This problem can be solved using Ohm's Law, which states the relationship between voltage, current, and resistance in an electrical circuit. Ohm's Law is given by \(V = IR\), where \(V\) is the voltage, \(I\) is the current, and \(R\) is the resistance. We are given the voltage and the current, and we need to find the resistance.
The prefix 'm' in mA stands for milli, which means \(10^{-3}\). So, 2 mA needs to be converted to Amperes.
\[ I = 2 \text{ mA} = 2 \times 10^{-3} \text{ A} \]Ohm's Law states \(V = IR\). To find the resistance \(R\), we can rearrange the formula:
\[ R = \frac{V}{I} \]Substitute the given voltage and the converted current into the formula:
\[ R = \frac{12 \text{ V}}{2 \times 10^{-3} \text{ A}} \]\[ R = \frac{12}{0.002} \text{ } \Omega \]\[ R = 6000 \text{ } \Omega \]This question tests the fundamental definitions related to spherical mirrors, specifically the center of curvature, aperture, and principal focus. We need to evaluate each statement based on standard optical definitions.
By definition, a spherical mirror is a part of a hollow sphere. The center of this sphere is called the center of curvature (C) of the mirror. This statement is a direct and correct definition.
The aperture of a spherical mirror is the diameter of its reflecting surface. It is the effective diameter of the mirror that reflects light. It is NOT the diameter of the entire sphere from which the mirror is cut. For example, if you cut a small cap from a large sphere, the aperture is the diameter of that cap, not the diameter of the original sphere. This statement is incorrect.
For spherical mirrors with a small aperture, the principal focus (F) is indeed located exactly midway between the pole (P) and the centre of curvature (C). This means the focal length \(f\) is half the radius of curvature \(R\), i.e., \(f = R/2\). This is a standard approximation used for paraxial rays and mirrors with small apertures, which is generally assumed unless stated otherwise in competitive exams. Therefore, this statement is considered true in the context of typical spherical mirror problems.
Based on the evaluation:
Therefore, statements (S - I) and (S - III) are true.
B) (S - I) and (S - III) only
A shopkeeper offers a scheme of either selling articles at successive discounts of 30% and 20% or one where a customer can “buy 2, get 1 free”. The positive difference in overall nominal percentage discounts offered in the two schemes is:
11 \( \frac{2}{3} \) %
10 \( \frac{1}{3} \) %
11 \( \frac{1}{3} \) %
10 \( \frac{2}{3} \)%
To find the positive difference in overall nominal percentage discounts, we need to calculate the effective discount for each scheme separately. For successive discounts, we use the formula for equivalent discount. For the "buy 2, get 1 free" scheme, we calculate the percentage discount based on the number of free items relative to the total number of items received.
Calculate the effective discount for the first scheme (successive discounts):
The shopkeeper offers successive discounts of 30% and 20%.
The formula for two successive discounts \(d_1\) and \(d_2\) is:
\[ \text{Effective Discount} = \left( d_1 + d_2 - \frac{d_1 \times d_2}{100} \right) \% \]Substituting \(d_1 = 30\) and \(d_2 = 20\):
\[ \text{Effective Discount}_1 = \left( 30 + 20 - \frac{30 \times 20}{100} \right) \% \] \[ \text{Effective Discount}_1 = \left( 50 - \frac{600}{100} \right) \% \] \[ \text{Effective Discount}_1 = (50 - 6) \% = 44 \% \]Calculate the effective discount for the second scheme ("buy 2, get 1 free"):
In this scheme, a customer buys 2 articles and gets 1 free. This means the customer receives a total of \(2+1=3\) articles, but only pays for 2 articles. The discount is on 1 article out of 3.
The percentage discount is calculated as:
\[ \text{Effective Discount} = \left( \frac{\text{Number of free articles}}{\text{Total number of articles received}} \times 100 \right) \% \] \[ \text{Effective Discount}_2 = \left( \frac{1}{2+1} \times 100 \right) \% \] \[ \text{Effective Discount}_2 = \left( \frac{1}{3} \times 100 \right) \% \] \[ \text{Effective Discount}_2 = 33 \frac{1}{3} \% \]Calculate the positive difference in overall nominal percentage discounts:
Difference \( = |\text{Effective Discount}_1 - \text{Effective Discount}_2|\)
\[ \text{Difference} = \left| 44 \% - 33 \frac{1}{3} \% \right| \] \[ \text{Difference} = \left| 44 - \frac{100}{3} \right| \% \] \[ \text{Difference} = \left| \frac{44 \times 3 - 100}{3} \right| \% \] \[ \text{Difference} = \left| \frac{132 - 100}{3} \right| \% \] \[ \text{Difference} = \frac{32}{3} \% \]Convert the improper fraction to a mixed number:
\[ \frac{32}{3} = 10 \frac{2}{3} \% \]D) The positive difference in overall nominal percentage discounts is \(10 \frac{2}{3} \%\).
