To find the single equivalent discount for two successive discounts, we can assume an initial price (e.g., ₹100) and apply each discount sequentially. Alternatively, a direct formula can be used for successive discounts.
Alternative Method (Formula):
For two successive discounts of \(x\%\) and \(y\%\), the equivalent single discount is given by the formula:
Given \(x = 40\%\) and \(y = 60\%\):
\[ \text{Equivalent Discount} = \left(40 + 60 - \frac{40 \times 60}{100}\right)\% \] \[ \text{Equivalent Discount} = \left(100 - \frac{2400}{100}\right)\% \] \[ \text{Equivalent Discount} = (100 - 24)\% \] \[ \text{Equivalent Discount} = 76\% \]B) The equivalent single discount is 76%.
The problem asks us to identify the logic connecting the letter-clusters in the given pairs and then apply the same logic to the options to find the correct one. We need to analyze the positional difference of each letter in the alphabet between the first and second clusters.
B) JKL: NOP
In a class of students, Radhika is 18th from the top and 32nd from the bottom. What is the total number of students in the class?
To find the total number of students in the class, we can use the formula that relates a person's rank from both the top and the bottom of a sequence. When a person's position is known from both ends, the total number of people can be calculated by adding the two ranks and subtracting 1 (because the person themselves is counted twice).
Correct Option: A) 49 is the total number of students in the class. This is derived by adding Radhika's rank from the top (18) and her rank from the bottom (32), and then subtracting 1 because Radhika herself is counted in both rankings.
What is the ranking of India in plant diversity in the world as on December 2010?
The question asks for India's ranking in plant diversity globally as of December 2010. This requires knowledge of biodiversity statistics and India's position in global biodiversity.
D) 10 th — India is known for its rich biodiversity and is ranked 10th globally in terms of plant diversity.
This is a direction sense problem. We need to determine the final direction of movement based on the initial conditions and a series of turns. The key information is the initial position of the shadow and the sequence of 90-degree turns.
D) South — Based on the step-by-step analysis, Surbhi's final direction of movement is South.
If Sec θ = \( \frac{5}{3} \) and θ is an acute angle, then the value of
\(\frac{3\tan\theta-5\cos\theta}{5\sin\theta-3\sec\theta}\) =?
We are given the value of \(\text{sec }\theta\) and that \(\theta\) is an acute angle. We need to find the value of a trigonometric expression. First, we will use the given \(\text{sec }\theta\) to find the values of other trigonometric ratios like \(\text{sin }\theta\), \(\text{cos }\theta\), and \(\text{tan }\theta\). Then, we will substitute these values into the given expression and simplify it.
Find \(\text{cos }\theta\): We are given \(\text{sec }\theta = \frac{5}{3}\). We know that \(\text{cos }\theta = \frac{1}{\text{sec }\theta}\).
\[ \text{cos }\theta = \frac{1}{\frac{5}{3}} = \frac{3}{5} \]Find \(\text{sin }\theta\): We use the identity \(\text{sin}^2\theta + \text{cos}^2\theta = 1\).
\[ \text{sin}^2\theta = 1 - \text{cos}^2\theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{25 - 9}{25} = \frac{16}{25} \] Since \(\theta\) is an acute angle (meaning \(0^\circ < \theta < 90^\circ\)), \(\text{sin }\theta\) will be positive. \[ \text{sin }\theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \]Find \(\text{tan }\theta\): We use the identity \[\text{tan }\theta = \frac{\text{sin }\theta}{\text{cos }\theta}\].
\[ \text{tan }\theta = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]Substitute the values into the expression: The given expression is \[\frac{3\text{tan }\theta - 5\text{cos }\theta}{5\text{sin }\theta - 3\text{sec }\theta}\].
Substitute the values we found:
Simplify the expression:
Numerator:
\[ 3\left(\frac{4}{3}\right) - 5\left(\frac{3}{5}\right) = 4 - 3 = 1 \]Denominator:
\[ 5\left(\frac{4}{5}\right) - 3\left(\frac{5}{3}\right) = 4 - 5 = -1 \]So, the expression becomes:
\[ \frac{1}{-1} = -1 \]Correct Option: C) -1
The 73rd Amendment of the Constitution enables the states with a population of less than 20 lakhs to have a minimum ______ structure in the local governance of the state.
The question asks about the structure of local governance mandated by the 73rd Amendment for states with a population of less than 20 lakhs. We need to recall the specific provisions of the 73rd Constitutional Amendment Act regarding the Panchayati Raj system.
Correct Option: C) two-tier. The 73rd Amendment allows states with a population of less than 20 lakhs to have a two-tier structure, exempting them from establishing the intermediate level Panchayat.
The monthly wages of 6 employees in a company are Rs 5,000, Rs 6,000, Rs 8,000, Rs 8,500, Rs 9,300, and Rs 9,500. Find the median of their wages.
To find the median of a set of numbers, first arrange the numbers in ascending order. If the total number of observations (\(n\)) is odd, the median is the middle value. If \(n\) is even, the median is the average of the two middle values.
The median of the wages is Rs 8,250.
Correct Option: C) 8,250
This problem involves setting up and solving a system of two linear equations with two variables. We will assign variables to the ages of the son and the father, translate the given conditions into equations, and then solve these equations simultaneously to find the father's age.
Correct Option: C) 28
The third proportional of (a+b) and (a+b)2 is
To find the third proportional of two numbers, say \(x\) and \(y\), we set up a proportion where \(x\) is to \(y\) as \(y\) is to the third proportional, let's call it \(z\). This can be written as \(x : y :: y : z\), or in fraction form as \[\frac{x}{y} = \frac{y}{z}\]. From this, we can solve for \(z\).
Let the two given numbers be \(x = (a+b)\) and \(y = (a+b)^2\).
Let the third proportional be \(z\).
According to the definition of third proportional, we have the relationship:
\[ \frac{x}{y} = \frac{y}{z} \]Substitute the given values of \(x\) and \(y\) into the equation:
\[ \frac{(a+b)}{(a+b)^2} = \frac{(a+b)^2}{z} \]Now, we need to solve for \(z\). Cross-multiply the terms:
\[ (a+b) \times z = (a+b)^2 \times (a+b)^2 \]Simplify the right side of the equation:
\[ (a+b)z = (a+b)^{2+2} \]\[ (a+b)z = (a+b)^4 \]Divide both sides by \((a+b)\) to find \(z\):
\[ z = \frac{(a+b)^4}{(a+b)} \]Using the rule of exponents \(\frac{m^p}{m^q} = m^{p-q}\), we get:
\[ z = (a+b)^{4-1} \]\[ z = (a+b)^3 \]A) \((a+b)^3\) is the correct answer, as derived from the calculation.