The sum of the interior angles of any quadrilateral is 360°. Given the ratio of the angles, we can represent each angle as a multiple of a common factor. Then, we sum these angles, equate them to 360°, and solve for the common factor. Once the common factor is known, we can find the measure of each angle and then calculate the difference between the largest and the smallest angle.
Let the angles of the quadrilateral be \(1x\), \(3x\), \(4x\), and \(7x\), where \(x\) is a common factor.
The sum of the interior angles of a quadrilateral is 360°.
Set up the equation:
\[ 1x + 3x + 4x + 7x = 360^\circ \]Combine the terms:
\[ (1 + 3 + 4 + 7)x = 360^\circ \]\[ 15x = 360^\circ \]Solve for \(x\):
\[ x = \frac{360^\circ}{15} \]\[ x = 24^\circ \]Now, find the measure of each angle:
Calculate the difference between the largest and the smallest angle:
\[ \text{Difference} = \text{Largest angle} - \text{Smallest angle} \]\[ \text{Difference} = 168^\circ - 24^\circ \]\[ \text{Difference} = 144^\circ \]D) 144° is the correct difference between the largest and the smallest angle.
This is a work and time problem. We will calculate the individual work rates of A, B, and C. Then, we'll consider the work done by each person in different phases of the task, especially noting when A and B left before completion. We can set up an equation based on the total work being 1 unit.
Calculate individual daily work rates:
Assume the total work: To simplify calculations, let's assume the total work is the Least Common Multiple (LCM) of 36, 54, and 72.
\[\text{LCM}(36, 54, 72) = 216 \text{ units}\]Calculate individual daily efficiencies (units/day):
Define the timeline: Let the total number of days to complete the task be \(T\) days.
Formulate the equation for total work: Total work done by A, B, and C must equal 216 units.
\[(\text{A's work}) + (\text{B's work}) + (\text{C's work}) = 216\] \[(\text{A's efficiency} \times \text{days A worked}) + (\text{B's efficiency} \times \text{days B worked}) + (\text{C's efficiency} \times \text{days C worked}) = 216\] \[6(T - 8) + 4(T - 12) + 3T = 216\]Solve for \(T\):
\[6T - 48 + 4T - 48 + 3T = 216\] \[13T - 96 = 216\] \[13T = 216 + 96\] \[13T = 312\] \[T = \frac{312}{13}\] \[T = 24 \text{ days}\]Determine the number of days C worked: C worked for the entire duration of the task, which is \(T\) days.
\[\text{Days C worked} = T = 24 \text{ days}\]A) 24 — The calculation shows that the total time taken to complete the task, and thus the number of days C worked, is 24 days.
The question asks to identify the author who created the fictional South Indian town of 'Malgudi' and featured the character 'Swami' in his stories. This is a direct recall question from Indian literature.
D) R. K. Narayan is the celebrated Indian author who created the fictional town of 'Malgudi' and wrote the famous novel "Swami and Friends", featuring the character 'Swami'. His works vividly depicted the life and culture of South India.
The question asks to identify the article of the Indian Constitution that deals with Panchayats and local body governments. This requires knowledge of the specific articles related to different parts of the Constitution.
D) Art. 243 — Article 243 is the starting point for Part IX of the Indian Constitution, which was added by the 73rd Constitutional Amendment Act of 1992. This part deals exclusively with Panchayats, providing for their establishment, powers, and responsibilities. Similarly, Article 243P onwards deals with Municipalities (local urban bodies) under Part IXA, introduced by the 74th Constitutional Amendment Act of 1992. Therefore, Article 243 is the correct answer as it encompasses the provisions for Panchayats and local body governments.
The question asks about the entity responsible for interconnecting all ATMs across India. This function is crucial for enabling seamless banking transactions regardless of the bank owning the ATM or the card. We need to identify the specific organization mandated with this role in the Indian financial ecosystem.
Correct Option: B) National Payments Corporation of India
The National Payments Corporation of India (NPCI) is the organization that operates the National Financial Switch (NFS), which is the largest network of shared automated teller machines (ATMs) in India. NFS facilitates inter-bank ATM transactions, allowing customers to use any bank's ATM with their debit/credit cards.
