To solve this problem, we need to track Pratap's movements step-by-step on a 2D plane. We can assign coordinates or simply visualize the path to determine his final position relative to the starting point. Then, calculate the shortest distance and direction needed to return to the starting point.
B) 5 km to the west
From the final position \((5,0)\), to reach the starting point \((0,0)\), Pratap must travel 5 km in the westward direction.
If the radius of a hemisphere is increased by 50%, then find the percentage increase in the volume. (Take π = \( \frac{22}{7} \))
To find the percentage increase in the volume of a hemisphere when its radius is increased by 50%, we will first calculate the initial volume, then the new volume after the radius increase, and finally use the formula for percentage increase.
C) 237.5% is the correct percentage increase in the volume of the hemisphere.
Sangeeta and Hemlata invest in a business in the ratio 1 : 3. If total profit is Rs. 2,568, then what is the difference between the profit (in Rs.) of Sangeeta and Hemlata?
To find the difference in profit between Sangeeta and Hemlata, we first need to determine their individual profit shares. The profit is distributed in the same ratio as their investments. We will calculate the total ratio parts, then find each person's share of the total profit, and finally compute the difference.
B) 1,284 is the correct difference between Sangeeta's and Hemlata's profits.
To find the area of an isosceles triangle, we first need to determine its height. We can do this by drawing an altitude from the apex to the base, which bisects the base. This creates two right-angled triangles. We can then use the Pythagorean theorem to find the height and subsequently calculate the area using the formula: Area = (1/2) × base × height.
Identify the given values:
Divide the isosceles triangle into two right-angled triangles:
When an altitude is drawn from the apex to the base of an isosceles triangle, it bisects the base. So, the base of each right-angled triangle will be half of the original base.
Use the Pythagorean theorem to find the height (h):
In a right-angled triangle, \((\text{hypotenuse})^2 = (\text{base})^2 + (\text{height})^2\).
Here, the hypotenuse is the equal side of the isosceles triangle (\(89 \text{ cm}\)), the base is \(39 \text{ cm}\), and the height is \(h\).
Calculate the area of the isosceles triangle:
The formula for the area of a triangle is \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\].
Correct Option: A) 3120
The problem asks us to identify a pair of numbers that shares the same relationship as the given pairs: (18, 111) and (13, 81). We need to find a mathematical operation or pattern that transforms the first number into the second number in each pair.
Let the first number be \(x\) and the second number be \(y\). We are looking for a relationship \(y = f(x)\).
We can try multiplication and addition/subtraction. If we multiply 18 by some factor, say 6, we get \(18 \times 6 = 108\). Then, to get 111, we need to add 3. So, the relationship could be \(y = 6x + 3\).
Using the proposed relationship \(y = 6x + 3\), let's check if it holds for the pair (13, 81).
\[ 6 \times 13 + 3 = 78 + 3 = 81 \]The relationship holds for both given pairs. Thus, the pattern is \(y = 6x + 3\).
Let \(x = 15\). Using the relationship \(y = 6x + 3\):
\[ y = 6 \times 15 + 3 = 90 + 3 = 93 \]This matches the given second number in the option (93). Therefore, option A shares the same relationship.
Let \(x = 19\). Using the relationship \(y = 6x + 3\):
\[ y = 6 \times 19 + 3 = 114 + 3 = 117 \]The second number in the option is 114, which is not 117.
Let \(x = 12\). Using the relationship \(y = 6x + 3\):
\[ y = 6 \times 12 + 3 = 72 + 3 = 75 \]The second number in the option is 69, which is not 75.
Let \(x = 17\). Using the relationship \(y = 6x + 3\):
\[ y = 6 \times 17 + 3 = 102 + 3 = 105 \]The second number in the option is 104, which is not 105.
This problem can be solved using Ohm's Law, which states the relationship between potential difference (voltage), current, and resistance in an electrical circuit. Ohm's Law is given by the formula \(V = IR\), where \(V\) is the potential difference, \(I\) is the current, and \(R\) is the resistance.
The current is given in milliamperes (mA), which needs to be converted to amperes (A) for consistency with Ohm's Law. 1 A = 1000 mA.
\[ I = 150 \text{ mA} = 150 \times 10^{-3} \text{ A} = 0.15 \text{ A} \]Use the formula \(V = IR\) to calculate the potential difference (\(V\)).
\[ V = I \times R \]\[ V = 0.15 \text{ A} \times 0.8 \Omega \]\[ V = 0.12 \text{ V} \]The potential difference across the conductor is 0.12 V.
Correct Option: D) 0.12 V
The question describes a fundamental relationship between potential difference and current in a metallic wire under constant temperature. We need to identify the scientific law that defines this relationship.
A) Ohm's law is the fundamental law in electrical circuits that states the potential difference across a conductor is directly proportional to the current flowing through it, provided all physical conditions (like temperature) remain constant. This perfectly matches the description given in the question.
