The problem asks us to identify the letter-cluster pair that does not follow the same pattern as the others. We need to analyze the relationship between the letters within each cluster and between the two clusters in each option, based on their positions in the English alphabet.
B) GJ−JP
This option is the odd one out because the difference between the letters in the second cluster (JP) is \(16 - 10 = 6\), not 3. Also, the last letter of the first cluster (J) and the first letter of the second cluster (J) are the same, meaning the gap is 0, not 1.
Which of the options below is equivalent to 216x3 + 27y3 + 324x2y + 162xy2?
The problem asks us to identify which of the given options is equivalent to the algebraic expression \(216x^3 + 27y^3 + 324x^2y + 162xy^2\). This expression resembles the expansion of a binomial cube, \((a+b)^3\). We will use the binomial expansion formula to check each option.
A) \((6x + 3y)^3\)
As shown in the step-by-step derivation, if we expand \((6x + 3y)^3\), we get:
\[ (6x + 3y)^3 = (6x)^3 + (3y)^3 + 3(6x)^2(3y) + 3(6x)(3y)^2 \]\[ = 216x^3 + 27y^3 + 3(36x^2)(3y) + 3(6x)(9y^2) \]\[ = 216x^3 + 27y^3 + 324x^2y + 162xy^2 \]This matches the given expression.
The value of (secθ − tanθ)2 (1 + sinθ)2 ÷ cos2θ is
To simplify the given trigonometric expression, we will convert all terms into sine and cosine functions. Then, we will use algebraic identities and fundamental trigonometric identities to simplify the expression to its simplest form.
The given expression is: \(( \text{sec}\theta - \text{tan}\theta )^2 (1 + \text{sin}\theta )^2 \div \text{cos}^2\theta\)
First, express \(\text{sec}\theta\) and \(\text{tan}\theta\) in terms of \(\text{sin}\theta\) and \(\text{cos}\theta\):
\[ \text{sec}\theta = \frac{1}{\text{cos}\theta} \]\[ \text{tan}\theta = \frac{\text{sin}\theta}{\text{cos}\theta} \]Substitute these into the first part of the expression:
\[ (\text{sec}\theta - \text{tan}\theta)^2 = \left( \frac{1}{\text{cos}\theta} - \frac{\text{sin}\theta}{\text{cos}\theta} \right)^2 \]\[ = \left( \frac{1 - \text{sin}\theta}{\text{cos}\theta} \right)^2 \]\[ = \frac{(1 - \text{sin}\theta)^2}{\text{cos}^2\theta} \]Now, substitute this back into the original expression:
\[ \frac{(1 - \text{sin}\theta)^2}{\text{cos}^2\theta} \cdot (1 + \text{sin}\theta)^2 \div \text{cos}^2\theta \]Rewrite the division as multiplication by the reciprocal:
\[ \frac{(1 - \text{sin}\theta)^2}{\text{cos}^2\theta} \cdot (1 + \text{sin}\theta)^2 \cdot \frac{1}{\text{cos}^2\theta} \]Combine the terms in the numerator and denominator:
\[ \frac{(1 - \text{sin}\theta)^2 (1 + \text{sin}\theta)^2}{\text{cos}^2\theta \cdot \text{cos}^2\theta} \]\[ = \frac{[(1 - \text{sin}\theta)(1 + \text{sin}\theta)]^2}{\text{cos}^4\theta} \]Use the algebraic identity \((a-b)(a+b) = a^2 - b^2\):
\[ (1 - \text{sin}\theta)(1 + \text{sin}\theta) = 1^2 - \text{sin}^2\theta = 1 - \text{sin}^2\theta \]Recall the fundamental trigonometric identity: \(\text{sin}^2\theta + \text{cos}^2\theta = 1\), which implies \((1 - \text{sin}^2\theta) = \text{cos}^2\theta\).
