LINEAR EQUATION IN TWO VARIABLES CLASS 9

1.     Find four different solutions of the equation x+2y=6.

2.     Find two solutions for each of the following equations:
(i) 4x + 3y = 12 
(ii) 2x + 5y = 0
(iii) 3y + 4=0

3.     Write four solutions for each of the following equations: 
(i) 2x + y = 7 
(ii) πx + y = 9 
(iii) x = 4y.

4.     Given the point (1, 2), find the equation of the line on which it lies. How many such equations are there?

5.     Draw the graph of the equation
(i) x + y = 7
(ii) 2y + 3 = 9
(iii) y - x = 2
(iv) 3x - 2y = 4
(v) x + y - 3 = 0

6.     Draw the graph of each of the following linear equations in two variables:
(i) x + y = 4
(ii) x - y = 2
(iii) y = 3x 
(iv) 3 = 2x + y
(v) x - 2 = 0
(vi) x + 5 = 0
(vii) 2x + 4 = 3x + 1.

7.     If the point (3, 4) lies on the graph of the equation 3y=ax+7, find the value of ‘a’.

8.     Solve the equations 2x + 1 = x - 3, and represent the solution(s) on
(i) the number line,
(ii) the Cartesian plane.

9.     Draw a graph of the line x - 2y = 3. From the graph, find the coordinates of the point when
(i) x = - 5 
(ii) y = 0.

10.   Draw the graph of y = x and y = - x in the same graph. Also, find the coordinates of the point where the two lines intersect.

Linear Equation in two variables class IX

HERON'S FORMULA FOR CLASS IX

1. Find the area of an equilateral triangle one of whose sides measure a cm, using heron’s formula

2.Find the area of a triangle, two sides of which are 18cm, and 10cm and the perimeter is 42cm

3. An isosceles triangle has perimeter 30cm and each of the equal sides is 12cm. Find the area of the triangle

4. Sides of a triangle are in the ratio 12: 17: 25 and its perimeter is 540 cm. Find its area

5. A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer  diagonal is 48 m, how much area of grass field will each cow be grazing

6. Two parallel sides of a trapezium are 60 cm and 77 cm and the other sides are 25 cm 26 cm. find the area of the trapezium

7. Two parallel sides of a trapezium are 120cm and 154cm and other sides are 50cm and 52cm. Find the area of the trapezium                                                                                                                                                                             

8. The perimeter of a right triangle is 12 cm and its hypotenuse is 5 cm. Find its area                                                 

9. The lengths of two adjacent sides of a parallelogram are 51cm and 37cm and one of its diagonal is 20cm.Find Its area                                                                                                                                                                                       

10. A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26cm, 28cm and 30cm and the parallelogram stands on the same base 28cm, find the height of the parallelogram

11. Find the area of a quadrilateral ABCD in which AB = 3cm, BC = 4cm, CD = 6cm, DA = 5cm and diagonal AC = 5cm                                                                                                                                                                                                                                                                                                                                                                     

12. Find the area of a quadrilateral PQRS if PQ = 8cm, QR = 6cm, RS = 14cm, PS = 16cm and diagonal PR= 10cm                                                                                                                                                                  

13. Find the semi-perimeter of a triangle whose sides are 26cm, 28cm and 30cm                 

14.The perimeter of a triangle is 240cm. If two of its sides are 78cm and 50cm. Find the length of perpendicular on the sides of length 50cm from opposite vertex                                                                                                         

                                                    

CH- POLYNOMIALS for class IX

        1) Find the zeros of the polynomial            (a) x² + 7x + 10                   (b) x² – 25                                                                       

        2) Using factor theorem, Show that (a - b) is the factor of  a (b²-c²) +b (c²-a²) +c (a²-b²)                      

3) Factorize:  (a) 4√3x² + 5x - 2√3                  (b) 21x² – 2x + 1/21            (c)   9(2a – b)² – 4(2a – b) –13  

4) Factorize:  a) 999² - 1           b) (10.2)³               c) 1002 X 998                                      

5) Factorize: a) x³ – 3x² – 9x – 5                              b) x³ + 7x² – 21x – 27

