2nd Year Math Chapter 4 Notes

Handwritten notes of 2nd Year Math Chapter 4 Notes written by Professor Asad Khalid Suib. These notes are very helpful in the preparation of 2nd year math chapter 4 solution for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.

• Important 2nd year math chapter 4 mcqs for Intermediate part-II students.
• 1. The symbols used for inequality are
• (a) 1 (b)  2 (c)  3 (d)   4 
• 2. An inequality with one or two variables has ___solutions.
• (a) One (b)  two (c)   three (d)   infinitely many 
• 3. ax+by
• (a)  a=0,b=0 (b) a≠0 ,b≠0 (c) a=0,b≠0 (d) a≠0,b=0,c=0
• 4. The graph of corresponding linear equation of the linear inequality is a line called____:
• (a) Boundary line (b) horizontal line (c) vertical line (d) inclined line
• Important 2nd Year Math Chapter 4 pdf for Intermediate part-II students.
• 5. The graph of a linear equation of the form ax+by=c is a line which divides the whole plane into ___disjoints parts.
• (a)  Two (b) four (c) more than four (d) infinitely many
• 6. The graph of the inequality x≤b is
• (a) Upper half plane (b) lower half plane (c) left half plane  (d) right half plane
• 7. The graph of the inequality y≤b is
• (a) Upper half plane (b) lower half plane  (c) left half plane (d) right half plane
• 8. The feasible solution which maximizes or minimizes the objective function is called
• (a) Exact solution          (b) optimal solution  (c) final solution (d) objective function
• 9. Solution space consisting of all feasible solutions of system of linear in inequalities is called
• (a) Feasible solution (b) Optimal solution (c) Feasible region  (d) General solution
• 10. Corner point is also called
• (a) Origin (b) Focus (c) Vertex            (d) Test point
• 11. For feasible region:
• (a)  x≥0,y≥0   (b)   x≥0,y≤0 (c) x≤0,y≥0 (d) x≤0,y≤0
• 12. x=0 is in the solution of the inequality 
• (a)  x<0              (b)   x+4<0 (c)   2x+3>0      (d)  2x+3<0
• 13. Linear inequality 2x-7y>3 is satisfied by the point
• (a) (5,1) (b)   (-5,-1) (c)  (0,0) (d)  (1,-1) 
• 14. The non-negative constraints are also called
• (a) Decision variable    (b) Convex variable     (c) Decision constraints (d) concave variable
• Important 2nd year maths chapter 4 mcqs with answers for Intermediate part-II students.
• 15. If the line segment obtained by joining any two points of a region lies entirely within the region , then the region is called
• (a) Feasible region (b) Convex region             (c) Solution region (d) Concave region
• 16. A  function which is to be maximized or minimized is called:
• (a) Linear function (b)  Objective function     (c) Feasible function   (d) None of these
• 17. For optimal solution we evaluate the objective function at 
• (a)  Origen (b) Vertex (c)  Corner Points (d) Convex points
• 18. We find corner points at
• (a)  Origen (b) Vertex (c)  Feasible region    (d) Convex region 
• 19. The set of points which are equal distance from a fixed point is called:
• (a)  Circle             (b) Parabola            (c) Ellipse          (d) Hyperbola
• 20. The circle whose radius is zero is called:
• (a)  Unit circle (b)  point circle      (c) circumcircle (d) in-circle
• 21. The circle whose radius is 1 is called:
• (a)  Unit circle    (b) point circle (c) circumcircle (d) in-circle
• Important 2nd year math chapter 4 important questions for Intermediate part-II students.
• Introduction: Geometry is one of the most ancient branches of mathematics. The Greeks systematically studied it about four centuries B.C. Most of the geometry taught in schools is due to Euclid who expounded thirteen books on the subject (300 B.C.). A French philosopher and mathematician Rene Descartes (1596-1650 A.D.) introduced algebraic methods in geometry which gave birth to analytical geometry (or coordinate geometry). Our aim is to present fundamentals of the subject in this book.
• Important 2nd year math exercise 4.1 solution pdf for Intermediate part-II students.
• Coordinate System: Draw in a plane two mutually perpendicular number lines x' * x and y' * y , one horizontal and the other vertical. Let their point of intersection be O which we call the origin and the real number 0 of both the lines is represented by O. The two lines are called the coordinate axes. The horizontal line x' * Ox is called the x-axis and the vertical line y' * 0y is called the y- axis.
• Important 2nd year math chapter 4 for Intermediate part-II students.
• As in the case of number line, we follow the convention that all points on the y-axis above x' * Ox are associated with positive real numbers, those below x' * Ox with negative real numbers. Similarly, all points on the x-axis and lying on the right of O will be positive and those on the left of O and lying on the x-axis will be negative.
• Important 2nd year math exercise 4.1 solution for Intermediate part-II students.
• Suppose P is any point in the plane. Then P can be located by using an ordered pair of real numbers. Through P draw lines parallel to the coordinates axes meeting x-axis at R and y-axis at S. Let the directed distance overline OR = x and the directed distance overline OS = y.
• Important 2nd year maths chapter 4 exercise 4.1 for Intermediate part-II students.
• Important 2nd year maths chapter 4 exercise 4.2 for Intermediate part-II students.
• Important 2nd year maths chapter 4 exercise 4.3 for Intermediate part-II students.
• Important 2nd year maths chapter 4 exercise 4.4 for Intermediate part-II students.