The question asks to identify the proponent of the 'Advaita' theory in Hinduism. This requires knowledge of the major philosophical schools (Darshanas) within Hinduism and their founders.
Correct Option: D) Adi Sankaracharya is credited with consolidating and propounding the Advaita Vedanta philosophy. His teachings emphasize the oneness of the individual soul (Atman) with the ultimate reality (Brahman), asserting that the world is an illusion (Maya) and only Brahman is real.
The problem involves calculating the principal amount (sum invested) given the simple interest, rate of interest, and time period. We will use the standard formula for simple interest to solve this problem.
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]where \(P\) is the Principal amount, \(R\) is the Rate of interest per annum, and \(T\) is the Time in years.To find the principal amount, we can rearrange the formula as:
\[ P = \frac{SI \times 100}{R \times T} \]Substitute \(SI = 19200\), \(R = 12\), and \(T = 5\) into the formula:
\[ P = \frac{19200 \times 100}{12 \times 5} \]First, calculate the product of \(R\) and \(T\) in the denominator:
\[ 12 \times 5 = 60 \]\[ P = \frac{19200 \times 100}{60} \]\[ P = \frac{1920000}{60} \]\[ P = 32000 \]The sum invested (Principal) is Rs 32,000.
Correct Option: A) Rs 32,000 is the correct sum invested, calculated using the simple interest formula.
This problem involves the refraction of light through an equilateral glass prism. We need to use Snell's Law and the geometry of the prism to relate the angle of incidence (θ) and the angle of emergence (β) under the given condition that the refracted ray inside the prism is parallel to its base.
Prism Geometry: An equilateral prism has all angles equal to 60°. So, the prism angle \(A = 60^\circ\).
Refraction at the first surface: Let the angle of incidence be \(\theta\) and the angle of refraction inside the prism be \(r_1\). According to Snell's Law at the first surface:
\[ n_1 \sin \theta = n_2 \sin r_1 \]where \(n_1\) is the refractive index of air (approximately 1) and \(n_2\) is the refractive index of the glass prism (let's denote it as \(n\)).
\[ \sin \theta = n \sin r_1 \quad \text{(Equation 1)} \]Refraction at the second surface: Let the angle of incidence inside the prism be \(r_2\) and the angle of emergence be \(\beta\). According to Snell's Law at the second surface:
\[ n \sin r_2 = 1 \sin \beta \]\[ n \sin r_2 = \sin \beta \quad \text{(Equation 2)} \]Prism Formula: For a prism, the relation between the prism angle and the angles of refraction is:
\[ A = r_1 + r_2 \]Since \(A = 60^\circ\), we have:
\[ 60^\circ = r_1 + r_2 \quad \text{(Equation 3)} \]Condition Given: The refracted ray inside the prism is parallel to its base. In an equilateral prism, if the refracted ray is parallel to the base, it means the prism is in a state of minimum deviation. In this specific case, the ray inside the prism is parallel to the base, which implies that the angle of incidence at the first face and the angle of emergence at the second face are equal, i.e., \(\theta = \beta\). Also, the angles of refraction inside the prism are equal, i.e., \(r_1 = r_2\).
Using the condition: If \(r_1 = r_2\), then from Equation 3:
\[ 60^\circ = r_1 + r_1 = 2r_1 \]\[ r_1 = 30^\circ \]Since \(r_1 = r_2\), then \(r_2 = 30^\circ\).
Relating \(\theta\) and \(\beta\): From Equation 1:
\[ \sin \theta = n \sin r_1 = n \sin 30^\circ = n \left(\frac{1}{2}\right) \]From Equation 2:
\[ \sin \beta = n \sin r_2 = n \sin 30^\circ = n \left(\frac{1}{2}\right) \]Comparing the expressions for \(\sin \theta\) and \(\sin \beta\), we get:
\[ \sin \theta = \sin \beta \]Since \(\theta\) and \(\beta\) are angles of incidence and emergence, they are positive and typically less than 90°. Therefore,
\[ \theta = \beta \]C) θ = β — This is true because when the refracted ray inside an equilateral prism is parallel to its base, the prism is in the condition of minimum deviation, which implies that the angle of incidence is equal to the angle of emergence.
This is a syllogism problem. We will use Venn diagrams to represent the given statements and then evaluate each conclusion based on these diagrams. We need to find the conclusions that logically follow from the statements.
Imagine three overlapping circles. The 'Ponds' circle is entirely inside the 'Streets' circle. The 'Cisterns' circle overlaps with the 'Ponds' circle (and thus also with the 'Streets' circle).
