NCERT Solutions

Miscellaneous Exercise Prove that: 1. Ans. L.H.S. = = = = = = 0 = R.H.S. 2. Ans. L.H.S. = = = = = 0 = R.H.S. 3. Ans. L.H.S. = = = = = = R.H.S. 4. Ans. L.H.S. = = = = = = R.H.S. 5. Ans. L.H.S. = = = = = … Read more

Exercise 8.2 Find the coefficient of: 1. in Ans. General form of the expansion is ………(i) Putting in eq. (i), Therefore, coefficient of on the expansion is = 1512. 2. in Ans. General form of the expansion is ………(i) Putting in eq. (i), Therefore, coefficient of on the expansion is = Write the general term … Read more

Exercise 11.4 In each of the Exercises 1 to 6, find the coordinates of the foci, and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. 1. Ans. Given: Equation of hyperbola in the form Since, the foci and vertices of the hyperbola lies on axis. and Coordinates of foci … Read more

Exercise 4.1 Prove the following by using the principle of mathematical induction for all N: 1. Ans. Let For 1 = 1 is true. Now, let be true for ……….(i) For [Using eq. (i)] is true. Therefore, is true. is true. Hence by Principle of Mathematical Induction, is true for all N. 2. Ans. Let … Read more

Miscellaneous Exercise 1. Find and in the expansion of if the first three terms of the expansion are 729, 7290 and 30375 respectively. Ans. Given: …..(i) …..(ii) And …..(iii) Multiplying eq. (i) and eq. (iii), …..(iv) Squaring both sides in eq. (ii), …..(v) Dividing eq. (iv) by eq. (v), From eq. (i), And From eq. … Read more

Miscellaneous Exercise 1. In a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus. Ans. A parabolic reflector with diameter PR = 20 cm and OQ = 5 cm Vertex of parabola is (0, 0) Let focus of the parabola be Now, PR = 20 cm PQ = 10 cm … Read more

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