To find the difference in interest, we need to calculate the simple interest for A and the compound interest for B separately for 3 years. Then, subtract the smaller interest from the larger one.
Calculate Simple Interest for A:
Principal \((P_A)\) = ₹8,000
Rate \((R_A)\) = 8% per annum
Time \((T)\) = 3 years
The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]Substitute the values for A:
\[ \text{SI}_A = \frac{8000 \times 8 \times 3}{100} = 80 \times 8 \times 3 = 640 \times 3 = \text{₹}1920 \]Calculate Compound Interest for B:
Principal \((P_B)\) = ₹6,000
Rate \((R_B)\) = 10% per annum
Time \((n)\) = 3 years
The formula for Amount (A) with compound interest is:
\[ \text{Amount} = P \left(1 + \frac{R}{100}\right)^n \]Substitute the values for B:
\[ \text{Amount}_B = 6000 \left(1 + \frac{10}{100}\right)^3 \] \[ \text{Amount}_B = 6000 \left(1 + \frac{1}{10}\right)^3 \] \[ \text{Amount}_B = 6000 \left(\frac{11}{10}\right)^3 \] \[ \text{Amount}_B = 6000 \times \frac{11 \times 11 \times 11}{10 \times 10 \times 10} \] \[ \text{Amount}_B = 6000 \times \frac{1331}{1000} \] \[ \text{Amount}_B = 6 \times 1331 = \text{₹}7986 \]Now, calculate the Compound Interest (CI) for B:
\[ \text{CI}_B = \text{Amount}_B - P_B \] \[ \text{CI}_B = 7986 - 6000 = \text{₹}1986 \]Calculate the difference between their interests:
Difference = \(\text{CI}_B - \text{SI}_A\)
Difference = \(1986 - 1920 = \text{₹}66\)
Correct Option: C) ₹66
The simple interest earned by A is ₹1920, and the compound interest earned by B is ₹1986. The difference between these two interests is ₹66.
The question asks about the administrative and judicial control over District and Sessions Judges in India. Understanding the hierarchy of the Indian judiciary, particularly at the state level, is crucial to answering this question.
Correct Option: C) The High Court of the state exercises direct control over the District and Sessions Judge, as per Article 235 of the Indian Constitution, ensuring the independence and proper functioning of the judiciary.
There is a regular hexagon of side 5 cm. Find its area.
\(\frac{50\sqrt{3}}{3}\) cm²
\(\frac{75\sqrt{3}}{2}\) cm²
36\(\sqrt{3}\) cm²
25\(\sqrt{3}\)cm²
To find the area of a regular hexagon, we can divide it into six equilateral triangles. The side length of each equilateral triangle will be equal to the side length of the hexagon. We then calculate the area of one equilateral triangle and multiply it by six.
B) \(\frac{75\sqrt{3}}{2}\) cm² is the correct area of the regular hexagon with a side of 5 cm.
We are given the initial ratio of two numbers and their ratio after a constant value is subtracted from each. We need to find the absolute difference between the original numbers. We can set up algebraic equations based on the given ratios and solve for the numbers.
Represent the initial numbers:
Let the two numbers be \(11x\) and \(19x\), since their ratio is \(11:19\).
Formulate the new numbers after subtraction:
If each number is subtracted by 7, the new numbers will be \((11x - 7)\) and \((19x - 7)\).
Set up the equation based on the new ratio:
The ratio of the new numbers is given as \(5:9\). So, we can write the equation:
Solve the equation for \(x\):
Cross-multiply to solve for \(x\):
Find the actual numbers:
Substitute the value of \(x\) back into the expressions for the original numbers:
Calculate the absolute difference:
The absolute difference between the actual numbers is:
Correct Option: B) 56
The problem asks us to find 40% of a number, given a fractional relationship of that number. We will first set up an equation to find the number, and then calculate 40% of it.
Let the unknown number be \(x\).
Translate the given information into an equation:
"Two-fifth of seven-seventeenth of three-fifth of a number is 84" can be written as:
\[ \frac{2}{5} \times \frac{7}{17} \times \frac{3}{5} \times x = 84 \]Multiply the fractions on the left side:
\[ \frac{2 \times 7 \times 3}{5 \times 17 \times 5} \times x = 84 \]\[ \frac{42}{425} \times x = 84 \]Solve for \(x\):
To find \(x\), multiply both sides by the reciprocal of \(\frac{42}{425}\):
\[ x = 84 \times \frac{425}{42} \]\[ x = 2 \times 425 \]\[ x = 850 \]Now, we need to find 40% of this number \(x\).
40% of \(x\) can be written as \(\frac{40}{100} \times x\) or \(0.40 \times x\).
\[ 40\% \text{ of } 850 = \frac{40}{100} \times 850 \]\[ = \frac{4}{10} \times 850 \]\[ = 4 \times 85 \]\[ = 340 \]Thus, 40% of the number is 340.
A) 340 is the correct answer, as calculated in the step-by-step solution.
