 2 years if the rate of interest for the first year is 10% and for the second year is 12%.

रु 20,000 को 2 वर्षको मिश्रित ब्याज पत्ता लगाउनुहोस् यदि पहिलो वर्षको ब्याजदर 10% र दोस्रो वर्षको 12% छ।

Hint: CA = P ( 1 + R1 / 100 ) ( 1 + R2 / 100 )

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SECTION 2: SEQUENCE AND SERIES (अनुक्रम र श्रेणी)

1. Find three arithmetic means between the terms 3 and 23.

3 र 23 को बीचमा तीनवटा समानान्तर मध्यमाहरू पत्ता लगाउनुहोस्।

Hint: Here, n = 5. Find common difference (d) using t5 = a + 4d.

2. If the third term of an arithmetic series is 0 and the tenth term is 42, find the sum of the first 15 terms.

यदि एउटा समानान्तर श्रेणीको तेस्रो पद 0 र दशौँ पद 42 छ भने, पहिलो 15 पदहरूको योगफल पत्ता लगाउनुहोस्।

Hint: Form two equations: a + 2d = 0 and a + 9d = 42 to find a and d, then use Sn formula.

3. Find the sum of the first 8 terms of the geometric series 64 + 32 + 16 + ...

64 + 32 + 16 + ... गुणोत्तर श्रेणीको पहिलो 8 पदहरूको योगफल पत्ता लगाउनुहोस्।

Hint: Here a = 64, r = 1/2. Use Sn = a(1 - r^n) / (1 - r).

4. The first term of an arithmetic sequence is 2. If the sum of the next five terms is four times the sum of the first five terms, prove that the 20th term (t20) is -112.

एउटा समानान्तर श्रेणीको पहिलो पद 2 छ। यदि पछिल्ला पाँच पदहरूको योगफल पहिलो पाँच पदहरूको योगफलको चार गुणा छ भने, 20औँ पद (t20) -112 हुन्छ भनी प्रमाणित गर्नुहोस्।

Hint: S10 - S5 = 4 * S5. Solve for common difference (d) first.

5. Find three geometric means between the numbers 5 and 405.

5 र 405 को बीचमा तीनवटा गुणोत्तर मध्यमाहरू पत्ता लगाउनुहोस्।

Hint: Here n = 5. Find common ratio (r) using t5 = ar^4.

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SECTION 3: ALGEBRAIC FRACTION (बीजगणितीय भिन्न)

1. Simplify the following expression: (a - b) / (a^2 - ab + b^2) + (a + b) / (a^2 + ab + b^2) - (2a^3) / (a^4 + a^2b^2 + b^4)

सरल गर्नुहोस्: (a - b) / (a^2 - ab + b^2) + (a + b) / (a^2 + ab + b^2) - (2a^3) / (a^4 + a^2b^2 + b^4)

Hint: Factorize the denominator (a^4 + a^2b^2 + b^4) into (a^2 + ab + b^2)(a^2 - ab + b^2).

2. If a / (2x + 1) + 1 / (x + 2) = (4x + 5) / (2x^2 + 5x + 2), find the value of 'a'.

यदि a / (2x + 1) + 1 / (x + 2) = (4x + 5) / (2x^2 + 5x + 2) भए, 'a' को मान पत्ता लगाउनुहोस्।

Hint: Take the LCM of the left side and compare the numerators with the right side.

3. Simplify: 1 / (1 - x) + 1 / (1 + x) + 2x / (1 - x^2) + 4x^3 / (1 - x^4)

सरल गर्नुहोस्: 1 / (1 - x) + 1 / (1 + x) + 2x / (1 - x^2) + 4x^3 / (1 - x^4)

Hint: Solve step by step by adding the first two terms first, then the result with the third term.

4. Simplify: (x^2 - 7x + 12) / (x^2 - 5x + 6) * (x - 2) / (x - 4)

सरल गर्नुहोस्: (x^2 - 7x + 12) / (x^2 - 5x + 6) * (x - 2) / (x - 4)

Hint: Factorize the quadratic expressions (x^2 - 7x + 12) and (x^2 - 5x + 6) using the mid-term break method.

5. Simplify: 1 / (x - y)(x - z) + 1 / (y - z)(y - x) + 1 / (z - x)(z - y)

सरल गर्नुहोस्: 1 / (x - y)(x - z) + 1 / (y - z)(y - x) + 1 / (z - x)(z - y)

Hint: Change the signs to make the cyclic order like (x - y), (y - z), and (z - x).

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SECTION 4: INDICES (घातांक)

1. Solve for x: 3^(x + 1) + 3^x = 4 / 27

x का लागि हल गर्नुहोस्: 3^(x + 1) + 3^x = 4 / 27

Hint: Factor out 3^x to get 3^x (3 + 1) = 4 / 27, then simplify to find x.

