12th Math

Chapter 2 (Differentiation) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 10
Type: Notes

Is there actually a way to “unlock math” for ourselves and learn the subject quickly? It is a universal and undebatable knowledge that mathematics is a darn difficult subject.

TITLESOLUTION
Exercise 2.1View
Exercise 2.2 View
Exercise 2.3 View
Exercise 2.4 View
Exercise 2.5 View
Exercise 2.6 View
Exercise 2.7 View
Exercise 2.8 View
Exercise 2.9 View
Exercise 2.10 View

In this chapter we will read these topics:

  • Name the mathematicians who invented different calculus.
  • Define dependent and independent varaibles.
  • Define the derivative of a function.
  • What is differentiation?
  • Differentiation by definition.
  • The first principle or AB-Initio Method.
  • Write different notations used for derivatives by different mathematicians.
  • Find derivative when n is a positive integer.
  • Find derivative when n is a negative integer.
  • Find derivative when n is zero.
  • Find from the definition, the differential coefficient of the given variable.
  • Find from the 1st principle.
  • Theorems on differentiation
  • State and prove the theorem of derivatives.
  • State and prove the difference theorem of the derivative.
  • State and prove the product theorem of derivatives.
  • State and prove the quotient theorem of derivatives.
  • State and prove the reciprocal rule of the derivative.
  • The Chain Rule.
  • A derivative of Parametric equations.
  • Define Parameter.
  • Define Parametric equations.
  • Define explicit and implicit functions.
  • A derivative of Implicit Relations.
  • Differentiation of trigonometric functions.
  • Prove by the ab-initio method.
  • Find derivatives from the first principle.
  • Derivation of inverse trigonometric functions.
  • Differentiation of exponential functions.
  • Define the exponential function.
  • Define general exponential function.
  • Define natural exponential function.
  • Differentiation of logarithmic functions.
  • Define the general logarithm function.
  • Define common logarithm function.
  • Define natural logarithm function.
  • Differentiation of hyperbolic functions.
  • Define hyperbolic functions.
  • A derivative of inverse hyperbolic functions.
  • Higher derivatives.
  • First derivative
  • Second derivative
  • Third derivative.
  • Fourth derivative.
  • Nth derivative.
  • Higher derivative.
  • Series expansions of functions
  • Define Maclaurin’s series expansion of a function.
  • State and prove Maclaurin’s series expansion of a function.
  • Find Maclaurin’s series for sinx.
  • State and prove Taylor’s series expansion of a function.
  • Geometrical meaning of derivative.
  • State the geometrical interpretation of derivative.
  • Define the tangent line.
  • Prove that geometrically derivative of a function gives the slope of a tangent line.
  • Discuss the tangent line to the graph of function |x| at x=0.
  • Define increasing function.
  • Define a decreasing function.
  • Define a constant function.
  • Define an increasing function in terms of derivatives.
  • Define decreasing function in terms of derivatives.
  • Define constant function in terms of derivatives.
  • Define stationary point.
  • Define critical value and critical point.
  • Define turning point of a function.
  • Define point of inflection.
  • Define relative minima of a function.
  • Define relative maxima of a function.
  • What is the relative extrema of a function.
  • Define first derivative test.
  • Applications of maxima and minima.
  • Find two positives integers whose sum is 9 and the product of one with the square of the other will be maximum.
  • What are dimensions of a box of a square base having largest volume if the sum of one side of the base and height is 12cm.
  • The perimeter of a triangle is 20cm. If one side is of the length 8cm, what are length of the other sides for the mum area of the triangle?
  • FInd the two positive integers whose sum is 30 and their product will be maximum.

Chapter 3 (Differentiation) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 8
Type: Notes

TITLESOLUTION
Exercise 3.1View
Exercise 3.2 View
Exercise 3.3 View
Exercise 3.4 View
Exercise 3.5 View
Exercise 3.6 View
Exercise 3.7 View
Exercise 3.8 View

In this chapter we will read these topics:

  • What is the differential of a function f(x) and what is the differential coefficient?
  • What is the difference between delta(y) and dy?
  • Use differential to find approximate value.
  • Find the approximate increase in the volume of a cube of the length its each edge changes from 5 to 5.02.
  • Find the approximate increase in the area of a circular disc if its diameter is increased from 44com to 44.4.
  • Define integration.
  • Some basic tips for integration.
  • Define the indefinite integral.
  • Write two theorems on anti-derivatives.
  • Some basic formulae.
  • Evaluate the problems.
  • Write a note on integration by substitution.
  • Write some important trigonometric substitutions.
  • Integration by parts.
  • Integration by partial fractions.
  • When denominator contains non-repeated linear factors.
  • When denominator contains repeated linear factors.
  • When denominator contains non-repeated irreducible quadratic factors.
  • Define definite integral.
  • Define the fundamental theorem of integral calculus.
  • Area under the curve.
  • Find the area bounded by the curve.
  • Find the area between the x-axis and the curve.
  • Define the differential equations.
  • Define the order of a differential equation.
  • Define the degree of a differential equation.
  • Define solution of a differential equation.
  • Define general solution of a differential eqaution.
  • Define particular solution of a differential equation.
  • How many constants are involved in general solution of first-order differential equation.
  • How many constants are involved in general solution of nth order.
  • Solve the following differential equations.

