To find the mode of the data, we first need to determine the value of 'y'. We are given that the mean of the data set \((9, 3, 5, 4, 4, 5, y)\) is 'y'. We can use the formula for the mean to set up an equation and solve for 'y'. Once 'y' is known, we can list all the data points and find the mode, which is the value that appears most frequently.
Identify the given data: The data set is \(9, 3, 5, 4, 4, 5, y\).
Count the number of data points: There are 7 data points in total.
Recall the formula for the mean:
\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \]Set up the equation using the given information: We are given that the mean is 'y'.
\[ y = \frac{9 + 3 + 5 + 4 + 4 + 5 + y}{7} \]Simplify the sum in the numerator:
\[ y = \frac{30 + y}{7} \]Solve for 'y': Multiply both sides by 7.
\[ 7y = 30 + y \]Subtract 'y' from both sides:
\[ 7y - y = 30 \]\[ 6y = 30 \]Divide by 6:
\[ y = \frac{30}{6} \]\[ y = 5 \]Determine the complete data set: Substitute \(y=5\) back into the original data set.
The data set becomes \(9, 3, 5, 4, 4, 5, 5\).
Find the mode of the data: The mode is the value that appears most frequently in the data set.
The value 5 appears most frequently.
B) 5 — After calculating the value of 'y' as 5, the complete data set is \(9, 3, 5, 4, 4, 5, 5\). In this set, the number 5 appears three times, which is more than any other number. Therefore, the mode of the data is 5.
The question asks to identify which of the given options is NOT a part of the Panchayati Raj system in India, as defined by the Constitution. The Panchayati Raj system is a three-tier structure of local self-governance in rural areas, established by the 73rd Constitutional Amendment Act of 1992.
B) Gram Nigam is not a part of the Panchayati Raj system according to the Constitution. The term "Gram Nigam" does not exist in the constitutional framework for rural local self-governance. "Nigam" is typically associated with urban local bodies or corporations.
To find the volume of a sphere, we need its radius. The problem provides the diameter, so the first step is to calculate the radius from the given diameter. Then, we use the standard formula for the volume of a sphere.
The diameter of the sphere is \(d = 12 \text{ cm}\).
The radius (\(r\)) of a sphere is half of its diameter.\[ r = \frac{d}{2} \]\[ r = \frac{12 \text{ cm}}{2} \]\[ r = 6 \text{ cm} \]
The volume (\(V\)) of a sphere with radius \(r\) is given by the formula:\[ V = \frac{4}{3}\pi r^3 \]
Substitute \(r = 6 \text{ cm}\) into the volume formula:\[ V = \frac{4}{3}\pi (6 \text{ cm})^3 \]\[ V = \frac{4}{3}\pi (216 \text{ cm}^3) \]
Now, perform the multiplication:\[ V = 4 \times \pi \times \frac{216}{3} \text{ cm}^3 \]\[ V = 4 \times \pi \times 72 \text{ cm}^3 \]\[ V = 288\pi \text{ cm}^3 \]
B) \(288\pi \text{ cm}^3\) is the correct volume calculated using the formula for a sphere.
This is a Syllogism question. We will use Venn diagrams to represent the given statements and then evaluate each conclusion based on these diagrams.
Only Conclusion I definitely follows from the given statements.
B) Only conclusion I follows
The question asks to identify a protein-splitting enzyme from the given options. This requires knowledge of the functions of various digestive enzymes in the human body.
Correct Option: D) Pepsin is a proteolytic enzyme (protein-splitting enzyme) secreted in the stomach. It breaks down proteins into smaller polypeptides in the acidic environment of the stomach.
Kidney is an organ of excretion and osmoregulation in humans. Regulation of which two substances is done by the kidneys?
Water and O2
Sugar and salts
CO2 and proteins
Water and salts
The question asks about the primary substances regulated by the kidneys, which are key organs for excretion and osmoregulation in humans. We need to identify the two substances from the given options that are directly and significantly regulated by kidney function.
D) Water and salts — The kidneys play a vital role in maintaining the body's water balance (hydration) and electrolyte balance (salt concentration). They adjust the amount of water and various salts (like sodium, potassium, calcium, chloride) reabsorbed from the filtered blood, thereby regulating blood volume, blood pressure, and the overall osmotic balance of the body fluids. This is a core function of osmoregulation.
The question asks to identify which of the given rivers is NOT a tributary of the Yamuna river system. To answer this, we need to recall the major tributaries of the Yamuna and identify the river that flows into a different river system.
Correct Option: D) Chenab is a tributary of the Indus River system, not the Yamuna River system.
The question asks to identify the incorrect statement regarding transportation in plants. We need to evaluate each option based on our knowledge of plant physiology, specifically the structure and function of xylem and phloem.
The statement "Movement of both water and food is bidirectional" is incorrect. Water movement through the xylem is primarily unidirectional, from the roots upwards to the leaves, driven by transpiration pull. Food (sugars) movement through the phloem, however, is bidirectional, moving from areas of production (source, usually leaves) to areas of consumption or storage (sink, e.g., roots, fruits, growing tips).
The question asks to identify the Indian cricketer who played his 100th Test match against Sri Lanka in Mohali in March 2022. This requires knowledge of significant milestones in Indian cricket, specifically Test match appearances.
C) Virat Kohli — Virat Kohli achieved the milestone of playing his 100th Test match on March 4, 2022, against Sri Lanka in Mohali. He became the 12th Indian cricketer to reach this landmark.
Simplify \(\frac{\sin A}{1+\cos A}+\frac{1+\cos A}{\sin A}\)
To simplify the given trigonometric expression, we will combine the two fractions by finding a common denominator. Then, we will use fundamental trigonometric identities such as \(\sin^2 A + \cos^2 A = 1\) and the reciprocal identity for cosecant.
The given expression is:
\[ \frac{\sin A}{1+\cos A} + \frac{1+\cos A}{\sin A} \]Find a common denominator, which is \((1+\cos A)(\sin A)\). Combine the fractions:
\[ \frac{\sin A \cdot \sin A + (1+\cos A) \cdot (1+\cos A)}{(1+\cos A)(\sin A)} \]\[ = \frac{\sin^2 A + (1+\cos A)^2}{(1+\cos A)(\sin A)} \]Expand the term \((1+\cos A)^2\):
\[ (1+\cos A)^2 = 1^2 + 2(1)(\cos A) + \cos^2 A = 1 + 2\cos A + \cos^2 A \]Substitute this back into the numerator:
\[ \frac{\sin^2 A + 1 + 2\cos A + \cos^2 A}{(1+\cos A)(\sin A)} \]Rearrange the terms in the numerator and use the identity \(\sin^2 A + \cos^2 A = 1\):
\[ \frac{(\sin^2 A + \cos^2 A) + 1 + 2\cos A}{(1+\cos A)(\sin A)} \]\[ = \frac{1 + 1 + 2\cos A}{(1+\cos A)(\sin A)} \]\[ = \frac{2 + 2\cos A}{(1+\cos A)(\sin A)} \]Factor out 2 from the numerator:
\[ \frac{2(1+\cos A)}{(1+\cos A)(\sin A)} \]Cancel out the common term \((1+\cos A)\) from the numerator and denominator:
\[ \frac{2}{\sin A} \]Recall that \(\frac{1}{\sin A} = \text{cosec } A\). Therefore:
\[ 2 \cdot \frac{1}{\sin A} = 2\text{cosec } A \]D) The simplified expression is \(2\text{cosec } A\).