The problem requires us to apply specific letter transformation rules to the word KINDLE and then identify which letter appears twice in the resulting group of letters. We need to categorize each letter in KINDLE as a vowel or a consonant and apply the respective rule.
B) J is the letter that appears twice in the group of letters thus formed (J, J, M, C, K, F).
The question asks about the specific district authority from whom producers must obtain permission to engage a child actor, according to the National Commission for Protection of Child Rights (NCPCR) draft regulatory guidelines 2022. This requires knowledge of the relevant legal and administrative provisions concerning child protection in the entertainment industry.
Correct Option: D) The National Commission for Protection of Child Rights (NCPCR) draft regulatory guidelines 2022 specify that producers must obtain permission from the District Magistrate of the district where the activity is to be conducted, or where the child resides, before engaging a child actor in the entertainment industry. This ensures a high-level administrative oversight for child protection.
The problem asks us to find the median of a dataset given its mean and mode, using the empirical relation between these three measures of central tendency. The empirical relation states that for a moderately skewed distribution, the difference between the mean and mode is approximately three times the difference between the mean and median.
D) 45.7 — By applying the empirical relation \( \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \) and substituting the given mean (46) and mode (45), we find the median to be approximately 45.7 after rounding to one decimal place.
A house and a garden together cost Rs 51 lakh and the cost of the garden is \( \frac{5}{12} \) of the cost of the house. Find the cost of the garden.
The problem involves setting up an equation based on the given information about the total cost and the ratio of the costs of the garden and the house. We will use variables to represent the costs and then solve for the cost of the garden.
Let the cost of the house be \(H\) and the cost of the garden be \(G\).
According to the problem, the house and garden together cost Rs 51 lakh. So, we can write the equation:
\[ H + G = 51 \quad (\text{Equation 1}) \]The cost of the garden is \( \frac{5}{12} \) of the cost of the house. This can be written as:
\[ G = \frac{5}{12} H \quad (\text{Equation 2}) \]Now, substitute the expression for \(G\) from Equation 2 into Equation 1:
\[ H + \frac{5}{12} H = 51 \]To combine the terms with \(H\), find a common denominator:
\[ \frac{12}{12} H + \frac{5}{12} H = 51 \]\[ \frac{12H + 5H}{12} = 51 \]\[ \frac{17H}{12} = 51 \]Solve for \(H\):
\[ 17H = 51 \times 12 \]\[ H = \frac{51 \times 12}{17} \]\[ H = 3 \times 12 \]\[ H = 36 \]So, the cost of the house is Rs 36 lakh.
Now, find the cost of the garden using Equation 2:
\[ G = \frac{5}{12} H \]\[ G = \frac{5}{12} \times 36 \]\[ G = 5 \times 3 \]\[ G = 15 \]Thus, the cost of the garden is Rs 15 lakh.
C) 15 lakh. As calculated, the cost of the garden is Rs 15 lakh.
The problem involves finding the measure of an exterior angle of a triangle given the ratio of its interior angles. We will use the properties that the sum of angles in a triangle is 180° and that an exterior angle of a triangle is equal to the sum of its two opposite interior angles.
Let the three angles of \(\triangle ABC\) be \(\angle A\), \(\angle B\), and \(\angle C\). Given that their ratio is \(3:4:5\), we can represent them as \(3k\), \(4k\), and \(5k\) respectively.
The sum of the angles in a triangle is \(180^\circ\). Therefore,
\[ 3k + 4k + 5k = 180^\circ \]Combine the terms:
\[ 12k = 180^\circ \]Solve for \(k\):
\[ k = \frac{180^\circ}{12} = 15^\circ \]Now, calculate the measure of each interior angle:
Verify that the sum is \(45^\circ + 60^\circ + 75^\circ = 180^\circ\).
We need to find the measure of the exterior angle \(\angle CBX\). An exterior angle of a triangle is equal to the sum of its two opposite interior angles. In this case, \(\angle CBX\) is exterior to \(\angle B\) and opposite to \(\angle A\) and \(\angle C\).
Therefore,
\[ \angle CBX = \angle A + \angle C \]Substitute the values of \(\angle A\) and \(\angle C\):
\[ \angle CBX = 45^\circ + 75^\circ = 120^\circ \]Alternatively, \(\angle CBX\) and \(\angle ABC\) (which is \(\angle B\)) form a linear pair, meaning their sum is \(180^\circ\). So,
\[ \angle CBX = 180^\circ - \angle ABC \]\[ \angle CBX = 180^\circ - 60^\circ = 120^\circ \]Given a sinθ = b cosθ and a sin3θ + b cos3θ = sinθ cosθ, for 0 < θ < 90°, find the value of a2 + b2
The question gives two relations, a sin θ = b cos θ and a sin³θ + b cos³θ = sin θ cos θ, for 0° < θ < 90°. We need to use the first relation to simplify the second, find a and b in terms of θ, and then calculate a² + b².
Identify the given information:
\[ a\sin\theta=b\cos\theta \] \[ a\sin^3\theta+b\cos^3\theta=\sin\theta\cos\theta \]0° < θ < 90°, and the value of a² + b² is to be determined.
Rewrite the first term of the second relation:
\[ a\sin^3\theta=(a\sin\theta)\sin^2\theta \]Using a sin θ = b cos θ:
\[ =b\cos\theta\sin^2\theta \]Simplify the second relation:
\[ b\cos\theta\sin^2\theta+b\cos^3\theta=\sin\theta\cos\theta \] \[ b\cos\theta(\sin^2\theta+\cos^2\theta)=\sin\theta\cos\theta \] \[ b\cos\theta=\sin\theta\cos\theta \]Since cos θ ≠0 for 0° < θ < 90°:
\[ b=\sin\theta \]Find the value of a:
\[ a\sin\theta=b\cos\theta=\sin\theta\cos\theta \]Since sin θ ≠0:
\[ a=\cos\theta \]Calculate a² + b²:
\[ a^2+b^2=\cos^2\theta+\sin^2\theta \] \[ =1 \]Correct Option: B) 1 — Substituting a sin θ = b cos θ into the second relation gives b = sin θ, and then a = cos θ. So, a² + b² = cos²θ + sin²θ = 1. Hence, Option B is the correct answer.
