We need to find the original sum. We are given the final sum after two successive percentage deductions. We can solve this by working backward from the final sum or by setting up an equation where the original sum is an unknown variable.
After deducting 10%, the remaining sum will be \(100\% - 10\% = 90\%\) of the original sum.
\[ \text{Sum after 1st deduction} = X \times \left(1 - \frac{10}{100}\right) = X \times \frac{90}{100} = 0.9X \]The remainder after the first deduction is \(0.9X\). Now, 20% is deducted from this remainder. So, the sum left will be \(100\% - 20\% = 80\%\) of this remainder.
\[ \text{Sum after 2nd deduction} = (0.9X) \times \left(1 - \frac{20}{100}\right) = (0.9X) \times \frac{80}{100} = 0.9X \times 0.8 \]We are given that the sum left after both deductions is ₹3,600.
\[ 0.9X \times 0.8 = 3600 \]\[ 0.72X = 3600 \]The original sum is ₹5,000.
C) 5,000 — This is the correct original sum calculated by reversing the percentage deductions.
The problem asks us to identify the odd one out from a given list of four words. This is a classification type of question where we need to find a common characteristic among three words and identify the one that does not share that characteristic.
Correct Option: B) Doctor is the odd one out because a Goldsmith, Blacksmith, and Carpenter are all tradesmen who work with raw materials to create physical objects, whereas a Doctor provides medical services.
The problem is an analogy question where we need to find the relationship between the first pair of numbers and apply the same relationship to the third number to find the fourth. We need to identify the mathematical operation or pattern connecting 36 and 216.
First, let's look at the numbers 36 and 216. We can try to express them as powers of some base numbers.
So, the relationship between the first and second numbers is that the second number is the cube of the base of which the first number is the square. In other words, if the first number is \(n^2\), the second number is \(n^3\).
The third number is 81. We need to find its base number, \(n\), such that \(n^2 = 81\).
So, for the third number, \(n = 9\).
Following the established pattern, the fourth number should be \(n^3\), where \(n=9\).
Thus, the missing number is 729.
B) 729 is the correct answer because it follows the pattern where the second number is the cube of the base whose square is the first number. For 36 (\(6^2\)), the related number is 216 (\(6^3\)). For 81 (\(9^2\)), the related number is 729 (\(9^3\)).
The question asks about the time difference between the Vikram Samvat calendar and the Gregorian calendar. To answer this, we need to know the starting year of the Vikram Samvat calendar relative to the Common Era (CE) or Gregorian calendar.
C) 56.7. The Vikram Samvat calendar is generally considered to be 57 years ahead of the Gregorian calendar. For example, if the current Gregorian year is 2024, the Vikram Samvat year would be \(2024 + 57 = 2081\). The option 56.7 accounts for the fact that the Vikram Samvat new year (Ugadi/Gudi Padwa) falls in March/April, meaning for the first few months of the Gregorian year, the Vikram Samvat year is effectively 56 years ahead of the current Gregorian year, and then becomes 57 years ahead once its new year begins. However, 57 is the full year difference. Among the given options, 56.7 is the closest and often used in competitive exams to represent this difference, especially when considering the current year's transition.
To find the value of \( x^{3} + y^{3} + z^{3} - 3xyz \), we need to use the algebraic identity relating it to \( (x+y+z) \) and \( (xy+yz+zx) \). The identity is \( x^{3} + y^{3} + z^{3} - 3xyz = (x+y+z)(x^2+y^2+z^2 - xy - yz - zx) \). We are given \( x+y+z \) and \( xy+yz+zx \). We need to find \( x^2+y^2+z^2 \) first.
