If tan θ = \( \frac{5}{12} \), 0 < θ < \( \frac{π}{2} \), then the value of \(\frac{\cos\theta+5\cot\theta}{\mathrm{cosec} \theta-\cos\theta}\) will be:
\(\frac{845}{109}\)
\(\frac{860}{190}\)
\(\frac{840}{109}\)
\(\frac{840}{190}\)
We are given the value of \(\tan \theta\) and the range of \(\theta\). We need to find the value of a trigonometric expression. We can use the given \(\tan \theta\) to find the values of other trigonometric ratios like \(\sin \theta\), \(\cos \theta\), \(\cot \theta\), and \(\text{cosec } \theta\). Since \(0 < \theta < \frac{\pi}{2}\), \(\theta\) is in the first quadrant, so all trigonometric ratios will be positive.
Given \(\tan \theta = \frac{5}{12}\). We know that \[\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\]. So, we can consider a right-angled triangle where the opposite side is 5 units and the adjacent side is 12 units.
Using the Pythagorean theorem, the hypotenuse \(h\) can be calculated as:
\[ h = \sqrt{(\text{Opposite})^2 + (\text{Adjacent})^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]Now, we can find the values of other trigonometric ratios:
Substitute these values into the given expression \[\frac{\cos\theta+5\cot\theta}{\mathrm{cosec} \theta-\cos\theta}\]:
\[ \frac{\frac{12}{13} + 5 \left(\frac{12}{5}\right)}{\frac{13}{5} - \frac{12}{13}} \]Simplify the numerator:
\[ \frac{12}{13} + 5 \times \frac{12}{5} = \frac{12}{13} + 12 = \frac{12 + 12 \times 13}{13} = \frac{12 + 156}{13} = \frac{168}{13} \]Simplify the denominator:
\[ \frac{13}{5} - \frac{12}{13} = \frac{13 \times 13 - 12 \times 5}{5 \times 13} = \frac{169 - 60}{65} = \frac{109}{65} \]Now, divide the simplified numerator by the simplified denominator:
\[ \frac{\frac{168}{13}}{\frac{109}{65}} = \frac{168}{13} \times \frac{65}{109} \]Cancel out common factors (13 and 65):
\[ \frac{168}{\cancel{13}} \times \frac{5 \times \cancel{13}}{109} = \frac{168 \times 5}{109} = \frac{840}{109} \]Correct Option: C) \(\frac{840}{109}\)
The question asks to identify which of the given options is NOT a part of computer hardware. To answer this, we need to understand the basic distinction between hardware and software in a computer system.
Correct Option: B) Microsoft Office is a suite of application software, not a physical component (hardware) of a computer.
The question asks to identify the cell organelle known as the "powerhouse of the cell." This is a fundamental concept in cell biology, referring to the organelle responsible for generating most of the cell's supply of adenosine triphosphate (ATP), used as a source of chemical energy.
Correct Option: A) Mitochondria are the organelles responsible for generating most of the cell's supply of adenosine triphosphate (ATP), which is used as a source of chemical energy. This process is known as cellular respiration.
To find the next number in the series, we need to identify the pattern or rule governing the given sequence of numbers: 0, 6, 25, 62.
A) 123 is the correct answer based on the identified pattern \(n^3 - 2\) for \(n \ge 2\), with the first term being \(1^3 - 1\).
The question asks to identify the river on which Independent India's first river dam was constructed. This requires knowledge of India's post-independence infrastructure projects, specifically the early river valley developments.
Correct Option: D) Damodar — The Damodar Valley Project (DVP) was the first major multi-purpose river valley project of independent India. The first dam under this project, the Tilaiya Dam, was constructed on the Barakar River, a tributary of the Damodar, making the Damodar river system the location of India's first river dam post-independence.
The question asks to identify the Indian envoy appointed to Bangladesh in October 2020. This is a factual question related to diplomatic appointments, which are common in current affairs sections of competitive exams.
In October 2020, Vikram Kumar Doraiswami was appointed as India's High Commissioner to Bangladesh. He succeeded Riva Ganguly Das.
C) Vikram Kumar Doraiswami — He was indeed appointed as the Indian High Commissioner to Bangladesh in October 2020. He is a 1992-batch Indian Foreign Service (IFS) officer.
The question asks to identify the builder of the Charminar in Hyderabad. This requires knowledge of Indian history, specifically the Deccan Sultanates and the Qutb Shahi dynasty.
Correct Option: B) Muhammad Quli Qutab Shah was the fifth ruler of the Qutb Shahi dynasty and the founder of the city of Hyderabad. He ordered the construction of the Charminar in 1591 CE to commemorate the end of a deadly plague in the city and as a mosque and madrasa.
The question asks to identify the Indian state where Ghoomar, a folk song and dance form, is popular. This requires knowledge of Indian folk arts and their regional associations.
Correct Option: C) Rajasthan
Ghoomar is indeed a very popular and iconic folk dance of Rajasthan. It originated among the Bhil tribe and was later adopted by other Rajasthani communities, including the Rajput royal families. It is performed by women in traditional ghagras, swirling gracefully to folk music, especially during festivals and special occasions.
The problem presents a series of letter-clusters. To find the missing cluster, we need to identify the pattern in the positions of the letters in the English alphabet.
The next letter-cluster in the series is formed by combining the next first letter (Q) and the next second letter (U).
Therefore, the missing cluster is QU.
C) QU is the correct option because it follows the established pattern for both the first and second letters of the clusters.
The problem asks us to find the next letter-cluster in the given series: DW, HS, LO, PK, ?. We need to identify the pattern in the positions of the letters in the English alphabet for both the first and second letters of each cluster.
C) TG — As determined from the step-by-step analysis, the first letters follow a pattern of adding 4 to their positional value, and the second letters follow a pattern of subtracting 4 from their positional value. This leads to TG.