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Punjab (PTB) · Quarter book test · Free

2nd Year Mathematics 4th Quarter Book Paper

28 marks · 1 hour · 24 questions. Covers: UNIT 10: SOLUTION OF TRIGONOMETRIC EQUATIONS, UNIT 11: VECTOR VALUED FUNCTIONS AND THEIR DERIVATIVES.

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2nd Year Mathematics 4th Quarter Book Paper

2nd Year · Mathematics · Punjab (PTB)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 28Time: 1 hour

Q1. Multiple choice questions (Attempt all · 12 × 1 = 12)

  1. What additional insight does the graphical method provide?

    1. The behavior of the functions and their interactions

    2. Only the exact angles

    3. Only the coefficients

    4. Only the constants

  2. A scalar function assigns a ...... to each point in its domain.

    1. Vector

    2. Single real number

    3. Matrix

    4. Complex vector

  3. What do the x-coordinates of the intersection points give?

    1. The slopes of the curves

    2. The maximum values

    3. The solutions satisfying f(x)=g(x)

    4. The y-intercepts

  4. Vector-valued functions are useful in:

    1. Engineering

    2. Physics

    3. Computer graphics

    4. All of these

  5. If one period is added repeatedly to a solution, we obtain:

    1. Only principal solutions

    2. All possible solutions

    3. No solution

    4. A different equation

  6. Vector-valued functions can be used to model:

    1. Position

    2. Velocity

    3. Force

    4. All of these

  7. What does the graphical solution of a trigonometric equation involve?

    1. Differentiating both sides

    2. Plotting the graphs of both sides

    3. Integrating both sides

    4. Factoring the equation

  8. Which of the following mathematical functions is classified as a scalar function?

    1. f(t)=(3t2-5)2

    2. f(t)=(3t2+5)2

    3. f(t)=(3t2-4)2

    4. f(t)=(3t2-3)2

  9. An unknown value of an angle that satisfies a trigonometric equation is called:

    1. Period

    2. Solution

    3. Amplitude

    4. Function

  10. The derivative of a vector-valued function provides the instantaneous:

    1. Position

    2. Rate of change

    3. Temperature

    4. Distance only

  11. Which coordinate of an intersection point represents the solution to f(x)=g(x)?

    1. y-coordinate

    2. x-coordinate

    3. Both coordinates separately

    4. Origin

  12. In the satellite example, the x-coordinate is:

    1. 4000 sint

    2. 3000t

    3. 4000 cos t

    4. 3000 cos t

Q2. Short questions (Attempt any 8 of 12 · 8 × 2 = 16)

  1. Find the Find the domain and range of the following vector-valued functions:

  2. A radio tower stands on flat ground with a small antenna mounted at its top. From a point 50 meters away from the base of the tower. The antenna subtends the same angle as a 1 meter vertical rod placed at the base of the tower. The total height of the tower and antenna is 10 meters.

    (a) Find the height of the tower (without the antenna).

    (b) Calculate the subtended angle.

  3. Given , construct r(t).

  4. Solve the following trigonometric eqautions:

  5. Find the first-order derivatives of the following vector-valued functions with respect to t.

  6. Solve the following trigonometric eqautions:

  7. A Ferris wheel of radius 20 m rotates with an angular speed of 0.2 rad/sec.
    (a) Construct its position vector r(t)
    (b) Find the passenger's velocity at t =10sec
    (c) Show that the acceleration is centripetal (toward the center)

  8. A projectile is launched from the ground with an initial velocity 30m/s. The range R of the projectile is given by the formula , where is the launch angle to the horizontal is the acceleration due to gravity.

    (a) Find the angle (in degrees) required for the projectile to hit a target exactly 80 meters away.

    (b) If the maximum range achievable is  determine the angle & that gives the maximum range. Justify your answer mathematically.

  9. Find the first-order derivatives of the following vector-valued functions with respect to t.

  10. Solve the equation graphically over the interval .

  11. A quarterback throws a football with a position vector .

    (a) Find the time of flight before it hits the ground.
    (b) What is the maximum height reached?
    (c) At which time did football reach its maximum height?

  12. Solve the following trigonometric equations: