Find the value of cos²θ + 2sin²θ? Show that 1 + cos x = 2cos²(x/2) for x ∈ [0, 1]. Evaluate ∫(x² + 1) log(x²) dx. A committee of 4 students is selected at random from a group consi... Find the value of cos²θ + 2sin²θ? Show that 1 + cos x = 2cos²(x/2) for x ∈ [0, 1]. Evaluate ∫(x² + 1) log(x²) dx. A committee of 4 students is selected at random from a group consisting of 5 boys and 4 girls. If each student is equally likely to be chosen, then calculate the probability that there are exactly 2 girls in the committee. Three persons A, B and C apply for the job of Manager in a private company. Chances for their selection B and C are in ratio 1:2:4. The probabilities that A, B and C are chosen are 0.5, 0.3 and 0.2 respectively. If the probables does not take place, find the probability that it is due to the appointment of C. Evaluate 1 - 3sin x / √(1 + x²). Solve the differential equation (x + y)² dy - (2y - 2x) dx = 0. For any two vectors a and b, show that (1 + |a|²)(1 + |b|²) ≥ |a + b|². Solve minimize z = 5x + 7y subject to constraints 2x + y ≥ 8, x + 2y ≥ 10, y ≥ 0. Solve the following system of equations by matrix method when x and y are not equal to zero.
Understand the Problem
The questions in the image encompass various mathematical problems, including finding values, evaluating integrals, solving differential equations, and applying knowledge related to geometry and probability. Each problem requires specific methods or formulas to solve, thus indicating a need for analytical and computational skills.
Answer
The evaluated expression for \( \cos^2 \theta + 2 \sin^2 \theta \) is \( 2 - \cos^2 \theta \). The solution strategy for the differential equation involves separation of variables and applying initial conditions.
Answer for screen readers
- For ( \cos^2 \theta + 2 \sin^2 \theta ), the simplified expression is ( 2 - \cos^2 \theta ).
- For ( j^2 + \log_x y ), the output depends on specific ( a ) and ( b ).
- The solution of the differential equation is found by integrating with constant.
Steps to Solve
- Evaluate the expression for the first question (21)
To find the value of ( \cos^2 \theta + 2 \sin^2 \theta ), we can use the identity ( \sin^2 \theta = 1 - \cos^2 \theta ).
Plugging this into the expression:
[ \cos^2 \theta + 2(1 - \cos^2 \theta) = \cos^2 \theta + 2 - 2\cos^2 \theta = 2 - \cos^2 \theta ]
Now, we need values for ( \theta ) to find a final numerical answer.
- Evaluate the logarithmic expression in question (22)
For the question ( j^2 + \log_x y ):
Using properties of logarithms, if ( x = a ) and ( y = b ):
[ \text{Let } j = k \implies j^2 + \log_a b ]
- Solve the differential equation in question (28)
Given the differential equation ( (x + y^2) \frac{dy}{dx} = 2y ):
To solve, separate the variables:
[ \frac{dy}{y} = \frac{2}{x + y^2} dx ]
Integrate both sides and apply the given initial condition ( y(0) = 0 ) to find the specific solution.
- Show the relationship in question (29)
If ( 1 + |a|^2 + |b|^2 ) indicates a scalar field:
Show associative properties for vectors ( a ) and ( b ):
Using the identity ( \text{arg}(a + b) = \text{arg}(a) + \text{arg}(b) ) when ( a ) and ( b ) are non-negative.
- For ( \cos^2 \theta + 2 \sin^2 \theta ), the simplified expression is ( 2 - \cos^2 \theta ).
- For ( j^2 + \log_x y ), the output depends on specific ( a ) and ( b ).
- The solution of the differential equation is found by integrating with constant.
More Information
The evaluation of ( \cos ) and ( \sin ) often leads to expressions involving Pythagorean identities. Understanding how to manipulate logarithmic properties directly affects the effectiveness of calculation. The method of separation of variables is crucial in solving differential equations.
Tips
- For question (21): Forgetting to apply the Pythagorean identity can lead to incorrect values.
- For question (28): Neglecting initial conditions when finding particular solutions.
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