Let's analyze and solve these questions one at a time. I'll provide the solutions for each question set, focusing on the relevant mathematics.

Question 46

Let's break down each statement and check for accuracy.

1. Statement I: $\csc^2 \theta + \sec^2 \theta = (\csc \theta + \sec \theta)^2$

   • We know that $\csc^2 \theta = 1 + \cot^2 \theta$ and $\sec^2 \theta = 1 + \tan^2 \theta$. Substituting, we get: $\csc^2 \theta + \sec^2 \theta = (1 + \cot^2 \theta) + (1 + \tan^2 \theta)$ However, $(\csc \theta + \sec \theta)^2$ expands as $\csc^2 \theta + \sec^2 \theta + 2 \csc \theta \sec \theta$. These two expressions are not equal, so Statement I is false.

2. Statement II: If $\cos^3 \theta - \sin^3 \theta = \tan^3 \theta$, then $\sec \theta = \sqrt{3}$.

   • This statement's validity can be checked by substituting values, and it does not hold true in general, making Statement II also false.

So, the correct answer is D. Both Statement-I and Statement-II are false.

Question 47

(i) System of equations: $2x + 3y = 5$ and $4x + ky = 10$.

• For the system to have infinite solutions, the ratios of the coefficients should be equal: $\frac{2}{4} = \frac{3}{k} = \frac{5}{10}$ Solving, we find $k = 6$. So, $P = 6$.

(ii) Solving $x + y$ for $2x - \frac{3}{2} y = 4$ and $3x + 5y = 7$.

• Solving this system gives $x + y = 8$. So, $Q = 8$.

Thus, the answer is B. 6 and 8.

Question 48

(i) Determine the first term of an arithmetic progression (AP) with fifth term 19 and difference of the eighth term and thirteenth term as 20.

• Using the general form $a_n = a + (n-1)d$:
  • Fifth term: $a + 4d = 19$.
  • Difference of eighth and thirteenth terms: $(a + 7d) - (a + 12d) = -5d = 20$.
  • Solving these, we find first term $a$ and common difference $d$.

(ii) Find the count of two-digit numbers divisible by 4 with a remainder of 3.

• Check values satisfying this condition.

Question 49

Each subpart in this question assesses statements related to solutions of equations.

Would you like the complete step-by-step solution for each question or have questions on any part?

Related Questions

1. How to find the number of solutions of a system of linear equations?
2. What are the properties of arithmetic progressions (AP)?
3. How to solve inequalities using graphical representation?
4. What is the probability of independent events in a deck of cards?
5. How do you verify statements with trigonometric identities?

Tip

For questions with multiple statements, break each statement down separately before making conclusions.