  6) Factorise: (a) 3x² + 27y² + z² - 18xy +6 √3yz -2 √3zx                   (b) 27 x³  + 125y³              ( c)  (2a – 3b + c )²                

                       (d)  [x – 1/x y]³                                            (e)  x⁴ y⁴ – x y                                                             

                        (f) 8x³ – (2x – y)³                                        (g) 27 a⁶  - 1                                                                                                                                                                           

7) For what value of a is 2x³ + ax² + 11x + a + 3 exactly divisible by (2x – 1)                                                                                                                                    

8) If x – 2 is a factor of a polynomial f(x) = x⁵ – 3x⁴ – ax³ + 3ax² + 2ax +4, then find the value of a 

9) Find the value of a and b so that x² – 4 is a factor of ax⁴ + 2x³ – 3x² + b x – 4                                                                                    

10) Find the value of a and b so that polynomial x³ – ax² -13x + b is exactly divisible by (x-1) as well as (x+3)                              

11)  The polynomial  x³ – mx² +4x + 6 when divided by (x+2) leaves remainder 14 find the value of m

12) If the polynomial ax³ + 3x² -13 and 2x³ – 5x + a when divided by (x – 2) leave the same remainder, find the Value of “a”                                                                                                                                                                        

13) If both (x – 2) and (x – ½) are factors of px² + 5x + r, show that p = r                                                                              

14) If f(x) = x⁴ – 2x³ + 3x² – ax + b is divided by x-1 and x+1 the remainders are 5 and 19 respectively, then find a and b   

15) Show that x + 1 and 2x – 3 are factors of 2x³ – 9x² + x + 12

16) By using suitable identity, find the value of:

 (a) (-6)³ + 13³+ (-7)³                    (b) (-21)³ + (28)³             (c) (9.8)³ – (11.3)³ + (1.5)³         (d) (8/15)³  +  (-1/3)³   +    (-1/5)³            

17) Find the remainder when f(x) = 4x³ – 12x² + 14x – 3 is divided by g(x) = x –  1                                                                         

18) Find the remainder when x⁵¹ + 51 is divided by x + 1                                                                                                                                

20) Find the remainder when x³ – px² + 6x – p is divided by (x–p)                                                                                                              

21) Find the value of x³ + y³ + 15xy – 125 when x + y = 5                                                                                                                                  

22) Find the value of p³ – q³ , if p – q  = 5/7 and p q = 7/3                                                                                                                  

23) If a + b + c =8, a² + b² + c² = 30. Find the value of a b + b c + c a                                                                                                                 

24) If 2x + 3y = 13 and x y = 6 then, find 8x³ + 27y³                                                                                                                                        

25) Find the value of a³+b³+c³–3abc, when a+b+c=8 and ab+bc+ca=25                                                                                    

26) Find the value of x³ + y³ + z³ – 3xyz, if x+y+z = 12 and x² + y² + z² =70                                                                                                

27) If x + y + z = 1, x y + y z + z x = -1 and xyz = -1, find the value of x³ + y³ +z³                                                                                             

28) If (a + b)² = 2a² + 2b², show that a = b                                              

29) If (a + b + c) = 0, then prove that a³ + b³ + c³ = 3abc

30) Prove that 2x³ +2y³ + 2z³ – 6xyz =(x + y + z) [(x – y )² + ( y – z )² + ( z – x )² ]

31) Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given   25a² – 35a + 12

32)Simplify : (a + b + c )² + (a – b + c )² + ( a + b  - c)²                                                        

33)What must be subtracted from 4x⁴ – 2x³ – 6x² + x – 5, so that the result is exactly  divisible by 2x²  +  x  -  1                               

34)Find the dimensions of a cuboid, whose volume is 2py² + 6py - 20p                                                                                   

35) If p(x) = x² – 4x + 3, evaluate p (2) – p (-1) + p (1/2)                                                   

CH- NUMBER SYSTEM CLASS IX

Test of algebra and linear equation

Algebra

Q .Solve and check the solution in following  LINEAR EQUATIONS :-