The difference between the ages of two sisters is 2 years when father's age was 52. Father is elder by 2 years to mother. Elder sister's age is half of mother's age. Find the age of younger sister.
We need to find the age of the younger sister by using the given relationships between the ages of the father, mother, and the two sisters. We will work backward from the father's age to find the mother's age, then the elder sister's age, and finally the younger sister's age.
Given that the father's age is 52 years, and the father is elder by 2 years to the mother.
\[ \text{Mother's age} = \text{Father's age} - 2 \] \[ \text{Mother's age} = 52 - 2 = 50 \text{ years} \]Given that the elder sister's age is half of the mother's age.
\[ \text{Elder sister's age} = \frac{\text{Mother's age}}{2} \] \[ \text{Elder sister's age} = \frac{50}{2} = 25 \text{ years} \]Given that the difference between the ages of the two sisters is 2 years. Since the elder sister is 25 years old, the younger sister must be 2 years younger.
\[ \text{Younger sister's age} = \text{Elder sister's age} - 2 \] \[ \text{Younger sister's age} = 25 - 2 = 23 \text{ years} \]The information about the father's age being 52 is the age at which the difference between the sisters' ages is 2 years. This implies that the age difference between the sisters is constant, regardless of the father's current age. The problem implies all ages are current or at the same point in time.
Correct Option: D) 23
This problem involves simple interest. We are given the maturity value (Amount), time period, and rate of interest. We need to find the principal amount. We can use the formula for simple interest and the relationship between Amount, Principal, and Interest.
Thus, the principal amount is Rs. 6,000.
Correct Option: B) Rs. 6,000
The problem is a coding-decoding type question. We need to identify the pattern or rule by which "PIXIE" is coded as "OHXHD" and then apply the same rule to decode "ELEANOR".
A) DKDAMNQ is the correct option. This is derived by applying the pattern where all letters are shifted back by one position, except for the middle letter of the word, which remains unchanged. For "PIXIE", the middle letter is 'X', which remains 'X'. For "ELEANOR", the middle letter is 'A', which remains 'A'. All other letters are shifted back by one position (e.g., E-1=D, L-1=K, N-1=M, O-1=N, R-1=Q).
The marked prices of big and small notebooks are Rs. 15 and Rs. 10 respectively. A student bought 5 dozen small and 10 dozen big notebooks at a total discount of 5%. Find the total discount amount.
To find the total discount amount, we first need to calculate the total marked price of all the notebooks purchased. This involves calculating the marked price for the small notebooks and the big notebooks separately, and then summing them up. Once the total marked price is known, we can apply the given total discount percentage to find the discount amount.
A student bought 5 dozen small notebooks. Since 1 dozen = 12 units:
\[ \text{Number of small notebooks} = 5 \times 12 = 60 \]The marked price of one small notebook is Rs. 10.
\[ \text{Total marked price of small notebooks} = 60 \times \text{Rs. } 10 = \text{Rs. } 600 \]A student bought 10 dozen big notebooks.
\[ \text{Number of big notebooks} = 10 \times 12 = 120 \]The marked price of one big notebook is Rs. 15.
\[ \text{Total marked price of big notebooks} = 120 \times \text{Rs. } 15 = \text{Rs. } 1800 \]Sum the total marked prices of small and big notebooks.
\[ \text{Total Marked Price} = \text{Rs. } 600 + \text{Rs. } 1800 = \text{Rs. } 2400 \]A total discount of 5% was given on the total marked price.
\[ \text{Discount Amount} = 5\% \text{ of Total Marked Price} \] \[ \text{Discount Amount} = \frac{5}{100} \times 2400 \] \[ \text{Discount Amount} = 5 \times 24 = \text{Rs. } 120 \]Correct Option: C) Rs. 120
The question asks for the term that describes an event happening once every four years. This requires knowledge of Latin-derived prefixes used to denote frequency or duration.
The prefix "quadr-" or "quadri-" means four. Therefore, a term incorporating this prefix would relate to four units of time.
Correct Option: B) Quadrennial is the correct term. The prefix "quadr-" means four, and "-ennial" relates to years. Thus, "quadrennial" means occurring every four years.
This is a blood relation problem where symbols represent relationships. We need to decode the given expression "L + M × N" step-by-step using the provided rules to determine the relationship between L and N.
Correct Option: D) Grandmother
Find the value of \(\frac{(525+252)^2-(525-252)^2}{525\times252}\).
This problem can be solved by recognizing and applying a fundamental algebraic identity. The expression resembles the expansion of \((a+b)^2 - (a-b)^2\). By simplifying this identity, we can quickly find the value of the given expression without performing complex arithmetic calculations.
Let \(a = 525\) and \(b = 252\).
The given expression is of the form \(\frac{(a+b)^2 - (a-b)^2}{a \times b}\).