If the interest earned during the 2nd year on a certain sum is ₹9,911, and the rate of interest is 10% per annum compounded annually, then the sum is:
The problem asks us to find the principal sum given the interest earned in the second year at a compound annual interest rate. We can use the formula for compound interest or calculate the interest for the first and second years separately and then find the difference to relate it to the given interest for the second year.
Calculate the interest for the 1st year (\(I_1\)):
\[ I_1 = P \times \frac{R}{100} = P \times \frac{10}{100} = 0.1P \]Amount at the end of the 1st year (\(A_1\)):
\[ A_1 = P + I_1 = P + 0.1P = 1.1P \]Interest for the 2nd year (\(I_2\)):
For compound interest, the principal for the second year is the amount at the end of the first year (\(A_1\)).
\[ I_2 = A_1 \times \frac{R}{100} = (1.1P) \times \frac{10}{100} = 1.1P \times 0.1 = 0.11P \]Equate \(I_2\) to the given value:
We are given that the interest earned during the 2nd year is ₹9,911.
\[ 0.11P = 9911 \]Solve for \(P\):
\[ P = \frac{9911}{0.11} \] \[ P = \frac{9911}{\frac{11}{100}} \] \[ P = \frac{9911 \times 100}{11} \] \[ P = 901 \times 100 \] \[ P = 90100 \]Thus, the sum is ₹90,100.
A) ₹90,100
What was the venue of the 23rd meeting of the Shanghai Cooperation Organisation (SCO) in 2024?
New Delhi, India
Shanghai, China
Islamabad, Pakistan
Dhaka, Bangladesh
The question asks for the venue of the 23rd meeting of the Shanghai Cooperation Organisation (SCO) in 2024. This requires knowledge of recent international summits and their host countries/cities.
Correct Option: C) Islamabad, Pakistan. Pakistan was slated to host the 23rd meeting of the SCO Council of Heads of State in 2024. The 22nd summit was hosted by India virtually in 2023.
By selling an article at \( \frac{2}{5} \) of its actual selling price, Manav incurs a loss of 27%. If he sells it at 84% of its actual selling price, then the profit percentage is:
We need to find the profit percentage when an article is sold at 84% of its actual selling price, given that selling it at \( \frac{2}{5} \) of its actual selling price results in a 27% loss. We will first establish a relationship between the cost price (CP) and the actual selling price (ASP) using the initial condition. Then, we will use this relationship to calculate the profit percentage for the new selling price.
Let the actual selling price be \( \text{ASP} \) and the cost price be \( \text{CP} \).
According to the first condition, the article is sold at \( \frac{2}{5} \) of its actual selling price. So, the selling price in this case is \( \text{SP}_1 = \frac{2}{5} \times \text{ASP} \).
Manav incurs a loss of 27%. This means the selling price \( \text{SP}_1 \) is 27% less than the cost price \( \text{CP} \).
\[ \text{SP}_1 = \text{CP} - 0.27 \times \text{CP} = (1 - 0.27) \times \text{CP} = 0.73 \times \text{CP} \]Equating the two expressions for \( \text{SP}_1 \):
\[ \frac{2}{5} \times \text{ASP} = 0.73 \times \text{CP} \]\[ 0.4 \times \text{ASP} = 0.73 \times \text{CP} \]From this, we can express \( \text{CP} \) in terms of \( \text{ASP} \):
\[ \text{CP} = \frac{0.4}{0.73} \times \text{ASP} \]Now, consider the second condition: the article is sold at 84% of its actual selling price. So, the new selling price is \( \text{SP}_2 = 0.84 \times \text{ASP} \).
To find the profit percentage, we need to calculate the profit, which is \( \text{SP}_2 - \text{CP} \), and then divide it by \( \text{CP} \) and multiply by 100.
\[ \text{Profit Percentage} = \frac{\text{SP}_2 - \text{CP}}{\text{CP}} \times 100 \% \]Substitute the expressions for \( \text{SP}_2 \) and \( \text{CP} \) in terms of \( \text{ASP} \):
\[ \text{Profit Percentage} = \frac{0.84 \times \text{ASP} - \frac{0.4}{0.73} \times \text{ASP}}{\frac{0.4}{0.73} \times \text{ASP}} \times 100 \% \]We can cancel out \( \text{ASP} \) from the numerator and denominator:
\[ \text{Profit Percentage} = \frac{0.84 - \frac{0.4}{0.73}}{\frac{0.4}{0.73}} \times 100 \% \]\[ \text{Profit Percentage} = \left( \frac{0.84 \times 0.73 - 0.4}{0.4} \right) \times 100 \% \]\[ \text{Profit Percentage} = \left( \frac{0.6132 - 0.4}{0.4} \right) \times 100 \% \]\[ \text{Profit Percentage} = \left( \frac{0.2132}{0.4} \right) \times 100 \% \]\[ \text{Profit Percentage} = 0.533 \times 100 \% \]\[ \text{Profit Percentage} = 53.3 \% \]Correct Option: D) 53.3%