Class 11

Exercise 3.3 Prove that: 1. Ans. 1. L.H.S. = = = R.H.S. 2. Ans. L.H.S. = = = = = 3. Ans. L.H.S. = = = = = = R.H.S. 4. Ans. L.H.S. = = = = = = R.H.S. 5. Find the value of: (i) (ii) Ans. (i) = (ii) = = Prove the … Read more

Miscellaneous Exercise 1. How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER? Ans. There are 8 letters in the word DAUGHTER. In this word 3 vowels and 5 consonants. 2 vowels and 3 consonants are to be selected. Number of … Read more

Exercise 11.2 In each of the following Exercises 1 to 6, find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum. 1. Ans. Given: Equation of parabola Comparing with we have Coordinates of focus are (3, 0). Axis of parabola is Equation of … Read more

Exercise 3.4 Find the principal and general solutions of the following equations: 1. Ans. Given: Here lies in first or third quadrant. or or or Therefore, the principal solutions are Now, where Z 2. Ans. Given: Here lies in first or fourth quadrant. or or or Therefore, the principal solutions are Now, where Z 3. … Read more

Exercise 8.1 Expand each of the expression in Exercises 1 to 5. 1. Ans. Using Binomial Theorem, = = 2. Ans. Using Binomial Theorem, = = 3. Ans. Using Binomial Theorem, = = 4. Ans. Using Binomial Theorem, = = 5. Ans. Using Binomial Theorem, = = Using binomial theorem evaluate each of the following: … Read more

Exercise 11.3 In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. 1. Ans. Given: Equation of ellipse: 36 > 16 Now = Coordinates of foci are Coordinates of … Read more

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