The question asks about the month in which Rama Navami, a Hindu festival celebrating Lord Rama's birthday, typically falls. This requires knowledge of the Hindu calendar and its correlation with the Gregorian calendar.
Correct Option: C) Rama Navami is a Hindu spring festival that commemorates God Rama’s birthday, which usually comes in the month of April.
The volume of a right circular cylinder is 814 cm³. If the radius and the height of the cylinder are equal, find the height (in cm) of the cylinder.
(Use π = \( \frac{22}{7} \))
\(\sqrt[5]{259}\)
\(2\sqrt[3]{259}\)
\(\sqrt[3]{259}\)
\(\sqrt[3]{295}\)
To find the height of the cylinder, we will use the formula for the volume of a right circular cylinder. We are given the volume and the condition that the radius and height are equal. We will substitute these values into the formula and solve for the height.
The volume \(V\) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]where \(r\) is the radius of the base and \(h\) is the height of the cylinder.
Since \(r = h\), we can replace \(r\) with \(h\) in the formula:
\[ V = \pi h^2 h = \pi h^3 \]Now, substitute the given volume and the value of \(\pi\):
\[ 814 = \frac{22}{7} \times h^3 \]Multiply both sides by \(\frac{7}{22}\):
\[ h^3 = 814 \times \frac{7}{22} \]Divide 814 by 22:
\[ 814 \div 22 = 37 \]So,
\[ h^3 = 37 \times 7 \] \[ h^3 = 259 \]To find \(h\), take the cube root of both sides:
\[ h = \sqrt[3]{259} \text{ cm} \]C) \(\sqrt[3]{259}\)
A shopkeeper finds that when he sells one kind of rice at Rs 50 per kg he loses 20% and when he sells another kind of rice at Rs 72 per kg he gains 20%. If he mixes them in the ratio 3 : 2 and sells the mixture at Rs 64 per kg, then what is his overall gain percentage (correct to one decimal place)?
To find the overall gain percentage, we first need to calculate the cost price (CP) of each type of rice. Then, we'll determine the total cost price and total selling price of the mixture based on the given ratio and selling price. Finally, we can calculate the overall gain percentage.
Selling Price (\(\text{SP}_1\)) = Rs 50 per kg
Loss = 20%
The formula for CP when there is a loss is:
\[ \text{CP} = \frac{\text{SP}}{1 - \frac{\text{Loss percentage}}{100}} \]So,
\[ \text{CP}_1 = \frac{50}{1 - \frac{20}{100}} = \frac{50}{1 - 0.20} = \frac{50}{0.80} = \text{Rs } 62.5 \text{ per kg} \]Selling Price (\(\text{SP}_2\)) = Rs 72 per kg
Gain = 20%
The formula for CP when there is a gain is:
\[ \text{CP} = \frac{\text{SP}}{1 + \frac{\text{Gain percentage}}{100}} \]So,
\[ \text{CP}_2 = \frac{72}{1 + \frac{20}{100}} = \frac{72}{1 + 0.20} = \frac{72}{1.20} = \text{Rs } 60 \text{ per kg} \]The rice types are mixed in the ratio 3:2. Let's assume 3 kg of the first kind and 2 kg of the second kind are mixed.
Total quantity = \(3 + 2 = 5\) kg
Cost of 3 kg of first kind = \(3 \times 62.5 = \text{Rs } 187.5\)
Cost of 2 kg of second kind = \(2 \times 60 = \text{Rs } 120\)
Total CP of 5 kg mixture = \(187.5 + 120 = \text{Rs } 307.5\)
Average CP of the mixture per kg = \(\frac{307.5}{5} = \text{Rs } 61.5 \text{ per kg}\)
The mixture is sold at Rs 64 per kg.
Total SP of 5 kg mixture = \(5 \times 64 = \text{Rs } 320\)
Overall Gain = Total SP - Total CP = \(320 - 307.5 = \text{Rs } 12.5\)
Overall Gain percentage = \(\frac{\text{Gain}}{\text{Total CP}} \times 100\)
\[ \text{Overall Gain percentage} = \frac{12.5}{307.5} \times 100 \]\[ \text{Overall Gain percentage} \approx 4.06504 \% \]
Rounding to one decimal place, the overall gain percentage is 4.1%.
B) 4.1% is the calculated overall gain percentage when rounded to one decimal place.
