To find the shortest distance and direction to return to the starting point, we need to determine the final position relative to the starting point. This can be done by tracking the net displacement in the East-West and North-South directions. We will consider North and East as positive directions and South and West as negative directions.
C) 56 km towards the west
The final position of Gyanesh is 56 km to the East of Point Y. Therefore, to return to Point Y, he must drive 56 km towards the West.
The question asks for the primary objective of 'Srjanam', a biomedical waste conversion system. To answer this, we need to understand the common problems associated with biomedical waste disposal and how a new conversion system would typically address them.
B) To reduce environmental hazards by eliminating incineration. Biomedical waste incineration is a significant source of air pollutants, including dioxins and furans, which are highly toxic. A new conversion system like 'Srjanam' would primarily aim to offer an environmentally safer alternative, thereby reducing these hazards by moving away from incineration.
This problem involves the property of intersecting chords outside a circle. When two chords intersect externally at a point P, the product of the segments of one chord is equal to the product of the segments of the other chord. We will use this property to find the unknown length.
Understand the property of external intersection of chords:
When two chords AB and CD of a circle intersect externally at a point P, the property states that:
Identify the given values:
We are given:
Express PC in terms of PD and CD:
Since P, C, and D are collinear and C is between P and D (as CD is a chord and P is external), we have:
Express PA in terms of PB and AB:
Similarly, P, A, and B are collinear and A is between P and B (as AB is a chord and P is external). Let \(PB = x\). Then:
Apply the external intersection property:
Substitute the expressions for PA, PB, PC, and PD into the formula:
Solve the quadratic equation:
Rearrange the equation into standard quadratic form:
Choose the valid solution:
Since \(x\) represents a length (PB), it must be a positive value. Therefore, \(x = 3\).
State the final answer:
The length of PB is \(3 \text{ cm}\).
A) 3
The problem asks us to identify the letter-cluster pair that does not follow the same pattern as the others, based on the English alphabetical order. We need to find a consistent rule or relationship between the letters within each pair and between the two clusters in each option.
Option A: OJ - NL
Option B: IK - PR
Option C: HJ - OQ
Option D: GI - NP
Comparing the patterns:
A) OJ - NL. This pair does not follow the established pattern where the letters within each cluster have a difference of 2, and the corresponding letters across clusters have a difference of 7.
To find the difference between the length of the train and the platform, we first need to calculate the length of the platform. We can do this by using the formula: \(\text{Distance} = \text{Speed} \times \text{Time}\). The total distance covered by the train while crossing the platform is the sum of the length of the train and the length of the platform. We need to ensure all units are consistent (meters and seconds).
The question asks about the geological feature that existed between the Himalayas and the Deccan Plateau before the formation of the Great Plain of India. This relates to the plate tectonics and geomorphological evolution of the Indian subcontinent.
Correct Option: C) A shallow trough or geosyncline. This accurately describes the geological depression that formed between the rising Himalayas and the stable Deccan Plateau, which was subsequently filled by sediments to form the Great Plain.
Find the value of \(\frac{\cos35^\circ}{\sin55^\circ}\) + \(\frac{\sin11^\circ}{\cos79^\circ}\) − 2 cos 27° cosec 63°.
This problem involves trigonometric identities, specifically the complementary angle identities. We will use the relations \(\sin(90^\circ - \theta) = \cos\theta\), \(\cos(90^\circ - \theta) = \sin\theta\), and \(\text{cosec}\theta = \frac{1}{\sin\theta}\) to simplify each term in the given expression.
The first term is \(\frac{\cos35^\circ}{\sin55^\circ}\).
We know that \(55^\circ = 90^\circ - 35^\circ\). So, \(\sin55^\circ = \sin(90^\circ - 35^\circ)\).
Using the complementary angle identity \(\sin(90^\circ - \theta) = \cos\theta\), we get \(\sin55^\circ = \cos35^\circ\).
