If sec θ − cosec θ = 0, then the value of 2 sin 2θ − 3 cos 2θ is:
The problem asks us to find the value of an expression involving trigonometric functions, given a condition relating secant and cosecant. We will first use the given condition to find the value of \(\theta\), and then substitute this value into the expression to be evaluated.
Analyze the given condition:
We are given the condition: \(\sec \theta - \text{cosec} \theta = 0\)
Rearranging this, we get:
\[ \sec \theta = \text{cosec} \theta \]Convert to sine and cosine:
Recall that \(\sec \theta = \frac{1}{\cos \theta}\) and \(\text{cosec} \theta = \frac{1}{\sin \theta}\).
Substituting these into the equation:
\[ \frac{1}{\cos \theta} = \frac{1}{\sin \theta} \]Solve for \(\theta\):
From the equation above, we can cross-multiply:
\[ \sin \theta = \cos \theta \]Find the value of \(\theta\):
Divide both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)):
\[ \frac{\sin \theta}{\cos \theta} = 1 \]\[ \tan \theta = 1 \]Determine \(\theta\):
The principal value of \(\theta\) for which \(\tan \theta = 1\) is \(\theta = 45^\circ\) or \(\theta = \frac{\pi}{4}\) radians. For this value, \(\sin \theta\) and \(\cos \theta\) are both \(\frac{1}{\sqrt{2}}\), so they are non-zero, validating our division by \(\cos \theta\).
Evaluate the expression:
Now we need to find the value of \(2 \sin 2\theta - 3 \cos 2\theta\).
Substitute \(\theta = 45^\circ\):
\[ 2 \sin (2 \times 45^\circ) - 3 \cos (2 \times 45^\circ) \]\[ 2 \sin (90^\circ) - 3 \cos (90^\circ) \]Use standard trigonometric values:
We know that \(\sin 90^\circ = 1\) and \(\cos 90^\circ = 0\).
Substitute these values:
\[ 2(1) - 3(0) \]\[ 2 - 0 \]\[ 2 \]B) The value of the expression is \(2\).
The question asks about the primary focus of India's first Five Year Plan. To answer this, one needs to recall the historical context and stated objectives of the plan.
B) Development of primary sector with focus on agriculture. The First Five Year Plan primarily focused on the rehabilitation of the economy from the effects of partition and World War II, with a strong emphasis on agriculture, irrigation, and power generation to achieve food self-sufficiency and improve the living standards of the rural population. It aimed to address the immediate challenges of food scarcity and inflation.
The question asks about the subject matter of the paintings found in the Ajanta Caves. To answer this, one needs to recall the primary themes depicted in the art of these historically significant Buddhist cave complexes.
C) Jataka tales — The Ajanta Caves are famous for their murals depicting the Jataka tales, which narrate the previous lives of the Buddha. These stories are central to Buddhist art and teachings.
The question asks for the ancient name of Iran. This is a factual question related to historical geography and nomenclature. We need to identify the historical name commonly associated with the region now known as Iran.
Correct Option: D) Persia is the ancient name for the country now known as Iran. The Persian Empire was one of the largest empires of the ancient world, originating from the region of Persis (modern-day Fars province in Iran).
If x = 12 and y = 7, then the value of \(\frac{x^2+y^2-xy}{x^3+y^3}\) is:
\( \frac{2}{19} \)
\( \frac{1}{19} \)
\( \frac{1}{2} \)
\( \frac{1}{5} \)
The problem asks us to evaluate an algebraic expression given specific values for variables \(x\) and \(y\). We can solve this by either directly substituting the values into the expression or by simplifying the expression first using algebraic identities and then substituting the values. Simplifying the expression often leads to fewer calculations and reduces the chance of errors.
Identify the given expression:
\[ \frac{x^2+y^2-xy}{x^3+y^3} \]Recall the algebraic identity for the sum of cubes:
\[ a^3+b^3 = (a+b)(a^2-ab+b^2) \]Apply this identity to the denominator of the given expression. Let \(a=x\) and \(b=y\):
\[ x^3+y^3 = (x+y)(x^2-xy+y^2) \]Substitute this expanded form of the denominator back into the original expression:
\[ \frac{x^2+y^2-xy}{(x+y)(x^2-xy+y^2)} \]Notice that the term \((x^2-xy+y^2)\) appears in both the numerator and the denominator. We can cancel this common term (assuming \(x^2-xy+y^2 \neq 0\)).
