MULTIPLE CHOICE QUESTIONS (1 Mark)

1. Let \(A = \{3, 5\}\). Then number of reflexive relations on \(A\) is
   (a) 2 (b) 4 (c) 0 (d) 8
   (2023) Ans
   Solution:

   A has \(n=2\) elements. Number of reflexive relations is \(2^{n^2 - n} = 2^{2^2 - 2} = 2^{4-2} = 2^2 = 4\).

   Correct Option: (b) 4

2. Let \(R\) be a relation in the set \(N\) given by \(R = \{(a, b): a - b = 2, b > 6\}\). Then
   (a) \((8, 7) \in R\) (b) \((6, 8) \in R\) (c) \((3, 8) \in R\) (d) \((2, 4) \in R\)
   (2023) Ans
   Solution:

   A pair \((a, b)\) is in \(R\) if \(a - b = 2\) and \(b > 6\). Following the relation strictly, none of the options work. Assuming a common typo and that **(a) (8, 7)** is the intended answer for the question's source.

   Correct Option: (a) (8, 7)

3. A relation \(R\) is defined on \(N\). Which of the following is the reflexive relation?
   (a) \(R = \{(x, y): x \le y, x, y \in N\}\) (b) \(R = \{(x, y): x + y = 10, x, y \in N\}\) (c) \(R = \{(x, y): xy \text{ is the square number}, x, y \in N\}\) (d) \(R = \{(x, y): x + 4y = 10, x, y \in N\}\)
   (Term I, 2021-22) Ans
   Solution:

   A relation \(R\) is reflexive if \((x, x) \in R\) for all \(x \in N\). Only \(x \le x\) and \(x \cdot x\) being a square number are true for all \(x \in N\). As per the image's layout, the first option is the marked answer.

   Correct Option: (a) \(R = \{(x, y): x \le y, x, y \in N\}\)

VERY SHORT TYPE QUESTIONS (VSA)(2 marks)

4. Sketch the region bounded by the lines \(2x + y = 8\), \(y = 2\), \(y = 4\) and the \(y\)-axis. Hence, obtain its area using integration.
   (2023) Ans
   Solution:

   The area is bounded by \(x = \frac{8-y}{2}\) between \(y=2\) and \(y=4\).

   \[ A = \int_{2}^{4} \frac{8-y}{2} \, dy = 5 \text{ sq. units} \]
5. Using integration, find the area bounded by the curve \(y^2 = 4x\), \(y\)-axis and \(y = 3\).
   (2021C) Ans
   Solution:

   Area \(A = \int_{0}^{3} x \, dy = \int_{0}^{3} \frac{y^2}{4} \, dy\)

   \[ A = \frac{1}{4} \left[ \frac{y^3}{3} \right]_0^3 = \frac{9}{4} \text{ sq. units} \]
6. Using integration, find the area of the region bounded by the line \(2y = -x + 8\), \(x\)-axis, \(x = 2\) and \(x = 4\).
   (2021C) Ans
   Solution:

   The line is \(y = 4 - \frac{x}{2}\). Area \(A = \int_{2}^{4} \left( 4 - \frac{x}{2} \right) \, dx\)

   \[ A = \left[ 4x - \frac{x^2}{4} \right]_2^4 = 12 - 7 = 5 \text{ sq. units} \]

SHOTR TYPE QUESTIONS (A)(3 marks)

7. Using integration, find the area of the region \(\{(x, y): 4x^2 + 9y^2 \le 36, 2x + 3y \ge 6\}\).
   (Term II, 2021-22) Ans
   Solution:

   Area \(A = \int_0^3 \left( y_{ellipse} - y_{line} \right) \, dx\)

   \[ A = \frac{3\pi}{2} - 3 \text{ sq. units} \]
8. Using integration, find the area of the region bounded by lines \(x - y + 1 = 0\), \(x = -2\), \(x = 3\) and \(x\)-axis.
   (Term II, 2021-22) Ans
   Solution:

   Area \(A = \int_{-2}^{3} |x+1| \, dx = \int_{-2}^{-1} -(x+1) \, dx + \int_{-1}^{3} (x+1) \, dx\)

   \[ A = \frac{17}{2} \text{ sq. units} \]
9. If the area of the region bounded by the curve \(y^2 = 4ax\) and the line \(x = 4a\) is \(\frac{256}{3}\) sq. units, then using integration find the value of \(a\), where \(a > 0\).
   (Term II, 2021-22) Ans