To find the value of \(\alpha^4 - \beta^4\), we first need to find the roots \(\alpha\) and \(\beta\) of the given quadratic equation \(x^2 - 10x + 16 = 0\). Once the roots are found, we can substitute them into the expression \(\alpha^4 - \beta^4\) and simplify. Alternatively, we can use algebraic identities to simplify \(\alpha^4 - \beta^4\) into terms of \((\alpha + \beta)\) and \((\alpha - \beta)\) or \((\alpha^2 + \beta^2)\) and \((\alpha^2 - \beta^2)\).
Correct Option: A) 4080
The volume of a cylinder having height 21 cm and radius 10 cm is:
(Use π = \( \frac{22}{7} \))
To find the volume of a cylinder, we use the standard formula \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. We are given the values for height, radius, and \(\pi\), so we substitute these values into the formula and calculate the result.
Correct Option: B) 6600 cm³
The problem asks us to identify a letter-cluster pair that follows the same logical relationship as the given pairs. We need to analyze the pattern in the provided examples and then apply that pattern to the options.
A) NLJ : VTR
N = 14, L = 12, J = 10 (Decreasing order: N > L > J)
V = 22, T = 20, R = 18 (Decreasing order: V > T > R)
Let's check the differences:
First letter: \(V - N = 22 - 14 = +8\)
Second letter: \(T - L = 20 - 12 = +8\)
Third letter: \(R - J = 18 - 10 = +8\)
This pair follows the same logic: each letter is shifted by +8, and letters within each cluster are in decreasing order.
Based on the following sequence of letters and numbers, answer the question given below:
(Left) Ω R 7 & S ∞ 5 ! D @ 4 Y 3 D # W A 6 & F O 7 % D C (Right)
Count from the left, how many such letters are there which are immediately preceded by another letter and immediately followed by a number?
The problem asks us to identify letters in a given sequence that are immediately preceded by another letter and immediately followed by a number. We need to scan the sequence from left to right and apply this condition to each letter.
Ω R 7 & S ∞ 5 ! D @ 4 Y 3 D # W A 6 & F O 7 % D CA) Two β This option is incorrect because we found three such letters.
When two objects move in opposite directions, their relative speed is the sum of their individual speeds. The total distance they need to cover to cross each other is the sum of their lengths. The time taken can then be calculated using the formula: Time = Distance / Speed.
Correct Option: A) 4
The question asks to identify the organization that launched the "Smart Event Tracking System" (SETS), a Google Map-based planning and analysis tool in March 2022. This requires knowledge of recent technological initiatives by major Indian government bodies.
Correct Option: A) Indian Railways
The problem requires us to find which pair of numbers, when interchanged in the given equation, makes the equation mathematically correct. We will test each option by interchanging the specified numbers and then evaluating the expression using the order of operations (BODMAS/PEMDAS).
The original equation is:
\[ 35 \times 13 \div 7 + 9 - 14 = 44 \]First, let's evaluate the left-hand side (LHS) of the original equation:
\[ \text{LHS} = 35 \times 13 \div 7 + 9 - 14 \]\[ \text{LHS} = (35 \div 7) \times 13 + 9 - 14 \]\[ \text{LHS} = 5 \times 13 + 9 - 14 \]\[ \text{LHS} = 65 + 9 - 14 \]\[ \text{LHS} = 74 - 14 \]\[ \text{LHS} = 60 \]The original equation \(60 = 44\) is incorrect. Now, we test the options.If we interchange 9 and 13, the equation becomes:
\[ 35 \times 9 \div 7 + 13 - 14 \]Now, let's evaluate the LHS:
\[ \text{LHS} = (35 \div 7) \times 9 + 13 - 14 \]\[ \text{LHS} = 5 \times 9 + 13 - 14 \]\[ \text{LHS} = 45 + 13 - 14 \]\[ \text{LHS} = 58 - 14 \]\[ \text{LHS} = 44 \]Since \(44 = 44\), the equation becomes correct after interchanging 9 and 13.
D) 9 and 13 β Interchanging 9 and 13 makes the equation \(35 \times 9 \div 7 + 13 - 14 = 44\), which simplifies to \(45 + 13 - 14 = 44\), or \(58 - 14 = 44\), which is \(44 = 44\). This is correct.
The question describes a fundamental economic principle related to production, specifically how output changes when one input is varied while others are kept constant. We need to identify the law that explains the phenomenon of diminishing returns to an input.
D) Law of Diminishing Marginal Product β This law states that as one input variable is increased, with all other inputs remaining fixed, a point will eventually be reached where the marginal product of that input will decrease. This perfectly matches the description in the question.
Find the value of \(\frac{(9.8\times9.8-5.8\times5.8)}{2\times(7.8)}\).
The problem requires us to evaluate a numerical expression. We can simplify the numerator using the algebraic identity \((a^2 - b^2) = (a - b)(a + b)\) and then perform the division.
Identify the structure of the numerator. The numerator is in the form \((a^2 - b^2)\), where \(a = 9.8\) and \(b = 5.8\).
Apply the algebraic identity \((a^2 - b^2) = (a - b)(a + b)\) to the numerator:
\[ 9.8^2 - 5.8^2 = (9.8 - 5.8)(9.8 + 5.8) \]Calculate the values inside the parentheses:
Substitute these values back into the numerator:
\[ (9.8 - 5.8)(9.8 + 5.8) = 4.0 \times 15.6 \]Now, substitute this back into the original expression:
\[ \frac{4.0 \times 15.6}{2 \times 7.8} \]Notice that \(2 \times 7.8 = 15.6\). So the expression becomes:
\[ \frac{4.0 \times 15.6}{15.6} \]Cancel out the common term \(15.6\) from the numerator and the denominator:
\[ \frac{4.0 \times \cancel{15.6}}{\cancel{15.6}} = 4.0 \]Thus, the value of the expression is \(4\).
The question asks for the term that describes a conductor's opposition to current flow. We need to identify the correct electrical property from the given options.
Correct Option: C) resistance
Resistance is the property of a material that opposes the flow of electric current. According to Ohm's Law, the resistance \(R\) is given by the ratio of the voltage \(V\) across the conductor to the current \(I\) flowing through it:
\[ R = \frac{V}{I} \]The unit of resistance is the Ohm (\(\Omega\)).