If ‘×’ means ‘−’, ‘−’ means ‘÷’, ‘÷’ means ‘+’, ‘+’ means ‘×’, what will come in place of the question mark (?) in the following equation?
27 ÷ 288 + 2 × 2 × 22 − 2 =?
This problem involves symbol substitution and then evaluating the resulting arithmetic expression using the order of operations (BODMAS/PEMDAS).
Given symbol transformations:
The original equation is:
\(27 \div 288 + 2 \times 2 \times 22 - 2 = ?\)
Applying the symbol transformations:
The new equation becomes:
\[ 27 + 288 \times 2 - 2 - 22 \div 2 = ? \]
Now, we solve this equation using the BODMAS/PEMDAS rule:
So, the value of the expression is \(590\).
C) 590 is the correct answer, as derived from the step-by-step calculation after applying the given symbol transformations.
In 10 years A will be twice as old as B was 10 years ago. If A is 9 years older than B, then the present age of A is:
39 years
13 years
21\( \frac{1}{2} \) years
48 years
This problem involves setting up a system of two linear equations based on the given age relationships between A and B. We will define variables for their present ages and then translate the conditions into algebraic equations to solve for the present age of A.
Let the present age of A be \(A\) years and the present age of B be \(B\) years.
The first condition states: "In 10 years A will be twice as old as B was 10 years ago."
The second condition states: "If A is 9 years older than B."
Now we have a system of two linear equations:
Equation 1: \(A - 2B = -30\)
Equation 2: \(A = B + 9\)
Substitute Equation 2 into Equation 1 to eliminate \(A\):
\[(B + 9) - 2B = -30\] \[9 - B = -30\] \[-B = -30 - 9\] \[-B = -39\] \[B = 39 \text{ years}\]Now that we have the value of \(B\), substitute it back into Equation 2 to find \(A\):
\[A = B + 9\] \[A = 39 + 9\] \[A = 48 \text{ years}\]Thus, the present age of A is 48 years.
Correct Option: D) 48 years
\(\sqrt{8836}\) + \(\sqrt{6084}\) – \(\sqrt{1849}\) equals:
To solve this problem, we need to find the square root of each number given and then perform the addition and subtraction operations as indicated. We can use methods like prime factorization or estimation to find the square roots of the given numbers.
We need to find a number whose square is \(8836\). Let's look at the last digit. Since the last digit is \(6\), the square root must end in \(4\) or \(6\). We can estimate that \(90^2 = 8100\) and \(100^2 = 10000\). So the number is between \(90\) and \(100\). Let's try \(94^2\) or \(96^2\).
So, \(\sqrt{8836} = 94\).
The last digit is \(4\), so the square root must end in \(2\) or \(8\). We know that \(70^2 = 4900\) and \(80^2 = 6400\). So the number is between \(70\) and \(80\). Let's try \(72^2\) or \(78^2\).
So, \(\sqrt{6084} = 78\).
The last digit is \(9\), so the square root must end in \(3\) or \(7\). We know that \(40^2 = 1600\) and \(50^2 = 2500\). So the number is between \(40\) and \(50\). Let's try \(43^2\) or \(47^2\).
So, \(\sqrt{1849} = 43\).
Now substitute the square root values back into the original expression:
\[ 94 + 78 - 43 \]\[ 172 - 43 \]\[ 129 \]C) 129 is the correct answer.
This is a syllogism problem. We will use Venn diagrams to represent the given statements and then evaluate each conclusion based on these diagrams.
B) All the conclusions I, II and III follow
As determined in the step-by-step analysis, all three conclusions (I, II, and III) logically follow from the given statements.
We can solve this problem by setting up a system of linear equations based on the given information. We will use variables to represent the present ages of the father and the son and then solve for the son's present age.
B) 30 years. By solving the system of equations derived from the problem statement, we find that the son's present age is 30 years.
The question asks about the factors affecting the resistance of a conductor. We need to recall the formula for electrical resistance and identify the direct proportionality.
where:
Correct Option: C) length — As derived from the formula \(R = \rho \frac{L}{A}\), the resistance \(R\) is directly proportional to the length \(L\) of the conductor. This means if you double the length of a wire, its resistance will also double (assuming the material and cross-sectional area remain constant).
Find the value of \(\left(x^{\frac{2}{3}}-x^{-\frac{2}{3}}\right)\left(x^{\frac{4}{3}}+1+x^{-\frac{4}{3}}\right)\).