Recall the algebraic identity for \( (x+y+z)^2 \):
\[ (x+y+z)^2 = x^2+y^2+z^2 + 2(xy+yz+zx) \]We are given \( x+y+z = 16 \) and \( xy+yz+zx = 78 \). Substitute these values into the identity:
\[ (16)^2 = x^2+y^2+z^2 + 2(78) \] \[ 256 = x^2+y^2+z^2 + 156 \]Solve for \( x^2+y^2+z^2 \):
\[ x^2+y^2+z^2 = 256 - 156 \] \[ x^2+y^2+z^2 = 100 \]Now, recall the main identity for \( x^{3} + y^{3} + z^{3} - 3xyz \):
\[ x^{3} + y^{3} + z^{3} - 3xyz = (x+y+z)(x^2+y^2+z^2 - (xy+yz+zx)) \]Substitute the known values: \( x+y+z = 16 \), \( x^2+y^2+z^2 = 100 \), and \( xy+yz+zx = 78 \):
\[ x^{3} + y^{3} + z^{3} - 3xyz = (16)(100 - 78) \] \[ x^{3} + y^{3} + z^{3} - 3xyz = (16)(22) \]Calculate the final product:
\[ 16 \times 22 = 352 \]Correct Option: A) 352
The question asks for the launch year of the Paryavaran Vahini Scheme by the Ministry of Environment and Forest, which aimed to encourage public participation in environmental protection. This requires knowledge of specific environmental initiatives and their timelines in India.
Correct Option: C) July 1992
The Paryavaran Vahini Scheme was launched in July 1992 by the Ministry of Environment and Forest. Its primary objective was to foster people's participation in environmental protection and management, particularly at the grassroots level. This scheme aimed to create environmental awareness and encourage local communities to report environmental violations.
If \(\sqrt{x}\) + \(\frac{1}{\sqrt{x}}\) = 3, then find the value of x² + \( \frac{1}{x^2} \).
The problem requires us to find the value of \(x^2 + \frac{1}{x^2}\) given the equation \(\sqrt{x} + \frac{1}{\sqrt{x}} = 3\). We can solve this by squaring the given equation twice to progressively raise the powers of \(x\).
We are given:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 3 \]Squaring both sides:
\[ \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^2 = 3^2 \]Using the identity \((a+b)^2 = a^2 + b^2 + 2ab\):
\[ (\sqrt{x})^2 + \left(\frac{1}{\sqrt{x}}\right)^2 + 2 \cdot \sqrt{x} \cdot \frac{1}{\sqrt{x}} = 9 \]\[ x + \frac{1}{x} + 2 \cdot 1 = 9 \]\[ x + \frac{1}{x} + 2 = 9 \]Subtract 2 from both sides:
\[ x + \frac{1}{x} = 9 - 2 \]\[ x + \frac{1}{x} = 7 \]Now we need to find \(x^2 + \frac{1}{x^2}\). Square both sides of the equation \(x + \frac{1}{x} = 7\):
\[ \left( x + \frac{1}{x} \right)^2 = 7^2 \]Using the identity \((a+b)^2 = a^2 + b^2 + 2ab\):
\[ x^2 + \left(\frac{1}{x}\right)^2 + 2 \cdot x \cdot \frac{1}{x} = 49 \]\[ x^2 + \frac{1}{x^2} + 2 \cdot 1 = 49 \]\[ x^2 + \frac{1}{x^2} + 2 = 49 \]Subtract 2 from both sides:
\[ x^2 + \frac{1}{x^2} = 49 - 2 \]\[ x^2 + \frac{1}{x^2} = 47 \]B) 47 is the correct answer, as derived from the step-by-step calculation.
Kunal was asked to find \( \frac{5}{6} \) times of a number. He multiplied it by \( \frac{6}{5} \). As a result, he got an answer which was more than the correct answer by 572. The number was:
Let the unknown number be \(x\). We need to set up an equation based on the given information: the difference between the incorrect result (multiplying by \(\frac{6}{5}\)) and the correct result (multiplying by \(\frac{5}{6}\)) is 572. Solving this equation for \(x\) will give us the number.
Let the number be \(x\).
