The value of
(1+cot2θ)(1+cosθ)(1−cosθ)+(1−sinθ)(1+sinθ)(1+tan2θ) is:
The problem requires simplifying a trigonometric expression. We will use fundamental trigonometric identities such as \( (a+b)(a-b) = a^2 - b^2 \), \( \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \), \( 1 + \text{cot}^2 \theta = \text{cosec}^2 \theta \), and \( 1 + \text{tan}^2 \theta = \text{sec}^2 \theta \).
Let the given expression be \(E\). The expression is:
\[ E = (1+\text{cot}^2 \theta)(1+\text{cos}\theta)(1-\text{cos}\theta) + (1-\text{sin}\theta)(1+\text{sin}\theta)(1+\text{tan}^2 \theta) \]Simplify the first part of the expression: \( (1+\text{cot}^2 \theta)(1+\text{cos}\theta)(1-\text{cos}\theta) \)
Using the identity \( (a+b)(a-b) = a^2 - b^2 \), we have \( (1+\text{cos}\theta)(1-\text{cos}\theta) = 1^2 - \text{cos}^2 \theta = 1 - \text{cos}^2 \theta \).
From the identity \( \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \), we know that \( 1 - \text{cos}^2 \theta = \text{sin}^2 \theta \).
Also, using the identity \( 1 + \text{cot}^2 \theta = \text{cosec}^2 \theta \).
So, the first part becomes:
\[ (\text{cosec}^2 \theta)(\text{sin}^2 \theta) \]Since \( \text{cosec}\theta = \frac{1}{\text{sin}\theta} \), we have \[\text{cosec}^2 \theta = \frac{1}{\text{sin}^2 \theta}\].
Therefore, the first part simplifies to:
\[ \frac{1}{\text{sin}^2 \theta} \cdot \text{sin}^2 \theta = 1 \]Now, simplify the second part of the expression: \( (1-\text{sin}\theta)(1+\text{sin}\theta)(1+\text{tan}^2 \theta) \)
Using the identity \( (a-b)(a+b) = a^2 - b^2 \), we have \( (1-\text{sin}\theta)(1+\text{sin}\theta) = 1^2 - \text{sin}^2 \theta = 1 - \text{sin}^2 \theta \).
From the identity \( \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \), we know that \( 1 - \text{sin}^2 \theta = \text{cos}^2 \theta \).
Also, using the identity \( 1 + \text{tan}^2 \theta = \text{sec}^2 \theta \).
So, the second part becomes:
\[ (\text{cos}^2 \theta)(\text{sec}^2 \theta) \]Since \( \text{sec}\theta = \frac{1}{\text{cos}\theta} \), we have \( \text{sec}^2 \theta = \frac{1}{\text{cos}^2 \theta} \).
Therefore, the second part simplifies to:
\[ \text{cos}^2 \theta \cdot \frac{1}{\text{cos}^2 \theta} = 1 \]Finally, add the simplified first and second parts:
\[ E = 1 + 1 = 2 \]The value of the given expression is 2.
Correct Option: D) 2
To find the compound interest when interest is compounded half-yearly, we need to adjust the annual interest rate and the time period. The formula for the amount when compounded half-yearly is given by \(A = P \left(1 + \frac{R/2}{100}\right)^{2n}\), where \(P\) is the principal, \(R\) is the annual rate of interest, and \(n\) is the time in years. The compound interest (CI) is then \(A - P\).
Since the interest is compounded half-yearly, the rate per compounding period will be half of the annual rate, and the number of compounding periods will be twice the number of years.
The formula for the amount is:
\[ A = P \left(1 + \frac{R'}{100}\right)^{N} \]\[ A = 80000 \left(1 + \frac{2}{100}\right)^{2} \]\[ A = 80000 \left(1 + 0.02\right)^{2} \]\[ A = 80000 \left(1.02\right)^{2} \]\[ A = 80000 \times 1.0404 \]\[ A = 83232 \]Compound Interest is the difference between the Amount and the Principal.
\[ \text{CI} = A - P \]\[ \text{CI} = 83232 - 80000 \]\[ \text{CI} = 3232 \]Correct Option: A) Rs 3,232
Eight friends, A, B, C, D, E, F, G and H, are sitting around a circular table facing the centre (but not necessarily in the same order).
