NCERT Solutions

Exercise 2.2 1. Let A = {1, 2, 3, ……., 14}. Define a relation R from A to A by R = where Write down its domain co-domain and range. Ans. Given: A = {1, 2, 3, ……….., 14} The ordered pairs which satisfy are (1, 3), (2, 6), (3, 9) and (4, 12). R … Read more

Exercise 5.1 Express each of the complex numbers given in the exercises 1 to 10 in the form : 1. Ans. Here = = =3 2. Ans. Here = 3. Ans. = = 4. Ans. Here = 5. Ans. Here = 6. Ans. Here = = 7. Ans. Here = = = = 8. Ans. … Read more

Exercise 9.1 Write the first five terms of each of the sequences in Exercises 1 to 6 whose terms are: 1. Ans. Given: Putting and 5, we get, Therefore, the first five terms are 3, 8, 15, 24 and 35. 2. Ans. Given: Putting and 5, we get, Therefore, the first five terms are and … Read more

Miscellaneous Exercise 1. The relation is defined by . The relation is defined by . Show that is a function and is not a function. Ans. Given: and At and It is observed that takes unique value at each point in its domain [0, 10]. Therefore, is a function. Now, and At and Therefore, does … Read more

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