To evaluate the given expression, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS (Brackets/Parentheses, Orders/Exponents, Division and Multiplication, Addition and Subtraction). We will start by solving the innermost bracket and work our way outwards.
The given expression is:
\[ 37 - [37 - \{37 - (37 - 37 \times 2)\}] \]First, evaluate the innermost parenthesis \((37 - 37 \times 2)\). According to BODMAS, multiplication comes before subtraction.
\[ 37 \times 2 = 74 \]Now, substitute this value back into the parenthesis:
\[ (37 - 74) = -37 \]Substitute this result back into the expression:
\[ 37 - [37 - \{37 - (-37)\}] \]Next, evaluate the curly braces \(\{37 - (-37)\}\). Subtracting a negative number is equivalent to adding the positive number.
\[ \{37 - (-37)\} = \{37 + 37\} = 74 \]Substitute this result back into the expression:
\[ 37 - [37 - 74] \]Now, evaluate the square bracket \([37 - 74]\).
\[ [37 - 74] = -37 \]Finally, substitute this result back into the expression:
\[ 37 - (-37) \]Again, subtracting a negative number is equivalent to adding the positive number.
\[ 37 - (-37) = 37 + 37 = 74 \]Thus, the value of the expression is 74.
D) 74 is the correct answer, as derived by following the order of operations.
The question asks to identify the author of the book 'Tuhfat-ul-Muwahhidin'. This requires knowledge of prominent figures in Indian history, particularly social and religious reformers, and their literary works.
Correct Option: B) Rammohan Roy
The problem involves compound interest. We are given the total amount Anand had, the interest rate, and the time periods for which Anil and Ashok invested their respective shares. The key information is that both Anil and Ashok received the same final amount. We need to find the initial amount given to Anil.
The question asks to identify the organization that collaborated with ISRO in the development of the Vikram 3201 and Kalpana 3201 microprocessors. This requires knowledge of specific collaborations in India's semiconductor and space technology sectors.
Correct Option: A) Semiconductor Laboratory (SCL) in Mohali is the correct answer. SCL is a unit under the Department of Space, Government of India, and has been instrumental in the design, development, and fabrication of very large-scale integration (VLSI) devices, including microprocessors, for ISRO's space applications.
The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 4.5 m away from the wall. The length of the ladder is:
This problem involves a right-angled triangle formed by the ladder, the wall, and the ground. We are given the angle of elevation and the distance of the foot of the ladder from the wall. We need to find the length of the ladder. We can use trigonometric ratios (specifically cosine) to solve this.
Visualize the setup: Imagine a right-angled triangle where:
Identify given values:
Identify what needs to be found:
Choose the appropriate trigonometric ratio: We know the adjacent side (distance from the wall) and want to find the hypotenuse (length of the ladder). The cosine function relates the adjacent side and the hypotenuse:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]Substitute the values into the formula: Let \(L\) be the length of the ladder.
\[ \cos(60^\circ) = \frac{4.5}{L} \]Recall the value of \(\cos(60^\circ)\):
\[ \cos(60^\circ) = \frac{1}{2} \]Solve for \(L\):
\[ \frac{1}{2} = \frac{4.5}{L} \]\[ L = 2 \times 4.5 \]\[ L = 9 \text{ m} \]D) 9 m is the correct length of the ladder, as calculated using the cosine trigonometric ratio.
The question provides two words, MANY and ARCS, along with their numerical codes, 5391 and 2458. We need to identify the code for the letter A by comparing the common letters and common digits in both coded words.
Identify the given words and their codes:
We need to determine the numerical code representing the letter A.
Compare the letters in both words:
The letters in MANY are M, A, N and Y.
The letters in ARCS are A, R, C and S.
The only common letter in both words is A.
Compare the digits in both codes:
The digits in 5391 are 5, 3, 9 and 1.
The digits in 2458 are 2, 4, 5 and 8.
The only common digit in both codes is 5.
Identify the code for A:
Since A is the only common letter and 5 is the only common digit in both coded words, the letter A corresponds to the digit 5.
\[ A=5 \]Therefore, the code for A is 5.
Correct Option: A) 5 — The letter A appears in both MANY and ARCS. Similarly, the digit 5 appears in both numerical codes, 5391 and 2458. Since A is the only common letter and 5 is the only common digit, A is coded as 5. Hence, Option A is the correct answer.
B) 3: This is incorrect because the digit 3 appears only in the code 5391, whereas A is present in both words.
