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Federal (FBISE) · Quarter book test · Free

2nd Year Mathematics 2nd Quarter Book Paper

40 marks · 1 hour · 28 questions. Covers: 04: Differential Equations, 05: Kinematics of Motion in a Straight Line, 06: Analytical Geometry.

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2nd Year Mathematics 2nd Quarter Book Paper

2nd Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 40Time: 1 hour

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

  1. 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
  2. The point of intersection of right bisectors of a triangle is called:
    1. incenter
    2. circumcenter
    3. centroid
    4. orthocenter
  3. Which of the following quantities is a vector?
    1. Charge
    2. Mass
    3. Momentum
    4. Time
  4. y = 8 is a solution of the differential equation:
    1. dydx+8y=32\frac{dy}{dx}+ 8y = 32
    2. dydx+6y=32\frac{dy}{dx}+ 6y = 32
    3. dydx+5y=32\frac{dy}{dx}+ 5y = 32
    4. dydx+4y=32\frac{dy}{dx}+ 4y = 32
  5. If the determinant of three points is zero then the points are:
    1. collinear
    2. non-collinear
    3. imaginary
    4. concurrent
  6. 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
  7. 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
  8. The lines represented by ax²+ 2hxy + by²= 0 are imaginary if:
    1. h²> ab
    2. h²= ab
    3. h²< ab
    4. h²≤ ab
  9. 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
  10. The general solution of y²dy - x²dx = 0 is:
    1. x² - y² = c
    2. x³ + y³ = c
    3. x³ - y³ = c
    4. x² + y² = c
  11. The point of intersection of three medians of a triangle is called:
    1. incenter
    2. circumcenter
    3. centroid
    4. orthocenter
  12. 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.

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

  1. Solve the initial value problems. (x²+ 2y²)dx = xy dy ; y (1) = 1
  2. Find the derivative of the vector function at f(t)=2t2i^+8j−1t3k^at t=5. f(t) = 2t^2\hat{i} + 8j - \frac{1}{t^3}\hat{k} \quad \text{at } t = 5.
  3. Find the lines represented by each of the following joint equation: 7x²− 2xy − 9y²= 0
  4. Solve the differential equation by separating the variable.xdydx=4yx\frac{dy}{dx} =4y
  5. Find the domain of r(t)=ln⁡∣t−1∣ t^+et j^+t k^ r(t) = \ln|t - 1| \, \hat{t} + e^t \, \hat{j} + \sqrt{t} \, \hat{k}
  6. Find the lines represented by each of the following joint equation. 5x²+ 3xy − 2y²= 0
  7. Identify linear and non-linear different equation.d2ydx2+12xy=0\frac{d^2y}{dx^2} + 12xy = 0
  8. Find the lines represented by each of the following joint equation: x²+ 6xy = 0
  9. Identify linear and non-linear different equation.dydx=a\frac{dy}{dx} = a
  10. Find the area of triangle bounded by the lines. 4x − 5y + 7 = 0, x − 2 = 0 and y + 1 = 0
  11. Check whether the function. f(x,y)=x2y+4f(x , y) = \frac{x}{2y}+ 4
  12. Find the area of triangle if points (4 , - 2) , (- 2 , 4) and (5 , 5) are vertices of a triangle.

Q3. Long questions (Attempt any 3 of 4 · 3 × 4 = 12)

  1. Verify that the indicated function is a solution of the given differential equation: 2y′ + y = 0 ;y=e−x2y= e^-\frac{x}{2}
  2. Find circumcenter of a triangle ABC with A(0 , 0) , B(6 , 0) and C(0 , 6). Also prove that circumcenter, centroid and orthocenter are collinear.
  3. Find the velocity and acceleration of the function at t = 0.r(t)=(3t+1)i+3tj+t2k r(t) = (3t + 1)i + \sqrt{3}tj + t^2k
  4. Verify y=x416 y = \frac{x^4}{16} that is a solution of the differential equation dydx−xy12=0\frac{dy}{dx} - xy^{\frac{1}{2}} = 0