The compound interest paid when a sum of ₹20,000 is invested for 1 year 4 months at 7\( \frac{1}{2} \)% compound interest per annum, compounded annually, is:
To calculate the compound interest for a period involving years and months, we first calculate the amount for the full years using the compound interest formula. Then, for the remaining months, we calculate simple interest on the amount obtained at the end of the full years. Finally, the compound interest is the total amount minus the principal.
Since the interest is compounded annually, for the first year, we can use the formula for amount:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]Here, \(n=1\) for the first year.
\[ A_1 = 20000 \left(1 + \frac{7.5}{100}\right)^1 \]\[ A_1 = 20000 \left(1 + 0.075\right) \]\[ A_1 = 20000 \times 1.075 \]\[ A_1 = \text{₹}21500 \]The remaining time is 4 months, which is \[\frac{4}{12} = \frac{1}{3}\] of a year. We calculate simple interest on the amount \(A_1\) for this period.
\[ \text{SI} = \frac{P' \times R \times T'}{100} \]Here, \(P' = A_1 = \text{₹}21500\), \(R = 7.5\)%, and \(T' = \frac{1}{3}\) year.
\[ \text{SI} = \frac{21500 \times 7.5 \times \frac{1}{3}}{100} \]\[ \text{SI} = \frac{21500 \times 7.5}{300} \]\[ \text{SI} = \frac{161250}{300} \]\[ \text{SI} = \text{₹}537.50 \]The total amount at the end of 1 year 4 months will be \(A_1 + \text{SI}\).
\[ \text{Total Amount} = 21500 + 537.50 \]\[ \text{Total Amount} = \text{₹}22037.50 \]Compound Interest is the Total Amount minus the Principal.
\[ \text{CI} = \text{Total Amount} - P \]\[ \text{CI} = 22037.50 - 20000 \]\[ \text{CI} = \text{₹}2037.50 \]B) ₹2,037.50 is the correct compound interest calculated.
The question asks to identify the river that originates near Mahabaleshwar in Maharashtra. This requires knowledge of the origin points of major Indian rivers.
D) Krishna — The Krishna River originates near Mahabaleshwar in the Western Ghats, Maharashtra. It flows generally east through Maharashtra, Karnataka, Telangana, and Andhra Pradesh before emptying into the Bay of Bengal.
A wire of length L and resistance R is reformed so that its length increases by 25% with no change in its volume. The resistance of the new wire is:
\( \frac{4}{5} \) R
\( \frac{25}{16} \) R
\( \frac{5}{4} \) R
\( \frac{16}{25} \) R
The problem involves calculating the new resistance of a wire after its length is increased while its volume remains constant. We will use the formula for resistance and the relationship between volume, length, and cross-sectional area to find the new resistance in terms of the original resistance.
Initial Resistance: The resistance \(R\) of a wire is given by the formula:
\[ R = \rho \frac{L}{A} \]where \(\rho\) is the resistivity of the material, \(L\) is the length, and \(A\) is the cross-sectional area.
Volume Conservation: The volume \(V\) of the wire remains constant. The volume is given by:
\[ V = A \times L \]Since the volume is constant, if the length changes, the cross-sectional area must also change accordingly.
New Length: The length of the wire increases by 25%. So, the new length \(L'\) is:
\[ L' = L + 0.25L = 1.25L = \frac{5}{4}L \]New Cross-sectional Area: Since the volume \(V\) is constant, \(V = A L = A' L'\). Therefore, the new cross-sectional area \(A'\) can be found:
\[ A' = \frac{V}{L'} = \frac{AL}{L'} = \frac{AL}{\frac{5}{4}L} = \frac{4}{5}A \]New Resistance: Now, we can calculate the new resistance \(R'\) using the new length \(L'\) and new area \(A'\):
\[ R' = \rho \frac{L'}{A'} \]Substitute the expressions for \(L'\) and \(A'\):
\[ R' = \rho \frac{\frac{5}{4}L}{\frac{4}{5}A} \]Simplify the expression:
\[ R' = \rho \frac{5}{4} \times \frac{5}{4} \frac{L}{A} \]\[ R' = \frac{25}{16} \left( \rho \frac{L}{A} \right) \]Since \(R = \rho \frac{L}{A}\), we can substitute \(R\) into the equation:
\[ R' = \frac{25}{16} R \]B) The new resistance is \( \frac{25}{16} \) R. This matches our calculation.
This problem involves calculating averages. We are given the initial average age of students and how it changes when a teacher's age is included. We can use the formula for average, which is the sum of all values divided by the number of values, to find the teacher's age.
