This problem involves compound interest. We need to find the time it takes for a sum of money to become four times itself, given that it doubles in 8 years. We can use the compound interest formula or a shortcut method based on the property of compound interest that the amount grows in geometric progression over equal periods.
Understand the given information: Let the principal amount be \(P\). After 8 years, the amount becomes \(2P\).
Use the compound interest formula (optional, for understanding): The formula for compound interest is \(A = P(1 + r/100)^t\), where \(A\) is the amount, \(P\) is the principal, \(r\) is the rate of interest per annum, and \(t\) is the time in years.
Given that the sum doubles in 8 years:
\[ 2P = P \left(1 + \frac{r}{100}\right)^8 \] \[ 2 = \left(1 + \frac{r}{100}\right)^8 \quad \text{(Equation 1)} \]Determine the target: We want to find the time \(T\) when the amount becomes four times the principal, i.e., \(4P\).
\[ 4P = P \left(1 + \frac{r}{100}\right)^T \] \[ 4 = \left(1 + \frac{r}{100}\right)^T \quad \text{(Equation 2)} \]Relate the equations: We know that \(4 = 2^2\). Substitute this into Equation 2:
\[ 2^2 = \left(1 + \frac{r}{100}\right)^T \]From Equation 1, we know that \(2 = \left(1 + \frac{r}{100}\right)^8\). Substitute this into the above equation:
\[ \left[\left(1 + \frac{r}{100}\right)^8\right]^2 = \left(1 + \frac{r}{100}\right)^T \] \[ \left(1 + \frac{r}{100}\right)^{8 \times 2} = \left(1 + \frac{r}{100}\right)^T \] \[ \left(1 + \frac{r}{100}\right)^{16} = \left(1 + \frac{r}{100}\right)^T \]Comparing the exponents, we get:
\[ T = 16 \text{ years} \]Shortcut Method: For compound interest, if a sum becomes \(x\) times in \(t\) years, it will become \(x^n\) times in \(n \times t\) years.
Given: Sum doubles (becomes 2 times) in 8 years.
We want it to become 4 times. We can write 4 as \(2^2\).
Here, \(x = 2\), \(t = 8\) years, and we want it to become \(x^n = 2^2\) times, so \(n = 2\).
The time taken will be \(n \times t = 2 \times 8 = 16\) years.
Correct Option: D) 16
The question asks to identify the monument built with Gujarati architectural influence to commemorate the visit of King George V and Queen Mary to India. We need to recall the historical context and architectural styles of the given options.
Correct Option: C) The Gateway of India was built to commemorate the landing of King George V and Queen Mary at Apollo Bunder, Mumbai, in December 1911. Its architectural style is Indo-Saracenic, incorporating elements of 16th-century Gujarati architecture, particularly in its intricate stone carvings and design details.
The question asks to identify the specific facility established by ISRO at Bharati station in Antarctica for receiving IRS (Indian Remote Sensing) data. This requires knowledge of ISRO's Antarctic ground stations and their functions.
C) AGEOS (Antarctic Ground Station for Earth Observation Satellites) is the correct answer. ISRO established AGEOS at Bharati station, Antarctica, for direct reception of data from its Indian Remote Sensing (IRS) satellites.
The question asks to identify the major hazard associated with nuclear power generation. We need to evaluate each option in the context of the risks and challenges posed by nuclear energy.
Correct Option: D) Storage and Dispersal. The safe and long-term storage and disposal of radioactive waste, which remains hazardous for millennia, is the most critical and unresolved challenge and hazard of nuclear power generation.
We are given a circle with diameter PQ and center O. A point R lies on the circle, and we know the angle \(\angle RPO\). We need to find \(\angle RQP\). We will use properties of circles and triangles, specifically that the radius of a circle is constant and the sum of angles in a triangle is 180°.
Identify key information:
Recognize properties of radii:
Since O is the center and P, Q, R are points on the circle, the segments OP, OQ, and OR are all radii of the circle. Therefore, their lengths are equal:
\[ OP = OQ = OR \]Analyze triangle \(\triangle POR\):
In \(\triangle POR\), we have \(OP = OR\) (both are radii). This means \(\triangle POR\) is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal.
So, \(\angle OPR = \angle ORP\).
Given \(\angle RPO = 39°\), which is the same as \(\angle OPR\), we have:
\[ \angle ORP = 39° \]Find \(\angle POQ\) or use the angle in a semicircle property:
Since PQ is the diameter, the angle subtended by the diameter at any point on the circumference is a right angle. Therefore, \(\angle PRQ = 90°\).
Calculate \(\angle RQP\):
Consider the triangle \(\triangle PQR\). The sum of angles in a triangle is 180°.
\[ \angle RPQ + \angle PQR + \angle QRP = 180° \]We know:
Substitute these values into the equation:
\[ 39° + \angle RQP + 90° = 180° \]\[ 129° + \angle RQP = 180° \]\[ \angle RQP = 180° - 129° \]\[ \angle RQP = 51° \]A) The measure of \(\angle RQP\) is 51°.
The problem asks us to identify the odd one out from a given list of four words. This is a classification-based question where we need to find a common characteristic among three words and identify the one that does not share that characteristic.
Correct Option: D) Delicious
Delicious is the odd one out because Salty, Bitter, and Sour are all fundamental taste sensations (basic tastes), while Delicious describes the overall pleasantness or enjoyable quality of food, which is a subjective evaluation rather than a specific taste category.
