NCERT Solutions

Exercise 5.2 Find the modulus and the argument of each of the complex numbers in exercises 1 to 2. 1. Ans. Given: and Squaring both sides and adding both the equations, we get and and [ lies in third quadrant] Therefore, and arg 2. Ans. Given: and Squaring both sides and adding both the equations, … Read more

Exercise 9.2 1. Find the sum of odd integers from 1 to 2001. Ans. Odd integers from 1 to 2001 are 1, 3, 5, 7, …….., 2001. Here, and Now, = 1002001 2. Find the sum of all natural numbers lying between 100 and 1000 which are multiples of 5. Ans. According to question, series … Read more

Exercise 2.3 1. Which of the following are functions? Give reasons. If it is a function determine its domain and range. (i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)} (ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)} (iii) {(1, 3), (1, 5), (2, … Read more

Exercise 5.3 Solve each of the following equations: 1. Ans. Given: 2. Ans. Given: Comparing with , and = = = Therefore, and 3. Ans. Given: Comparing with , and = = = Therefore, and 4. Ans. Given: Comparing with , and = = = Therefore, and 5. Ans. Given: Comparing with , and = … Read more

Exercise 9.3 1. Find the 20th and terms of the G.P. Ans. Here and = And = = 2. Find the 12th term of a G.P. whose 8th term is 192 and the common ratio if 2. Ans. Let be the first term of given G.P. Here and = == 3072 3. The 5th, 8th … Read more

Exercise 1.1 1. Which of the following are sets? Justify your answer. (i) The collection of all the months of a year beginning with the letter J. (ii) The collection of ten most talented writers of India. (iii) A team of eleven best-Cricket batmen of the world. (iv) The collection of all boys in your … Read more

Miscellaneous Exercise 1. Evaluate: Ans. Given: = = = = = = = = 2. For any two complex numbers and prove that Ans. Let and Then and Now, = = = 3. Reduce to the standard form. Ans. Here: = = = = = = = = = 4. If prove that Ans. Given: … Read more

16. Find a G.P. for which sum of the first two terms is and the fifth term is 4 times the third term. Ans. Let be the first term and be the common ratio of given G.P. Given: ……..(i) And Putting in eq. (i), we get Therefore, required G.P. is Putting in eq. (i), we … Read more

Exercise 1.2 1. Which of the following are examples of the null set: (i) Set of odd natural numbers divisible by 2. (ii) Set of even prime numbers. (iii) { is a natural number, and } (iv) { is a point common to any two parallel lines} Ans. (i) Set of odd natural numbers divisible … Read more

Exercise 6.1 1. Solve when: (i) is a natural number (ii) is an integer Ans. Given: Dividing both sides by 24, (i) When is a natural number then values of that make statement true are 1, 2, 3, 4. The solution set of inequality is {1, 2, 3, 4}. (ii) When is an integer then … Read more

Exercise 9.4 Find the sum to terms in each of the series in Exercises 1 to 7. 1. Ans. Given:to terms = [ term of 1, 2, 3, ………..][ term of 2, 3, 4, 5, ………] = = = = = = = = = 2. Ans. Given: to terms = [ term of 1, … Read more

Exercise 1.4 1. Find the union of each of the following pairs of sets: (i) X = {1, 3, 5} and Y = {1, 2, 3} (ii) A = and B = (iii) A = { is a natural number and multiple of 3} and B = { is a natural number less than 6} … Read more

Exercise 6.2 Solve the following inequalities graphically in two dimensional planes: 1. Ans. Given: Table of values satisfying the equation 1 2 4 3 Putting (0, 0) in the given in equation, 0 < 5 which is true. Therefore, Half plane of towards origin. 2. Ans. Given: Table of values satisfying the equation 1 2 … Read more

Miscellaneous Exercise 1. Show that the sum of  and  terms of an A.P. is equal to twice the  terms. Ans. Here,  …..(i) and   …..(ii) To prove: Adding eq. (i) and (ii), we get                  2. If the sum of three numbers in A.P., is 24 and their product is 440. … Read more

Exercise 1.6 1. If X and Y are two sets such that and find Ans. Given: and 2. If X and Y are two sets such that has 18, X has 8 elements and Y has 15 elements; how many elements has ? Ans. Given: and 3. In a group of 400 people, 250 can … Read more

Exercise 6.3 Solve the following systems of inequalities graphically: 1. Ans. Given: Putting (0, 0) in which is false. Half plane of is away from origin. Again Putting (0, 0) in which is false. Half plane of is away from origin. 2. Ans. Given: Table of values satisfying the equation 2 4 3 0 Putting … Read more

17. If  are in G.P., prove that  are in G.P. Ans. Given:  are in G.P. To prove:  are in G.P.    Let       And    Also    Now,         Therefore,  are in G.P. 18. If  and  are the roots  and  are roots of  where  form a G.P. Prove that   Ans. … Read more