The question asks to identify which of the given hill ranges in India are known as the "Blue Mountains". This requires knowledge of the geographical names and common aliases of prominent Indian hill ranges.
C) Nilgiri Hills — The Nilgiri Hills are famously known as the "Blue Mountains" due to the bluish haze caused by the Strobilanthes kunthiana flowers and the general appearance of the mountains.
The problem asks us to find the value of \( x^2 + y^2 + z^2 \) given the expressions for \( x \), \( y \), and \( z \) in terms of \( r \), \( A \), and \( C \). We will substitute the given expressions into the required sum of squares and use trigonometric identities to simplify the result.
A) \( r^2 \) is the correct answer, as derived from the step-by-step simplification using trigonometric identities.
If '+' means '×', '×' means '-', and '÷' means '+', then which of the following would be the value of 14 ÷ 2 + 8 × 5?
This is a problem of symbolic substitution and arithmetic evaluation. We need to replace the given mathematical operators with new definitions and then evaluate the expression following the standard order of operations (BODMAS/PEMDAS).
The original expression is \(14 \div 2 + 8 \times 5\).
Applying the substitutions:
So, \(14 \div 2 + 8 \times 5\) becomes \(14 + 2 \times 8 - 5\).
The order of operations is Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
The expression is \(14 + 2 \times 8 - 5\).
First, perform multiplication:
\[ 2 \times 8 = 16 \]The expression becomes:
\[ 14 + 16 - 5 \]Next, perform addition and subtraction from left to right:
\[ 14 + 16 = 30 \]Then,
\[ 30 - 5 = 25 \]The value of the expression is 25.
C) 25 is the correct value obtained after substituting the operators and applying the order of operations.
The problem asks us to find the fourth term in an analogy, given the relationship between the first and second terms. We need to identify the pattern or rule that transforms "BLCM" into "FHGI" and then apply the same rule to "KHWS" to find the missing term.
The pattern is: \(+4, -4, +4, -4\).
Correct Option: D) ODAO
The problem is a coding-decoding question. We need to identify the pattern used to code letters into numbers. The given examples are G is 20 and E is 22. We will analyze these to find the rule and then apply it to the word GREAT.
Let's consider the reverse alphabetical order. In the reverse alphabetical order (Z=1, Y=2, ..., A=26):
Alternatively, the pattern can be found by subtracting the letter's forward alphabetical position from 27. For example:
This confirms that each letter is coded by its position from the end of the alphabet (or \(27 - \text{forward position}\)).
Putting these values together for GREAT, we get 20922267.
Correct Option: B) 20922267
The problem is a coding-decoding question. We need to identify the pattern used to code the given words and then apply the same pattern to find the code for the word "COMMON". The pattern involves assigning numerical values to letters and then concatenating them.
Correct Option: C) 31513141214
Mahesh sold \( \frac{3}{4} \) of his stock at a gain of 20% and the remaining stock at a gain of 15%. Find the total gain percentage (approximate).
To find the total gain percentage, we can assume a total cost price for the stock. Then, we calculate the selling price for each part of the stock based on the given gain percentages and finally determine the overall gain percentage.
Let the total cost price of the stock be \( \$100 \). This makes calculations with percentages straightforward.
Mahesh sold \( \frac{3}{4} \) of his stock at a gain of 20%.
The remaining stock is \[1 - \frac{3}{4} = \frac{1}{4}\].
The total gain percentage is approximately 19%.
A) 19% is the correct option as the calculated total gain percentage is 18.75%, which approximates to 19%.
If \( \frac{154}{0.154} \)= \( \frac{15.4}{x} \), then find the value of x.
The problem involves solving an equation with fractions and decimals. The key is to simplify the left side of the equation first, then set it equal to the right side and solve for the unknown variable \(x\). We will use properties of fractions and decimal manipulation to achieve this.
The given equation is:
\[ \frac{154}{0.154} = \frac{15.4}{x} \]Let's simplify the left-hand side (LHS):
\[ \text{LHS} = \frac{154}{0.154} \]To remove the decimal from the denominator, multiply both the numerator and the denominator by 1000 (since there are three decimal places in 0.154):
\[ \text{LHS} = \frac{154 \times 1000}{0.154 \times 1000} = \frac{154000}{154} \]Now, divide 154000 by 154:
\[ \text{LHS} = 1000 \]Now we have:
\[ 1000 = \frac{15.4}{x} \]To solve for \(x\), multiply both sides by \(x\):
\[ 1000x = 15.4 \]Now, divide both sides by 1000:
\[ x = \frac{15.4}{1000} \]Dividing by 1000 means moving the decimal point three places to the left:
\[ x = 0.0154 \]Correct Option: B) 0.0154
The question asks for the specific term that describes the eye's ability to change its focal length to view objects at different distances. This is a fundamental concept in human vision.
Correct Option: A) accommodation is the term used to describe the ability of the eye lens to adjust its focal length to see objects clearly at various distances.
To find the smallest number by which 720 should be multiplied to make it a perfect square, we need to perform prime factorization of 720. A number is a perfect square if all the exponents in its prime factorization are even. We will identify the prime factors with odd exponents and multiply 720 by the product of these factors to make all exponents even.
First, we find the prime factors of 720.
\[ 720 = 10 \times 72 \]\[ 720 = (2 \times 5) \times (8 \times 9) \]\[ 720 = (2 \times 5) \times (2^3 \times 3^2) \]\[ 720 = 2^1 \times 5^1 \times 2^3 \times 3^2 \]\[ 720 = 2^{(1+3)} \times 3^2 \times 5^1 \]\[ 720 = 2^4 \times 3^2 \times 5^1 \]For a number to be a perfect square, all the exponents in its prime factorization must be even. In the prime factorization of 720:
To make the exponent of 5 even, we need to multiply \(5^1\) by \(5^1\), which will result in \(5^2\). Therefore, the smallest number by which 720 should be multiplied is 5.
If we multiply 720 by 5:
\[ 720 \times 5 = (2^4 \times 3^2 \times 5^1) \times 5^1 \]\[ = 2^4 \times 3^2 \times 5^{(1+1)} \]\[ = 2^4 \times 3^2 \times 5^2 \]\[ = (2^2 \times 3 \times 5)^2 \]\[ = (4 \times 3 \times 5)^2 \]\[ = (60)^2 \]\[ = 3600 \]Since 3600 is \(60^2\), it is a perfect square.
Correct Option: D) 5