Chapter 4 (Introduction to Analytic Geometry) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 5
Type: Notes

TITLESOLUTION
Exercise 4.1View
Exercise 4.2 View
Exercise 4.3 View
Exercise 4.4 View
Exercise 4.5 View

In this chapter we will read these topics:

  • What are the meanings of the word geometry?
  • Who was the founder of geometry.
  • What is Plane geometry.
  • What is Solid geometry.
  • What is analytic geometry?
  • Define a coordinate plane and coordinate axes.
  • Define abscissa and ordinate of a point.
  • Define quadrants.
  • State and prove distance formula.
  • Distance of point P(x,y) from x-axis is |y| and from y-axis is |x|.
  • Distance measured non-parallel to coordinate axes is called undirected distance.
  • Distance measured parallel to the coordinate axes is called directed distance.
  • State and prove the Ration formula or Division formula.
  • State and prove the Ration formula/Division formula for external division.
  • Derive mid-point formula from ratio formula.
  • Define Median of a triangle.
  • What do you mean by concurrent lines.
  • Define Centroid of a triangle.
  • Show that medians of a triangle are concurrent.
  • What are coordinates of centroid of a triangle.
  • Define angle bisector and in-center of a triangle.
  • Define the translation of axes.
  • Write the equations of translation of axes.
  • Define the rotation of axes.
  • Write the equation of axes.
  • Write the equation of rotation of axes.
  • Define the inclination of a line.
  • Define the slope or gradient of a line.
  • Write formula for slope of a line passing through two points.
  • What is the condition of two parallel lines.
  • What is the condition of two perpendicular lines.
  • When three points are said to be colinear.
  • What is the equation of line parallel to x-axis.
  • What is the equation of line parallel to y-axis.
  • What do you mean by intercepts.
  • State and prove the equation of line in slop-intercept form.
  • Types of lines.
  • State and prove equation of line in point-slope form.
  • Derive the symmetric form of a line.
  • State and prove equation of line in two-points form.
  • Equation of non-vertical straight line passing through two points.
  • State and prove equation of line in intercept form.
  • State and prove equation of line in normal form.
  • Describe the position of a point with respect to line.
  • Point of intersection and condition of concurrent.
  • Prove that altitudes of a triangle are concurrent.
  • Prove that the right bisector of triangle are concurrent.
  • Angle between two lines.
  • Two lines l1 and l2 are parallel if and only if m1=m2.
  • Find point of intersection of the lines.
  • Find equation of line.
  • Find acute angle.
  • Find interior angles of triangles.
  • Find interior angle of the quadrilateral.
  • Find the center of the circumcircle of the triangle.
  • Define the homogeneous equation of degree n in x and y variables.
  • Define general homogeneous equation of second degree in x and y variables.
  • Find a joint equation of the lines through the origin and perpendicular to the lines.
  • Find the area of the bounded region.
  • Area of tapezium

Chapter 7 (Vectors) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 5
Type: Notes

TITLESOLUTION
Exercise 7.1View
Exercise 7.2 View
Exercise 7.3 View
Exercise 7.4 View
Exercise 7.5 View

In this chapter we will read these topics:

  • Define scalar quantity.
  • Define the vector quantity.
  • Give the geometrical interpretation of a vector quantity.
  • Define the magnitude of a vector.
  • Define unit vector.
  • Define null vector.
  • Define negative of a vector.
  • Define the multiplication of a vector by a scalar.
  • Define equal vectors.
  • Define parallel vectors.
  • Define triangular law of the addition of vectos.
  • What is the parallelogram law of vector addition?
  • Describe the subtraction of two vectors.
  • Define the position vector.
  • Describe vector in a plane.
  • Discuss the addition and subtraction of two vectors in a plane.
  • Define the magnitude of a vector in R2.
  • State and prove ratio formula.
  • Introduction of a vector in space.
  • Discuss the addition and subtraction of two vectors in space.
  • Find the magnitude, length, a norm of a vector in space.
  • Properties of a vector in space.
  • State and prove a commutative property of addition of vector in space.
  • State and prove the associative property of addition of vectors in space.
  • State and prove the additive inverse of a vector in space.
  • State and prove the distributive property of a vector in space.
  • State and prove scalar multiplication property of a vector in space.
  • Write the form of the vector in space.
  • State and prove distance formula.
  • Define direction angles and direction cosines of a vector.
  • Scalar product of two vectors.
  • Define scalar and dot product.
  • Define the scalar product of vectors in a plane.
  • Define the scalar product of vectors in a space.
  • Define perpendicular vectors.
  • Show that the zero vector is perpendicular to every vector.
  • Properties of dot product.
  • Define the angle between two vectors.
  • Projection of a vector upon another.
  • Cross product or vector product.
  • Define the cross product of two vectors.
  • Define the right-hand rule.
  • Important results of the cross product.
  • The analytical expression of cross products.
  • Properties of cross product.
  • Define the area of a parallelogram.
  • Define area of a triangle.
  • Scalar triple product of vectors.
  • Define the scalar triple product of three vectors.
  • Define parallelepiped.
  • Define tetrahedron.
  • Give the geometrical interpretation of the scalar triple product.
  • Define the volume of a tetrahedron.
  • Properties of scalar triple product.
  • Define coplanar lines.
  • Find the volume of a tetrahedron.
  • Define words done by a force.
  • Define moment of arm.