The correct answer Kunal should have obtained is \( \frac{5}{6} \) times the number:
\[ \text{Correct Answer} = \frac{5}{6}x \]The answer Kunal actually obtained is \( \frac{6}{5} \) times the number:
\[ \text{Obtained Answer} = \frac{6}{5}x \]According to the problem, the obtained answer was 572 more than the correct answer. So, we can write the equation:
\[ \text{Obtained Answer} - \text{Correct Answer} = 572 \] \[ \frac{6}{5}x - \frac{5}{6}x = 572 \]To solve for \(x\), find a common denominator for the fractions, which is 30:
\[ \frac{6 \times 6}{5 \times 6}x - \frac{5 \times 5}{6 \times 5}x = 572 \] \[ \frac{36}{30}x - \frac{25}{30}x = 572 \]Combine the terms:
\[ \frac{36 - 25}{30}x = 572 \] \[ \frac{11}{30}x = 572 \]Isolate \(x\) by multiplying both sides by \(\frac{30}{11}\):
\[ x = 572 \times \frac{30}{11} \]Perform the division and multiplication:
\[ x = (572 \div 11) \times 30 \] \[ x = 52 \times 30 \] \[ x = 1560 \]Correct Option: C) 1560
This problem involves calculating speeds and time taken for different modes of transport. We are given the total distance travelled by steamer, train, and horse, and the total time taken. We are also given relationships between the speeds of the train, horse, and steamer. The goal is to find the speed of the train. We will set up equations based on the given information and solve for the unknown speeds.
Define variables:
Formulate speed relationships:
Express other speeds in terms of \(S_t\):
Calculate time taken for each part of the journey:
Recall that \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).
Set up the total time equation:
Total time = 13 h 30 min = \(13.5\) hours.
\[ T_s + T_t + T_h = 13.5 \] \[ \frac{180}{S_t} + \frac{450}{S_t} + \frac{180}{S_t} = 13.5 \]Solve for \(S_t\):
\[ \frac{180 + 450 + 180}{S_t} = 13.5 \] \[ \frac{810}{S_t} = 13.5 \] \[ S_t = \frac{810}{13.5} \] \[ S_t = \frac{810}{\frac{27}{2}} = \frac{810 \times 2}{27} \] \[ S_t = \frac{1620}{27} \] \[ S_t = 60 \text{ km/h} \]Correct Option: C) 60 km/h
This problem involves compound interest. We are given the principal amount, the amount after a certain period, and that the interest is compounded at a constant rate. We need to find the amount after a longer period. The key insight here is to recognize the multiplicative factor by which the amount grows over a fixed period. If an amount becomes \(x\) times its initial value in \(T\) years, it will become \(x^n\) times its initial value in \(nT\) years.
Understand the Compound Interest Formula:
The formula for compound interest is given by:
where \(A\) is the amount, \(P\) is the principal, \(R\) is the annual interest rate, and \(T\) is the time in years.
Calculate the growth factor for the first 4 years:
Given: Principal \(P = \text{₹}4,800\), Amount \(A_1 = \text{₹}6,000\), Time \(T_1 = 4\) years.
Let the growth factor for 4 years be \(x\). So, \(A_1 = P \times x\).
\[ x = \frac{A_1}{P} = \frac{6000}{4800} = \frac{60}{48} = \frac{5}{4} \]This means that in 4 years, the amount becomes \(\frac{5}{4}\) times the principal.
Determine the total time and number of growth cycles:
We need to find the amount after 12 years. The total time is \(T_2 = 12\) years.
Since the growth factor \(x\) applies for every 4-year period, we can find how many 4-year periods are there in 12 years:
\[ \text{Number of cycles} = \frac{12 \text{ years}}{4 \text{ years/cycle}} = 3 \text{ cycles} \]Calculate the final amount:
The amount after 12 years will be the initial principal multiplied by the growth factor for 4 years, raised to the power of the number of cycles.
Let \(A_2\) be the amount after 12 years.
\[ A_2 = P \times x^3 \]\[ A_2 = 4800 \times \left(\frac{5}{4}\right)^3 \]\[ A_2 = 4800 \times \frac{5^3}{4^3} \]\[ A_2 = 4800 \times \frac{125}{64} \]Now, perform the multiplication:
\[ A_2 = \frac{4800 \times 125}{64} \]\[ A_2 = \frac{600000}{64} \]To simplify, divide both numerator and denominator by common factors. For example, divide by 8:
\[ A_2 = \frac{75000}{8} \]Now, divide by 8 again:
\[ A_2 = 9375 \]So, the amount after 12 years will be ₹9,375.
Correct Option: D) ₹9,375