C) 2: This is incorrect because the digit 2 appears only in the code 2458, whereas A appears in both words.
D) 8: This is incorrect because the digit 8 appears only in the code 2458, whereas the code for A must be common to both numerical codes.
If the mode of the following data is 160, then what is the value of x?
| Class | 145-150 | 150-155 | 155-160 | 160-165 | 165-170 |
| Frequency | 10 | 18 | 31 | x | 24 |
The mode for grouped data can be calculated using the formula: \[\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h\]. Given the mode is 160, we first identify the modal class. Then, we substitute the known values into the formula and solve for the unknown frequency \(x\).
Identify the Modal Class: The mode is given as 160. This value falls into the class interval 160-165. Therefore, the modal class is 160-165.
Extract Values from the Data:
Apply the Mode Formula:
The formula for the mode of grouped data is:
\[ \text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \]Substitute the values into the formula:
Given \(\text{Mode} = 160\):
\[ 160 = 160 + \left(\frac{x - 31}{2x - 31 - 24}\right) \times 5 \]Solve for \(x\):
Subtract 160 from both sides:
\[ 0 = \left(\frac{x - 31}{2x - 55}\right) \times 5 \]Divide by 5:
\[ 0 = \frac{x - 31}{2x - 55} \]For this equation to be true, the numerator must be zero (assuming the denominator is not zero):
\[ x - 31 = 0 \]\[ x = 31 \]Correct Option: B) 31
What positive value of x satisfies the equation \( \frac{x}{147} \) = \( \frac{48}{x} \)?
The problem requires finding the positive value of \(x\) that satisfies the given equation. This can be solved by cross-multiplication to eliminate the denominators, followed by isolating \(x^2\) and then taking the square root to find \(x\).
\[ \frac{x}{147} = \frac{48}{x} \]
To eliminate the denominators, multiply both sides by \(147x\), or simply cross-multiply:
\[ x \times x = 147 \times 48 \]
\[ x^2 = 147 \times 48 \]
Multiply \(147\) by \(48\):
\[ 147 \times 48 = 7056 \]
So, the equation becomes:
\[ x^2 = 7056 \]
We need to find the positive value of \(x\), so we take the positive square root:
\[ x = \sqrt{7056} \]
To find the square root of \(7056\), we can use prime factorization or estimation. Since \(80^2 = 6400\) and \(90^2 = 8100\), the number is between 80 and 90. The last digit is 6, so the unit digit of \(x\) must be 4 or 6. Let's try \(84\):
\[ 84 \times 84 = (80+4)(80+4) = 80^2 + 2 \times 80 \times 4 + 4^2 = 6400 + 640 + 16 = 7056 \]
Thus, \(x = 84\).
A) 84 is the correct value for \(x\) that satisfies the given equation.
This question asks about the fundamental reason for connecting a voltmeter in parallel within an electrical circuit. Understanding the function of a voltmeter and how it interacts with a circuit is key to answering this.
Correct Option: D) To measure the difference in potential between two points
A voltmeter is specifically designed to measure the voltage, which is the potential difference, across two points in a circuit. Connecting it in parallel allows it to sense the potential at both points without significantly drawing current from the main path, thus providing an accurate reading of the voltage drop across that part of the circuit.
The given equation is a biquadratic equation. To reduce it to a quadratic equation, we can use a substitution method. By letting \( y = x^2 \), the equation can be transformed into a standard quadratic form \( ay^2 + by + c = 0 \). We then solve this quadratic equation for \( y \) to find its roots.
Identify the given equation:
The given equation is \( x^{4} - 13x^{2} + 36 = 0 \).
Perform substitution to reduce to a quadratic equation:
Notice that \( x^4 = (x^2)^2 \). Let \( y = x^2 \). Substituting this into the equation, we get:
\[ (x^2)^2 - 13(x^2) + 36 = 0 \]\[ y^2 - 13y + 36 = 0 \]Solve the quadratic equation for \( y \):
We need to find two numbers that multiply to 36 and add up to -13. These numbers are -9 and -4.
So, we can factor the quadratic equation as:
\[ (y - 9)(y - 4) = 0 \]Find the roots of the reduced quadratic equation:
Setting each factor to zero, we get:
\[ y - 9 = 0 \implies y = 9 \]\[ y - 4 = 0 \implies y = 4 \]The roots of the reduced quadratic equation are 9 and 4.
A) 9, 4. The roots of the reduced quadratic equation \( y^2 - 13y + 36 = 0 \) are \( y=9 \) and \( y=4 \).