To find A's age after 9 years, we first need to determine A's current age. We can do this by using the given average ages. The total age of a group is the average age multiplied by the number of people in the group. We will use this principle to find the sum of ages for A, B, and C, and then for B and C. Subtracting the latter from the former will give us A's current age.
Correct Option: B) 25 years
To find the digit in the hundreds place of \(19!\), we need to calculate the value of \(19!\) and then identify the third digit from the right. A key observation for factorials is that they grow very quickly, and the presence of factors of 10 (which come from pairs of 2 and 5) determines the number of trailing zeros. If there are enough trailing zeros, the digits in the hundreds place (and tens and units place) will be zero.
Understand Factorial: The factorial of a non-negative integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). So, \(19! = 1 \times 2 \times 3 \times \dots \times 19\).
Count Trailing Zeros: Trailing zeros in a factorial are determined by the number of times 10 is a factor in its prime factorization. Since \(10 = 2 \times 5\), we need to count the number of pairs of 2 and 5 in the prime factorization of \(19!\). The number of factors of 5 is usually less than the number of factors of 2, so we only need to count the factors of 5.
The number of factors of 5 in \(19!\) can be calculated using Legendre's formula:
\[ N_5 = \left\lfloor \frac{19}{5} \right\rfloor + \left\lfloor \frac{19}{25} \right\rfloor + \dots \]For \(19!\):
So, the total number of factors of 5 is \(3\).
The number of factors of 2 is much higher:
\[ N_2 = \left\lfloor \frac{19}{2} \right\rfloor + \left\lfloor \frac{19}{4} \right\rfloor + \left\lfloor \frac{19}{8} \right\rfloor + \left\lfloor \frac{19}{16} \right\rfloor = 9 + 4 + 2 + 1 = 16 \]Since there are 3 factors of 5 and 16 factors of 2, there are 3 pairs of \((2 \times 5)\), meaning there are 3 factors of 10. Therefore, \(19!\) will end with 3 trailing zeros.
Determine the Hundreds Digit: If a number ends with 3 trailing zeros, it means its last three digits are 000. For example, a number like 123000 has 0 in the units place, 0 in the tens place, and 0 in the hundreds place.
Since \(19!\) ends with 3 zeros, its units digit is 0, its tens digit is 0, and its hundreds digit is also 0.
D) 0 — As calculated, \(19!\) has 3 trailing zeros, which means the units, tens, and hundreds digits are all 0.
If 19 \( \frac{2}{3} \) − 7 \( \frac{1}{4} \) = x + 2 \( \frac{1}{2} \), then what will be the value of x?
9 \( \frac{11}{12} \)
9 \( \frac{1}{12} \)
\( \frac{11}{12} \)
11 \( \frac{9}{12} \)
To find the value of \(x\), we need to simplify the given equation involving mixed fractions. The strategy is to convert all mixed fractions into improper fractions, perform the subtraction on the left side, and then isolate \(x\) by moving the term \(2 \frac{1}{2}\) to the left side.
Convert mixed fractions to improper fractions:
Rewrite the equation with improper fractions:
The given equation is \[19 \frac{2}{3} - 7 \frac{1}{4} = x + 2 \frac{1}{2}\]. Substituting the improper fractions:
\[ \frac{59}{3} - \frac{29}{4} = x + \frac{5}{2} \]Perform the subtraction on the left side:
To subtract \(\frac{59}{3}\) and \(\frac{29}{4}\), find a common denominator, which is 12.
\[ \frac{59}{3} - \frac{29}{4} = \frac{59 \times 4}{3 \times 4} - \frac{29 \times 3}{4 \times 3} = \frac{236}{12} - \frac{87}{12} = \frac{236 - 87}{12} = \frac{149}{12} \]Substitute the result back into the equation:
\[ \frac{149}{12} = x + \frac{5}{2} \]Isolate \(x\):
Subtract \(\frac{5}{2}\) from both sides of the equation:
\[ x = \frac{149}{12} - \frac{5}{2} \]Perform the subtraction to find \(x\):
Find a common denominator for \(\frac{149}{12}\) and \(\frac{5}{2}\), which is 12.
\[ x = \frac{149}{12} - \frac{5 \times 6}{2 \times 6} = \frac{149}{12} - \frac{30}{12} = \frac{149 - 30}{12} = \frac{119}{12} \]Convert the improper fraction back to a mixed fraction:
Divide 119 by 12:
\[ 119 \div 12 = 9 \text{ with a remainder of } 11 \]\[ x = 9 \frac{11}{12} \]A) \(9 \frac{11}{12}\) is the correct value for \(x\), as derived from the step-by-step calculation.
The problem asks us to identify the odd one out among four letter-clusters. This typically involves finding a pattern or rule that applies to three of the clusters and not to the fourth. We will analyze the positional values of the letters in the English alphabet for each cluster and look for a consistent mathematical relationship or sequence.
From the analysis, options A, C, and D follow the pattern where the difference between the 1st and 3rd letter's positional value is \(+4\), and the difference between the 2nd and 4th letter's positional value is \(-4\). Option B does not follow this pattern, as its differences are \((+4, +3)\).
B) NARD — This cluster is the odd one out because the difference between its second and fourth letters is \(+3\), while in the other clusters, this difference is \(-4\).