To solve this problem, we need to translate the given statement into a mathematical equation. We will represent the unknown number as a variable, form a quadratic equation, and then solve it to find the value of the number.
Define the variable: Let the number be \(x\).
Express its reciprocal: The reciprocal of the number \(x\) is \(\frac{1}{x}\).
Formulate the equation: The problem states that the sum of the number and its reciprocal is \(-2\). So, we can write the equation as:
\[ x + \frac{1}{x} = -2 \]Convert to a quadratic equation: To eliminate the fraction, multiply the entire equation by \(x\):
\[ x \cdot x + x \cdot \frac{1}{x} = -2 \cdot x \] \[ x^2 + 1 = -2x \]Rearrange the terms to form a standard quadratic equation \(ax^2 + bx + c = 0\):
\[ x^2 + 2x + 1 = 0 \]Solve the quadratic equation: We can solve this quadratic equation by factoring or using the quadratic formula. This equation is a perfect square trinomial:
\[ (x+1)^2 = 0 \]Taking the square root of both sides:
\[ x+1 = 0 \] \[ x = -1 \]Verify the solution: Substitute \(x = -1\) back into the original statement:
\[ (-1) + \frac{1}{(-1)} = -1 - 1 = -2 \]This matches the given condition.
Correct Option: C) The quadratic equation is \(x^2 + 2x + 1 = 0\), and the number is \(x = -1\).
The question asks about the Right-Hand Thumb Rule, a fundamental concept in electromagnetism used to determine the direction of the magnetic field produced by a current-carrying conductor or the direction of the current itself in relation to the magnetic field. We need to identify which part of the hand represents the direction of the current according to this rule.
D) the thumb. According to the Right-Hand Thumb Rule, if you point your right thumb in the direction of the current, your curled fingers will show the direction of the magnetic field lines. Therefore, the thumb represents the direction of the current.
This is a floor-based puzzle. We need to deduce the living arrangement of six people (H, N, P, T, W, Y) on six different floors (1 to 6) based on the given clues. We will build a floor plan step-by-step.
Correct Option: A) H
In a right-angled triangle ABC, right-angled at B, if tan A = \(\frac{1}{\sqrt{3}}\), the value of (sin A cos C + cos A sin C) is:
1
\( \frac{3}{4} \)
\( \frac{1}{2} \)
\( \frac{1}{4} \)
The problem involves a right-angled triangle and trigonometric ratios. We are given the value of \(\tan A\) and need to find the value of \(\sin A \cos C + \cos A \sin C\). This expression is the expansion of \(\sin(A+C)\). We can use the given \(\tan A\) to find angle A, and then use the properties of a right-angled triangle to find angle C, or directly use the trigonometric identity.
Identify the given information:
We have a right-angled triangle ABC, right-angled at B. This means \(\angle B = 90^\circ\).
We are given \(\tan A = \frac{1}{\sqrt{3}}\).
Determine angle A:
We know that \(\tan 30^\circ = \frac{1}{\sqrt{3}}\). Therefore, \(\angle A = 30^\circ\).
Determine angle C:
In a triangle, the sum of angles is \(180^\circ\).
\[ \angle A + \angle B + \angle C = 180^\circ \]
\[ 30^\circ + 90^\circ + \angle C = 180^\circ \]
\[ 120^\circ + \angle C = 180^\circ \]
\[ \angle C = 180^\circ - 120^\circ \]
\[ \angle C = 60^\circ \]
Evaluate the expression \(\sin A \cos C + \cos A \sin C\):
Substitute the values of angles A and C into the expression:
\[ \sin A \cos C + \cos A \sin C = \sin 30^\circ \cos 60^\circ + \cos 30^\circ \sin 60^\circ \]
We know the standard trigonometric values:
\[ \sin 30^\circ = \frac{1}{2} \]
\[ \cos 60^\circ = \frac{1}{2} \]
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Substitute these values:
\[ \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \]
\[ = \frac{1}{4} + \frac{3}{4} \]
\[ = \frac{1+3}{4} \]
\[ = \frac{4}{4} \]
\[ = 1 \]
Alternative Method using Trigonometric Identity:
The expression \(\sin A \cos C + \cos A \sin C\) is the expansion of the trigonometric identity \(\sin(A+C)\).
Since \(\angle B = 90^\circ\) in a right-angled triangle, we know that \(\angle A + \angle C = 180^\circ - \angle B = 180^\circ - 90^\circ = 90^\circ\).
Therefore, the expression becomes:
\[ \sin(A+C) = \sin(90^\circ) \]
\[ = 1 \]
A) The value of the expression is 1, as determined by both direct substitution of angles and using the trigonometric identity \(\sin(A+C)\).