1. **Multiple Choice Questions (6 marks)** 1. The slope of the line passing through points $(1,2)$ and $(3,6)$ is: - (a) 2 - (b) 3 - (c) 4 - (d) 1 2. The equation of a circle with center at $(0,0)$ and radius 5 is: - (a) $$x^2 + y^2 = 25$$ - (b) $$x^2 + y^2 = 5$$ - (c) $$x^2 - y^2 = 25$$ - (d) $$x^2 + y^2 = 10$$ 3. The eccentricity of a parabola is: - (a) 0 - (b) 1 - (c) Between 0 and 1 - (d) Greater than 1 4. The direction ratios of a line are proportional to: - (a) Its slope - (b) Its direction cosines - (c) Its intercepts - (d) None of these 5. The equation of the plane passing through the origin and perpendicular to vector $$\vec{n} = 3\hat{i} - 2\hat{j} + 4\hat{k}$$ is: - (a) $$3x - 2y + 4z = 0$$ - (b) $$3x + 2y + 4z = 0$$ - (c) $$x - y + z = 0$$ - (d) $$3x - 2y - 4z = 0$$ 6. The standard form of the hyperbola with transverse axis along x-axis is: - (a) $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ - (b) $$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$$ - (c) $$x^2 + y^2 = 1$$ - (d) $$x^2 - y^2 = 1$$ 2. **Assertion-Reason Questions (4 marks)** 1. Assertion: The sum of the distances from any point on an ellipse to the two foci is constant. Reason: The eccentricity of an ellipse is always less than 1. 2. Assertion: The equation of a line in 3D can be represented using parametric form. Reason: Parametric form uses a parameter to express coordinates of points on the line. 3. **Short Answer Questions (2 marks each, 6 marks)** 1. Find the slope of the line perpendicular to the line $$2x - 3y + 5 = 0$$. 2. Write the equation of the parabola with vertex at origin and focus at $(0,2)$. 3. Find the direction cosines of a line whose direction ratios are 3, 4, 12. 4. **Short Answer Questions (3 marks each, 12 marks)** 1. Find the equation of the plane passing through points $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. 2. Find the coordinates of the foci of the ellipse $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$. 3. Find the angle between the lines $$\frac{x-1}{2} = \frac{y+3}{-1} = \frac{z}{4}$$ and $$\frac{x}{1} = \frac{y}{2} = \frac{z-1}{-2}$$. 4. Find the equation of the hyperbola with foci at $(\pm 5,0)$ and eccentricity 2. 5. **Long Answer Question (5 marks)** Find the equation of the line of intersection of the planes $$2x - y + z = 3$$ and $$x + y - 2z = 4$$. 6. **Case Based Question (4 marks)** A plane passes through the point $(2, -1, 3)$ and is perpendicular to the vector $$\vec{n} = 4\hat{i} - 2\hat{j} + 5\hat{k}$$. (a) Write the equation of the plane. (b) Find the distance of the point $(1, 2, -1)$ from this plane. (c) Find the foot of the perpendicular from the point $(1, 2, -1)$ to the plane. (d) Verify if the point $(3, 0, 4)$ lies on the plane.