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\[ \text { If } \cos ^4 A-\sin ^4 A=p \text {, then find the value of } p \text {. } \]
Solution Given: \[ \cos^4 A - \sin^4 A = p \] Step 1: Use the difference of squares formula \[ \cos^4 A - \sin^4 A = (\cos^2 A + \sin^2 A)(\cos^2 A - \sin^2 A) \] Step 2: Use the Pythagorean identity Since \(\sin^2 A + \cos^2 A = 1\), we have: \[ (\cos^2 A + \sin^2 A)(\cos^2 A - \sin^2 A) = (1)(\cosRead more
Solution
Given:
\[ \cos^4 A – \sin^4 A = p \]
Step 1: Use the difference of squares formula
\[ \cos^4 A – \sin^4 A = (\cos^2 A + \sin^2 A)(\cos^2 A – \sin^2 A) \]
Step 2: Use the Pythagorean identity
Since \(\sin^2 A + \cos^2 A = 1\), we have:
\[ (\cos^2 A + \sin^2 A)(\cos^2 A – \sin^2 A) = (1)(\cos^2 A – \sin^2 A) \]
Step 3: Use the double angle formula
The double angle formula for cosine is \(\cos 2A = \cos^2 A – \sin^2 A\), so:
\[ \cos^2 A – \sin^2 A = \cos 2A \]
Step 4: Find the value of \(p\)
Therefore, we have:
\[ p = \cos 2A \]
Conclusion
The value of \(p\) is \(\cos 2A\).
See lessIf x + y + z = 12 , then find the maximum value of (x – 1) (y – 2) (z – 3).
To find the maximum value of the product \((x-1)(y-2)(z-3)\) subject to the constraint \(x + y + z = 12\), we can use the method of Lagrange multipliers or directly apply the AM-GM inequality. Let's use the AM-GM inequality: By the Arithmetic Mean-Geometric Mean inequality, for non-negative numbersRead more
To find the maximum value of the product \((x-1)(y-2)(z-3)\) subject to the constraint \(x + y + z = 12\), we can use the method of Lagrange multipliers or directly apply the AM-GM inequality.
Let’s use the AM-GM inequality:
By the Arithmetic Mean-Geometric Mean inequality, for non-negative numbers \(a\), \(b\), and \(c\):
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
Equality holds when \(a = b = c\).
Applying this to our problem, we first rewrite \((x-1)(y-2)(z-3)\) in a form that allows us to use the AM-GM inequality.
Let \(a = x – 1\), \(b = y – 2\), and \(c = z – 3\). Then \(a + b + c = x + y + z – 6 = 12 – 6 = 6\).
By AM-GM, we have:
\[ \frac{a + b + c}{3} = \frac{6}{3} = 2 \geq \sqrt[3]{abc} \]
\[ \sqrt[3]{abc} \leq 2 \]
\[ abc \leq 8 \]
So the maximum value of \((x-1)(y-2)(z-3) = abc\) is \(8\), and this maximum is achieved when \(a = b = c = 2\), or equivalently, when \(x = 3\), \(y = 4\), and \(z = 5\).
See lessA can do a piece of work in 15 days, B can do the same work in 10 days, and C do the same work in 12 days. All three of them do the same work together, then they collectively get Rs. 9000. If B’s share is divided among three new persons D, E and F in the ratio of 1 : 5 : 3 respectively then find the share of F.
Solution Given: - A can complete the work in 15 days. - B can complete the work in 10 days. - C can complete the work in 12 days. - The total payment for the work is Rs. 9000. Step 1: Calculate the total work done by A, B, and C Let the total work be \(W\) units. - A's work rate: \(\frac{W}{15}\) unRead more
Solution
Given:
– A can complete the work in 15 days.
– B can complete the work in 10 days.
– C can complete the work in 12 days.
– The total payment for the work is Rs. 9000.
Step 1: Calculate the total work done by A, B, and C
Let the total work be \(W\) units.
