Express 0.\(\overline{64}\) in the form \( \frac{p}{q} \), where p and q are integers and q ≠ 0
\(\frac{7}{9}\)
\(\frac{9}{11}\)
\(\frac{67}{95}\)
\(\frac{64}{99}\)
To express a repeating decimal (non-terminating recurring decimal) in the form \(\frac{p}{q}\), we use an algebraic method. We assign the given decimal to a variable, multiply it by a power of 10 to shift the repeating block to the left of the decimal point, and then subtract the original equation from the new one to eliminate the repeating part.
Thus, \(0.\overline{64}\) can be expressed as \(\frac{64}{99}\), where \(p=64\) and \(q=99\) are integers and \(q \neq 0\).
D) \(\frac{64}{99}\) is the correct representation of \(0.\overline{64}\) in the form \(\frac{p}{q}\).
A man receives Rs 6,500 as monthly salary. He saves 20% of his salary every month. His expenditure per month is (in Rs):
To find the man's monthly expenditure, we first need to calculate the amount he saves. Since he saves 20% of his salary, we can find the savings amount and then subtract it from his total salary to get the expenditure.
Savings is 20% of the monthly salary.
\[ \text{Savings} = 20\% \text{ of } \text{Rs } 6,500 \]\[ \text{Savings} = \frac{20}{100} \times 6,500 \]\[ \text{Savings} = 20 \times 65 \]\[ \text{Savings} = \text{Rs } 1,300 \]Expenditure is the total salary minus the savings.
\[ \text{Expenditure} = \text{Monthly Salary} - \text{Savings} \]\[ \text{Expenditure} = 6,500 - 1,300 \]\[ \text{Expenditure} = \text{Rs } 5,200 \]Thus, the man's monthly expenditure is Rs 5,200.
C) 5,200 — As calculated, the man's monthly expenditure is Rs 5,200.
The question asks for the establishment year of the High Court of Delhi. This is a factual question related to the history of the Indian judiciary. To answer this, one needs to recall the specific year of its formation.
Therefore, the correct year is 1966.
B) 1966 is the correct answer. The High Court of Delhi was established on October 31, 1966, under the Delhi High Court Act, 1966.
The question asks about the number of times light undergoes refraction when passing through a prism. Refraction is the bending of light as it passes from one medium to another. We need to trace the path of light through a prism to determine how many times this bending occurs.
Correct Option: A) two times. Light refracts once when entering the prism from the air and a second time when exiting the prism back into the air.
Ten years ago, Krishna was one-third as old as he will be 20 years from now. Krishna’s present age (in years) is:
This problem can be solved by setting up an algebraic equation based on the given information. We will define Krishna's present age as a variable and then express his age ten years ago and his age 20 years from now in terms of that variable.
Let Krishna's present age be \(P\) years.
Ten years ago, Krishna's age was \(P - 10\) years.
20 years from now, Krishna's age will be \(P + 20\) years.
According to the problem statement, ten years ago, Krishna was one-third as old as he will be 20 years from now. So, we can write the equation:
\[ P - 10 = \frac{1}{3} (P + 20) \]Multiply both sides by 3 to eliminate the fraction:
\[ 3(P - 10) = P + 20 \]Distribute the 3 on the left side:
\[ 3P - 30 = P + 20 \]Subtract \(P\) from both sides:
\[ 2P - 30 = 20 \]Add 30 to both sides:
\[ 2P = 50 \]Divide by 2:
\[ P = \frac{50}{2} \] \[ P = 25 \]So, Krishna's present age is 25 years.
Correct Option: B) 25
The value of the discriminant of the quadratic equation
2x2−4x−3=0 is:
To find the discriminant of a quadratic equation, we need to compare the given equation with the standard form of a quadratic equation and then use the discriminant formula.
The given quadratic equation is \(2x^2 - 4x - 3 = 0\).
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).
By comparing the given equation with the standard form, we can identify the coefficients:
The discriminant of a quadratic equation is given by the formula \(\Delta = b^2 - 4ac\).
Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:
\[ \Delta = (-4)^2 - 4(2)(-3) \]Calculate the terms:
Substitute these values back into the discriminant formula:
\[ \Delta = 16 - (-24) \]\[ \Delta = 16 + 24 \]\[ \Delta = 40 \]Thus, the value of the discriminant is \(40\).
C) 40 — The calculation \(b^2 - 4ac = (-4)^2 - 4(2)(-3) = 16 - (-24) = 16 + 24 = 40\) is correct.
This question relates to electromagnetic induction. When the current in coil A changes (switched on or off), it produces a changing magnetic flux through coil B, inducing an electromotive force (EMF) and thus a current in coil B, which is detected by the galvanometer. To increase the deflection, we need to increase the induced current, which means increasing the induced EMF. According to Faraday's law, induced EMF is proportional to the rate of change of magnetic flux. We need to find a material that can enhance the magnetic flux.
Correct Option: B) an iron rod
Inserting an iron rod (a ferromagnetic material) into coil B will significantly increase the magnetic flux linking coil B because iron has a high magnetic permeability. This increased flux will lead to a larger change in flux when the current in coil A is switched, resulting in a larger induced EMF and thus a greater deflection in the galvanometer.
The question asks to identify the social reformer who authored the book 'Barno-Porichoy'. This requires knowledge of prominent social reformers in India and their literary contributions, particularly in the Bengali language context.
Correct Option: C) Ishwar Chandra Vidyasagar
Ishwar Chandra Vidyasagar (1820–1891) was a towering figure of the Bengal Renaissance. He was a philosopher, academic educator, writer, translator, printer, publisher, entrepreneur, reformer, and philanthropist. His most significant contribution to Bengali literature and education was 'Barno-Porichoy' (Introduction to the Alphabet), published in two parts in 1855. This book simplified the Bengali alphabet and made learning the language accessible to a wider audience, playing a crucial role in modernizing Bengali prose.
The question asks about the reasons behind carbon's catenation property. Catenation is the ability of an element to form bonds with atoms of the same element, creating long chains or rings. We need to identify the atomic characteristics of carbon that contribute to this unique property.
Correct Option: C) Carbon exhibits catenation due to its small size, which allows for effective orbital overlap and strong covalent bond formation, and the inherently high strength of the carbon-carbon single, double, and triple bonds. This combination leads to stable, long chains and complex structures.
Find the value of
[(12 × 4) × {6 ÷ 4 × \(\left(\frac{7-4}{6}\right)\)}]
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This means we perform operations in the following order: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).
Thus, the value of the entire expression is 36.
D) 36 is the correct answer, as derived by following the order of operations.