ALGEBRA : ALGEBRA
Learning Objectives : How to handle statement questions.
How to solve Inequalities.
How to solve Progressions.
How to form Quadratic equations.
Some actual CAT questions. Learning Objectives
Q1. If a number is as large from the smallest odd prime no. as it is smaller than the greatest two digit prime number. Find the no. : Q1. If a number is as large from the smallest odd prime no. as it is smaller than the greatest two digit prime number. Find the no.
Slide 4 : Answer is 50.
Q2. In a class, the no. of students is 300. If each boy gives Rs. 3 & each girl gives Rs. 1.50, the total collection is Rs. 600. Find the no. of boys in the class. : Q2. In a class, the no. of students is 300. If each boy gives Rs. 3 & each girl gives Rs. 1.50, the total collection is Rs. 600. Find the no. of boys in the class.
Slide 6 : Boys = 100, Girls = 200.
Q3. If x + 1/x = 11, what is the value of x2 + 1/x2 ? : Q3. If x + 1/x = 11, what is the value of x2 + 1/x2 ?
Slide 8 : Answer is 119.
Q4. If x + 1/x = 29, what is the value of x - 1/x? : Q4. If x + 1/x = 29, what is the value of x - 1/x?
Slide 10 : Answer is ± 5
Q5. Find the value of p3 + q3, given pq = 100, (p + q) = 20. : Q5. Find the value of p3 + q3, given pq = 100, (p + q) = 20.
Slide 12 : Answer is 2000.
Q6. If p = 21/3 – 2–1/3, find the value of 2p3 + 6p. : Q6. If p = 21/3 – 2–1/3, find the value of 2p3 + 6p.
Slide 14 : Answer is 3.
Q7.Solve: x3 - 2x2 + x = - 5(x – 1)2: : Q7.Solve: x3 - 2x2 + x = - 5(x – 1)2:
Slide 16 : Ans: x = -5 or x = 1.
Q8. Solve : 2x2 – 17x + 21 = 0 : Q8. Solve : 2x2 – 17x + 21 = 0
Slide 18 : Factors are 14 & 3, solve & get answers as 3/2 & 7
General Solution of ax2 + bx + c = 0 (a 0) : General Solution of ax2 + bx + c = 0 (a 0)
Slide 20 : x = [ -b ± (b2 - 4ac)] /2a
In ax2 + bx + c = 0 , what is the Sum of the roots? : In ax2 + bx + c = 0 , what is the Sum of the roots?
Slide 22 : -b/a
In ax2 + bx + c = 0 , what is the Product of the roots? : In ax2 + bx + c = 0 , what is the Product of the roots?
Slide 24 : c/a
In ax2 + bx + c = 0 , what happens if b2 < 4ac? : In ax2 + bx + c = 0 , what happens if b2 < 4ac?
Slide 26 : Roots become imaginary
In ax2 + bx + c = 0, when are the 2 roots of a quadratic equal? : In ax2 + bx + c = 0, when are the 2 roots of a quadratic equal?
Slide 28 : When
b2 = 4ac
In ax2 + bx + c = 0, when are the roots reciprocal of each other? : In ax2 + bx + c = 0, when are the roots reciprocal of each other?
Slide 30 : When c = a
Q9. If and are the roots of the equation, x2 – 7x + 12 = 0, find an equation whose roots are / and /. (CAT 1989). : Q9. If and are the roots of the equation, x2 – 7x + 12 = 0, find an equation whose roots are / and /. (CAT 1989).
Slide 32 : The equation is 12x2 – 25x + 12 = 0
Q10. A farmer decides to fence along 1 side of property. So, he plans to place several posts at 6 mts. gaps, with posts fixed at both ends. After buying the material, he found that the no. of posts with him was 5 less than needed. Then, he finds that no. of posts he had bought would be just sufficient if he puts them 8 mts apart. What is the length & the no. he buys? 1. 100, 15 2. 100, 16 3. 120, 15 4. 120, 16 (CAT 2000) : Q10. A farmer decides to fence along 1 side of property. So, he plans to place several posts at 6 mts. gaps, with posts fixed at both ends. After buying the material, he found that the no. of posts with him was 5 less than needed. Then, he finds that no. of posts he had bought would be just sufficient if he puts them 8 mts apart. What is the length & the no. he buys? 1. 100, 15 2. 100, 16 3. 120, 15 4. 120, 16 (CAT 2000)
Slide 34 : Its 4th option.
Q11. M, K, L and J bought a motorbike for $60.00. M paid 1/2 of the sum of the amounts paid by the other boys, K paid 1/3rd of the sum of the amounts paid by the other boys; and L paid 1/4th of the sum of the amounts paid by the other boys. How much did J have to pay (CAT 2002)? 1. 15 2. 13 3. 17 4. None of these : Q11. M, K, L and J bought a motorbike for $60.00. M paid 1/2 of the sum of the amounts paid by the other boys, K paid 1/3rd of the sum of the amounts paid by the other boys; and L paid 1/4th of the sum of the amounts paid by the other boys. How much did J have to pay (CAT 2002)? 1. 15 2. 13 3. 17 4. None of these
Slide 36 : Answer is 2nd option.
Q12. S visited R on vocation. In the mornings, they both go for yoga. In the evenings they play tennis. To enjoy, they indulge only in 1 activity/day, i.e., either they do yoga or play tennis. There were days, when they stayed home all day long. There were 24 mornings when they did nothing, 14 evenings when they did nothing, & a total of 22 days when they did yoga or tennis. For how many days S stayed with R (CAT 2002)?1. 32 2. 24 3. 30 4. None of these : Q12. S visited R on vocation. In the mornings, they both go for yoga. In the evenings they play tennis. To enjoy, they indulge only in 1 activity/day, i.e., either they do yoga or play tennis. There were days, when they stayed home all day long. There were 24 mornings when they did nothing, 14 evenings when they did nothing, & a total of 22 days when they did yoga or tennis. For how many days S stayed with R (CAT 2002)?1. 32 2. 24 3. 30 4. None of these
Slide 38 : Its 3rd option.
Q13. The number of solutions of the equation 2x + y = 40 where both x and y are positive integers and x ≤ y is (CAT 2006):1.7 2.13 3.144.18 5.20 : Q13. The number of solutions of the equation 2x + y = 40 where both x and y are positive integers and x ≤ y is (CAT 2006):1.7 2.13 3.144.18 5.20
Slide 40 : Answer is 2nd option.
Q14. In a group, there are only rabbits and parrots. The total number of legs was 300 and the total number of eyes were 200. What is the number of parrots in the group? : Q14. In a group, there are only rabbits and parrots. The total number of legs was 300 and the total number of eyes were 200. What is the number of parrots in the group?
Slide 42 : Answer is 50 parrots.
Q15. What is the largest possible value of natural number ‘k’, for which the following expression will also be a natural number? (12k3 + 32k2 + 24k + 12) 4k. : Q15. What is the largest possible value of natural number ‘k’, for which the following expression will also be a natural number? (12k3 + 32k2 + 24k + 12) 4k.
Slide 44 : Answer is 3.
Slide 45 : THANK YOU