– A’s work rate: \(\frac{W}{15}\) units/day
– B’s work rate: \(\frac{W}{10}\) units/day
– C’s work rate: \(\frac{W}{12}\) units/day
Step 2: Calculate the total work done by A, B, and C together in one day
\[ \text{Total work rate} = \frac{W}{15} + \frac{W}{10} + \frac{W}{12} = \frac{4W}{60} + \frac{6W}{60} + \frac{5W}{60} = \frac{15W}{60} = \frac{W}{4} \text{ units/day} \]
Step 3: Calculate the share of B
Since the total payment is for 1 day of work, B’s share is proportional to his work rate:
\[ \text{B’s share} = \frac{\text{B’s work rate}}{\text{Total work rate}} \times \text{Total payment} = \frac{\frac{W}{10}}{\frac{W}{4}} \times 9000 = \frac{4}{10} \times 9000 = Rs. 3600 \]
Step 4: Calculate F’s share
B’s share is divided among D, E, and F in the ratio of 1:5:3. The total parts in the ratio are \(1 + 5 + 3 = 9\).
– F’s share = \(\frac{3}{9} \times \text{B’s share} = \frac{3}{9} \times 3600 = Rs. 1200\)
Conclusion
The share of F is Rs. 1200.
See lessA shopkeeper marked up the price of an item by 96% on the actual cost price and allows the discount of 25%. If he gave 2 items free on every dozen purchase, then find the profit percent on sale of 1 dozen items.
Solution Let's assume the cost price (CP) of one item is Rs. 100. Step 1: Calculate the selling price after markup and discount - After a 96% markup, the price becomes Rs. 196. - After a 25% discount, the selling price (SP) becomes \(196 \times \frac{3}{4} = Rs. 147\). Step 2: Calculate the effectivRead more
Solution
Let’s assume the cost price (CP) of one item is Rs. 100.
Step 1: Calculate the selling price after markup and discount
– After a 96% markup, the price becomes Rs. 196.
– After a 25% discount, the selling price (SP) becomes \(196 \times \frac{3}{4} = Rs. 147\).
Step 2: Calculate the effective selling price for a dozen items with 2 items free
– When 2 items are given free on a dozen, the effective number of items sold is 10.
– Therefore, the effective selling price for a dozen items is \(10 \times 147 = Rs. 1470\).
Step 3: Calculate the cost price for a dozen items
– The cost price for a dozen items is \(12 \times 100 = Rs. 1200\).
Step 4: Calculate the profit and profit percent
– Profit = Selling price for a dozen – Cost price for a dozen = Rs. 1470 – Rs. 1200 = Rs. 270.
– Profit percent = \(\frac{\text{Profit}}{\text{Cost price for a dozen}} \times 100 = \frac{270}{1200} \times 100 = 22.5\%\).
Conclusion
The profit percent on the sale of 1 dozen items is 22.5%.
See lessLagaan is levied on the 40% of the total land in Village Sitapur. The revenue department collected total Rs. 4,48,800 through the lagaan from the village. Hari, a very rich farmer, paid only Rs. 580 as lagaan for his piece of land. The percentage of total land of Hari over the total taxable land of the village (approx) is:
Solution Given: - Lagaan is levied on 40% of the total land in Village Sitapur. - Total lagaan collected from the village is Rs. 4,48,800. - Hari paid Rs. 580 as lagaan for his piece of land. Step 1: Find the land corresponding to Rs. 1 of lagaan If Rs. 4,48,800 is collected from 40% of the total laRead more
Solution
Given:
– Lagaan is levied on 40% of the total land in Village Sitapur.
– Total lagaan collected from the village is Rs. 4,48,800.
– Hari paid Rs. 580 as lagaan for his piece of land.
Step 1: Find the land corresponding to Rs. 1 of lagaan
If Rs. 4,48,800 is collected from 40% of the total land, then the land corresponding to Rs. 1 of lagaan is:
\[ \frac{0.4x}{448800} \text{ (where } x \text{ is the total land)} \]
Step 2: Calculate the taxable land of Hari
Hari paid Rs. 580 as lagaan, so the taxable land of Hari is:
\[ 580 \times \frac{0.4x}{448800} \]
Step 3: Convert taxable land of Hari to total land of Hari
Since the taxable land is only 40% of the total land, the total land of Hari is:
\[ \frac{100}{40} \times 580 \times \frac{0.4x}{448800} \]
Step 4: Calculate the percentage of Hari’s total land over the total taxable land of the village
The percentage of Hari’s total land over the total taxable land of the village is:
\[ \frac{\frac{100 \times 580 \times 0.4x}{40 \times 448800}}{0.4x} \times 100 = \frac{580 \times 100}{448800} \times 100 \approx 0.323\% \]
Conclusion
The percentage of total land of Hari over the total taxable land of the village is approximately 0.323%.