The expression simplifies to:
\[ \frac{1}{x+y} \]Now, substitute the given values \(x=12\) and \(y=7\) into the simplified expression:
\[ \frac{1}{12+7} \]Perform the addition in the denominator:
\[ \frac{1}{19} \]B) \( \frac{1}{19} \) is the correct value obtained after simplifying the expression and substituting the given values of \(x\) and \(y\).
The problem asks us to find the rate of interest given the principal, time, and the difference between compound interest (CI) and simple interest (SI). We can use the formula for the difference between CI and SI for 2 years, or calculate CI and SI separately and then find the difference.
Identify the given values:
Recall the formula for the difference between CI and SI for 2 years:
For 2 years, the difference between compound interest and simple interest is given by:
\[ CI - SI = P \left( \frac{R}{100} \right)^2 \]Substitute the given values into the formula:
We have \(P = 24000\), \(CI - SI = 60\). Let \(R\) be the rate of interest.
\[ 60 = 24000 \left( \frac{R}{100} \right)^2 \]Solve for \(R\):
First, isolate the term \(\left( \frac{R}{100} \right)^2\):
\[ \left( \frac{R}{100} \right)^2 = \frac{60}{24000} \]\[ \left( \frac{R}{100} \right)^2 = \frac{1}{400} \]Take the square root of both sides:
\[ \frac{R}{100} = \sqrt{\frac{1}{400}} \]\[ \frac{R}{100} = \frac{1}{20} \]Calculate \(R\):
\[ R = \frac{100}{20} \]\[ R = 5 \]Thus, the rate of interest is 5%.
D) 5% is the correct rate of interest as calculated using the formula for the difference between compound interest and simple interest for 2 years.
To find the total surface area of the four walls of a room, we need to calculate the lateral surface area of the cuboid. The formula for the lateral surface area of a cuboid is \(2 \times (\text{length} + \text{breadth}) \times \text{height}\).
The total surface area of the four walls of a room is given by the perimeter of the base multiplied by the height.
\[ \text{Area} = 2 \times (L + B) \times H \]C) 3150 m² is the correct answer, as calculated above.
To find the largest angle of a parallelogram, we need to use the properties of parallelograms: opposite angles are equal, and consecutive angles are supplementary (sum up to 180°). We will set up an equation based on the given relationship between one angle and the smallest angle, solve for the smallest angle, and then determine the largest angle.
Let the smallest angle of the parallelogram be \(x\).
According to the problem statement, one angle is 28° less than thrice the smallest angle. So, this angle can be expressed as \(3x - 28^\circ\).
In a parallelogram, consecutive angles are supplementary. This means that if \(x\) is the smallest angle, then the angle adjacent to it must be \(180^\circ - x\). The angle \(3x - 28^\circ\) must be one of the angles of the parallelogram. Since \(x\) is the smallest angle, \(3x - 28^\circ\) must be the angle adjacent to it (or equal to it if \(x\) is not the smallest, but we will confirm this later).
So, we can set up the equation:
\[ x + (3x - 28^\circ) = 180^\circ \]Solve the equation for \(x\):
\[ 4x - 28^\circ = 180^\circ \]\[ 4x = 180^\circ + 28^\circ \]\[ 4x = 208^\circ \]\[ x = \frac{208^\circ}{4} \]\[ x = 52^\circ \]So, the smallest angle of the parallelogram is \(52^\circ\).
Now, calculate the other angle using the expression \(3x - 28^\circ\):
\[ 3(52^\circ) - 28^\circ \]\[ 156^\circ - 28^\circ \]\[ 128^\circ \]The two distinct angles of the parallelogram are \(52^\circ\) and \(128^\circ\). We can verify that their sum is \(52^\circ + 128^\circ = 180^\circ\), which is correct for consecutive angles in a parallelogram.
The largest angle of the parallelogram is the larger of these two angles, which is \(128^\circ\).
D) 128° is the largest angle of the parallelogram, as calculated.
The problem asks us to decode a pattern from a given word transformation and then apply that pattern to a new word. We need to identify how the letters in "FURNITURE" are rearranged to form "RUFTINERU".
B) UOFATNSNI is the correct option, as it matches the pattern derived from the example.
This is a coding-decoding question where letters are assigned numerical values. We need to identify the pattern or rule used to assign these values and then apply it to find the value of "BAT".
The standard alphabetical position of Z is 26. This suggests that each letter is assigned its serial number in the English alphabet (A=1, B=2, C=3, ..., Z=26).
If we sum these values, we get: \(1 + 3 + 20 = 24\). This matches the given value for ACT.
Now, we apply the same rule to find the value of BAT:
Summing these values:
\[ \text{BAT} = \text{B} + \text{A} + \text{T} = 2 + 1 + 20 = 23 \]Correct Option: A) 23