The problem asks us to identify a set of three numbers that follows the same pattern as the given sets. We need to find the relationship between the numbers in the example sets and then apply that relationship to the options.
D) (27, 33, 36)
The question asks for the total projected infrastructure spending in India over the next seven years, according to CRISIL's Infrastructure Yearbook 2023. This is a factual question based on a specific report. To answer correctly, one needs to recall the key figures mentioned in the CRISIL report.
B) ₹143 lakh crore is the correct figure as per CRISIL's Infrastructure Yearbook 2023 for the projected infrastructure spending in India over the next seven years (FY24-FY30).
The problem provides a relationship between three variables \(a\), \(b\), and \(c\) in the form of a proportion, along with two specific conditions: \(c = 25a\) and \(b = 20\). We need to find the positive value of \(c\). The core idea is to use the given proportion to establish an equation relating \(a\), \(b\), and \(c\), and then substitute the given conditions to solve for \(c\).
Understand the proportion: The given proportion is \(a : b :: b : c\). This can be written as a ratio:
\[ \frac{a}{b} = \frac{b}{c} \]Cross-multiply the proportion: From the ratio, we can derive the relationship:
\[ b^2 = ac \]Substitute the given values: We are given \(b = 20\) and \(c = 25a\). Substitute these into the equation \(b^2 = ac\):
\[ (20)^2 = a(25a) \]Simplify and solve for \(a\):
\[ 400 = 25a^2 \]\[ a^2 = \frac{400}{25} \]\[ a^2 = 16 \]\[ a = \pm\sqrt{16} \]\[ a = \pm 4 \]Determine the value of \(a\): The question asks for the positive value of \(c\). Since \(c = 25a\), if \(a\) were negative, \(c\) would also be negative. To ensure \(c\) is positive, \(a\) must be positive. Therefore, \(a = 4\).
Calculate \(c\): Now, use the relation \(c = 25a\) and the positive value of \(a\):
\[ c = 25 \times 4 \]\[ c = 100 \]C) 100 — By using the given proportion \(a:b::b:c\) to form the equation \(b^2 = ac\), and then substituting the given values \(b=20\) and \(c=25a\), we found that \(a=4\). Substituting \(a=4\) into \(c=25a\) yields \(c=100\).
This is a syllogism problem. We need to analyze the given statements and determine which conclusions logically follow. We can use Venn diagrams or logical deduction to solve this.
Analyze Statement 1: "All sparrows are parrots."
Analyze Statement 2: "All parrots are macau."
Analyze Statement 3: "Some sparrows are dove."
Evaluate Conclusion (I): "Some sparrows are macau."
Evaluate Conclusion (II): "Some doves are parrots."
A) Both conclusions (I) and (II) follow
The question asks for the theme of National Tourism Day celebrated in January 2025. This requires knowledge of current affairs and specific themes designated for national observances.
Correct Option: B) Tourism for Inclusive Growth
The problem presents an analogy between letter-clusters. We need to find the pattern or relationship between the letters in the given pair (INJ : %) and apply the same logic to find the missing cluster (# : RWS).
Correct Option: C) # = UZV, % = FKG. This option correctly follows the pattern where each letter in the first cluster is shifted back by 3 positions to get the corresponding letter in the second cluster.
To understand what motion generates waves in a string, we need to consider how energy is transferred through the medium. Waves are disturbances that propagate through a medium, and this propagation requires a specific type of motion from the particles of the medium.
A) Periodic oscillatory motion — This is the correct description. When a string vibrates to produce a wave, each point on the string undergoes a repetitive, back-and-forth movement around its equilibrium position. This is precisely what periodic oscillatory motion entails, allowing for the propagation of energy in the form of a wave.
The question asks to identify a National Park located in the state of Rajasthan from the given options. This requires knowledge of the geographical locations of prominent National Parks in India.
Correct Option: D) Ranthambore National Park is a well-known National Park and tiger reserve located in the Sawai Madhopur district of Rajasthan, India.