See lessThere are 351 gold coins that are supposed to be divided among Abhay, Vishal and Kishore in the ratio 2 : 3 : 4 but by mistake it was divided in the ratio of 1/2 : 1/3 : 1/4. The number of extra/deficit gold coins incurred to Abhay due to this mistake is?
Solution Given: - The correct ratio for dividing the gold coins among Abhay, Vishal, and Kishore is 2:3:4. - The mistaken ratio used for division is 1/2:1/3:1/4. - There are a total of 351 gold coins. Step 1: Calculate the number of coins each person should have received Total parts in the correct rRead more
Solution
Given:
– The correct ratio for dividing the gold coins among Abhay, Vishal, and Kishore is 2:3:4.
– The mistaken ratio used for division is 1/2:1/3:1/4.
– There are a total of 351 gold coins.
Step 1: Calculate the number of coins each person should have received
Total parts in the correct ratio = 2 + 3 + 4 = 9 parts
– Abhay’s share in the correct ratio: \(\frac{2}{9} \times 351 = 78\) coins
– Vishal’s share in the correct ratio: \(\frac{3}{9} \times 351 = 117\) coins
– Kishore’s share in the correct ratio: \(\frac{4}{9} \times 351 = 156\) coins
Step 2: Calculate the number of coins each person received due to the mistake
Total parts in the mistaken ratio = 1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12 parts
– Abhay’s share in the mistaken ratio: \(\frac{1/2}{13/12} \times 351 = \frac{6}{13} \times 351 = 162\) coins
– Vishal’s share in the mistaken ratio: \(\frac{1/3}{13/12} \times 351 = \frac{4}{13} \times 351 = 108\) coins
– Kishore’s share in the mistaken ratio: \(\frac{1/4}{13/12} \times 351 = \frac{3}{13} \times 351 = 81\) coins
Step 3: Calculate the number of extra/deficit gold coins incurred to Abhay due to the mistake
Extra/deficit coins for Abhay = Abhay’s share in the mistaken ratio – Abhay’s share in the correct ratio
\[ = 162 – 78 = 84 \text{ coins} \]
Conclusion
The number of extra gold coins incurred to Abhay due to the mistake is 84 coins.
See lessThe difference between the time taken by 2 cars to cover 450 Km is 1 hour 30 minutes. If the difference between their speeds is 15 Kmph, find the speed of the slower car?
Solution Let the speed of the slower car be \(v\) km/h, and the speed of the faster car be \(v + 15\) km/h. Given: - The difference in time taken by the two cars to cover 450 km is 1 hour 30 minutes (which is 1.5 hours). - The difference in their speeds is 15 km/h. Step 1: Express the time taken byRead more
Solution
Let the speed of the slower car be \(v\) km/h, and the speed of the faster car be \(v + 15\) km/h.
Given:
– The difference in time taken by the two cars to cover 450 km is 1 hour 30 minutes (which is 1.5 hours).
– The difference in their speeds is 15 km/h.
Step 1: Express the time taken by each car to cover 450 km
– Time taken by the slower car: \(\frac{450}{v}\) hours
– Time taken by the faster car: \(\frac{450}{v + 15}\) hours
Step 2: Set up the equation based on the difference in time taken
\[ \frac{450}{v} – \frac{450}{v + 15} = 1.5 \]
Step 3: Solve the equation
Multiplying all terms by \(v(v + 15)\) to clear the denominators:
\[ 450(v + 15) – 450v = 1.5v(v + 15) \]
\[ 450v + 6750 – 450v = 1.5v^2 + 22.5v \]
\[ 6750 = 1.5v^2 + 22.5v \]
\[ v^2 + 15v – 4500 = 0 \]
Factoring the quadratic equation:
\[ (v + 75)(v – 60) = 0 \]
Since the speed cannot be negative, we take the positive value:
\[ v = 60 \]
Conclusion
The speed of the slower car is 60 km/h.
See lessThe length of the sides of a triangle are 9 cm, 12 cm and 15 cm. Find the length of the perpendicular from the opposite vertex to the side whose length is 15 cm.
Solution Given: - The lengths of the sides of the triangle are 9 cm, 12 cm, and 15 cm. We can use Heron's formula to find the area of the triangle and then use the area to find the length of the perpendicular from the opposite vertex to the side of length 15 cm. Step 1: Calculate the semi-perimeterRead more
Solution
Given:
– The lengths of the sides of the triangle are 9 cm, 12 cm, and 15 cm.
We can use Heron’s formula to find the area of the triangle and then use the area to find the length of the perpendicular from the opposite vertex to the side of length 15 cm.