Since 'All ponds are streets' (Statement 2), it logically follows that 'Some streets are ponds'. If all ponds are within streets, then the part of streets that contains ponds is indeed ponds. This conclusion is definitely true.
We know 'Some cisterns are ponds' and 'All ponds are streets'. This means some cisterns are definitely streets. However, we cannot conclude that ALL cisterns are streets. There might be cisterns that are not ponds and also not streets. This conclusion is not necessarily true.
We know 'Some cisterns are ponds'. This does not mean that all ponds are cisterns. There could be ponds that are not cisterns. This conclusion is not necessarily true.
From Statement 1 ('Some cisterns are ponds') and Statement 2 ('All ponds are streets'), we deduce that some cisterns are streets. Therefore, it is false to say 'No street is a cistern'. This conclusion is definitely false.
Based on the evaluation, only Conclusion I logically follows.
D) Only conclusion I follows.
To determine the nature of washing soda, we need to understand its chemical composition and how it behaves when dissolved in water. Washing soda is chemically known as sodium carbonate, and its properties in aqueous solution will reveal its acidic, basic, neutral, or amphoteric nature.
Correct Option: A) Washing soda (sodium carbonate) is a salt formed from a strong base (NaOH) and a weak acid (H₂CO₃). When dissolved in water, the carbonate ion hydrolyzes to produce hydroxide ions, making the solution basic.
The value of \(\mathrm{\left(\frac{1}{\sec^2\theta-\cos^2\theta}+\frac{1}{\operatorname{cosec}^2\theta-\sin^2\theta}\right)}\)(sin²θ cos²θ) is:
\(\frac{1+\cos^{2}\theta\sin^{2}\theta}{2-\cos^{2}\theta\sin^{2}\theta}\)
\(\frac{1+\cos^{2}\theta\sin^{2}\theta}{2+\cos^{2}\theta\sin^{2}\theta}\)
\(\frac{1-\cos^{2}\theta\sin^{2}\theta}{2+\cos^{2}\theta\sin^{2}\theta}\)
\(\frac{1-\cos^{2}\theta\sin^{2}\theta}{2-\cos^{2}\theta\sin^{2}\theta}\)
The problem asks us to simplify a trigonometric expression. We will use fundamental trigonometric identities to rewrite the expression in terms of \(\sin\theta\) and \(\cos\theta\), and then perform algebraic manipulations to simplify it.
Let the given expression be \(E\):
\[ E = \left(\frac{1}{\sec^2\theta-\cos^2\theta}+\frac{1}{\operatorname{cosec}^2\theta-\sin^2\theta}\right)(\sin^2\theta \cos^2\theta) \]Recall the identities: \(\sec\theta = \frac{1}{\cos\theta}\) and \(\operatorname{cosec}\theta = \frac{1}{\sin\theta}\).
Substitute these identities into the expression:
\[ E = \left(\frac{1}{\frac{1}{\cos^2\theta}-\cos^2\theta}+\frac{1}{\frac{1}{\sin^2\theta}-\sin^2\theta}\right)(\sin^2\theta \cos^2\theta) \]Simplify the denominators inside the parentheses:
\[ \frac{1}{\frac{1}{\cos^2\theta}-\cos^2\theta} = \frac{1}{\frac{1-\cos^4\theta}{\cos^2\theta}} = \frac{\cos^2\theta}{1-\cos^4\theta} \]\[ \frac{1}{\frac{1}{\sin^2\theta}-\sin^2\theta} = \frac{1}{\frac{1-\sin^4\theta}{\sin^2\theta}} = \frac{\sin^2\theta}{1-\sin^4\theta} \]Substitute these simplified terms back into the expression for \(E\):
\[ E = \left(\frac{\cos^2\theta}{1-\cos^4\theta}+\frac{\sin^2\theta}{1-\sin^4\theta}\right)(\sin^2\theta \cos^2\theta) \]Factor the denominators using the difference of squares formula \((a^2-b^2) = (a-b)(a+b)\):
\[ 1-\cos^4\theta = (1-\cos^2\theta)(1+\cos^2\theta) = \sin^2\theta(1+\cos^2\theta) \]\[ 1-\sin^4\theta = (1-\sin^2\theta)(1+\sin^2\theta) = \cos^2\theta(1+\sin^2\theta) \]Substitute these factored forms back into the expression:
\[ E = \left(\frac{\cos^2\theta}{\sin^2\theta(1+\cos^2\theta)}+\frac{\sin^2\theta}{\cos^2\theta(1+\sin^2\theta)}\right)(\sin^2\theta \cos^2\theta) \]Distribute the term \((\sin^2\theta \cos^2\theta)\) into the parentheses:
\[ E = \frac{\cos^2\theta}{\sin^2\theta(1+\cos^2\theta)}(\sin^2\theta \cos^2\theta) + \frac{\sin^2\theta}{\cos^2\theta(1+\sin^2\theta)}(\sin^2\theta \cos^2\theta) \]Cancel out common terms:
\[ E = \frac{\cos^4\theta}{1+\cos^2\theta} + \frac{\sin^4\theta}{1+\sin^2\theta} \]We know that \(\cos^2\theta = 1-\sin^2\theta\) and \(\sin^2\theta = 1-\cos^2\theta\). Let's try to simplify by using these relations or by finding a common denominator.
