When two quadratic equations have both roots common, their corresponding coefficients must be in proportion. We will use this property to set up ratios of coefficients and solve for the unknown variables \(b\) and \(c\).
Given the two quadratic equations:
\( 4x^{2} + bx + 3 = 0 \) (Equation 1)
\( 8x^{2} + 4x + c = 0 \) (Equation 2)
If both roots are common, the ratio of the corresponding coefficients must be equal. Let the coefficients of the first equation be \(a_1 = 4\), \(b_1 = b\), \(c_1 = 3\), and for the second equation, \(a_2 = 8\), \(b_2 = 4\), \(c_2 = c\).
Therefore, we can write:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]Substitute the given coefficients into the ratio:
\[ \frac{4}{8} = \frac{b}{4} = \frac{3}{c} \]Simplify the first ratio:
\[ \frac{4}{8} = \frac{1}{2} \]Now, equate the first and second ratios to find \(b\):
\[ \frac{1}{2} = \frac{b}{4} \]Multiply both sides by 4:
\[ b = \frac{4}{2} \] \[ b = 2 \]Next, equate the first and third ratios to find \(c\):
\[ \frac{1}{2} = \frac{3}{c} \]Cross-multiply:
\[ 1 \cdot c = 2 \cdot 3 \] \[ c = 6 \]Thus, the values for \(b\) and \(c\) are 2 and 6, respectively.
Correct Option: D) 2, 6
The question asks to identify the agreement associated with the Citizenship (Amendment) Act, 1985. This requires knowledge of key historical agreements in India and their legislative outcomes.
B) Assam Accord
The Citizenship (Amendment) Act, 1985, was passed to implement certain clauses of the Assam Accord. The Accord, signed on August 15, 1985, aimed to resolve the long-standing agitation against illegal immigrants in Assam. The Act made special provisions for the citizenship of persons covered by the Assam Accord, particularly those who came to Assam from Bangladesh on or after January 1, 1966, but before March 25, 1971.
To find the arithmetic mean of a set of observations, we need to sum all the observations and then divide by the total number of observations.
Correct Option: A) 56 is the arithmetic mean of the given observations.
The question asks about the founder of the Chalukyas of Badami. This requires knowledge of the early history and key figures of this South Indian dynasty. Identifying the correct founder is a fundamental historical fact.
Correct Option: A) Pulakesin I is widely regarded as the founder of the Chalukya dynasty of Badami. He established his capital at Badami (Vatapi) in the mid-6th century CE and performed the Ashwamedha sacrifice to assert his imperial sovereignty.
The question asks for the measure that indicates the percentage of a country's population living below the poverty line. We need to evaluate each option to determine which one directly corresponds to this definition.
Correct Option: D) Poverty Headcount Ratio — This measure directly calculates the proportion of a country's population that falls below the nationally or internationally defined poverty line. It is typically expressed as a percentage.
We need to find the length of the longer side of a rectangle. We are given two conditions: the relationship between the shorter and longer sides, and a relationship between its area and perimeter. We will set up equations based on these conditions and solve for the dimensions of the rectangle.
Define variables:
Let the length of the longer side be \(L\) cm.
Let the length of the shorter side be \(W\) cm.
Formulate the first equation from the given condition:
"The shorter side of a rectangle is 16 cm less than the longer side."
\[ W = L - 16 \quad \text{(Equation 1)} \]
Write expressions for Area and Perimeter:
The area of a rectangle is given by \(A = L \times W\).
The perimeter of a rectangle is given by \(P = 2(L + W)\).
Formulate the second equation from the given condition:
"The numerical value of its area is equal to 3 times the numerical value of its perimeter."
\[ A = 3P \]\[ L \times W = 3 \times 2(L + W) \]\[ LW = 6(L + W) \quad \text{(Equation 2)} \]
Substitute Equation 1 into Equation 2:
Substitute \(W = L - 16\) into \(LW = 6(L + W)\):
\[ L(L - 16) = 6(L + (L - 16)) \]\[ L^2 - 16L = 6(2L - 16) \]\[ L^2 - 16L = 12L - 96 \]
Rearrange into a quadratic equation:
Move all terms to one side to form a quadratic equation:
\[ L^2 - 16L - 12L + 96 = 0 \]\[ L^2 - 28L + 96 = 0 \]
Solve the quadratic equation for L:
We need to find two numbers that multiply to 96 and add up to -28. These numbers are -4 and -24.
\[ (L - 4)(L - 24) = 0 \]This gives two possible values for \(L\):
\[ L = 4 \quad \text{or} \quad L = 24 \]
Check the validity of the solutions:
If \(L = 4\) cm, then \(W = L - 16 = 4 - 16 = -12\) cm. A side length cannot be negative, so \(L = 4\) is not a valid solution.
If \(L = 24\) cm, then \(W = L - 16 = 24 - 16 = 8\) cm. Both lengths are positive, so this is a valid solution.
State the final answer:
The length of the longer side is 24 cm.
D) 24 is the correct answer. The calculation shows that the longer side \(L\) must be 24 cm for the conditions to be met, as \(L=4\) leads to a negative shorter side.
The question asks about the natural material used in the 'Lipane' step of Warli painting to prepare the wall. This requires knowledge of the traditional techniques and materials used in Warli art.
C) Geru is a natural red ochre pigment. In Warli painting, it is traditionally mixed with cow dung and mud to prepare the wall surface during the 'Lipane' step, providing a reddish-brown base for the white paint.
The question asks about the context in which the Supreme Court of India described itself as the "sentinel on the qui vive." This phrase signifies an alert and watchful guardian. We need to identify which of the given options aligns with the Supreme Court's primary role as a protector of constitutional values and individual rights.
Correct Option: A) Protection of Fundamental Rights. The Supreme Court has consistently emphasized its role as the protector and guarantor of Fundamental Rights enshrined in Part III of the Constitution. This phrase aptly describes its vigilant stance in safeguarding these rights against any encroachment by the state or other entities.
The question asks for the quickest way to open an existing MS Excel spreadsheet. This involves understanding common operating system and application shortcuts/methods for file access.
Correct Option: B) Double-click the .xlsx file in File Explorer. This is the most direct and quickest method. When you double-click an Excel file (.xlsx), the operating system automatically launches MS Excel and opens that specific file in one action.