Step 1: Calculate the semi-perimeter (s) of the triangle
\[ s = \frac{9 + 12 + 15}{2} = 18 \text{ cm} \]
Step 2: Use Heron’s formula to find the area (A) of the triangle
\[ A = \sqrt{s(s – 9)(s – 12)(s – 15)} \]
\[ A = \sqrt{18(18 – 9)(18 – 12)(18 – 15)} \]
\[ A = \sqrt{18 \times 9 \times 6 \times 3} \]
\[ A = \sqrt{2916} \]
\[ A = 54 \text{ cm}^2 \]
Step 3: Find the length of the perpendicular (h) from the opposite vertex to the side of length 15 cm
Using the formula for the area of a triangle (\(A = \frac{1}{2} \times \text{base} \times \text{height}\)):
\[ 54 = \frac{1}{2} \times 15 \times h \]
\[ h = \frac{54 \times 2}{15} \]
\[ h = \frac{108}{15} \]
\[ h = 7.2 \text{ cm} \]
Conclusion
The length of the perpendicular from the opposite vertex to the side whose length is 15 cm is 7.2 cm.
See lessA pump can be used for filling as well emptying a tank. The capacity of the tank is 2400 m^3 . The emptying tank capacity is 10 m^3 per minute higher than its filling capacity and the pump needs 8 minutes lesser to empty the tank than it needs to fill it. What is the filling capacity of the pump?
Solution Let the filling capacity of the pump be \(x\) m³ per minute. Then, the emptying capacity of the pump is \(x + 10\) m³ per minute. Given: - The capacity of the tank is 2400 m³. - The pump needs 8 minutes lesser to empty the tank than it needs to fill it. Step 1: Express the time taken to filRead more
Solution
Let the filling capacity of the pump be \(x\) m³ per minute. Then, the emptying capacity of the pump is \(x + 10\) m³ per minute.
Given:
– The capacity of the tank is 2400 m³.
– The pump needs 8 minutes lesser to empty the tank than it needs to fill it.
Step 1: Express the time taken to fill and empty the tank
– Time taken to fill the tank: \(\frac{2400}{x}\) minutes
– Time taken to empty the tank: \(\frac{2400}{x + 10}\) minutes
Step 2: Set up the equation based on the given information
The pump needs 8 minutes lesser to empty the tank than to fill it:
\[ \frac{2400}{x} – \frac{2400}{x + 10} = 8 \]
Step 3: Solve the equation
\[ 2400(x + 10) – 2400x = 8x(x + 10) \]
\[ 24000 = 8x^2 + 80x \]
\[ x^2 + 10x – 3000 = 0 \]
Factoring the quadratic equation:
\[ (x + 60)(x – 50) = 0 \]
Since the filling capacity cannot be negative, we take the positive value:
\[ x = 50 \]
Conclusion
The filling capacity of the pump is 50 m³ per minute.
See lessA shop of electronic goods remains closed on Monday. The average sales per day for remaining six days of a week is Rs. 13240 & the average sale of Tuesday to Saturday is Rs. 13924. The sales on Sunday is:
Solution Given: - The shop remains closed on Monday. - The average sales per day from Tuesday to Sunday is Rs. 13,240. - The average sales per day from Tuesday to Saturday is Rs. 13,924. Step 1: Calculate the total sales from Tuesday to Sunday Total sales from Tuesday to Sunday = Average sales per dRead more
Solution
Given:
– The shop remains closed on Monday.
– The average sales per day from Tuesday to Sunday is Rs. 13,240.
– The average sales per day from Tuesday to Saturday is Rs. 13,924.
Step 1: Calculate the total sales from Tuesday to Sunday
Total sales from Tuesday to Sunday = Average sales per day (Tuesday to Sunday) × Number of days
\[ = 13240 \times 6 \]
\[ = Rs. 79,440 \]
Step 2: Calculate the total sales from Tuesday to Saturday
Total sales from Tuesday to Saturday = Average sales per day (Tuesday to Saturday) × Number of days
\[ = 13924 \times 5 \]
\[ = Rs. 69,620 \]
Step 3: Calculate the sales on Sunday
Sales on Sunday = Total sales from Tuesday to Sunday – Total sales from Tuesday to Saturday
\[ = 79,440 – 69,620 \]
\[ = Rs. 9,820 \]
Conclusion
The sales on Sunday are Rs 9,820.
See less