2. Solve the equation: 16^x - 5 * 4^(x + 1) + 64 = 0

समीकरण हल गर्नुहोस्: 16^x - 5 * 4^(x + 1) + 64 = 0

Hint: Let 4^x = 'a', then 16^x = a^2. This forms a quadratic equation: a^2 - 20a + 64 = 0.

3. If x = 2^(1/3) + 2^(-1/3), prove that 2x^3 - 6x = 5.

यदि x = 2^(1/3) + 2^(-1/3) भए, 2x^3 - 6x = 5 हुन्छ भनी प्रमाणित गर्नुहोस्।

Hint: Cube both sides using the identity (a + b)^3 = a^3 + b^3 + 3ab(a + b).

4. Solve for x: 5^x + 5^(2 - x) = 26

x का लागि हल गर्नुहोस्: 5^x + 5^(2 - x) = 26

Hint: This can be written as 5^x + 25 / 5^x = 26. Use substitution 5^x = a.

5. Simplify the expression: [ (x^a / x^b)^(a + b) ] * [ (x^b / x^c)^(b + c) ] * [ (x^c / x^a)^(c + a) ]

सरल गर्नुहोस्: [ (x^a / x^b)^(a + b) ] * [ (x^b / x^c)^(b + c) ] * [ (x^c / x^a)^(c + a) ]

Hint: Use the laws of indices: (x^a / x^b) = x^(a - b). Then apply (x^m)^n = x^(mn).

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SECTION 5: CURRENCY AND EXCHANGE RATE (मुद्रा र विनिमय दर)

1. A businessman exchanged Rs 8,40,000 into Pound Sterling (£) at the rate of £1 = Rs 150. After 5 days, the Nepali currency devalued by 5%. How much profit or loss did he make if he exchanged his pounds back into Nepali Rupees?

एक व्यवसायीले रु 8,40,000 लाई £1 = रु 150 को दरमा पाउण्ड स्टर्लिङमा साटे। 5 दिनपछि नेपाली मुद्राको 5% अवमूल्यन भयो। यदि उनले सो पाउण्डलाई पुनः नेपाली रुपैयाँमा साटे भने उनलाई कति नाफा वा घाटा भयो?

Hint: First find the pounds he has. Then increase the exchange rate by 5% (150 + 5% of 150) before converting back.

2. If $176 American Dollars = 100 Pound Sterling and 1 Pound Sterling = Rs 151, find how many Nepali Rupees can be exchanged for 132 American Dollars.

यदि 176 अमेरिकी डलर = 100 पाउण्ड स्टर्लिङ र 1 पाउण्ड स्टर्लिङ = रु 151 भए, 132 अमेरिकी डलरसँग कति नेपाली रुपैयाँ साट्न सकिन्छ?

Hint: Use the Chain Rule: 132 USD * (100 GBP / 176 USD) * (151 NPR / 1 GBP).

3. A person bought a laptop in London for £600. When he brought it to Nepal, he had to pay 15% customs duty and 13% VAT. If £1 = Rs 155, find the total cost of the laptop in Nepali Rupees.

एक व्यक्तिले लण्डनमा £600 मा एउटा ल्यापटप किने। उनले नेपाल ल्याउँदा 15% भन्सार शुल्क र 13% भ्याट तिर्नुपर्‍यो। यदि £1 = रु 155 भए, सो ल्यापटपको कुल मूल्य नेपाली रुपैयाँमा पत्ता लगाउनुहोस्।

Hint: Calculate the price in NPR first, then add 15% duty on that price, and finally add 13% VAT on the sum of price and duty.

4. The exchange rate of 1 US Dollar is Rs 132. If the Nepali currency is revalued by 10%, find the new exchange rate of 1 US Dollar.

1 अमेरिकी डलरको विनिमय दर रु 132 छ। यदि नेपाली मुद्राको 10% अधिमूल्यन (Revaluation) भयो भने, 1 अमेरिकी डलरको नयाँ विनिमय दर पत्ता लगाउनुहोस्।

Hint: Revaluation makes the foreign currency cheaper. New Rate = 132 - (10% of 132).

5. A merchant imported goods worth 5,00,000 Indian Rupees (IC). If the exchange rate is fixed at IC 100 = Rs 160, and he spent Rs 15,000 on transportation and paid 10% customs duty, find his total expenditure in NPR.