To determine the type of triangle formed by the given vertices, we need to calculate the lengths of all three sides of the triangle using the distance formula. Once we have the side lengths, we can classify the triangle based on these lengths (equilateral, isosceles, scalene) and also check for the right-angled property using the Pythagorean theorem.
Given points \(A(1, 2)\) and \(B(3, 4)\).
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]\[ AB = \sqrt{(3 - 1)^2 + (4 - 2)^2} \]\[ AB = \sqrt{(2)^2 + (2)^2} \]\[ AB = \sqrt{4 + 4} \]\[ AB = \sqrt{8} \]Given points \(B(3, 4)\) and \(C(4, 1)\).
\[ BC = \sqrt{(4 - 3)^2 + (1 - 4)^2} \]\[ BC = \sqrt{(1)^2 + (-3)^2} \]\[ BC = \sqrt{1 + 9} \]\[ BC = \sqrt{10} \]Given points \(C(4, 1)\) and \(A(1, 2)\).
\[ CA = \sqrt{(1 - 4)^2 + (2 - 1)^2} \]\[ CA = \sqrt{(-3)^2 + (1)^2} \]\[ CA = \sqrt{9 + 1} \]\[ CA = \sqrt{10} \]We have the side lengths: \(AB = \sqrt{8}\), \(BC = \sqrt{10}\), and \(CA = \sqrt{10}\).
Since \(BC = CA = \sqrt{10}\), two sides of the triangle are equal in length. Therefore, the triangle is an isosceles triangle.
For a right-angled triangle, the square of the longest side must be equal to the sum of the squares of the other two sides.
The squares of the side lengths are:
The longest side is either BC or CA (both are \(\sqrt{10}\)). Let's check if \(BC^2 = AB^2 + CA^2\) or \(CA^2 = AB^2 + BC^2\).
If it were right-angled, the hypotenuse would be \(\sqrt{10}\). So, we would check if \((\sqrt{10})^2 = (\sqrt{8})^2 + (\sqrt{10})^2\), which means \(10 = 8 + 10\), or \(10 = 18\), which is false.
Alternatively, if the longest side were \(\sqrt{10}\), then \((\sqrt{10})^2 = (\sqrt{8})^2 + (\sqrt{10})^2\) or \((\sqrt{10})^2 = (\sqrt{8})^2 + (\sqrt{10})^2\). This is not possible as it would imply \(10 = 8 + 10\), which is \(10 = 18\), which is false. The hypotenuse must be strictly greater than the other two sides.
Let's check if \(BC^2 = AB^2 + CA^2\): \(10 = 8 + 10\), which is \(10 = 18\) (False).
Let's check if \(CA^2 = AB^2 + BC^2\): \(10 = 8 + 10\), which is \(10 = 18\) (False).
Let's check if \(AB^2 = BC^2 + CA^2\): \(8 = 10 + 10\), which is \(8 = 20\) (False).
None of the conditions for a right-angled triangle are met. Therefore, the triangle is not right-angled.
Correct Option: C) isosceles
The triangle has two sides of equal length (\(BC = CA = \sqrt{10}\)), making it an isosceles triangle.
This question tests a fundamental property of triangles in Euclidean geometry. We need to recall the basic theorem regarding the sum of interior angles of any triangle.
A) 180° — The sum of the interior angles of any triangle in Euclidean geometry is always 180 degrees.
The problem involves a circle, a chord, and its distance from the center. We need to find the radius of the circle. We can use the property that a perpendicular drawn from the center to a chord bisects the chord, forming a right-angled triangle. Then, the Pythagorean theorem can be applied to find the radius.
Let the circle have its center at point O and a chord AB. The length of the chord AB is given as 24 cm.
The distance of the chord from the center is given as 5 cm. Let M be the midpoint of the chord AB, such that OM is perpendicular to AB. So, \(OM = 5\) cm.
A perpendicular from the center to a chord bisects the chord. Therefore, M is the midpoint of AB, and \(AM = MB = \frac{AB}{2}\).
Calculate the length of AM:
\[ AM = \frac{24 \text{ cm}}{2} = 12 \text{ cm} \]Now, consider the right-angled triangle OMA, where O is the center, M is the midpoint of the chord, and A is an endpoint of the chord. The sides of this triangle are OM (distance from center to chord), AM (half the length of the chord), and OA (the radius of the circle).
According to the Pythagorean theorem in triangle OMA:
\[ OA^2 = OM^2 + AM^2 \]Substitute the known values:
\[ OA^2 = (5 \text{ cm})^2 + (12 \text{ cm})^2 \]\[ OA^2 = 25 \text{ cm}^2 + 144 \text{ cm}^2 \]\[ OA^2 = 169 \text{ cm}^2 \]Calculate the radius OA:
\[ OA = \sqrt{169 \text{ cm}^2} \]\[ OA = 13 \text{ cm} \]Thus, the radius of the circle is 13 cm.
Correct Option: A) 13
This problem involves profit and loss concepts. We are given the selling price and a condition that the profit percentage is equal to the cost price. We need to find the cost price. We can set up an equation based on the profit formula and solve for the cost price.
Correct Option: D) 90