If \(\frac{\sec\theta-\tan\theta}{\sec\theta+\tan\theta}\) = \( \frac{3}{5} \), then 2sinθ is equal to:
\( \frac{1}{2} \)
\( \frac{1}{4} \)
\( \frac{1}{8} \)
1
The problem asks us to find the value of \(2\sin\theta\) given the equation \[\frac{\sec\theta-\tan\theta}{\sec\theta+\tan\theta} = \frac{3}{5}\]. We will use algebraic manipulation and trigonometric identities to simplify the given expression and find the value of \(\sin\theta\).
Given the equation:
\[ \frac{\sec\theta-\tan\theta}{\sec\theta+\tan\theta} = \frac{3}{5} \]Apply componendo and dividendo rule. If \[\frac{a}{b} = \frac{c}{d}\], then \[\frac{a+b}{a-b} = \frac{c+d}{c-d}\]. Here, let \( a = \sec\theta-\tan\theta \) and \( b = \sec\theta+\tan\theta \).
\[ \frac{(\sec\theta-\tan\theta) + (\sec\theta+\tan\theta)}{(\sec\theta-\tan\theta) - (\sec\theta+\tan\theta)} = \frac{3+5}{3-5} \]Simplify both sides of the equation:
Numerator of LHS: \( \sec\theta-\tan\theta + \sec\theta+\tan\theta = 2\sec\theta \)
Denominator of LHS: \( \sec\theta-\tan\theta - \sec\theta-\tan\theta = -2\tan\theta \)
RHS: \( \frac{8}{-2} = -4 \)
So, the equation becomes:
\[ \frac{2\sec\theta}{-2\tan\theta} = -4 \] \[ -\frac{\sec\theta}{\tan\theta} = -4 \]Simplify the trigonometric expression:
We know that \( \sec\theta = \frac{1}{\cos\theta} \) and \( \tan\theta = \frac{\sin\theta}{\cos\theta} \).
\[ \frac{\sec\theta}{\tan\theta} = \frac{\frac{1}{\cos\theta}}{\frac{\sin\theta}{\cos\theta}} = \frac{1}{\sin\theta} \]Substitute this back into the equation:
\[ -\frac{1}{\sin\theta} = -4 \]Solve for \(\sin\theta\):
\[ \frac{1}{\sin\theta} = 4 \] \[ \sin\theta = \frac{1}{4} \]Finally, calculate \(2\sin\theta\):
\[ 2\sin\theta = 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]We are given the amount of a sum of money at simple interest for two different time periods. We can use this information to find the simple interest earned over a specific period, then determine the principal amount, and finally calculate the rate of interest.
Let the principal be \(P\) and the rate of interest be \(R\) per annum. The formula for amount \(A\) on simple interest is:
\[ A = P + \frac{P \times R \times T}{100} \]Given that the sum amounts to ₹1,250 in 2 years:
\[ 1250 = P + \frac{P \times R \times 2}{100} \quad \text{(Equation 1)} \]Given that the sum amounts to ₹2,000 in 4 years:
\[ 2000 = P + \frac{P \times R \times 4}{100} \quad \text{(Equation 2)} \]Subtract Equation 1 from Equation 2 to find the simple interest earned in \(4 - 2 = 2\) years:
\[ (2000 - 1250) = \left(P + \frac{P \times R \times 4}{100}\right) - \left(P + \frac{P \times R \times 2}{100}\right) \]\[ 750 = \frac{P \times R \times 4}{100} - \frac{P \times R \times 2}{100} \]\[ 750 = \frac{P \times R \times (4 - 2)}{100} \]\[ 750 = \frac{P \times R \times 2}{100} \]This means the simple interest for 2 years is ₹750. Now, substitute this value back into Equation 1:
\[ 1250 = P + 750 \]\[ P = 1250 - 750 \]\[ P = 500 \]So, the principal amount is ₹500. Now we can use the simple interest for 2 years and the principal to find the rate \(R\). The simple interest \(SI\) for 2 years is ₹750.
\[ SI = \frac{P \times R \times T}{100} \]\[ 750 = \frac{500 \times R \times 2}{100} \]\[ 750 = 5 \times R \times 2 \]\[ 750 = 10 \times R \]\[ R = \frac{750}{10} \]\[ R = 75 \]Therefore, the rate of interest per annum is 75%.
B) 75% is the correct rate of interest per annum, as calculated.