Only D is sitting between A and E. E is sitting second to the right of F. Only B is sitting between C and G. Only F is sitting between G and H.
What is F’s position with respect to D?
To the immediate left
Third to the left
To the immediate right
Third to the right
This is a circular seating arrangement problem. We need to deduce the positions of eight friends (A, B, C, D, E, F, G, H) sitting around a circular table based on the given clues. We will build the arrangement step-by-step and then determine F's position with respect to D.
B) Third to the left
As determined in step 6, F is third to the left of D.
To find the heat generated per second in an electric circuit, we can use Joule's Law of Heating. This law states that the heat generated is directly proportional to the square of the current, the resistance of the circuit, and the time for which the current flows. The formula for heat generated is \(H = I^2 R t\).
The formula for heat generated (\(H\)) is:
\[ H = I^2 R t \]The standard unit for heat or energy in the International System of Units (SI) is the joule.
Correct Option: A) 250 joule — The calculation using Joule's law \(H = I^2 R t\) with the given values \(I = 5 \text{ A}\), \(R = 10 \text{ } \Omega\), and \(t = 1 \text{ s}\) yields \(250 \text{ joules}\).
This problem involves relative speeds in upstream and downstream motion. We need to set up a system of two linear equations based on the given information (distance, time, and speed relationships) and then solve for the speed of the stream.
Let the speed of the boat in still water be \(u\) km/h and the speed of the stream be \(v\) km/h.
When the boat goes upstream, its effective speed is \( (u - v) \) km/h.
When the boat goes downstream, its effective speed is \( (u + v) \) km/h.
Recall the formula: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).
From the first condition: 30 km upstream and 44 km downstream in 10 hours.
\[ \frac{30}{u - v} + \frac{44}{u + v} = 10 \quad \text{(Equation 1)} \]
From the second condition: 40 km upstream and 55 km downstream in 13 hours.
\[ \frac{40}{u - v} + \frac{55}{u + v} = 13 \quad \text{(Equation 2)} \]
To solve these equations, let \(x = \frac{1}{u - v}\) and \(y = \frac{1}{u + v}\). The equations become:
\[ 30x + 44y = 10 \quad \text{(Equation 3)} \]
\[ 40x + 55y = 13 \quad \text{(Equation 4)} \]
Multiply Equation 3 by 4 and Equation 4 by 3 to eliminate \(x\):
Equation 3 \(\times 4\): \(120x + 176y = 40\)
Equation 4 \(\times 3\): \(120x + 165y = 39\)
Subtract the new Equation 4 from the new Equation 3:
\[ (120x + 176y) - (120x + 165y) = 40 - 39 \]
\[ 11y = 1 \]
\[ y = \frac{1}{11} \]
Substitute \(y = \frac{1}{11}\) into Equation 3:
\[ 30x + 44 \left(\frac{1}{11}\right) = 10 \]
\[ 30x + 4 = 10 \]
\[ 30x = 6 \]
\[ x = \frac{6}{30} = \frac{1}{5} \]
Now, substitute back the values of \(x\) and \(y\):
\[ \frac{1}{u - v} = x = \frac{1}{5} \implies u - v = 5 \quad \text{(Equation 5)} \]
\[ \frac{1}{u + v} = y = \frac{1}{11} \implies u + v = 11 \quad \text{(Equation 6)} \]
Add Equation 5 and Equation 6:
\[ (u - v) + (u + v) = 5 + 11 \]
\[ 2u = 16 \]
\[ u = 8 \text{ km/h} \]
Substitute \(u = 8\) into Equation 6:
\[ 8 + v = 11 \]
\[ v = 11 - 8 \]
\[ v = 3 \text{ km/h} \]
The speed of the stream is \(v = 3\) km/h.
Correct Option: B) 3 km/h
To find the minimum number of rows, we first need to determine the maximum number of participants that can be seated in each row. This number must be a common factor of the participants in all three subjects (A, B, and C) because all participants in a row are from the same subject and each row has the same number of participants. Therefore, we need to find the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of the number of participants in subjects A, B, and C. Once we have the HCF, we can divide the number of participants in each subject by the HCF to find the number of rows required for each subject and then sum them up to get the total minimum number of rows.
