A wire of a given material has length 'l' and resistance 'R'. Another wire of the same material having nine times the length and the same area of cross section will have a resistance equal to:
The resistance of a wire depends on its material, length, and cross-sectional area. We will use the formula for resistance and compare the resistance of the two wires given their respective properties.
Recall the formula for resistance:
The resistance \(R\) of a wire is given by the formula:
\[ R = \rho \frac{l}{A} \]where:
Analyze the first wire:
For the first wire, we are given:
So, the resistance of the first wire is:
\[ R = \rho \frac{l}{A} \quad \text{(Equation 1)} \]Analyze the second wire:
For the second wire, we are given:
Let the resistance of the second wire be \(R'\). Using the formula:
\[ R' = \rho \frac{9l}{A} \quad \text{(Equation 2)} \]Compare the resistances:
We can rewrite Equation 2 by factoring out the constant terms:
\[ R' = 9 \left( \rho \frac{l}{A} \right) \]From Equation 1, we know that \(R = \rho \frac{l}{A}\). Substitute this into the expression for \(R'\):
\[ R' = 9R \]Therefore, the resistance of the second wire will be 9R.
Correct Option: C) 9R
The problem asks us to find the largest of three consecutive multiples of 9 whose sum is 2457. We can represent these multiples algebraically and set up an equation to solve for them.
Correct Option: A) 828
To find the approximate value, we will round each number in the equation to its nearest whole number or a convenient decimal place before performing the calculations. This simplifies the arithmetic while maintaining a reasonable level of accuracy for approximation questions.
The equation becomes:
\[ ? = 126 - 36 + 4 \times 16 \]C) 154 is the closest approximate value.
Study the diagram and answer the question that follows. The numbers in different sections indicate the number of persons.
How many Indian industrialists make autos but are NOT steel producers?
To determine how many Indian industrialists make autos but are not steel producers, we need to analyze the Venn diagram provided.
Correct Option: A) 6
The question asks to identify the state where India's largest brackish water lake is located. This requires knowledge of major geographical features of India, specifically the largest brackish water lake and its location.
Correct Option: B) Odisha
Chilika Lake, located in the state of Odisha, is the largest brackish water lagoon in India and the second-largest lagoon in the world. It is a highly productive ecosystem with rich fishery resources and is a designated Ramsar site, indicating its international importance as a wetland habitat.
P, Q, R and S are four points on the surface of a concave mirror as shown in the figure.
If r1, r2, r3 and r4 are the distances of centre of curvature from points P, Q, R and S, respectively, then the correct relation between r1, r2, r3 and r4 is:
r1 = r2, r3 = r4, r2 ≠ r3
r1 = r2 = r3 = r4
r1 > r2 > r3 > r4
r1 < r2 < r3 < r4
To solve this problem, we need to understand the properties of a concave mirror and the concept of the center of curvature.
Therefore, the relation between the distances is \(r_1 = r_2 = r_3 = r_4\).
Correct Option: B) \(r_1 = r_2 = r_3 = r_4\)
An object placed 30 cm from a concave lens forms a virtual erect image at a distance of 10 cm from the lens. The magnification produced by the lens is:
To find the magnification produced by a concave lens, we need to use the lens formula and the magnification formula. We are given the object distance and the image distance. We must pay attention to the sign conventions for concave lenses and virtual images.
Identify the given values and sign conventions:
For a concave lens, the focal length is negative. Object distance (\(u\)) is always negative as per convention when placed in front of the lens.
Apply the magnification formula:
The magnification (\(m\)) produced by a lens is given by the ratio of the image distance to the object distance:
Substitute the values into the formula:
\[ m = \frac{-10 \text{ cm}}{-30 \text{ cm}} \]Calculate the magnification:
\[ m = \frac{1}{3} \]Interpret the result:
A positive magnification indicates that the image is erect. A magnification less than 1 indicates that the image is diminished. Both are consistent with the properties of a concave lens forming a virtual, erect, and diminished image.
Correct Option: D) \( \frac{1}{3} \)
The magnification is \( \frac{1}{3} \). This positive value indicates an erect image, and a value less than 1 indicates a diminished image, which is characteristic of a concave lens forming a virtual image.
This problem involves calculating the actual travel time based on varying speeds and corresponding delays. We can use the relationship between distance, speed, and time, and the given delays to find the actual time required to cover the distance.
Let the distance be \(D\) km and the actual time to reach on time be \(T\) hours. The actual time in minutes is \(60T\).
When the bike runs at 50 km/h, it is 10 minutes late. So, the time taken is \(T + \frac{10}{60}\) hours. Using the formula \(\text{Distance} = \text{Speed} \times \text{Time}\), we get:
\[ D = 50 \left( T + \frac{10}{60} \right) \quad \text{(Equation 1)} \]When the bike runs at 60 km/h, it is 5 minutes late. So, the time taken is \(T + \frac{5}{60}\) hours. Using the same formula, we get:
\[ D = 60 \left( T + \frac{5}{60} \right) \quad \text{(Equation 2)} \]Since the distance \(D\) is the same in both cases, we can equate Equation 1 and Equation 2:
\[ 50 \left( T + \frac{10}{60} \right) = 60 \left( T + \frac{5}{60} \right) \]Simplify the equation:
\[ 50T + 50 \times \frac{10}{60} = 60T + 60 \times \frac{5}{60} \] \[ 50T + \frac{500}{60} = 60T + \frac{300}{60} \] \[ 50T + \frac{50}{6} = 60T + 5 \]Rearrange the terms to solve for \(T\):
\[ \frac{50}{6} - 5 = 60T - 50T \] \[ \frac{50 - 30}{6} = 10T \] \[ \frac{20}{6} = 10T \] \[ T = \frac{20}{6 \times 10} \] \[ T = \frac{20}{60} \] \[ T = \frac{1}{3} \text{ hours} \]Convert the actual time \(T\) from hours to minutes:
\[ \text{Time in minutes} = \frac{1}{3} \text{ hours} \times 60 \frac{\text{minutes}}{\text{hour}} = 20 \text{ minutes} \]So, the bike should take 20 minutes to complete the journey on time.
