2nd Year Mathematics First half Book Paper

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

1. If $f$ is differentiable for every x, then $f$ is
   1. Discontinuous
   2. Continuous
   3. inite
   4. Infinite
2. 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
3. $\text{If } f \text{ is integrable, then it is .}$
   1. discontinous
   2. unbounded
   3. continuous
   4. linear
4. Which of the following quantities is a vector?
   1. Charge
   2. Mass
   3. Momentum
   4. Time
5. The logarithmic function $f(x) = \log_b x$
   1. all real numbers
   2. x < 0
   3. x > 0
   4. x ≥ 0
6. 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. $\frac{3}{2}x^2$
7. 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. $\frac{dy}{dx}+ 4y = 32$
8. $\int \sqrt{x} \, dx$ is:
   1. $x^{3/2} + C$
   2. $\frac{2}{3} x^{3/2} + C$
   3. $\frac{3}{2} x^{3/2} + C$
   4. $x^{1/2} + C$
9. 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
10. 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.
11. If $f$ is differentiable for every value of x, then $f$ is continuous for:
    1. Some value of x
    2. [0 , ∞)
    3. Every value of x
    4. [−∞ , 0]
12. The solution of cosx siny dx + sinx cosy = 0 is:
    1. sinx cosy = c
    2. cosx cosy = c
    3. cosx siny = c
    4. sinx siny = c
13. If ' f(x) = 3x² + 2x , then f(x) is :
    1. 6x + 2 + c
    2. x³ + x² + c
    3. 3x³ + 2x² + c
    4. $1.5x³ + x^2 + c$
14. A car travels 100 km. The journey takes two hours. The highest speed of the car is 80 km/h, and the lowest speed 40 km/h. What is the average speed for the journey?
    1. 40 km/h
    2. 50 km/h
    3. 60 km/h
    4. 120 km/h
15. Which of the following is an example of exponential growth function?
    1. f(x) = 3x + 4
    2. $f(x) = 3^x \times 5$
    3. f(x) = x³
    4. f(x) = x²
16. The general solution of y²dy - x²dx = 0 is:
    1. x² - y² = c
    2. x³ + y³ = c
    3. x³ - y³ = c
    4. x² + y² = c
17. ∫ sin2x dx is:
    1. $\frac{\cos 2x}{2} + c$
    2. 2cos2x + c
    3. $-\frac{\sin 2x}{2} + c$
    4. $-\frac{\cos 2x}{2} + c$
18. A heavy object is released near the surface of earth and falls freely. Air resistance can be ignored. Which statement about the acceleration of the object due to gravity is correct?
    1. The acceleration depends on mass of the object.
    2. The acceleration depends on volume of the object.
    3. The acceleration is constant.
    4. The acceleration is initially zero and increases as the object falls.
19. The function f: R → [0 , ∞) defined by f(x) = x²+ 1, is:
    1. onto but not one-to-one
    2. one-to-one but not onto
    3. neither one-to-one nor onto
    4. both one-to-one and onto
20. The differential equation $\frac{dy}{dx}= e^{x +y}$ has solution.
    1. $e^{-x -y}= c$
    2. $e^{-x} + e^y = c$
    3. $e^x + e^y = c$
    4. $e^{-x} + e⁻y = c^{-y}$