The problem requires simplifying an algebraic expression involving fractional exponents. We can recognize the expression as a product that resembles the difference of cubes formula, \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\). By identifying appropriate values for 'a' and 'b', we can simplify the given expression.
Let the given expression be \(E\):
\[ E = \left(x^{\frac{2}{3}}-x^{-\frac{2}{3}}\right)\left(x^{\frac{4}{3}}+1+x^{-\frac{4}{3}}\right) \]Recall the algebraic identity for the difference of cubes:
\[ a^3 - b^3 = (a-b)(a^2+ab+b^2) \]Compare the given expression with the right-hand side of the identity. Let's try to identify 'a' and 'b'.
From the first parenthesis, we can set:
\[ a = x^{\frac{2}{3}} \quad \text{and} \quad b = x^{-\frac{2}{3}} \]Now, let's check if the second parenthesis matches \(a^2+ab+b^2\):
\[a^2 = \left(x^{\frac{2}{3}}\right)^2 = x^{\frac{2}{3} \times 2} = x^{\frac{4}{3}}\]
\[b^2 = \left(x^{-\frac{2}{3}}\right)^2 = x^{-\frac{2}{3} \times 2} = x^{-\frac{4}{3}}\]
\[ab = \left(x^{\frac{2}{3}}\right)\left(x^{-\frac{2}{3}}\right) = x^{\frac{2}{3} - \frac{2}{3}} = x^0 = 1\]
Substitute these values back into \(a^2+ab+b^2\):
\[ a^2+ab+b^2 = x^{\frac{4}{3}} + 1 + x^{-\frac{4}{3}} \]This exactly matches the second parenthesis in the given expression.
Since the expression fits the form \((a-b)(a^2+ab+b^2)\), we can write it as \(a^3 - b^3\):
\[ E = \left(x^{\frac{2}{3}}\right)^3 - \left(x^{-\frac{2}{3}}\right)^3 \]Simplify the terms:
\[ \left(x^{\frac{2}{3}}\right)^3 = x^{\frac{2}{3} \times 3} = x^2 \] \[ \left(x^{-\frac{2}{3}}\right)^3 = x^{-\frac{2}{3} \times 3} = x^{-2} \]Substitute these back into the expression for \(E\):
\[ E = x^2 - x^{-2} \]Recall that \(x^{-2} = \frac{1}{x^2}\). So, the expression becomes:
\[ E = x^2 - \frac{1}{x^2} \]If a = (0.\(\overline{29}\) + 0.\(\overline{32}\)) + (0.\(\overline{2}\) + 0.\(\overline{16}\)) + 0.\(\overline{6}\), then the value of a is :
1\( \frac{2}{3} \)
2 \( \frac{1}{3} \)
3\( \frac{2}{3} \)
1 \( \frac{1}{3} \)
The problem requires us to sum several repeating decimals. The most straightforward approach is to convert each repeating decimal into its equivalent fractional form. Once all terms are in fractions, we can sum them up and simplify the result to find the value of 'a'.
A repeating decimal of the form \(0.\overline{d}\) can be written as \(\frac{d}{9}\). A repeating decimal of the form \(0.\overline{d_1 d_2}\) can be written as \(\frac{d_1 d_2}{99}\).
A) \(1 \frac{2}{3}\) is the correct value of 'a' as calculated by converting the repeating decimals to fractions and summing them up.
The question asks about the managing body of the Development Fund for Microfinance Institutions (MFIs). This requires knowledge of the regulatory and developmental roles of various financial institutions in India, particularly concerning the rural and microfinance sectors.
A) NABARD (National Bank for Agriculture and Rural Development) is indeed responsible for managing the Development Fund for MFIs. NABARD plays a crucial role in the development and regulation of the microfinance sector in India, including providing financial support and capacity building to MFIs.
The question asks to identify the mineral for which the Ballari-Chitradurga-Chikkamagaluru-Tumakuru belt in Karnataka is famous. This requires knowledge of the major mineral belts in India and their associated resources.
The Ballari-Chitradurga-Chikkamagaluru-Tumakuru belt is renowned for its extensive deposits of iron ore. Ballari district, in particular, is one of the largest iron ore producing regions in India.
B) Iron-ore. The Ballari-Chitradurga-Chikkamagaluru-Tumakuru belt in Karnataka is famously known for its rich reserves of iron ore. This region is a major contributor to India's iron ore production.