C) 20 minutes is the correct answer. This is the calculated actual time needed to complete the journey on time.
The problem describes a seating arrangement around a square table and then changes in the arrangement. We need to determine the final position of individuals and identify who is sitting to the immediate left of a specific person after the changes.
B) Shalaka. The question asks who is sitting to the immediate left of Tara. Based on the final arrangement (Shalaka, Mitali, Kamini, Tara in clockwise order), if you are Tara and facing the center, the person to your immediate left (anti-clockwise) is Kamini. Oh, wait, I made a mistake in my step 8. Let's re-evaluate step 8.
Let's re-evaluate step 8 carefully. The clockwise order is Shalaka, Mitali, Kamini, Tara. If Tara is at a position, and everyone is facing the center, then to Tara's immediate left (anti-clockwise direction) is Kamini. This means the correct answer should be Kamini. However, the provided correct answer is B) Shalaka. Let's re-read the question and my interpretation.
Let's re-draw the initial arrangement to be absolutely sure. Assume the positions are 1, 2, 3, 4 clockwise.
1. Himani is to the immediate right of Brinda. (B -> H clockwise)
2. Mitali is to the immediate left of Shalaka. (S -> M anti-clockwise)
3. Himani is second to the left of Shalaka. (S -> _ -> H anti-clockwise)
Let's place Shalaka (S) at position 1.
From (3), Himani (H) is at position 3 (second to the left of S, which is anti-clockwise).
From (2), Mitali (M) is to the immediate left of Shalaka. So M is at position 2.
So far: S (pos 1), M (pos 2), H (pos 3), _ (pos 4)
From (1), Himani is to the immediate right of Brinda. This means Brinda is to the immediate left of Himani. So Brinda (B) is at position 4.
Initial arrangement (clockwise): Shalaka (1), Mitali (2), Himani (3), Brinda (4).
Now for the changes:
New arrangement (clockwise): Shalaka (1), Mitali (2), Kamini (3), Tara (4).
The question asks: "Now who is sitting to the immediate left of Tara?"
If Tara is at position 4, and everyone is facing the center, then her immediate left is the person at position 3. The person at position 3 is Kamini.
Therefore, the correct answer should be Kamini.
Let's re-check the provided correct answer (B) Shalaka. If Shalaka is to the immediate left of Tara, it implies the order is Tara, Shalaka, ... (anti-clockwise). This would mean Tara is at position 1 and Shalaka at position 4, which contradicts our derived arrangement.
Let's assume the interpretation of "left" and "right" might be reversed in my initial understanding or the question's intended meaning, or the provided answer is incorrect. However, standard seating arrangement problems define "left" as anti-clockwise and "right" as clockwise when facing the center.
Let's try another way to build the initial arrangement, assuming "right" is clockwise and "left\
To find the number of sides of a regular polygon given its interior angle, we can use the formula for the measure of an interior angle of a regular polygon, or we can first find the exterior angle and then use the relationship between the exterior angle and the number of sides.
The measure of each interior angle of a regular polygon with \(n\) sides is given by:
\[ \text{Interior Angle} = \frac{(n-2) \times 180^{\circ}}{n} \]We are given that the interior angle is \(120^{\circ}\). So,
\[ 120^{\circ} = \frac{(n-2) \times 180^{\circ}}{n} \]Multiply both sides by \(n\):
\[ 120n = (n-2) \times 180 \]Distribute \(180\) on the right side:
\[ 120n = 180n - 360 \]Subtract \(180n\) from both sides:
\[ 120n - 180n = -360 \] \[ -60n = -360 \]Divide by \(-60\):
\[ n = \frac{-360}{-60} \] \[ n = 6 \]The interior angle and exterior angle at any vertex of a polygon sum to \(180^{\circ}\).
\[ \text{Interior Angle} + \text{Exterior Angle} = 180^{\circ} \]Given Interior Angle = \(120^{\circ}\), so:
\[ 120^{\circ} + \text{Exterior Angle} = 180^{\circ} \] \[ \text{Exterior Angle} = 180^{\circ} - 120^{\circ} = 60^{\circ} \]For any regular polygon, the sum of its exterior angles is \(360^{\circ}\). If there are \(n\) sides, then each exterior angle is \(\frac{360^{\circ}}{n}\).
\[ \text{Exterior Angle} = \frac{360^{\circ}}{n} \]Substitute the calculated exterior angle:
\[ 60^{\circ} = \frac{360^{\circ}}{n} \]Solve for \(n\):
\[ n = \frac{360^{\circ}}{60^{\circ}} \] \[ n = 6 \]Correct Option: B) 6