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Punjab (PTB) · Quarter book test · Free

2nd Year Mathematics 2nd Quarter Book Paper

28 marks · 1 hour · 24 questions. Covers: UNIT 4: DIFFERENTIAL EQUATIONS, UNIT 5: ANALYTICAL GEOMETRY, UNIT 6: CONICS.

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2nd Year Mathematics 2nd 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. The separate linear equations for the pair of lines  are:

    1. 6x+y=0 and x-4y=0

    2. 2x+y=0 and 3x-4y=0

    3. 2x-y=0 and 3x+4y=0

    4. 3x+y=0 and 2x-4y=0

  2. If y=cx-1. then its corresponding differential equation is:

    1. x=1+y

    2. =x+y

    3. x=y-1

    4. =cy

  3. During the derivation of the standard equation of a horizontal ellipse, what is the defined algebraic relationship between constants a,b and c?

    1. b2=a2 +c2

    2. b2=c2 -a2

    3. b2=a2 - c2

    4. b2=a-c

  4. The point where all three medians of a triangle intersect each other is called the:

    1. Orthocenter

    2. Circumcenter

    3. Incenter

    4. Centroid

  5. Inflation in economics means that the price of goods is:

    1. Decreasing over time

    2. Increasing over time

    3. Remaining constant

    4. Becoming zero

  6. According to Kepler's first law, the orbits of the planets in the solar system follow which geometric shape?

    1. Circular

    2. Parabolic

    3. Elliptical

    4. Hyperbolic

  7. Who first developed analytical geometry in the 17th century?

    1. Isaac Newton

    2. René Descartes

    3. Pythagoras

    4. Euclid

  8. The differential equation =ex-y is:

    1. Homogeneous but not separable

    2. Separable but not homogeneous

    3. Both separable and homogeneous

    4. Neither separable nor homogeneous

  9. For the elliptical flower bed with a major axis of 10 m and minor axis of 6 m, what is its semi-major axis (a) and semi-minor axis (b)?

    1. a=10, b=6

    2. a=5, b=3

    3. a=25, b=9

    4. a=6, b=10

  10. In a triangle PQR with vertices P(1, 2), Q(5, -6), and R(7, -2), what are the coordinates of the midpoint of side QR through which the median from P passes?

    1. (3,-4)

    2. (6,-2)

    3. (6,-4)

    4. (1,-4)

  11. A differential equation of degree 1 is specifically called a:

    1. First-order differential equation

    2.  First-degree differential equation

    3. Linear differential equation

    4. Homogeneous differential equation

  12. A line that is perpendicular to a tangent line at its exact point of contact is defined as the:

    1. Normal

    2. Asymptote

    3. Directrix

    4. Axis

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

  1. Write the order and degree (if ddefined) of the following differential equation: 

  2. Find the equations of the circles passing through the points A(3,4) and B(3,-4) and touching the line x+2y+9=0.

  3. Define analytical geometry.

  4.  Solve the initial value problem: 

  5. An ellipse has the following properties: The centre is at the point (3,-2). Find the standard equation of this ellipse.

  6. Write down an equation of the straight line which cuts the x-axis at (3,0) and y-axis (0,-6).

  7. A cup of coffee is served at  in a room where the temperature is . After 10 minutes the coffee cools to . Find the temperature of the coffee after 20 minutes.

  8. Find the equations of the tangent and normal to the circles: .

  9. Write the practical applications of analytical geometry.

  10. A radioactive isotopes has a half-life of 12 hours. The initial mass is 160 grams. How much mass will remain after 36 hours? 

  11. A landscaper wants to create an elliptical flower bed with a major axis of 10 meters and a minor axis of 6 meters. Using the gardener's method, two stakes are placed at the foci, and a loop of string is stretched around them to trace the ellipse: (a) How far apart should the two stakes be placed? (b) What should be the total length of the string loop?

  12. Triangle ABC has vertices A(2,3), B(4,5) and C(-2,7). Find the equations of medians of the triangle ABC.