This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD

1. LINEAR AND QUADRATIC EQUATIONS IN QUANTITATIVE APTITUDE by : DR. T.K. JAIN AFTERSCHO ☺ OL centre for social entrepreneurship sivakamu veterinary hospital road bikaner 334001 rajasthan, india FOR – PGPSE / CSE PARTICIPANTS mobile : 91+9414430763

2. My words.... Here I present a few basic questions on linear and quadratic equations. I wish that more people should become entrepreneurs. An ordinary Indian entrepreneur wishes to remain an honest entrepreneur and contribute to the development of nation but we have to strengthen those institutions which truly promote entrepreneurship, not just degree granting institutions. Let us work together to promote knowledge, wisdom, social development and education. We believe in free education for all, free support for all, entrepreneurship opportunities and training for all. Let us work together for these goals. ... I alone cant do much, I need support of perosns like you .......... ...

3. What is an equation? Equal = the two sides must be equal, so equation generally puts a mathematical structure equal to 0 or equal to Y or some other variable The beginning letters of the alphabet a, b, c, etc. are typically used to denote constants, while the letters x, y, z , are typically used to denote variables. For example, if we write y = ax² + bx + c, we mean that a, b, c are constants (i.e. fixed numbers), and that x and y are variables.

4. What is quadratic equation? Generally, when the highest power of x is 2 in an equation, it is called quadratic equation, and when the highest power of x is 1, it is linear equation. In a linear equation, we have relationship between two variables (x and Y). y = 2x + 6 This is called an equation of the first degree. It is called that because the highest exponent is 1. (exponent = power of x)

5. Which of the following ordered pairs solve this equation: y = 3x − 4 ? Options (0, −4) (1, 2) (1, −1) (2, −3) solution (0, −4) and (1, −1). Because when x and y have those values, the equation is truegive you solution as when you put the value of x =0 you get y =-4 and when you put value of x =1, you get y=2.

6. Which of these ordered pairs solves the equation y = 5x − 6 ? (1, −2) (1, −1) (2, 3) (2, 4) solution : (1,-1) and (2,4) give you solution as when you put the value of x =1 you get y =-1 and when you put value of x =2, you get y=4.

7. Form an equation, where roots are 7 and -3? Format of equation is : x^2 – (sum of roots) x+ (product of roots) = 0 sum of roots = 4 product of roots = -21 thus answer = =x^2 -4x-21 = 0

8. Form an equation, where roots are 3 and -2? Format of equation is : x^2 – (sum of roots) x+ (product of roots) = 0 sum of roots = 1 product of roots = -6 thus answer = =x^2 -x-6 = 0

9. Form an equation, where roots are -12 and -7? Format of equation is : x^2 – (sum of roots) x+ (product of roots) = 0 sum of roots = -19 product of roots = 84 thus answer = =x^2 +19x +84= 0

10. Form an equation, where roots are 2 and 27? Format of equation is : x^2 – (sum of roots) x+ (product of roots) = 0 sum of roots = 29 product of roots = 54 thus answer = =x^2 -29x +54= 0

11. Which of these is a root of this equation : x^4 – 10x^2 +9 =0? Options : 21, 0, -4 ,- 3 try with options, when we put the value of X as -3, we get the answer, so answer = -3

12. Which of these is a root of this equation : 8x^2-22x-21=0? Options : 3/4, 5/2, 7/2, 9/2 try with options, when we put the value of X as 7/2 , we get the answer, so answer = 7/2

13. What is the discriminant of this equation : x^4 – 10x^2 +9 =0? Discriminant =b^2 – 4ac in this equation a = 1, b=-10 and c = 9 =100 – 4(1*9) =64 answer

14. What are the roots of 2x^2 -5x -4 =0 There are two roots in quadratic equations alpha and beta alpha = (-b-sqrt(discriminant)) / 2a beta = (-b+sqrt(discriminant)) / 2a discriminant = 25 – 4(-8) = 57 alpha = (5- (sqrt(57)) / (2*2) =3.11 BETA = 6.89 ANSWER

15. What are the roots of x^2 -8x -21 =0 There are two roots in quadratic equations alpha and beta alpha = (-b-sqrt(discriminant)) / 2a beta = (-b+sqrt(discriminant)) / 2a discriminant = 64 – 4(-21) =148 alpha = (8- (sqrt(148)) / (2) alpha = 2 and beta = 14 (approximate)

16. What are the roots of x^2 -8x+21 =0 There are two roots in quadratic equations alpha and beta alpha = (-b-sqrt(discriminant)) / 2a beta = (-b+sqrt(discriminant)) / 2a discriminant = 64 – 4(21) =-20 here both the roots are imaginary because the discriminant is negative. (-20) answer

17. How many different roots are possible in this equation: x^2 -8x+16 =0 There are two roots in quadratic equations alpha and beta alpha = (-b-sqrt(discriminant)) / 2a beta = (-b+sqrt(discriminant)) / 2a discriminant = 64 – 4(16) =0 here we have only one root because discriminant is zero, so both the roots are same numbers.

18. How many solutions are possible from the following equations : 3x-2y=1 and 6x-4y=2 We know that : a1/a2 = b1/b2 = c1/c2, then there are infinite number of solution, in these two equations, we have =3/6 = 2/4 = 1/2, so we have infinite number of solutions

19. How many solutions are possible from the following equations : 3x-2y=1 and 6x-4y=9 We know that : a1/a2 = b1/b2 >< c1/c2, (c1/c2 is not equal to the first two ) then there is no solution , in these two equations, we have =3/6 = 2/4 not equal to 1/9 so we dont have any solution

20. How many solutions are possible from the following equations : 3x-2y=1 and 6x-3y=9 We know that : a1/a2 is not equal to b1/b2 then there is only one solution let us multiply the first equation by 2 and find the difference : y = 7 and X = 5

21. Kx^2 +2x +3k = 0 clues : 1. sum of roots is equal to product of roots. Can you guess k ? Sum of roots = -b/a product of roots = c / a here sum of root is = -2/k and product of root is 3k/k -2/k = 3k/k =-2/k = 3 k = -2/3 answer

22. There are two numbers. Their product is 782 and their sum is 57, can you guess the numbers ? Options : 43 & 14, 44 & 13 , 34 & 23 24&33 answer : 34 & 23

23. 3x^2 +11x+k=0 the roots are reciprocal to each other. Can you guess what is k ? Product of two reciprocals is always 1. We know that product of roots = c/a, which is k/3 roots are reciprocal so c/a = 1 k/3 =1, so k =3 answer

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