Sujeet Kumar

Chapter 4 Pair of Straight Lines Ex 4.1 Question 1.Find the combined equation of the following pairs of lines:(i) 2x + y = 0 and 3x – y = 0Solution:The combined equation of the lines 2x + y = 0 and 3x – y = 0 is(2x + y)( 3x – y) = 0∴ 6×2 – … Read more

Chapter 3 Trigonometric Functions Miscellaneous Exercise 3 I) Select the correct option from the given alternatives.Question 1.The principal of solutions equation sinθ = −1/2 are ________.Solution:(b) 7π6,11π6 Question 2.The principal solution of equation cot θ = √3 ___________.Solution: Question 3.The general solution of sec x = √2 is __________. Question 4.If cos pθ = cosqθ, p ≠ q rhen ________. Question 5.If … Read more

Question 1.Find the principal values of the following : Question 2.Evaluate the following : Solution: Solution: (iii) tan-1√3 – sec-1(-2)Solution: (iv) cosec-1(−√2) + cot-1(√3)Solution: Question 3.Prove the following : Solution: Solution: Solution: Solution: Solution: = θ/2 …[∵ tan-1(tanθ) = θ]= RHS.

Question 1.Find the Cartesian co-ordinates of the point whose polar co-ordinates are : Let the cartesian coordinates be (x, y) Let the cartesian coordinates be (x, y) Question 2.Find the of the polar co-ordinates point whose Cartesian co-ordinates are. Question 4. Solution:By the sine rule, Question 5. Solution: Question 6.In ∆ABC, prove that a3sin(B – … Read more

Chapter 3 Trigonometric Functions Ex 3.1 Question 1.Find the principal solutions of the following equations : We know that,Hence, the required principal solution are (iv) cot θ = 0.Solution: Question 2.Find the principal solutions of the following equations:(i) sinθ = −1/2Solution:We know that,sin π/6 = 1/2 and sin (π + θ) = -sinθ,sin(2π – θ) = -sinθHence, the required … Read more

I) Choose the correct answer from the given alterna-tives in each of the following questions :Question 1. (a) a = – 2, b = 1(b) a = 2, b = 4(c) a = 2, b = –1(d) a = 1, b = –2Solution:(a) a = – 2, b = 1 Question 2. Solution: Question 3. … Read more

Question 1. Question 2.By R1 – R2, we get,By R1 – R3 and By R2 – R3, we get Question 3.Check whether the following matrices are invertible or not: ∴ A is a non-singular matrix.Hence, A-1 exist. = sec2θ – tan2θ = 1 ≠ 0.∴ A is a non-singular matrix.Hence, A-1 exist. = 3(5 – 0) – 4(5 – 0) + … Read more

Question 1.Solve the following equations by inversion method.(i) x + 2y = 2, 2x + 3y = 3Solution:The given equations can be written in the matrix form as :This is of the form AX = B, where ∴ A-1 = [−322−1]Now, premultiply AX = B by A-1, we get,A-1(AX) = A-1B∴ (A-1A)X = A-1B∴ IX = A-1B … Read more

Question 1.Find the co-factors of the elements of the following matricesHere, a11 = -11, M11 = 4∴ A11 = (-1)1+1(4) = 4a12 = 2, M12 = -3∴ A12 = (-1)1+2(- 3) = 3a21 = – 3, M21 = -2∴ A21 = (- 1)2+1(2) = -2a22 = 4, M22 = -1∴ A22 = (-1)2+2(-1) = -1. The co-factor of aij is given by Aij = (-1)i+jMij Question 2.Find the matrix … Read more

Question 1.Apply the given elementary transformation on each of the following matrices. Question 3. Question 4. Question 5. Question 6.We conclude from Ex. 5 and Ex. 6 that the matrix remains same by interchanging the order of the elementary transformations. Hence, the transformations are commutative. Question 7. Question 8. Question 9.