2nd Year Mathematics Second half Book Paper

Q1. Multiple choice questions (Attempt all · 20 × 1 = 20)

1. $\text{For what value of } x \in (0, 2\pi) \text{, } \sin x = -\frac{1}{2}?$
   1. $$\frac{7\pi}{6}, \frac{\pi}{6}$$
   2. $$\frac{7\pi}{6}, \frac{11\pi}{6}$$
   3. $$\frac{5\pi}{6}, \frac{11\pi}{6}$$
   4. $$\frac{\pi}{6}, \frac{5\pi}{6}$$
2. The point of intersection of right bisectors of a triangle is called:
   1. incenter
   2. circumcenter
   3. centroid
   4. orthocenter
3. $\text{What results } \sec^{-1}(1) - \sec^{-1}(\sqrt{2})$
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{2}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{4}$
4. What does the Bisection method do?
   1. Finds the maximum of a function
   2. Estimates the area under the curve
   3. Finds the root of the function
   4. Differentiates a function
5. The focus of the parabola x²= − 16y is:
   1. (4 , 0)
   2. (– 4 , 0)
   3. (0 , 4)
   4. (0 , – 4)
6. $\text{If } \cos^2 x = \frac{1}{2} \text{ with } x \in [0, \pi] \text{ then value of } x \text{ is :}$
   1. $$\frac{\pi}{4}, \frac{7\pi}{4}$$
   2. $$\frac{\pi}{4}, \frac{3\pi}{4}$$
   3. $$\frac{3\pi}{4}, \frac{5\pi}{4}$$
   4. $$\frac{5\pi}{4}, \frac{7\pi}{4}$$
7. If the determinant of three points is zero then the points are:
   1. collinear
   2. non-collinear
   3. imaginary
   4. concurrent
8. For what value of x ∈ (0 , 2π) , Sin x =$-\frac{1}{2}$
   1. $\frac{7\pi}{6}, \frac{\pi}{6}$
   2. $\frac{7\pi}{6}, \frac{11\pi}{6}$
   3. $\frac{5\pi}{6}, \frac{11\pi}{6}$
   4. $\frac{\pi}{6}, \frac{5\pi}{6}$
9. Numerical techniques more commonly involve:
   1. Elimination
   2. Reduction
   3. Iterative method
   4. Direct methods
10. Equation of conjugate axis of the hyperbola $\frac{(x-1)^2}{4} - \frac{(y+3)^2}{12} = 1 \text{ is:}$
    1. x = 1
    2. x = - 1
    3. y = 3
    4. y = - 3
11. $\text{What is the solution of } \sec x = 2 \text{, if } x \in [0, \pi]?$
    1. $$\left\{ -\frac{\pi}{6} \right\}$$
    2. $$\left\{ -\frac{\pi}{3} \right\}$$
    3. $$\left\{ \frac{\pi}{3} \right\}$$
    4. $$\left\{ \frac{\pi}{6} \right\}$$
12. The lines represented by ax²+ 2hxy + by²= 0 are imaginary if:
    1. h²> ab
    2. h²= ab
    3. h²< ab
    4. h²≤ ab
13. $\text{The value of } \cos\left[\frac{\pi}{6} + \cos^{-1}\left(-\frac{1}{2}\right)\right] \text{ is :}$
    1. $\frac{1}{2}$
    2. $-\frac{1}{2}$
    3. $-\frac{\sqrt{3}}{2}$
    4. $\frac{\sqrt{3}}{2}$
14. Which method splits the half to find the root?
    1. Newton’s method
    2. Regula Falsi method
    3. Bisection method
    4. Trapezoidal rule
15. $\text{The chord joining the two points } (at_1^2, 2at_1) \text{ and } (at_2^2, 2at_2) \text{ on the parabola } y^2 = 4ax \text{ is focal chord if:}$
    1. $$t_1 + t_2 = 1$$
    2. $$t_1 + t_2 = -1$$
    3. $$t_1 t_2 = 1$$
    4. $$t_1 t_2 = -1$$
16. $\text{One solution of a trigonometric equation } \cos x + \sin x = \sqrt{2} \text{, if } x \in [0, 2\pi] \text{ ?}$
    1. $$\left\{ \frac{\pi}{4} \right\}$$
    2. $$\left\{ -\frac{\pi}{4} \right\}$$
    3. $$\left\{ \frac{2\pi}{4} \right\}$$
    4. $$\left\{ \frac{3\pi}{2} \right\}$$
17. The point of intersection of three medians of a triangle is called:
    1. incenter
    2. circumcenter
    3. centroid
    4. orthocenter
18. , $\text{If } \sin^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{2} - x \text{, then value of x is:}$:
    1. $\frac{\pi}{6}$
    2. $\frac{\pi}{3}$
    3. $\frac{\pi}{4}$
    4. $\frac{\pi}{2}$
19. Using Bisection method, one root of x⁴− x − 1 = 0 lies between 1 and 2. After $2^{nd}$ iteration the root may lie in the interval:
    1. (1.25, 1.5)
    2. (1, 1.25)
    3. (1, 1.5)
    4. None of these
20. Exactly one tangent can be drawn to a circle if point lies:
    1. Outside the circle
    2. On the circle
    3. Inside the circle
    4. Centre of circle