Algebra - II

Algebra - II : Algebra - II

Learning Objectives : Learning Objectives How to handle statement questions. How to solve inequalities. How to solve Progressions. How to form quadratic equations. Some actual CAT questions.

Show an inequality on the number line : x  3 : Show an inequality on the number line : x  3 0 1 2 3

x  3 Is x + 2  3 + 2 ? : x  3 Is x + 2  3 + 2 ?

Slide 5 : Seems to be ..

x  3 Is x - 4  3 - 4 ? : x  3 Is x - 4  3 - 4 ?

Slide 7 : The same number can be added/subtracted without changing the truth of the inequality.

x  3 Is x  2  3  2 ? : x  3 Is x  2  3  2 ?

Slide 9 : So, the same positive number can be multiplied without changing the truth of the inequality.

x  3 Is x  (-2)  3  (-2)?  : x  3 Is x  (-2)  3  (-2)? 

Slide 11 : A negative number multiplied on both sides, reverses the sign of inequality. So it should be x  (-2)  3  (-2) ? 

Q. 1 : Solve the inequality (5x – 1) / -2 < 3 : Q. 1 : Solve the inequality (5x – 1) / -2 < 3

Slide 13 : Answer is x>-1.

Q. 2 : Solve: x2 – 7x – 8 < 0 : Q. 2 : Solve: x2 – 7x – 8 < 0

Slide 15 : - 1 < x < 8.

Q.3 : Find the value of x, which satisfies the following conditions:x2 – 16x + 55 < 0 and x > 7. : Q.3 : Find the value of x, which satisfies the following conditions:x2 – 16x + 55 < 0 and x > 7.

Slide 17 : Range is 7 < x < 11.

Q. 4 : Solve the given inequality for x : Q. 4 : Solve the given inequality for x

Slide 19 : Answer is -1 < x < 9/7

Q 5. Find the sum of the first 50 terms of the series 12, 14, 16….? : Q 5. Find the sum of the first 50 terms of the series 12, 14, 16….?

Slide 21 : Answer is 3050.

Q. 6: What will be sum of first 12 terms of the series 3, 6, 12, 24,48…? : Q. 6: What will be sum of first 12 terms of the series 3, 6, 12, 24,48…?

Slide 23 : Answer is 12285.

Q. 7: If a person gives you Rs. 150 today, Rs. 75 tomorrow, Rs. 37.50 day after tomorrow and so on. What sum of money, he will pay in total, if he lives for ever and keeps on paying you in this manner? : Q. 7: If a person gives you Rs. 150 today, Rs. 75 tomorrow, Rs. 37.50 day after tomorrow and so on. What sum of money, he will pay in total, if he lives for ever and keeps on paying you in this manner?

Slide 25 : Answer is Rs. 300.

Q 8. Find the sum of all the multiples of 4 from 11 to 200, inclusive, excluding those, which are multiples of 12. : Q 8. Find the sum of all the multiples of 4 from 11 to 200, inclusive, excluding those, which are multiples of 12.

Slide 27 : Answer is 3456.

Q9. Find the sum of all the odd numbers between 20 and 200, excluding those, which are multiples of 12. : Q9. Find the sum of all the odd numbers between 20 and 200, excluding those, which are multiples of 12.

Slide 29 : Answer is 9900.

Q 10: A ball is thrown from a height of 600 metres, everytime it touches the ground, it goes 5/6th of the dis. it has come from. What is total dis. traveled by the ball, before it comes to rest? : Q 10: A ball is thrown from a height of 600 metres, everytime it touches the ground, it goes 5/6th of the dis. it has come from. What is total dis. traveled by the ball, before it comes to rest?

Slide 31 : Is the answer 3600 mt. It is wrong. Answer is 6600 mt. Think Why.

Q11 : What is the value of the following expression (CAT 2000)?1/(22 – 1) + 1/(42 – 1) + 1/(62 – 1) +……………….+ 1/(202 – 1) =? 1. 9/19 2. 10/19 3. 10/21 4. 11/21 : Q11 : What is the value of the following expression (CAT 2000)?1/(22 – 1) + 1/(42 – 1) + 1/(62 – 1) +……………….+ 1/(202 – 1) =? 1. 9/19 2. 10/19 3. 10/21 4. 11/21

Slide 33 : Answer is 3rd option.

Q 12 : If both a & b belong to the set {1, 2, 3, 4}, then the number of equations of the form ax2 + bx + 1 = 0 having real roots is (CAT 2003) 1. 10 2. 7 3. 6 4. 12 : Q 12 : If both a & b belong to the set {1, 2, 3, 4}, then the number of equations of the form ax2 + bx + 1 = 0 having real roots is (CAT 2003) 1. 10 2. 7 3. 6 4. 12

Slide 35 : Answer is 2nd option.

Q13 : A real number x satisfying 1 – 1/n < x  3 + 1/n , for everypositive integer n, is best described by (CAT 2003): 1. 1 < x < 4 2.1 < x  3 3. 0 < x  4 4. 1  x  3 : Q13 : A real number x satisfying 1 – 1/n < x  3 + 1/n , for everypositive integer n, is best described by (CAT 2003): 1. 1 < x < 4 2.1 < x  3 3. 0 < x  4 4. 1  x  3

Slide 37 : Answer is 3rd option.

Q14 : If n is such that 36  n  72, then satisfies (CAT 2003):1. 20< x<54 2.23Q14 : If n is such that 36  n  72, then satisfies (CAT 2003):1. 20< x<54 2.23

Slide 39 : Answer is 4th option.

Q15 : A colony of bacteria in a container grows by each bacterium splitting into 8 next generation bacteria. But only 50% of the bacteria in a generation split in the this manner. A 1st generation was put in the container & the 7th generation bacteria was 4096 mn. What was size of the 1st gen. Bacteria in mn (CAT 1998)?1. 2 2. 8 3. 1 4. 4 : Q15 : A colony of bacteria in a container grows by each bacterium splitting into 8 next generation bacteria. But only 50% of the bacteria in a generation split in the this manner. A 1st generation was put in the container & the 7th generation bacteria was 4096 mn. What was size of the 1st gen. Bacteria in mn (CAT 1998)?1. 2 2. 8 3. 1 4. 4

Slide 41 : Its 3rd option.

Q 16: A group of 630 children is arranged in rows for a group photograph session. Each row contains three fewer children than the row in front of it. What number of rows is not possible? 1.3 2.4 3.5 4.6 5.7 : Q 16: A group of 630 children is arranged in rows for a group photograph session. Each row contains three fewer children than the row in front of it. What number of rows is not possible? 1.3 2.4 3.5 4.6 5.7

Slide 43 : Its 4th option.

Slide 44 : THANK YOU