To solve this problem, we need to use the divisibility rule for 11. The rule states that a number is divisible by 11 if the difference between the sum of its digits at odd places (from the right) and the sum of its digits at even places (from the right) is either 0 or a multiple of 11. Once we find the value of K, we will substitute it into the expression \(K^2 - 3K\) to find the final answer.
The given number is 58K764.
Sum of digits at odd places (from the right): \(4 + 7 + 8 = 19\)
Sum of digits at even places (from the right): \(6 + K + 5 = 11 + K\)
According to the divisibility rule, the difference between these two sums must be 0 or a multiple of 11.
\[ (19) - (11 + K) = 0 \text{ or a multiple of } 11 \]Let's set the difference to 0 first:
\[ 19 - (11 + K) = 0 \]\[ 19 - 11 - K = 0 \]\[ 8 - K = 0 \]\[ K = 8 \]Since K is a digit, its value must be between 0 and 9. \(K=8\) is a valid digit.
If we consider the difference to be a multiple of 11, for example, 11:
\[ 8 - K = 11 \]\[ K = 8 - 11 \]\[ K = -3 \]This is not a valid digit. If we consider -11:
\[ 8 - K = -11 \]\[ K = 8 + 11 \]\[ K = 19 \]This is also not a valid digit. Thus, \(K=8\) is the only possible value.
Substitute \(K = 8\) into the expression \(K^2 - 3K\):
\[ K^2 - 3K = (8)^2 - 3(8) \]\[ = 64 - 24 \]\[ = 40 \]Correct Option: A) 40 is the value of \(K^2 - 3K\) when K=8.
This problem involves similar triangles. When an object casts a shadow, the sun's rays create a right-angled triangle with the object's height and its shadow length. At any given time, the angle of elevation of the sun is the same for all objects in the vicinity. Therefore, the triangles formed by the electric pole and its shadow, and the tree and its shadow, are similar. We can use the property of similar triangles that the ratio of corresponding sides is equal to find the unknown height of the pole.
Identify the given information:
Understand the principle of similar triangles:
At the same time of day, the angle of elevation of the sun is constant. This means that the angle formed by the top of the object, the tip of its shadow, and the ground is the same for both the tree and the pole. Since both objects are perpendicular to the ground, they form right-angled triangles with their shadows. Therefore, the triangle formed by the pole and its shadow is similar to the triangle formed by the tree and its shadow.
Set up the proportion:
For similar triangles, the ratio of corresponding sides is equal. So, the ratio of the height to the shadow length will be the same for both the tree and the pole.
Substitute the known values into the proportion:
\[ \frac{H_{\text{pole}}}{42 \text{ m}} = \frac{12 \text{ m}}{16 \text{ m}} \]Solve for the height of the pole (\(H_{\text{pole}}\)):
\[ H_{\text{pole}} = \frac{12}{16} \times 42 \]\[ H_{\text{pole}} = \frac{3}{4} \times 42 \]\[ H_{\text{pole}} = 3 \times \frac{42}{4} \]\[ H_{\text{pole}} = 3 \times 10.5 \]\[ H_{\text{pole}} = 31.5 \text{ m} \]Thus, the height of the electric pole is 31.5 m.
A) 31.5 m — This is the correct height calculated using the properties of similar triangles.
The question asks to identify a viral infection that can be transmitted through sexual contact. We need to evaluate each option based on whether it is a viral infection and if it is sexually transmissible.
Correct Option: C) HIV-AIDS is a viral infection that is primarily transmitted through sexual contact.
The problem requires simplifying a trigonometric expression using fundamental trigonometric identities. We will use the identity \((a+b)(a-b) = a^2 - b^2\) and the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\), along with the identity \(\sec^2 A = 1 + \tan^2 A\).
Consider the given expression:
\[ (1 + \sin A)(1 - \sin A)(1 + \tan^2 A) \]Apply the algebraic identity \((a+b)(a-b) = a^2 - b^2\) to the first two terms \((1 + \sin A)(1 - \sin A)\). Here, \(a=1\) and \(b=\sin A\):
\[ (1 + \sin A)(1 - \sin A) = 1^2 - \sin^2 A = 1 - \sin^2 A \]Recall the fundamental trigonometric Pythagorean identity: \(\sin^2 A + \cos^2 A = 1\). From this, we can deduce that \(1 - \sin^2 A = \cos^2 A\).
Substitute this back into the expression:
\[ (\cos^2 A)(1 + \tan^2 A) \]Recall another fundamental trigonometric identity: \(1 + \tan^2 A = \sec^2 A\).
Substitute this into the expression:
\[ (\cos^2 A)(\sec^2 A) \]Recall the reciprocal identity: \(\sec A = \frac{1}{\cos A}\). Therefore, \(\sec^2 A = \frac{1}{\cos^2 A}\).
Substitute this into the expression:
\[ (\cos^2 A) \left(\frac{1}{\cos^2 A}\right) \]Cancel out the \(\cos^2 A\) terms:
\[ \cos^2 A \cdot \frac{1}{\cos^2 A} = 1 \]C) The simplified expression is \(1\).
We need to determine the arrangement of seven boxes (D, E, F, R, S, T, U) stacked one over the other based on the given clues. Once the arrangement is finalized, we can find the number of boxes between R and F.
Correct Option: D) Three
The boxes between R (position 3) and F (position 7) are S (position 6), U (position 5), and T (position 4). There are three boxes.
We need to determine who is the second eldest among five friends (S, T, U, V, W) using the given statements. We will analyze each statement individually and then combine them if necessary to see if the question can be answered.
Correct Option: D) Data in statement II alone is sufficient to answer the question while data in statement I is not.
Match the following rivers with their tributaries.
| Rivers | Tributaries |
| 1. Brahmaputra | a. Betwa |
| 2. Indus | b. Chenab |
| 3. Yamuna | c. Dibang |
1-a, 2-b, 3-c
1-c, 2-a, 3-b
1-b, 2-c, 3-a
1-c, 2-b, 3-a
To answer this question, we need to match each major river with one of its significant tributaries from the given options. This requires knowledge of the major river systems of India and their tributaries.
Correct Option: D) 1-c, 2-b, 3-a
The question asks for the relationship between current (I), power (P), and voltage (V). This is a fundamental concept in electrical engineering and physics, often described by the power formula for DC circuits or instantaneous power in AC circuits.
Correct Option: D) I = P ÷ V
The question asks to identify the articles in the Indian Constitution that deal with the Vice-President's election, qualification, and removal. This requires knowledge of the specific articles related to the Vice-President.
B) Article 63 - 71: This range of articles in the Indian Constitution specifically covers the office of the Vice-President, including his existence (Article 63), role as ex-officio Chairman of the Rajya Sabha (Article 64), acting as President (Article 65), election (Article 66), term of office and removal (Article 67), filling vacancies (Article 68), oath (Article 69), and matters relating to election (Article 71).