Chapter 6 (Conic Section) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 9
Type: Notes

TITLESOLUTION
Exercise 6.1View
Exercise 6.2 View
Exercise 6.3 View
Exercise 6.4 View
Exercise 6.5 View
Exercise 6.6 View
Exercise 6.7 View
Exercise 6.8 View
Exercise 6.9 View

In this chapter we will read these topics:

  • Define cone, vertex of cone and axis of cone.
  • Define nappes of cone and types of cone.
  • Define Conic or Conic section.
  • Define Circle.
  • Define Point Circle.
  • Define Ellipse.
  • Define Parabola.
  • Define Hyperbola.
  • What is the difference between the diameter and chord of a circle?
  • Derive the standard equation of circle.
  • Find the equation of circle having center at origin and radius r.
  • Find the parametric equation of circle having radius r and center at origin.
  • When two circles touch each other externally.
  • When two circles touch each other internally.
  • Tangent to a circle is perpendicular to radius.
  • Tangents and normals.
  • Define tangent and normal to a curve.
  • Find equation of tangent and normal to circle.
  • Normal line is perpendicular to the line.
  • Condition of tangency.
  • Tangential distance.
  • Equation of chord of contact.
  • Analytic proofs of important properties of circle.
  • Perpendicular dropped from the center of a circle on a chord bisects the chord.
  • The perpendicular bisector of any chord of a circle passes through the center of the circle.
  • The line joining the center of a circle to the mid-point of a chord is perpendicular to the chord.
  • The slope of chord.
  • Congruent chords of a circle are equidistance from center.
  • Show that the measure of the central angle of minor arc is double the measure of the angle subtended in the corresponding major arc.
  • An angle in a semi-circle is a right angle.
  • The tangent to a circle at any point of the circle is perpendicular to the radial segment at that point.
  • Perpendicular at outer end of a radial segment is tangent to circle.
  • Define parabola.
  • Derive the standard equation of parabola y2=4ax.
  • Derive the general equation of a parabola.
  • Write the relationship between the second-degree equation and parabola.
  • Define focus, directrix, axis and vertex of parabola.
  • Define tangent at vertex and chord of parabola.
  • Define the focal chord and latus rectum of the parabola.
  • What do you mean by the eccentricity of a parabola?
  • Prove that length of latus rectum of a parabola is 4a.
  • Point of a parabola which is closest to the focus is the vertex of parabola.
  • Length of latus rectum.
  • Define Reflecting property of parabola.
  • Define ellipse.
  • The fixed point is called the focus.
  • Define the diameters of an ellipse.
  • Define major axis of an ellipse and length of major axis.
  • Define minor axis of an ellipse and length of minor axis.
  • Define directrices of an ellipse.
  • Define the Foci of an ellipse.
  • Define vertices of an ellipse.
  • Define co-vertices of an ellipse.
  • Define chord of an ellipse.
  • Define focal chord of an ellipse.
  • Define latus rectum and length of latus rectum of an ellipse.
  • Define center of an ellipse.
  • Define semi-major axis and semi-minor axis of an ellipse.
  • Write the parametric equations of an ellipse.
  • Derive the equation of directrix of an ellipse.
  • Define hyperbola.
  • Define center of Hyperbola.
  • Define focal axis or transvers axis of Hyperbola.
  • Define the conjugate axis of Hyperbola.
  • Define the vertices of Hyperbola.
  • What is length of latus rectum of Hyperbola?
  • Write the parametric equation of Hyperbola.
  • Which conics re called conics.
  • Discuss the sketch graph of the equation.
  • Tangents and normals.
  • Condition of tangency.
  • Find the point of intersection ellipse and hyperbola.
  • Find the point of intersection of conics.
  • Define the translation of axes.
  • Write the equation of the translation of axes.
  • Define the rotation of axes.
  • Write the equation of rotation of axes.
  • Transformed equation.
  • Define the general equation of second degree.
  • When the general equation of second degree represents a circle.
  • When the general equation of second degree represents a parabola.
  • When the general equation of second degree represents a ellipse.
  • When the general equation of second degree represents a hyperbola.
  • Classification of conics by discriminant.
  • Discuss the nature of conics.
  • What is degenerate conic?
  • Elements of the ellipse.
  • Equation of major axis.
  • Equation of minor axis.
  • Equation of transformation.
  • Elements of the hyperbola.
  • Equation of focal axis.
  • Equation of the conjugate axis.