Question 1. Evaluate ∫1/ a2-b2x2 dx Solution: Let us assume I = ∫ 1/ (a2-b2x2)dx take 1/b2 common from above equation = 1/b2 ∫ 1/… Read More
Tag Archives: RD Sharma Class-12
Question 1(i): Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses line through yz-plane. Solution: We… Read More
Question 11. Evaluate ∫ sin2x/ √cos4x-sin2x+2 dx Solution: Let us assume I =∫ sin2x/ √cos4x-sin2x+2 dx =∫sin2x/ √cos4x-(1-cos2x)+2 dx (i) Put cos2x = t -2sinxcosx… Read More
Evaluate the following integrals. Question 1. ∫(ex(cosx -sinx))dx Solution: Given expression is∫(excosx)-(exsinx)dx=∫(excosx) dx -∫(exsinx)dx=ex(cosx )-∫(exd(cosx)/dx-∫exsinx dx=ex(cosx )+∫exsinx dx-∫exsinx dx=ex(cosx) + c Question 2. ∫ex(x-2+2x-3)dx Solution: Given… Read More
Question 1. Evaluate ∫ x/ √x4+a4 dx Solution: Let us assume I = ∫ x/ √x4+a4 dx = ∫ x/ √(x2)2+(a2)2 dx (i) Put x2… Read More
Question 33. Find the angle between the two vectors and , if (i) =√3, = 2 and = √6 Solution: We know, ⇒ √6 = 2√3 cos θ ⇒ cos… Read More
Find the inverse of each of the following matrices by using elementary row transformation(Questions 1- 16): Question 1. Solution: Here, A = A = AI … Read More
Question 10. For the following parts of matrices verify that (AB)-1 = B-1A-1. (i) A = and B = Solution: To prove (AB)-1= B-1A-1 We take LHS… Read More
Question 25. Show that the matrix A = satisfies the equation A3 – A2 – 3A – I3 = 0. Hence, find A-1. Solution: Here, A… Read More
Evaluate the following integrals. Question 11. ∫ex(sin4x-4)/(2sin22x)dx Solution: We have, ∫ex(sin4x-4)/(2sin22x)dx =∫ex(2sin2xcos2x-4)/(2sin22x)dx =∫ex(((2sin2xcos2x)/(2sin22x))-4/(2sin22x))dx =∫ex(cot2x-2cosec22x)dx =∫excot2xdx-2∫excosec22xdx Integrating by parts, excot2x-2∫exd(cot2x)/dx-2∫excosec22xdx = excot2x+2∫excosec22xdx-2∫excosec22xdx = excot2x+C Question 12. ∫ex(2-x)/(1-x)2dx… Read More
Question 23. At every point on a curve, the slope is the sum of the abscissa and the product of the ordinate and the abscissa,… Read More
Question 12. Experiments show that radium disintegrates at a rue proportional to the amount of radium present at the moment. Its half-life is 1590 years.… Read More
Question 1. The surface area of a balloon being inflated changes at a rate proportional to time t. If initially its radius is 1 unit… Read More
Question 1. Find the adjoint of the following matrices: Verify that (adj A)A = |A|I = A(adj A) for the above matrices: (i) Solution: Here,… Read More
Solve the following differential equations: Question 24. (2x – 10y3)(dy/dx) + y = 0 Solution: We have, (2x – 10y3)(dy/dx) + y = 0 (2x… Read More
