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Punjab (PTB) · Half book test (board pattern) · Free

2nd Year Mathematics First half Book Paper

70 marks · 3 hours · 57 questions. Covers: 6 chapters (UNIT 1: GRAPHICAL REPRESENTATION OF FUNCTIONS to UNIT 6: CONICS).

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2nd Year Mathematics First half Book Paper

2nd Year · Mathematics · Punjab (PTB)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 70Time: 3 hours

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

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

    1. x=1+y

    2. =x+y

    3. x=y-1

    4. =cy

  2. The logarithmic function y=loga x is continuous for:

    1. All x>0

    2. All real numbers

    3. All negative real numbers

    4. Integers only

  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 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

  5. When finding the global extrema of a function on a closed interval [a,b], at which points should you first evaluate the function?

    1. Only at the point x=0

    2. Only at the midpoint of the interval

    3. At all critical points and endpoints of the interval

    4. At infinity

  6. The value of a definite integral is always a:

    1. Family of functions

    2. Real number

    3. Variable quantity

    4. Imaginary number

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

    1. Decreasing over time

    2. Increasing over time

    3. Remaining constant

    4. Becoming zero

  8. A sound has an intensity of  and a reference intensity of  Find the sound level in decibels (dB):

    1. 50 dB

    2. 60 dB

    3. 70 dB

    4. 80 dB

  9. 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

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

    1. Orthocenter

    2. Circumcenter

    3. Incenter

    4. Centroid

  11. Find for the parametric equations x=sin2 t and y= cos 2t:

    1. =-2

    2. =2

    3. = - tan t

    4. =- 4sint cot t

  12. What is the primary objective of using a trigonometric substitution in integration?

    1. To increase the degree of the polynomial

    2. To convert a definite integral into an indefinite one

    3. To eliminate the radical sign in the integrand

    4. To find the derivative of the function

  13. 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

  14. Which of the following simplifies to sech2 x?

    1. 1+tanh2x

    2. 1 - tanh2x

    3. coth2x - 1

    4. cosh x + sinh x

  15. 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

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

    1. Isaac Newton

    2. René Descartes

    3. Pythagoras

    4. Euclid

  17. Differentiation will be applied to identify which behavior of functions?

    1. ncreasing and decreasing functions

    2. Domain and range

    3. Periodic behavior

    4. Continuity and limits

  18. The rule that corresponds to the Product Rule for differentiation is called:

    1. Substitution method

    2. Integration by parts

    3. Partial fractions

    4. Chain rule

  19. 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

  20. The modulus or absolute value of any real number is always:

    1. Strictly negative

    2. Zero only

    3. Non-negative

    4. An integer

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

  1. Using the second derivative test, find local extreme values of the following function: 

  2. Is the following function is transcendental? 

  3. Find  if .

  4. Define the term "Shifting of Graphs".

  5. Find  if: 

  6. Solve the equation: 

  7. Derive the formula for the derivative of the following function w.r.t. x: sec x

  8. Let . Find the function obtained after: a vertical stretch by factor 2.

  9. Find the linear approximation of the following function at the given points: 

  10. Is the following function is algebraic? 

  11. What are higher-order derivatives?

  12. Let : Verify graphically that the graphs of f(x) and Let  are symmetric about the line y=x.

Q3. Short questions (Attempt any 8 of 12 · 8 × 2 = 16)

  1. Evaluate the following integrals: 

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

  3. A patient receives an IV bolus dose of a drug. The concentration in the blood is modeled by  where t is the time in hours since that does was taken. the minimum effective concentraction (MEC) is 3 mg/L. The drug is considered effective if total AUC in a 24-hour period is at least 150 mg-h/L. Find the time T (in hours) when the concentration reaches the MEC.

  4.  Solve the initial value problem: 

  5. Evaluate the following: 

  6. 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.

  7. A patient is given an intravenous dose oof an antibiotic. The concentration in the bloodstream over time is modeled by:  where t is an hours. Find the total drug exposure (AUC) from t=0 to t=6 hours.

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

  9. Evaluate the following integrals: Evaluate: 

  10. Find the general solution of the differential equation: 

  11. Evaluate 

  12. Name any three real-life situations that can be modeled using first-order differential equations.

Q4. Short questions (Attempt any 9 of 13 · 9 × 2 = 18)

  1. Define analytical geometry.

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

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

  5. Write the practical applications of analytical geometry.

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

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

  8. 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?

  9. The vertices of  are P(1,2), Q(5,-6) and R(7,-2). Find the equation of all three medians.

  10. Find the eccentricity of the hyperbola whose asymptotes are perpendicular.

  11. Convert the following equation into: x-2y+4=0 (i) Slope intercept form (ii) Two point form (iii) Intercept form (iv) Point slope form (v) Normal form (vi) Symmetric form.

  12. Find an equation of the ellipse with foci (2,1), (2,7) major axis of length 10 and sketch its graph.

  13. Convert the following equation into: 4x+5y-20=0 (i) Slope intercept form (ii) Two point form (iii) Intercept form (iv) Point slope form (v) Normal form (vi) Symmetric form.