To find the value of A that satisfies the given trigonometric equation, we will use the double angle identity for sine, which is \(\sin(2A) = 2 \sin(A) \cos(A)\). We will then rearrange the equation and solve for A.
The given equation is:
\[ \sin(2A) = 2 \sin(A) \]Apply the double angle identity for sine, which states \(\sin(2A) = 2 \sin(A) \cos(A)\). Substitute this into the equation:
\[ 2 \sin(A) \cos(A) = 2 \sin(A) \]Rearrange the equation to bring all terms to one side:
\[ 2 \sin(A) \cos(A) - 2 \sin(A) = 0 \]Factor out the common term, \(2 \sin(A)\):
\[ 2 \sin(A) (\cos(A) - 1) = 0 \]For the product of two terms to be zero, at least one of the terms must be zero. So, we have two possible cases:
Case 1: \(2 \sin(A) = 0\)
\[ \sin(A) = 0 \]The general solution for \(\sin(A) = 0\) is \(A = n\pi\), where \(n\) is an integer. For the options given in degrees, the principal value is \(A = 0^\circ\).
Case 2: \(\cos(A) - 1 = 0\)
\[ \cos(A) = 1 \]The general solution for \(\cos(A) = 1\) is \(A = 2n\pi\), where \(n\) is an integer. For the options given in degrees, the principal value is \(A = 0^\circ\).
Both cases lead to \(A = 0^\circ\) as a possible solution. Comparing this with the given options, \(0^\circ\) is present.
Correct Option: C) 0°
This is a syllogism problem. We need to determine which conclusions logically follow from the given statements. We can use Venn diagrams or direct logical deduction to solve this.
Correct Option: B) Only II
If \(\frac{1}{5\sqrt{2}-7}\) = a\(\sqrt{2}\) + b, then the value of a and b are:
The question gives an expression containing a surd in the denominator and asks us to express it in the form \(a\sqrt{2}+b\). We need to rationalise the denominator using its conjugate and then compare the resulting expression with the given form to determine the values of a and b.
Identify the given expression:
\[ \frac{1}{5\sqrt{2}-7}=a\sqrt{2}+b \]We need to find the values of a and b.
Recall the rationalisation method:
To remove a surd from the denominator, multiply both the numerator and denominator by the conjugate of the denominator.
The denominator is:
\[ 5\sqrt{2}-7 \]Its conjugate is:
\[ 5\sqrt{2}+7 \]Rationalise the denominator:
Multiply the numerator and denominator by the conjugate:
\[ \frac{1}{5\sqrt{2}-7} \times \frac{5\sqrt{2}+7}{5\sqrt{2}+7} \] \[ =\frac{5\sqrt{2}+7} {(5\sqrt{2}-7)(5\sqrt{2}+7)} \]Apply the difference of squares identity:
Recall the identity:
\[ (x-y)(x+y)=x^2-y^2 \]Applying this identity to the denominator:
\[ (5\sqrt{2})^2-7^2 \] \[ =25\times2-49 \] \[ =50-49 \] \[ =1 \]Therefore, the expression becomes:
\[ \frac{5\sqrt{2}+7}{1} \] \[ =5\sqrt{2}+7 \]Compare with the given form:
The question states:
\[ \frac{1}{5\sqrt{2}-7}=a\sqrt{2}+b \]After rationalisation:
\[ 5\sqrt{2}+7=a\sqrt{2}+b \]Comparing the coefficients:
\[ a=5 \] \[ b=7 \]Therefore, the values of a and b are 5 and 7, respectively.
Correct Option: B) 5, 7 — Rationalising the denominator:
\[ \frac{1}{5\sqrt{2}-7} =\frac{5\sqrt{2}+7}{50-49} \] \[ =5\sqrt{2}+7 \]
Comparing this with \(a\sqrt{2}+b\), we obtain:
\[ a=5,\quad b=7 \]
Hence, Option B is the correct answer.
A) -7, 5: This is incorrect because rationalisation gives a positive coefficient of 5 for \(\sqrt{2}\) and a constant term of 7.
C) 7, 5: This is incorrect because it interchanges the coefficient of \(\sqrt{2}\) and the constant term. The correct values are a = 5 and b = 7.
D) 5, -7: This is incorrect because the constant term after rationalisation is +7, not -7.
To find the ratio of the volume of a cylinder to that of a cone, we need to recall the formulas for the volumes of these two geometric shapes. The problem states that both have the same base radius and the same height. We will use these conditions to simplify the ratio.
Correct Option: A) 3 : 1
This problem involves calculating the future population based on an annual percentage increase. This is a compound growth problem, similar to compound interest. We can calculate the population after each year sequentially or use the compound interest formula.
Increase in 1st year = \(50,000 \times 4\% = 50,000 \times 0.04 = 2,000\)
Population after 1st year = \(50,000 + 2,000 = 52,000\)
The increase for the second year is calculated on the population at the end of the first year.
Increase in 2nd year = \(52,000 \times 4\% = 52,000 \times 0.04 = 2,080\)
Population after 2 years = \(52,000 + 2,080 = 54,080\)
The formula for population after \(n\) years with an annual growth rate \(r\) is:
\[ P_n = P \left(1 + \frac{r}{100}\right)^n \]Substitute the given values:
\[ P_2 = 50,000 \left(1 + \frac{4}{100}\right)^2 \] \[ P_2 = 50,000 (1 + 0.04)^2 \] \[ P_2 = 50,000 (1.04)^2 \] \[ P_2 = 50,000 \times 1.0816 \] \[ P_2 = 54,080 \]Both methods yield the same result.
B) 54,080 is the correct population after 2 years, as calculated by both sequential and compound growth methods.
The United Nations General Assembly commemorates 21st June as:
International Education Day
International Babies Day
International Day of Yoga
International Health Day
The question asks to identify the specific international day commemorated by the United Nations General Assembly on June 21st. This requires knowledge of significant dates and their associated UN observances.
Correct Option: C) International Day of Yoga
The question asks to identify an antivirus software from the given options. Antivirus software is designed to detect, prevent, and remove malicious software like viruses, worms, and Trojans from computer systems. We need to evaluate each option based on its primary function.
Correct Option: C) Norton — Norton is a widely recognized brand for antivirus and cybersecurity software, offering protection against various types of malware.
The question asks to identify the hill range that has the Indian Desert (Thar Desert) to its western margin. This requires knowledge of the geographical location of the Indian Desert and the major hill ranges in India.
Correct Option: C) The Aravalli Range is an ancient mountain range in Western India. It runs approximately 692 km in a southwest direction, starting near Delhi, passing through southern Haryana and Rajasthan, and ending in Ahmedabad, Gujarat. The Thar Desert lies to the west of the Aravalli Range, making the Aravallis the eastern boundary of the desert.
The question asks to identify the river in which Majuli, India's largest river island, is located. This is a factual question related to Indian geography.
Correct Option: D) Majuli, the largest river island in India and one of the largest in the world, is located in the Brahmaputra River in the state of Assam.
The question asks to identify which of the given groups did not attend the First Round Table Conference in London. This requires knowledge of the participants and non-participants of this historical event.
D) Indian National Congress — The Indian National Congress boycotted the First Round Table Conference as part of its Civil Disobedience Movement, demanding complete independence or Purna Swaraj.