Substitute this back into the expression:
\[ \frac{[\text{cos}^2\theta]^2}{\text{cos}^4\theta} \]\[ = \frac{\text{cos}^4\theta}{\text{cos}^4\theta} \]Assuming \(\text{cos}\theta \neq 0\), the expression simplifies to:
\[ 1 \]D) The value of the given expression simplifies to 1.
This problem can be solved using a system of linear equations. We are given the total quantity of fruits, the total cost, and the per-kilogram prices of apples and mangoes. We need to find the amount spent on mangoes.
Let \(A\) be the quantity of apples in kg and \(M\) be the quantity of mangoes in kg.
From the problem statement, the total quantity of fruits is 45 kg:
\[ A + M = 45 \quad \text{(Equation 1)} \]
The price of apples is ₹25/kg and mangoes is ₹30/kg. The total cost is ₹1,240:
\[ 25A + 30M = 1240 \quad \text{(Equation 2)} \]
From Equation 1, we can express \(A\) in terms of \(M\):
\[ A = 45 - M \]
Substitute this expression for \(A\) into Equation 2:
\[ 25(45 - M) + 30M = 1240 \]
Now, solve for \(M\):
\[ 1125 - 25M + 30M = 1240 \]
\[ 1125 + 5M = 1240 \]
\[ 5M = 1240 - 1125 \]
\[ 5M = 115 \]
\[ M = \frac{115}{5} \]
\[ M = 23 \text{ kg} \]
The quantity of mangoes is 23 kg. We need to find the money spent on mangoes. The price of mangoes is ₹30/kg.
\[ \text{Cost of mangoes} = \text{Quantity of mangoes} \times \text{Price per kg of mangoes} \]
\[ \text{Cost of mangoes} = 23 \text{ kg} \times \text{₹}30/\text{kg} \]
\[ \text{Cost of mangoes} = \text{₹}690 \]
Correct Option: C) ₹690
This is a blood relation question where symbols represent relationships. We need to decode the given expression 'M × N − O ÷ P + Q' step-by-step to determine the relationship between M and Q. We will build a family tree or list the relationships as we decode each part of the expression.
Now, let's combine these relationships to find how M is related to Q:
From these, we can deduce:
So, M is the Father's Brother's Wife of Q.
A) Father's Brother's Wife. This matches our deduction that M is the wife of Q's paternal uncle (father's brother).
The question asks to identify the leader who collaborated with Mahatma Gandhi during the Kheda Satyagraha in 1918. This requires knowledge of key figures involved in early Gandhian movements in India.
Correct Option: C) Sardar Vallabhbhai Patel was a key associate of Mahatma Gandhi during the Kheda Satyagraha. He played a crucial role in organizing the peasants and leading the movement alongside Gandhi.
To solve this problem, we need to identify the highest and lowest numbers from the given set of three-digit numbers. Then, we extract specific digits from these identified numbers as per the question's instructions and perform the required arithmetic operation (addition).
The resultant value is 18.
Correct Option: D) 18
The question asks for the total number of Gram Panchayats in India as per the Union Budget 2025-26. This requires knowledge of the latest official figures related to local self-governance bodies, often mentioned in economic surveys or budget documents.
A) 2.68 lakh
As per the latest available data and figures often referenced in Union Budget documents and economic surveys, the total number of Gram Panchayats in India is approximately 2.68 lakh. This figure represents the foundational units of local self-governance in rural areas.
The problem presents a number series, and we need to find the next term. We will analyze the differences between consecutive terms to identify the pattern.
D) 962 — The pattern of differences is -2, -4, -6, -8. The next difference should be -10. So, \(972 - 10 = 962\).
The question asks to identify the Harappan site where evidence of ploughed fields has been found. This requires knowledge of the specific archaeological findings associated with major Indus Valley Civilization sites.
Correct Option: A) Kalibangan is the Harappan site where archaeologists have found evidence of ploughed fields. This discovery is significant as it provides insights into the agricultural practices of the Harappan people, showing a grid pattern of furrows indicating the simultaneous cultivation of two different crops.