If x + y + z = 0, then what will be the value of \(\frac{x^2}{yz}\) +\(\frac{y^2}{zx}\)+ \(\frac{z^2}{xy}\) ?
\(\frac{x^2y^2z^2}{x}\)
3
\(\frac{3x^2+3y^2+3z^2}{x}\)
\( x^{2} + y^{2} + z^{2} \)
The problem asks us to find the value of an algebraic expression given the condition \(x + y + z = 0\). We will simplify the given expression by finding a common denominator and then substitute the condition.
The given expression is:
\[ E = \frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy} \]To add these fractions, we find a common denominator, which is \(xyz\). We multiply the numerator and denominator of each term by the missing variable:
\[ E = \frac{x^2 \cdot x}{yz \cdot x} + \frac{y^2 \cdot y}{zx \cdot y} + \frac{z^2 \cdot z}{xy \cdot z} \]This simplifies to:
\[ E = \frac{x^3}{xyz} + \frac{y^3}{xyz} + \frac{z^3}{xyz} \]Combine the terms over the common denominator:
\[ E = \frac{x^3 + y^3 + z^3}{xyz} \]We are given the condition \(x + y + z = 0\). A well-known algebraic identity states that if \(x + y + z = 0\), then \(x^3 + y^3 + z^3 = 3xyz\).
Substitute this identity into the expression for \(E\):
\[ E = \frac{3xyz}{xyz} \]Assuming \(x, y, z \neq 0\), we can cancel \(xyz\) from the numerator and the denominator:
\[ E = 3 \]B) The value of the expression is 3.
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This means we first perform operations inside Brackets/Parentheses, then Orders/Exponents, then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).
D) The final calculated value of the expression is 84.
Study the given pattern carefully and select the number from the given options that can replace the question mark (?).
To solve this pattern-based question, observe the relationship between the central number in each shape and the resulting number. The pattern involves multiplying the central number by a specific factor.
Thus, the number that should replace the question mark is 32.
Correct Option: A) 36
To find the length of the diagonal of a rectangle, we first need to determine its length and breadth. We are given the ratio of the sides and the perimeter. Using these, we can calculate the actual dimensions and then apply the Pythagorean theorem to find the diagonal.
Represent the sides: Let the length of the rectangle be \(l\) and the breadth be \(b\). Given that the ratio of the two unequal sides is \(3:4\), we can write:
\[ \frac{b}{l} = \frac{3}{4} \quad \text{or} \quad \frac{l}{b} = \frac{4}{3} \]Let's assume the sides are \(3x\) and \(4x\). So, \(b = 3x\) and \(l = 4x\).
Use the perimeter to find \(x\): The perimeter of a rectangle is given by the formula \(P = 2(l + b)\). We are given \(P = 42 \text{ cm}\). Substitute the values of \(l\) and \(b\):
\[ 42 = 2(4x + 3x) \] \[ 42 = 2(7x) \] \[ 42 = 14x \] \[ x = \frac{42}{14} \] \[ x = 3 \]Calculate the actual dimensions: Now substitute the value of \(x\) back into the expressions for length and breadth:
\[ l = 4x = 4 \times 3 = 12 \text{ cm} \] \[ b = 3x = 3 \times 3 = 9 \text{ cm} \]Calculate the length of the diagonal: The diagonal of a rectangle forms a right-angled triangle with the length and breadth as its legs. By the Pythagorean theorem, if \(d\) is the diagonal:
\[ d^2 = l^2 + b^2 \] \[ d^2 = (12)^2 + (9)^2 \] \[ d^2 = 144 + 81 \] \[ d^2 = 225 \] \[ d = \sqrt{225} \] \[ d = 15 \text{ cm} \]Correct Option: C) 15 cm
The question asks to identify the British Prime Minister who dispatched the Cripps Mission to India in 1942. This requires knowledge of key historical figures and events during India's independence movement, specifically during World War II.
Correct Option: D) Winston Churchill was the Prime Minister of the United Kingdom when the Cripps Mission was sent to India in March 1942. The mission was an attempt to secure full Indian cooperation and support for their efforts in World War II.