Therefore, the first term becomes:
\[ \frac{\cos35^\circ}{\sin55^\circ} = \frac{\cos35^\circ}{\cos35^\circ} = 1 \]The second term is \(\frac{\sin11^\circ}{\cos79^\circ}\).
We know that \(79^\circ = 90^\circ - 11^\circ\). So, \(\cos79^\circ = \cos(90^\circ - 11^\circ)\).
Using the complementary angle identity \(\cos(90^\circ - \theta) = \sin\theta\), we get \(\cos79^\circ = \sin11^\circ\).
Therefore, the second term becomes:
\[ \frac{\sin11^\circ}{\cos79^\circ} = \frac{\sin11^\circ}{\sin11^\circ} = 1 \]The third term is \(-2 \cos27^\circ \text{cosec}63^\circ\).
We know that \(63^\circ = 90^\circ - 27^\circ\). So, \(\text{cosec}63^\circ = \text{cosec}(90^\circ - 27^\circ)\).
Using the complementary angle identity \(\text{cosec}(90^\circ - \theta) = \sec\theta\), we get \(\text{cosec}63^\circ = \sec27^\circ\).
Also, we know that \(\sec\theta = \frac{1}{\cos\theta}\).
Therefore, the third term becomes:
\[ -2 \cos27^\circ \text{cosec}63^\circ = -2 \cos27^\circ \sec27^\circ = -2 \cos27^\circ \left(\frac{1}{\cos27^\circ}\right) = -2 \]Now, substitute the simplified values back into the original expression:
\[ \frac{\cos35^\circ}{\sin55^\circ} + \frac{\sin11^\circ}{\cos79^\circ} - 2 \cos27^\circ \text{cosec}63^\circ = 1 + 1 - 2 \]\[ = 2 - 2 = 0 \]A) The final calculated value of the expression is 0.
To determine the greatest ratio, we can convert each ratio into a decimal or a common fraction. Converting to decimals is often the easiest way to compare them directly.
Comparing these decimal values, \(0.6667\) is the greatest.
B) \(10 : 15\)
This ratio simplifies to \(2:3\), which is approximately \(0.6667\), the largest value among the given options.
The problem asks us to find the missing letter-clusters, # and %, based on a pattern and relationship established between the given letter-cluster pair. We need to analyze the transformation from the first cluster to the second in the known pair and apply the same logic to find the missing clusters.
To find the smallest number by which 6760 must be multiplied to make it a perfect square, we need to perform prime factorization of 6760. A number is a perfect square if all the exponents in its prime factorization are even. We will identify the prime factors with odd exponents and multiply 6760 by the missing factors to make all exponents even.
First, we find the prime factors of 6760.
\[ 6760 = 10 \times 676 \]\[ 6760 = (2 \times 5) \times (26 \times 26) \]\[ 6760 = (2 \times 5) \times (2 \times 13) \times (2 \times 13) \]\[ 6760 = 2 \times 5 \times 2^2 \times 13^2 \]\[ 6760 = 2^3 \times 5^1 \times 13^2 \]For a number to be a perfect square, all the exponents in its prime factorization must be even. In the prime factorization of 6760:
To make the exponents of 2 and 5 even, we need to multiply 6760 by \(2^1\) and \(5^1\).
The smallest number to multiply by is \(2 \times 5\).
\[ \text{Multiplier} = 2 \times 5 = 10 \]If we multiply 6760 by 10:
\[ 6760 \times 10 = (2^3 \times 5^1 \times 13^2) \times (2^1 \times 5^1) \]\[ = 2^{(3+1)} \times 5^{(1+1)} \times 13^2 \]\[ = 2^4 \times 5^2 \times 13^2 \]\[ = (2^2 \times 5 \times 13)^2 \]\[ = (4 \times 5 \times 13)^2 \]\[ = (20 \times 13)^2 \]\[ = (260)^2 \]Since \(260^2\) is a perfect square, the smallest number to multiply 6760 by is 10.
D) 10 is the smallest number by which 6760 must be multiplied to make it a perfect square, as it provides the missing prime factors (2 and 5) to make all exponents even in the prime factorization.