2nd Year Mathematics Full Book Paper

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

1. The logarithmic function $f(x) = \log_b x$
   1. all real numbers
   2. x < 0
   3. x > 0
   4. x ≥ 0
2. Which of the following quantities is a vector?
   1. Charge
   2. Mass
   3. Momentum
   4. Time
3. The point of intersection of right bisectors of a triangle is called:
   1. incenter
   2. circumcenter
   3. centroid
   4. orthocenter
4. $\text{What results } \sec^{-1}(1) - \sec^{-1}(\sqrt{2})$
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{2}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{4}$
5. $\text{For what value of } x \in (0, 2\pi) \text{, } \sin x = -\frac{1}{2}?$
   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. $$\frac{\pi}{6}, \frac{5\pi}{6}$$
6. If $f$ is differentiable for every x, then $f$ is
   1. Discontinuous
   2. Continuous
   3. inite
   4. Infinite
7. $\text{If } f \text{ is integrable, then it is .}$
   1. discontinous
   2. unbounded
   3. continuous
   4. linear
8. 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
9. The solution of the differential equation dy = dx is:
   1. y = x + c
   2. y = x² + c
   3. y² = x² + c
   4. y² = x + c
10. The focus of the parabola x²= − 16y is:
    1. (4 , 0)
    2. (– 4 , 0)
    3. (0 , 4)
    4. (0 , – 4)
11. Scaling refers to:
    1. increasing the size of an object.
    2. decreasing the size of an object.
    3. maintaining the properties while resizing an object.
    4. changing the shape of an object.
12. Which of following can be used to determine the magnitude of velocity?
    1. Area under acceleration–time graph
    2. Area under velocity-time graph
    3. Gradient of an acceleration–time graph
    4. Gradient of a velocity–time graph
13. If the determinant of three points is zero then the points are:
    1. collinear
    2. non-collinear
    3. imaginary
    4. concurrent
14. 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. $\frac{\pi}{6}, \frac{5\pi}{6}$
15. $\text{If } \cos^2 x = \frac{1}{2} \text{ with } x \in [0, \pi] \text{ then value of } x \text{ is :}$
    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. $$\frac{5\pi}{4}, \frac{7\pi}{4}$$
16. If $f(2) = 2$, $g(x) = x^2$, then $\frac{d}{dx} \left[ \frac{3g(x)}{f(2)} \right]$
    1. 2x
    2. 3x
    3. $\frac{3}{2}x$
    4. $\frac{3}{2}x^2$
17. $\int \sqrt{x} \, dx$ is:
    1. $x^{3/2} + C$
    2. $\frac{2}{3} x^{3/2} + C$
    3. $\frac{3}{2} x^{3/2} + C$
    4. $x^{1/2} + C$
18. Numerical techniques more commonly involve:
    1. Elimination
    2. Reduction
    3. Iterative method
    4. Direct methods
19. y = 8 is a solution of the differential equation:
    1. $\frac{dy}{dx}+ 8y = 32$
    2. $\frac{dy}{dx}+ 6y = 32$
    3. $\frac{dy}{dx}+ 5y = 32$
    4. $\frac{dy}{dx}+ 4y = 32$
20. 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