एक व्यापारीले रु 5,00,000 भारतीय रुपैयाँ (IC) बराबरको सामान आयात गरे। यदि विनिमय दर IC 100 = रु 160 छ, र उनले ढुवानीमा रु 15,000 र 10% भन्सार शुल्क तिरेका छन् भने, उनको कुल खर्च नेपाली रुपैयाँमा कति भयो?

Hint: Convert IC to NPR first, then calculate 10% duty on that NPR value, and add the transportation cost at the end.

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SECTION 6: ALGEBRAIC FRACTION - PART 2 (बीजगणितीय भिन्न - भाग २)

1. Simplify the following expression by maintaining cyclic order: 1 / (x - y)(x - z) + 1 / (y - z)(y - x) + 1 / (z - x)(z - y)

चक्रीय क्रम मिलाएर सरल गर्नुहोस्: 1 / (x - y)(x - z) + 1 / (y - z)(y - x) + 1 / (z - x)(z - y)

Hint: Change (x - z) to -(z - x) and (y - x) to -(x - y) to get a common denominator of (x - y)(y - z)(z - x).

2. Simplify: (x + 2) / (x - 2) - (x - 2) / (x + 2) - 8x / (x^2 + 4)

सरल गर्नुहोस्: (x + 2) / (x - 2) - (x - 2) / (x + 2) - 8x / (x^2 + 4)

Hint: Subtract the first two fractions first using (a + b)^2 - (a - b)^2 = 4ab. Then subtract the third term from the result.

3. Simplify the complex fraction: 1 / (a^2 - a - 6) + 1 / (a^2 + a - 2) + 1 / (a^2 - 4a + 3)

सरल गर्नुहोस्: 1 / (a^2 - a - 6) + 1 / (a^2 + a - 2) + 1 / (a^2 - 4a + 3)

Hint: Factorize each quadratic denominator into linear factors like (a - 3)(a + 2), (a + 2)(a - 1), etc., then find the LCM.

4. Simplify the expression using algebraic identities: (a^2 + ab + b^2) / (a + b) + (a^2 - ab + b^2) / (a - b) - 2a^3 / (a^2 - b^2)

बीजगणितीय सूत्रहरू प्रयोग गरी सरल गर्नुहोस्: (a^2 + ab + b^2) / (a + b) + (a^2 - ab + b^2) / (a - b) - 2a^3 / (a^2 - b^2)

Hint: Find the LCM of (a + b) and (a - b), which is (a^2 - b^2), then combine the first two terms before subtracting the third.

5. Prove that: 1 / (1 + x^(a-b)) + 1 / (1 + x^(b-a)) = 1

प्रमाणित गर्नुहोस्: 1 / (1 + x^(a-b)) + 1 / (1 + x^(b-a)) = 1

Hint: Express x^(a-b) as x^a / x^b and x^(b-a) as x^b / x^a, then simplify each fraction by finding the common denominator.

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SECTION 7: INDICES - PART 2 (घातांक - भाग २)

1. Solve for x: 5^x + 125 / 5^x = 30

x का लागि हल गर्नुहोस्: 5^x + 125 / 5^x = 30

Hint: Let 5^x = a. Solve the quadratic equation a + 125 / a = 30, which simplifies to a^2 - 30a + 125 = 0.

2. Solve: 4 * 3^(x + 1) - 9^x = 27

हल गर्नुहोस्: 4 * 3^(x + 1) - 9^x = 27

Hint: Express 9^x as (3^x)^2 and 3^(x+1) as 3 * 3^x. Let 3^x = m and solve the quadratic m^2 - 12m + 27 = 0.

3. If x = 1 + 2^(1/3) + 2^(2/3), prove that x(x^2 - 3x - 3) = 1.

यदि x = 1 + 2^(1/3) + 2^(2/3) भए, x(x^2 - 3x - 3) = 1 हुन्छ भनी प्रमाणित गर्नुहोस्।

Hint: Start with (x - 1) = 2^(1/3) + 2^(2/3) and cube both sides of the equation.

4. Solve for the value of x: x^(x * sq-root(x)) = (x * sq-root(x))^x

x को मान पत्ता लगाउनुहोस्: x^(x * sq-root(x)) = (x * sq-root(x))^x

Hint: Express sq-root(x) as x^(1/2). The equation becomes x^(x^(3/2)) = (x^(3/2))^x. Equate the exponents.

5. Solve the exponential equation: 2^(x + 2) + 2^(x + 1) + 2^x = 7 / 32

घातांक समीकरण हल गर्नुहोस्: 2^(x + 2) + 2^(x + 1) + 2^x = 7 / 32

Hint: Factor out 2^x to get 2^x (2^2 + 2^1 + 1) = 7 / 32. Then divide both sides by 7.

End of Assignment 1. Ensure all 7 sections are completed in your copy for Sabal Sir.

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