The question asks about the authority responsible for the final approval of Five-Year Plans in India. This requires knowledge of the institutional framework for economic planning that existed in India.
Correct Option: A) National Development Council
The problem asks us to find the value of 'x' given a percentage relationship. We need to translate the given statement "x% of 10.8 is equal to 32.4" into a mathematical equation and then solve for x.
Correct Option: D) 300
The question asks to identify the individual responsible for assassinating Michael O'Dwyer, who was the Lieutenant Governor of Punjab during the Jallianwala Bagh Massacre. This requires knowledge of key figures and events in India's freedom struggle.
Correct Option: D) Sardar Udham Singh — Sardar Udham Singh assassinated Michael O'Dwyer in London on March 13, 1940, to avenge the Jallianwala Bagh Massacre. He was present at the massacre and was deeply affected by it.
The question asks to identify the plateau where the city of Hyderabad is located. This requires knowledge of the major physiographic divisions of India, specifically the location of prominent plateaus and key cities within them.
C) Deccan. Hyderabad is located in the Deccan Plateau. The Deccan Plateau is a large plateau in peninsular India, and the city of Hyderabad falls within its northern part, specifically in the Telangana Plateau region, which is a part of the larger Deccan Plateau.
Simplify \(\left(\frac{2+\sqrt{3}}{2-\sqrt{3}}\right)^2\).
\( \frac{16}{3} \)
97 − 56\(\sqrt{3}\)
97 + 56\(\sqrt{3}\)
48
To simplify the given expression, we first rationalize the denominator of the fraction inside the parenthesis. Then, we square the resulting expression.
The given expression is \(\left(\frac{2+\sqrt{3}}{2-\sqrt{3}}\right)^2\). Let's first simplify the term inside the parenthesis: \(\frac{2+\sqrt{3}}{2-\sqrt{3}}\).
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is \(2+\sqrt{3}\).
\[ \frac{2+\sqrt{3}}{2-\sqrt{3}} = \frac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})} \]Using the identity \((a-b)(a+b) = a^2 - b^2\) for the denominator and \((a+b)^2 = a^2 + 2ab + b^2\) for the numerator:
Numerator: \((2+\sqrt{3})^2 = 2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}\)
Denominator: \((2-\sqrt{3})(2+\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1\)
So, the simplified fraction is:
\[ \frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3} \]Now, we need to square this result:
\[ (7 + 4\sqrt{3})^2 \]Using the identity \((a+b)^2 = a^2 + 2ab + b^2\) where \(a=7\) and \(b=4\sqrt{3}\):
\[ (7 + 4\sqrt{3})^2 = 7^2 + 2(7)(4\sqrt{3}) + (4\sqrt{3})^2 \] \[ = 49 + 56\sqrt{3} + (4^2 \times (\sqrt{3})^2) \] \[ = 49 + 56\sqrt{3} + (16 \times 3) \] \[ = 49 + 56\sqrt{3} + 48 \] \[ = (49 + 48) + 56\sqrt{3} \] \[ = 97 + 56\sqrt{3} \]Correct Option: C) \(97 + 56\sqrt{3}\)
The question asks to identify the main river to which the Mechi River, a trans-boundary river between India and Nepal, is a tributary. This requires knowledge of the river systems in the region, particularly the tributaries of major rivers flowing through India and Nepal.
Correct Option: C) Mahananda — The Mechi River is a significant tributary of the Mahananda River. The Mahananda itself is a tributary of the Ganga (or Padma in Bangladesh).
What are the values of x in the following equation?
\( 3^{2x+1} - 3^{x} = 3^{x+3} - 3^{2} \)
To solve this exponential equation, we will use the properties of exponents to simplify the terms, group similar terms, and then make a substitution to convert the equation into a quadratic form. Finally, we will solve the quadratic equation for the substituted variable and then find the values of \(x\).