Calculate the initial total age of students:
Given that the average age of 35 students is 15 years.
Determine the new number of people and new average age:
When the teacher's age is included, the number of people becomes \(35 + 1 = 36\).
The average age increases by one year, so the new average age is \(15 + 1 = 16\) years.
Calculate the new total age (students + teacher):
\[ \text{New Average Age} = \frac{\text{Sum of Ages (Students + Teacher)}}{\text{New Number of People}} \]\[ 16 = \frac{\text{Sum of Ages (Students + Teacher)}}{36} \]\[ \text{Sum of Ages (Students + Teacher)} = 16 \times 36 = 576 \text{ years} \]Find the teacher's age:
The teacher's age is the difference between the new total age and the initial total age of students.
B) 51 years. As calculated, the teacher's age is 51 years.
This is a syllogism problem. We will use Venn diagrams to represent the given statements and then evaluate the conclusions based on these diagrams. We need to find conclusions that logically follow from the statements.
Since 'Some hats are goggles' (Statement 3) and 'All hats are caps' (Statement 1), it implies that the part of 'Goggles' that overlaps with 'Hats' must also be within 'Caps'. Therefore, some 'Goggles' are definitely 'Caps'. This conclusion follows.
From the combined diagram, we know that 'Hats' are inside 'Gloves'. We also know that 'Some hats are goggles'. This means there is an overlap between 'Hats' and 'Goggles'. Since 'Hats' are entirely within 'Gloves', the portion of 'Goggles' that overlaps with 'Hats' must also be within 'Gloves'. Therefore, some 'Gloves' are definitely 'Goggles'. This conclusion follows.
As established in the evaluation of Conclusion II, some 'Gloves' are 'Goggles'. This directly contradicts Conclusion III. Therefore, Conclusion III does not follow.
Based on our analysis, Conclusion I (Some goggles are caps) and Conclusion II (Some gloves are goggles) logically follow from the given statements. Conclusion III (No goggle is a glove) does not follow.
This problem involves analyzing logical statements and conclusions, a common type of question in competitive exams. We will use Venn diagrams to represent the relationships given in the statements and then check if the conclusions logically follow from these diagrams.
B) Neither conclusion I nor II follows
As analyzed in the step-by-step explanation, neither Conclusion I ("Some angles are thumbs") nor Conclusion II ("Some trees are fingers") can be logically derived with certainty from the given statements. There exist possible Venn diagram configurations where both conclusions are false while the statements remain true.
The question asks to identify the chief guest for India's Republic Day celebrations in 2020. This is a factual question related to current affairs and international relations, specifically concerning high-profile diplomatic visits.
Correct Option: B) Jair Bolsonaro was the President of Brazil and was the chief guest at India's Republic Day parade on January 26, 2020.
To find the heat produced in a resistor, we can use Joule's Law of Heating. This law states that the heat produced (H) is directly proportional to the square of the current (I), the resistance (R), and the time (t) for which the current flows.
The formula for heat produced is:
\[ H = I^2 R t \]Correct Option: D) The heat produced in the resistor is \(0.4 \, \text{J}\).
The question asks about the nature of washing soda. To answer this, we need to recall the chemical composition of washing soda and its properties when dissolved in water.
This reaction produces hydroxide ions (\(\text{OH}^{-}\)), which makes the solution basic.
C) basic — Washing soda (sodium carbonate) forms a basic solution in water due to the hydrolysis of the carbonate ion, which produces hydroxide ions.
A quadratic equation with roots \(\alpha\) and \(\beta\) can be written in the form \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\). We will use this standard form to construct the equation from the given roots.
Identify the roots:
Given roots are \(\alpha = 15\) and \(\beta = -16\).
Calculate the sum of the roots:
The sum of the roots is \(\alpha + \beta\).
\[ \alpha + \beta = 15 + (-16) = 15 - 16 = -1 \]
Calculate the product of the roots:
The product of the roots is \(\alpha\beta\).
\[ \alpha\beta = 15 \times (-16) = -240 \]
Formulate the quadratic equation:
The general form of a quadratic equation with roots \(\alpha\) and \(\beta\) is:
\[ x^2 - (\alpha + \beta)x + \alpha\beta = 0 \]
Substitute the calculated sum and product of the roots into this equation:
\[ x^2 - (-1)x + (-240) = 0 \]
\[ x^2 + x - 240 = 0 \]
B) The equation \(x^2 + x - 240 = 0\) matches our derived equation.