Mathematics Part-1

Trigonometric Functions (Ex – 3.4) Find the principal and general solutions of the following equations: Question 1.tanx = √3 Solution. Question 2.sec x = 2 Solution. Question 3.cotx = −√3 Solution. Question 4.cosec x = -2 Solution. Find the general solution for each of the following equations: Question 5.cos 4x = cos 2x Solution. Question […]

Trigonometric Functions (Ex – 3.3) Question 1.Prove that: sin2π/6+cos2π/3−tan2π/4=−1/2 Solution.L.H.S. = sin2π/6+cos2π/3−tan2π/4=[(1/2)2+(1/2)2−(1)2]=1/4+1/4−1=−1/2=R.H.S. Question 2. Solution. Question 3. Solution. Question 4. Solution. Question 5.Find the value of:(i) sin 75°(ii) tan 15°Solution.(i) sin (75°) = sin (30° + 45°) (ii) tan 15° = tan (45° – 30°) Prove the following: Question 6. Solution.We have, Question 7. Solution.We have, Question

Trigonometric Functions (Ex – 3.2) Find the values of other five trigonometric functions in Exercises 1 to 5. Question 1.cos x = -1/2, x lies in third quadrant. Solution.  Question 2.sin x = 3/5, x lies in second quadrant. Solution. Question 3.cot x = 3/4, xlies in third quadrant. Solution. Question 4.sec x = 13/5,

Trigonometric Functions (Ex – 3.1) Question 1.Find the radian measures corresponding to the following degree measures:(i) 25°(ii) -47°30′(iii) 240°(iv) 520° Solution.We have, 180° = π Radians Question 2.Find the degree measures corresponding to the following radian measures (Use π=22/7)(i) 11/16(ii) -4(iii) 5π/3(iv) 7π/6 Solution.We have π Radians = 180° Question 3.A wheel makes 360 revolutions in one minute. Through

Relations and Functions (Ex – 2.3) Question 1.Which of the following relations 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,

Relations and Functions (Ex – 2.2) Question 1.Let 4 = {1,2,3, ,14}. Define a relation R from A to A by R = {(x,y): 3x – y = 0, where x, y ∈ 4}.Write down its domain, codomain and range. Solution.We have A = (1, 2, 3,……..,14)Given relation R = {(x, y) : 3x –

Relations and Functions (Ex – 2.1) Question 1.If (x/3+1,y−2/3)=(5/3,1/3), find the values of x and y. Solution.Since the ordered pairs are equal. So, the corresponding elements are equal∴ x/3+1=5/3 and y−2/3=1/3⇒ x/3=5/3−1 and y=1/3+2/3 ⇒ x = 2 and y = 1. Question 2.If the set A has 3 elements and the set B {3, 4, 5}, then find the number of elements

Sets (Miscellaneous Exercise) Question 1.Decide, among the following sets, which sets are subsets of one and another:A = {x : x ∈R and x satisfy x2 – 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8,…}, D = {6}. Solution.Here A = {x : x ∈ R and x

Sets (Ex – 1.6) Question 1.If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n (X ∪ Y) = 38, find n(X ∩ Y). Solution.Here n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38We know thatn(X ∪ Y) = n(X) + n(Y) -n(X ∩ Y)⇒

Sets (Ex – 1.5) Question 1.Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = { 1, 2, 3, 4}, B = (2,4,6,8} and C = {3,4,5,6}. Find(i) A’(ii) B’(iii) (A ∪ C)’(iv) (A ∪B)’(v) (A’)’(vi) (B – C)’ Solution.Here U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4},