Let's find a common denominator:
\[ E = \frac{\cos^4\theta(1+\sin^2\theta) + \sin^4\theta(1+\cos^2\theta)}{(1+\cos^2\theta)(1+\sin^2\theta)} \]\[ E = \frac{\cos^4\theta+\cos^4\theta\sin^2\theta + \sin^4\theta+\sin^4\theta\cos^2\theta}{1+\sin^2\theta+\cos^2\theta+\sin^2\theta\cos^2\theta} \]Recall \(\sin^2\theta+\cos^2\theta=1\). Substitute this into the denominator:
\[ \text{Denominator} = 1+1+\sin^2\theta\cos^2\theta = 2+\sin^2\theta\cos^2\theta \]Now, let's simplify the numerator:
\[ \text{Numerator} = (\cos^4\theta+\sin^4\theta) + \sin^2\theta\cos^2\theta(\cos^2\theta+\sin^2\theta) \]\[ \text{Numerator} = (\cos^4\theta+\sin^4\theta) + \sin^2\theta\cos^2\theta(1) \]\[ \text{Numerator} = \cos^4\theta+\sin^4\theta + \sin^2\theta\cos^2\theta \]We know that \((a^2+b^2) = (a+b)^2 - 2ab\). So, \(\cos^4\theta+\sin^4\theta = (\cos^2\theta+\sin^2\theta)^2 - 2\sin^2\theta\cos^2\theta\).
\[ \cos^4\theta+\sin^4\theta = (1)^2 - 2\sin^2\theta\cos^2\theta = 1 - 2\sin^2\theta\cos^2\theta \]Substitute this back into the numerator:
\[ \text{Numerator} = (1 - 2\sin^2\theta\cos^2\theta) + \sin^2\theta\cos^2\theta \]\[ \text{Numerator} = 1 - \sin^2\theta\cos^2\theta \]Combine the simplified numerator and denominator:
\[ E = \frac{1 - \sin^2\theta\cos^2\theta}{2 + \sin^2\theta\cos^2\theta} \]C) The simplified expression is \[\frac{1-\cos^{2}\theta\sin^{2}\theta}{2+\cos^{2}\theta\sin^{2}\theta}\]. This matches option C.
This is a logical reasoning question where we need to evaluate conclusions based solely on the given statement. We must not assume any outside information. The statement says "Sale up to 70% on all clothing items". This implies a maximum discount, not a uniform one.
A) Neither conclusion I nor II follows
Select the option that is related to the seventh number in the same way as the second number is related to the first number, the fourth number is related to the third number and the sixth number is related to the fifth number.
Number series:
56 : 8 :: 84 : 12 :: 42 : 6 :: 14 :?
The problem presents a series of number pairs and asks us to find the missing number in the last pair based on the relationship observed in the preceding pairs. We need to identify the mathematical operation or pattern that connects the first number to the second number in each given pair.
C) 2 — The pattern established is that the second number is the first number divided by 7. Applying this to \(14 : ?\), we get \(14 \div 7 = 2\).
The question asks for the most likely reason behind the higher maternal mortality rate (MMR) in young adolescents (10-14 years) compared to other women, as highlighted by the WHO. We need to evaluate each option based on biological and medical understanding of adolescent pregnancy.
Correct Option: A) Their body size is often too small to deliver a baby
The question describes a scenario where an object is placed at a distance such that light rays from it are parallel to the principal axis of a concave mirror. We need to determine the position of the image formed under these conditions. This is a fundamental concept in ray optics for concave mirrors.
C) at the principal focus — As per the ray tracing rules for concave mirrors, light rays coming parallel to the principal axis from a distant object converge at the principal focus after reflection, forming the image there.