A) 0: This is incorrect because the correct value works out to 1; this value cannot be obtained mathematically from the given data.
C) 3: This is incorrect because the correct value works out to 1; this value cannot be obtained mathematically from the given data.
D) 2: This is incorrect because the correct value works out to 1; this value cannot be obtained mathematically from the given data.
Fleming's Right Hand Rule is used to determine the direction of induced current in a conductor moving in a magnetic field. This rule is based on the principle of electromagnetic induction. It establishes a relationship between the direction of the magnetic field, the direction of motion of the conductor, and the direction of the induced current. The rule specifically states that these three directions must be mutually perpendicular to each other.
Correct Option: B) The directions of the applied magnetic field, the motion of the conductor, and the induced current must be mutually at 90° with respect to each other for Fleming's Right Hand Rule to be applied. This is a defining characteristic of the rule.
To find the magnification of the image formed by a convex lens, we will first use the lens formula to determine the image distance. Once the image distance is known, we can calculate the magnification using the magnification formula.
Identify the given values:
Apply the lens formula:
The lens formula is given by:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]Substitute the given values:
\[ \frac{1}{v} - \frac{1}{(-1.5f)} = \frac{1}{f} \] \[ \frac{1}{v} + \frac{1}{1.5f} = \frac{1}{f} \]Solve for the image distance (\(v\)):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{1.5f} \]To subtract the fractions, find a common denominator, which is \(1.5f\):
\[ \frac{1}{v} = \frac{1.5}{1.5f} - \frac{1}{1.5f} \] \[ \frac{1}{v} = \frac{1.5 - 1}{1.5f} \] \[ \frac{1}{v} = \frac{0.5}{1.5f} \] \[ \frac{1}{v} = \frac{1}{3f} \]Therefore, the image distance is:
\[ v = 3f \]Calculate the magnification (\(m\)):
The magnification formula for a lens is:
\[ m = \frac{v}{u} \]Substitute the values of \(v\) and \(u\):
\[ m = \frac{3f}{-1.5f} \] \[ m = -2 \]The negative sign indicates that the image is inverted. The magnitude of the magnification is 2.
C) The magnification is 2.0. The magnitude of the magnification is 2, indicating that the image is twice the size of the object. The negative sign in the calculation indicates that the image is inverted and real, which is consistent with an object placed between \(F\) and \(2F\) for a convex lens (since \(1.5f\) is between \(f\) and \(2f\)).
Eight friends are sitting around a square table facing the centre. Four of them are sitting at the corners and the remaining four are sitting at the exact middle of the four sides. Aman is sitting fourth to the left of Priya. Lakshay is an immediate neighbour of both Priya and Sakshi. Utkarsh is sitting fourth to the left of Sakshi. Riddhi is sitting to the immediate right of Utkarsh. Priya is sitting to the immediate left of Vidhi. Vidhi is sitting fourth to the right of Tejaswi. Who is sitting to the immediate right of Aman?
The question gives the seating positions of eight friends around a square table, all facing the centre. We need to place each friend step by step using the given conditions, and then find who is sitting to the immediate right of Aman.
Identify the given information:
Recall the direction rule:
When people sit around a table facing the centre, moving to a person's left means moving clockwise, and moving to a person's right means moving anticlockwise.
Place Priya, Aman, Vidhi and Tejaswi:
Fix Priya at seat 1 and number the seats clockwise from 1 to 8.
Aman is fourth to the left of Priya, so Aman is at seat 5.
Priya is to the immediate left of Vidhi, so Vidhi is one seat anticlockwise from Priya, at seat 8.
Vidhi is fourth to the right of Tejaswi, so Tejaswi is four seats clockwise from Vidhi, at seat 4.
Place Lakshay, Sakshi, Utkarsh and Riddhi:
Lakshay is a neighbour of Priya. Seat 8 is taken by Vidhi, so Lakshay is at seat 2 and Sakshi is at seat 3.
Utkarsh is fourth to the left of Sakshi, so Utkarsh is at seat 7.
Riddhi is to the immediate right of Utkarsh, so Riddhi is at seat 6.
Final arrangement (clockwise): Priya (1), Lakshay (2), Sakshi (3), Tejaswi (4), Aman (5), Riddhi (6), Utkarsh (7), Vidhi (8).
Find the person to the immediate right of Aman:
Aman is at seat 5. His immediate right is one seat anticlockwise, which is seat 4.
Tejaswi sits at seat 4.
Correct Option: A) Tejaswi — The clockwise arrangement is Priya, Lakshay, Sakshi, Tejaswi, Aman, Riddhi, Utkarsh, Vidhi. One seat anticlockwise from Aman is Tejaswi, so Tejaswi is to his immediate right. Hence, Option A is the correct answer.
B) Riddhi: This is incorrect because Riddhi sits to the immediate left of Aman, not the right.
C) Lakshay: This is incorrect because Lakshay sits third to the right of Aman.
D) Vidhi: This is incorrect because Vidhi sits third to the left of Aman.
The given series is a sequence of numbers. We need to identify the pattern or rule governing this sequence to find the missing number. We will examine the differences between consecutive terms or look for a common mathematical property.
Correct Option: A) 29 is the prime number that fits into the decreasing sequence of prime numbers.