Recall the algebraic identities:
Now, subtract the second identity from the first:
\[ (a+b)^2 - (a-b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) \]\[ = a^2 + 2ab + b^2 - a^2 + 2ab - b^2 \]\[ = (a^2 - a^2) + (b^2 - b^2) + (2ab + 2ab) \]\[ = 0 + 0 + 4ab \]\[ = 4ab \]Substitute this simplified form back into the original expression:
\[ \frac{(a+b)^2 - (a-b)^2}{a \times b} = \frac{4ab}{ab} \]Cancel out the common term \(ab\) from the numerator and the denominator (since \(a\) and \(b\) are non-zero):
\[ \frac{4\cancel{ab}}{\cancel{ab}} = 4 \]Therefore, the value of the given expression is 4.
B) The value of the expression is 4, as derived from the algebraic identity \((a+b)^2 - (a-b)^2 = 4ab\).
If 2cosθ = \(\sqrt{3}\), cosθ × tanθ = ?
1
\(\frac{\sqrt{3}}{3}\)
\(\frac{\sqrt{3}}{2}\)
\( \frac{1}{2} \)
The problem asks us to find the value of \(\cos\theta \times \tan\theta\) given that \(2\cos\theta = \sqrt{3}\). We will first find the value of \(\cos\theta\) from the given equation. Then, we will use the trigonometric identity \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) to simplify the expression \(\cos\theta \times \tan\theta\) and substitute the known values.
Find the value of \(\cos\theta\):
Given the equation:
\[ 2\cos\theta = \sqrt{3} \]
Divide both sides by 2:
\[ \cos\theta = \frac{\sqrt{3}}{2} \]
Simplify the expression \(\cos\theta \times \tan\theta\):
We know the trigonometric identity \(\tan\theta = \frac{\sin\theta}{\cos\theta}\).
Substitute this into the expression we need to evaluate:
\[ \cos\theta \times \tan\theta = \cos\theta \times \frac{\sin\theta}{\cos\theta} \]
Assuming \(\cos\theta \neq 0\), we can cancel \(\cos\theta\) from the numerator and denominator:
\[ \cos\theta \times \tan\theta = \sin\theta \]
Find the value of \(\sin\theta\):
We know that \(\cos\theta = \frac{\sqrt{3}}{2}\). This value corresponds to \(\theta = 30^\circ\) or \(\frac{\pi}{6}\) radians in the first quadrant. For this angle, \(\sin\theta\) is:
\[ \sin\theta = \sin(30^\circ) = \frac{1}{2} \]
Alternatively, we can use the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\):
\[ \sin^2\theta = 1 - \cos^2\theta \]
Substitute the value of \(\cos\theta\):
\[ \sin^2\theta = 1 - \left(\frac{\sqrt{3}}{2}\right)^2 \]
\[ \sin^2\theta = 1 - \frac{3}{4} \]
\[ \sin^2\theta = \frac{4-3}{4} \]
\[ \sin^2\theta = \frac{1}{4} \]
Taking the square root of both sides (assuming \(\theta\) is in a quadrant where \(\sin\theta\) is positive, which is usually implied in such problems unless specified otherwise):
\[ \sin\theta = \sqrt{\frac{1}{4}} = \frac{1}{2} \]
Final result:
Since \(\cos\theta \times \tan\theta = \sin\theta\), and we found \(\sin\theta = \frac{1}{2}\), the value of the expression is \(\frac{1}{2}\).
Correct Option: D) \( \frac{1}{2} \)
This question tests fundamental knowledge about energy conversion devices. We need to identify the primary function of a generator based on its energy transformation process.
Correct Option: A) mechanical energy into electrical energy.
A generator works on the principle of electromagnetic induction. Mechanical energy (e.g., from rotating a coil in a magnetic field or rotating magnets around a coil) is used to induce an electric current, thereby converting the mechanical motion into electrical energy. This is the fundamental function of all types of electrical generators, whether they are AC (alternating current) or DC (direct current).
A man has an equal number of five, ten and twenty rupee notes amounting to Rs. 385. Find the number of notes of each denomination.
The problem states that a man has an equal number of five, ten, and twenty rupee notes, and the total amount is Rs. 385. We need to find the number of notes of each denomination. We can set up an equation where 'x' represents the number of notes of each denomination and then solve for 'x'.
Let \(x\) be the number of notes of each denomination. This means the man has \(x\) five-rupee notes, \(x\) ten-rupee notes, and \(x\) twenty-rupee notes.
Calculate the total value contributed by each denomination:
The total amount is given as Rs. 385. So, we can form an equation:
\[ 5x + 10x + 20x = 385 \]Combine the terms on the left side of the equation:
\[ (5 + 10 + 20)x = 385 \]\[ 35x = 385 \]Solve for \(x\) by dividing both sides by 35:
\[ x = \frac{385}{35} \]\[ x = 11 \]Thus, the number of notes of each denomination is 11.
Correct Option: B) 11