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7. If cos A = 1/2, then the value of (cos A - sec A) is:

• The secant function is the reciprocal of the cosine function. If cos A = 1/2, then sec A = 2. Now, we can calculate the expression: $\text{cos A} - \text{sec A} = \frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2}$ Answer: (B) -3/2

8. Find the value of K for which the equation $4x^2 - 8Kx + 9 = 0$ has equal roots:

• For equal roots, the discriminant of the quadratic equation must be zero. The discriminant $\Delta$ is given by: $\Delta = b^2 - 4ac$ For the equation $4x^2 - 8Kx + 9 = 0$, a = 4, b = -8K, and c = 9. The discriminant is: $\Delta = (-8K)^2 - 4(4)(9) = 64K^2 - 144$ For equal roots, $\Delta = 0$: $64K^2 - 144 = 0 \quad \Rightarrow \quad 64K^2 = 144 \quad \Rightarrow \quad K^2 = \frac{144}{64} = \frac{9}{4}$ $K = \pm \frac{3}{2}$ Answer: (C) $\frac{3}{2}$

9. According to Basic Proportionality Theorem, which of the following is true?

• The Basic Proportionality Theorem (also known as Thales’ Theorem) states that if a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those two sides in the same ratio. From the given options: $\frac{AB}{BD} = \frac{AE}{EC}$ Answer: (D) All of these (Since all are valid in the context of the theorem)

10. If $\left(\frac{a}{2}, 4\right)$ is the midpoint of the line segment joining the points A(-6, 5) and B(-2, 3), then the value of a is:

• The midpoint formula is given by: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Here, the midpoint is $\left(\frac{a}{2}, 4\right)$, and the coordinates of A and B are (-6, 5) and (-2, 3). Using the midpoint formula for the x-coordinate: $\frac{-6 + (-2)}{2} = \frac{a}{2} \quad \Rightarrow \quad \frac{-8}{2} = \frac{a}{2} \quad \Rightarrow \quad a = -8$ Answer: (A) -8

11. For the following distribution, the lower limit of the median class is:

• To find the median class, we need to determine the cumulative frequency and locate the median position. The median class is the class where the cumulative frequency exceeds half of the total frequency. Let's calculate it step-by-step:
  • Total number of students = 3 + 12 + 27 + 57 + 75 + 80 = 254
  • Half of the total = 254/2 = 127
  • Now, find the class interval where the cumulative frequency first exceeds 127:
    • Below 10: 3
    • Below 20: 3 + 12 = 15
    • Below 30: 15 + 27 = 42
    • Below 40: 42 + 57 = 99
    • Below 50: 99 + 75 = 174 (this exceeds 127) The median class is the one where the cumulative frequency exceeds 127, which is the "Below 50" class. Therefore, the lower limit is 40. Answer: (C) 40

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Here are a few more related questions to explore:

1. How do we calculate the discriminant for any quadratic equation?
2. Can you derive the Basic Proportionality Theorem from similar triangles?
3. What are other properties of secant and cosine functions?
4. How can you determine the midpoint of a line segment in a 3D coordinate plane?
5. What is the significance of cumulative frequency in statistics?

Tip: When dealing with quadratic equations, always check if the discriminant is zero to determine if roots are equal!