Identify the number of participants for each subject:
Find the HCF of 96, 160, and 224:
We can use the prime factorization method.
The common prime factors are \(2^5\). So, the HCF is \(2^5 = 32\).
This means that a maximum of 32 participants can be seated in each row.
Calculate the number of rows for each subject:
Calculate the total minimum number of rows:
Total rows = Rows for A + Rows for B + Rows for C
Total rows = \(3 + 5 + 7 = 15\) rows.
D) 15
In this question, the statement is followed by four conclusions. Find out which conclusion is NOT true based on the given statement.
Statement: P ≥ T = W > L < S ≤ B
This question involves analyzing a statement with inequality and equality signs to determine which of the given conclusions is NOT true. We need to break down the given statement into simpler relationships and then evaluate each conclusion against these relationships.
Let's evaluate option A: \(P \le W\).
Thus, conclusion A is NOT true.
If α and β are the roots of the equation 2x2−3x−5=0, then the quadratic equation whose roots are \(\frac{\beta}{\alpha}\) and \(\frac{\alpha}{\beta}\) is:
10x2+29x−10=0
10x2-29x+10=0
10x2-29x−10=0
10x2+29x+10=0
To find the new quadratic equation, we first determine the sum and product of the roots of the given equation. Then, we calculate the sum and product of the new roots \(\frac{\beta}{\alpha}\) and \(\frac{\alpha}{\beta}\). Finally, we form the quadratic equation using the formula \(x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0\).
Identify coefficients of the given quadratic equation:
The given equation is \(2x^2 - 3x - 5 = 0\).
Comparing it with the standard form \(ax^2 + bx + c = 0\), we have:
Calculate the sum and product of roots for the given equation:
If \(\alpha\) and \(\beta\) are the roots of \(ax^2 + bx + c = 0\), then:
For the given equation:
Calculate the sum of the new roots:
The new roots are \(\frac{\beta}{\alpha}\) and \(\frac{\alpha}{\beta}\).
Sum of new roots \[S' = \frac{\beta}{\alpha} + \frac{\alpha}{\beta}\]
To add these fractions, find a common denominator:
\[ S' = \frac{\beta^2 + \alpha^2}{\alpha \beta} \]We know that \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta\). Substitute the values:
\[ \alpha^2 + \beta^2 = \left(\frac{3}{2}\right)^2 - 2\left(-\frac{5}{2}\right) \]\[ \alpha^2 + \beta^2 = \frac{9}{4} + 5 \]\[ \alpha^2 + \beta^2 = \frac{9 + 20}{4} = \frac{29}{4} \]Now substitute this back into the sum of new roots formula:
\[ S' = \frac{\frac{29}{4}}{-\frac{5}{2}} = \frac{29}{4} \times \left(-\frac{2}{5}\right) \]\[ S' = -\frac{29 \times 2}{4 \times 5} = -\frac{29}{10} \]Calculate the product of the new roots:
Product of new roots \[P' = \left(\frac{\beta}{\alpha}\right) \times \left(\frac{\alpha}{\beta}\right)\]
Form the new quadratic equation:
The general form of a quadratic equation with roots \(r_1\) and \(r_2\) is \(x^2 - (r_1 + r_2)x + r_1 r_2 = 0\).
Using \(S' = -\frac{29}{10}\) and \(P' = 1\):
To eliminate the fraction, multiply the entire equation by 10:
\[ 10x^2 + 29x + 10 = 0 \]Correct Option: D) \(10x^2 + 29x + 10 = 0\)
Suraj starts walking 9 m towards the east, then turns left. He walks 5 m, turns right and again walks 5 m. Then he turns right and walks 8 m. He then again turns right and walks 5 m. Again, he turns right and walks 3 m. Which direction is he facing now? (All turns are 90° turns only.)
This problem involves tracking a person's movements and turns to determine their final facing direction. We will visualize each step and turn, assuming standard cardinal directions (North, South, East, West).
At the end of all movements, Suraj is facing North.
A) North is the correct direction Suraj is facing at the end of his movements.
To find the overall percentage change, we can use two methods: assuming an initial salary and calculating the changes sequentially, or using the effective percentage change formula for successive percentage changes.
A) 3.04% increase. Both methods confirm that the overall change is a 3.04% increase in Ram's salary.