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Trigonometry (2 Questions – Height and Distance)

Q17. A person standing 50 meters away from the base of a tower observes the top of the tower at an angle of elevation of 60∘. What is the height of the tower?

1. 50 meters
2. 86.6 meters
3. 100 meters
4. 115.5 meters

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Q18. A man observes the top of a lighthouse at an angle of elevation of 30∘. If the distance between the man and the base of the lighthouse is 100 meters, what is the height of the lighthouse?

1. 33.33 meters
2. 50 meters
3. 57.733 meters
4. 75 meters

Read the following pie chart carefully and answer the questions given below. The following pie chart shows the distribution of four different types (Education, Rent, Transport and Others) of monthly expenditures and savings of Ravi.

Q19. If amount spend by Ravi on Rent is Rs.12000, then find the monthly income of Ravi? (in Rs.)

(a) 60000

(b) 48000

(c) 72000

(d) 54000

(e) 84000

Modern Math (1 Question)

Q20. In how many ways can 5 people be seated on a circular table?

1. 24
2. 60
3. 120
4. 720

Question 15

Q15: Solve for \(x\) in the quadratic equation: \[ 2x^2 – 7x + 3 = 0 \]

Solution:

We are solving: \[ 2x^2 – 7x + 3 = 0 \] Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}, \text{ where } a = 2, b = -7, c = 3. \] Substitute values: \[ x = \frac{-(-7) \pm \sqrt{(-7)^2 – 4(2)(3)}}{2(2)} = \frac{7 \pm \sqrt{49 – 24}}{4} = \frac{7 \pm \sqrt{25}}{4}. \] \[ x = \frac{7 + 5}{4} \text{ or } \frac{7 – 5}{4}. \] Roots: \[ x = 3 \text{ or } x = 0.5. \]

Question 16

Q16: If \(x + \frac{1}{x} = 3\), find the value of \(x^3 + \frac{1}{x^3}\).

Solution:

We are given: \[ x + \frac{1}{x} = 3. \] To find \(x^3 + \frac{1}{x^3}\), use the identity: \[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 – 3\left(x + \frac{1}{x}\right). \] Substitute \(x + \frac{1}{x} = 3\): \[ x^3 + \frac{1}{x^3} = 3^3 – 3(3) = 27 – 9 = 18. \]

Trigonometry

Q17 Solution: Using \( \tan \theta = \frac{\text{height}}{\text{distance}} \): \[ \sqrt{3} = \frac{h}{50} \implies h = 50 \times \sqrt{3} = 86.6 \, \text{meters}. \]

Q18 Solution: Using \( \tan 30^\circ = \frac{h}{100} \): \[ \frac{1}{\sqrt{3}} = \frac{h}{100} \implies h = \frac{100}{\sqrt{3}} \approx 33.33 \, \text{meters}. \]

Modern Math

Q19 Solution: Arrangements in a circle: \((n – 1)!\). For \(n = 5\): \[ (5 – 1)! = 4! = 24. \]

DI Solution:

Given that the Rent percentage is 20% of Ravi’s monthly income and its value is ₹12,000, we calculate the total monthly income using the formula:

\[ \text{Monthly Income} = \frac{\text{Rent Value} \times 100}{\text{Rent Percentage}}. \]

Substitute the given values:

\[ \text{Monthly Income} = \frac{12,000 \times 100}{20}. \]

Simplify:

\[ \text{Monthly Income} = \frac{1,200,000}{20} = 60,000. \]

Final Answer: Ravi’s monthly income is ₹60,000.

Correct Option: (a) ₹60,000