The given equation is: \( 3^{2x+1} - 3^{x} = 3^{x+3} - 3^{2} \)
Using the property \(a^{m+n} = a^m \cdot a^n\), we can rewrite the terms:
\[ 3^{2x} \cdot 3^1 - 3^x = 3^x \cdot 3^3 - 3^2 \] \[ 3 \cdot (3^x)^2 - 3^x = 27 \cdot 3^x - 9 \]Bring all terms to the left side to form a standard quadratic-like equation:
\[ 3 \cdot (3^x)^2 - 3^x - 27 \cdot 3^x + 9 = 0 \] \[ 3 \cdot (3^x)^2 - 28 \cdot 3^x + 9 = 0 \]Let \(y = 3^x\). Since \(3^x\) must be positive, \(y > 0\). Substitute \(y\) into the equation:
\[ 3y^2 - 28y + 9 = 0 \]We can solve this quadratic equation by factoring or using the quadratic formula. By factoring, we look for two numbers that multiply to \(3 \times 9 = 27\) and add up to \(-28\). These numbers are \(-27\) and \(-1\).
\[ 3y^2 - 27y - y + 9 = 0 \] \[ 3y(y - 9) - 1(y - 9) = 0 \] \[ (3y - 1)(y - 9) = 0 \]This gives two possible values for \(y\):
\[ 3y - 1 = 0 \implies 3y = 1 \implies y = \frac{1}{3} \] \[ y - 9 = 0 \implies y = 9 \]Case 1: \(y = \frac{1}{3}\)
\[ 3^x = \frac{1}{3} \] \[ 3^x = 3^{-1} \] \[ x = -1 \]Case 2: \(y = 9\)
\[ 3^x = 9 \] \[ 3^x = 3^2 \] \[ x = 2 \]For \(x = -1\):
\[ 3^{2(-1)+1} - 3^{-1} = 3^{-2+1} - 3^{-1} = 3^{-1} - 3^{-1} = 0 \] \[ 3^{-1+3} - 3^2 = 3^2 - 3^2 = 0 \]So, \(0 = 0\), which is true.
For \(x = 2\):
\[ 3^{2(2)+1} - 3^2 = 3^{4+1} - 9 = 3^5 - 9 = 243 - 9 = 234 \] \[ 3^{2+3} - 3^2 = 3^5 - 9 = 243 - 9 = 234 \]So, \(234 = 234\), which is true.
The values of \(x\) are \(2\) and \(-1\).
Correct Option: C) 2, −1
The problem asks us to identify the odd one out from a set of four letter-clusters. This typically involves finding a common pattern or rule that applies to three of the clusters, while the fourth one deviates from that rule. We will analyze the positional values of the letters in the English alphabet for each cluster.
C) HJK is the odd one out because the difference between its first two letters (H to J) is \(+2\), whereas for options A, B, and D, the difference between the first two letters is \(+3\).
Select the option that can replace the question mark (?) in the following series.
\( \frac{K}{11} \), \( \frac{15}{O} \),\( \frac{S}{19} \), \( \frac{23}{W} \), \( \frac{A}{1} \), \( \frac{?}{?} \)
\( \frac{5}{E} \)
\( \frac{6}{E} \)
\( \frac{E}{5} \)
\( \frac{5}{F} \)
The problem presents a series of fractions where each term consists of an English alphabet letter and a number. We need to identify the pattern for both the letters and the numbers separately to find the missing term.
The letters in the numerators/denominators are: K, O, S, W, A, ?
Let's find their positions in the English alphabet:
Observe the pattern of the letter positions:
Following this pattern, the next letter should be 4 positions after A (1): \(1 + 4 = 5\). The 5th letter in the alphabet is E.
The numbers in the denominators/numerators are: 11, 15, 19, 23, 1, ?
Observe the pattern of the numbers:
The terms are \( \frac{\text{Letter}}{\text{Number}} \) or \( \frac{\text{Number}}{\text{Letter}} \).
It is clear that for each term, the letter's alphabetical position is equal to the number. The position of the letter and number alternates between numerator and denominator.
For the next term, the letter should be E (5th letter) and the number should be 5. Since the previous term \( \frac{A}{1} \) had the letter in the numerator, the next term should have the number in the numerator and the letter in the denominator, following the alternating pattern.
So, the next term should be \( \frac{5}{E} \).
A) \( \frac{5}{E} \) is the correct option because the letter sequence advances by 4 positions (K, O, S, W, A, E) and the number sequence matches the alphabetical position of the letter. The position of the letter and number also alternates between numerator and denominator. The next letter after A (1st) is E (5th). The corresponding number is 5. Since the previous term was \( \frac{\text{Letter}}{\text{Number}} \), the next term should be \( \frac{\text{Number}}{\text{Letter}} \).
To find when the traffic lights will change simultaneously again, we need to find the least common multiple (LCM) of the time intervals at which each light changes. The LCM will give us the total time after which all lights will synchronize again. We will then add this time to the initial simultaneous change time.
The next time they will change simultaneously is 9:37:12 am.
Correct Option: A) 9:37:12 am