The LCM of 15x3y4 and 12x2y5 is:
To find the Least Common Multiple (LCM) of two algebraic expressions, we need to find the LCM of their numerical coefficients and the highest power of each common and uncommon variable present in the expressions.
D) \(60x^3y^5\) is the correct LCM, as calculated by finding the LCM of the numerical coefficients (60) and taking the highest power of each variable (\(x^3\) and \(y^5\)).
An object of mass 1 kg has a potential energy of 2 J relative to the ground, when it is at a height of (take g = 10 m/s2):
To find the height at which an object has a certain potential energy, we use the formula for gravitational potential energy, which is \(PE = mgh\), where \(PE\) is potential energy, \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is the height.
The gravitational potential energy is given by:
\[ PE = mgh \]To find the height, we can rearrange the formula as:
\[ h = \frac{PE}{mg} \]Now, plug in the values:
\[ h = \frac{2 \text{ J}}{(1 \text{ kg})(10 \text{ m/s}^2)} \]Perform the calculation:
\[ h = \frac{2}{10} \text{ m} \]\[ h = 0.2 \text{ m} \]A) 0.2 m — The calculation shows that the height is 0.2 m, which matches this option.
To find the next number in the series, we need to identify the pattern or the rule governing the sequence of numbers. This often involves looking at the differences between consecutive terms.
Therefore, the next number in the series is 88.
Correct Option: B) 88
84÷[50−{43−(30−128÷\( \overline{8 \times 4} \))}]=?
The question asks for the year when Babur established the Mughal dynasty in India. This is a direct factual question related to Indian history, specifically the early Mughal period. Recall the key battle that marked the beginning of Mughal rule.
Correct Option: A) 1526 AD is the correct year when Babur established the Mughal dynasty in India after defeating Ibrahim Lodi in the First Battle of Panipat.
This problem involves the relationship between distance, speed, and time. Since the distance is constant, we can relate the two scenarios (driving at normal speed vs. driving at increased speed due to delay) using the formula \(\text{Distance} = \text{Speed} \times \text{Time}\). We will set up equations based on the given speeds and the time difference to find the total distance.
Let the distance covered be \(D\) km.
Let the usual time taken to cover the distance be \(T\) hours.
The usual speed is 45 km/h. So, we have the equation:
\[ D = 45T \quad \text{(Equation 1)} \]Rahim started 15 minutes late. 15 minutes in hours is \[\frac{15}{60} = \frac{1}{4}\] hour.
To reach on time, he had to drive at a speed of 54 km/h. This means the actual time taken for the journey was \(T - \frac{1}{4}\) hours.
Using the new speed and the actual time, we get another equation for the same distance \(D\):
\[ D = 54 \left(T - \frac{1}{4}\right) \quad \text{(Equation 2)} \]Now, we equate Equation 1 and Equation 2 since the distance \(D\) is the same:
\[ 45T = 54 \left(T - \frac{1}{4}\right) \]Expand the right side of the equation:
\[ 45T = 54T - 54 \times \frac{1}{4} \] \[ 45T = 54T - \frac{54}{4} \] \[ 45T = 54T - 13.5 \]Rearrange the terms to solve for \(T\):
\[ 54T - 45T = 13.5 \] \[ 9T = 13.5 \] \[ T = \frac{13.5}{9} \] \[ T = 1.5 \text{ hours} \]Now substitute the value of \(T\) back into Equation 1 to find the distance \(D\):
\[ D = 45T \] \[ D = 45 \times 1.5 \] \[ D = 67.5 \text{ km} \]Correct Option: B) 67.5 km
To find the total cost of gravelling the paths, we first need to calculate the total area of the paths. The paths run through the center of a square park, forming a cross shape. We will calculate the area of each path and subtract the area of the overlapping central square to avoid double-counting. Finally, we will convert the area to dm² and multiply by the given rate.
This path will have a length equal to the side of the park and a width of 3 m.
\[ \text{Area of path 1} = S \times w = 23 \text{ m} \times 3 \text{ m} = 69 \text{ m}^2 \]This path will also have a length equal to the side of the park and a width of 3 m.
\[ \text{Area of path 2} = S \times w = 23 \text{ m} \times 3 \text{ m} = 69 \text{ m}^2 \]When two paths intersect at the center, they form a square overlap. The side of this square will be equal to the width of the path.
\[ \text{Area of overlap} = w \times w = 3 \text{ m} \times 3 \text{ m} = 9 \text{ m}^2 \]The total area of the paths is the sum of the areas of the two paths minus the area of the overlap (because the overlap is counted twice).
\[ \text{Total area of paths} = (\text{Area of path 1} + \text{Area of path 2}) - \text{Area of overlap} \] \[ \text{Total area of paths} = (69 \text{ m}^2 + 69 \text{ m}^2) - 9 \text{ m}^2 \] \[ \text{Total area of paths} = 138 \text{ m}^2 - 9 \text{ m}^2 = 129 \text{ m}^2 \]We know that \(1 \text{ m} = 10 \text{ dm}\). Therefore, \(1 \text{ m}^2 = (10 \text{ dm})^2 = 100 \text{ dm}^2\).
\[ \text{Total area in dm}^2 = 129 \text{ m}^2 \times 100 \frac{\text{dm}^2}{\text{m}^2} = 12900 \text{ dm}^2 \]Correct Option: C) 12,900
This is a question based on logical deduction, often solved using Venn diagrams or by analyzing the relationships between categories. We need to determine which conclusions logically follow from the given statements.
Correct Option: D) Only conclusion 1 follows.
The question asks to identify the first female Chief Justice of the Balochistan High Court in Pakistan, appointed in 2018. This is a factual question related to current affairs and judicial appointments.
Correct Option: A) Syeda Tahira Safdar was appointed as the Chief Justice of the Balochistan High Court on September 1, 2018. This appointment marked a significant milestone as she became the first woman to hold the position of Chief Justice of any High Court in Pakistan's history.
The question asks about the number of banks nationalised in India in 1969. This is a factual question related to India's economic history and banking reforms. To answer this, one needs to recall the key events of bank nationalisation.
Correct Option: D) 14