CSS Applied Mathematics Past Paper 2021

Instructions specific to this paper

• Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
• Use of Calculator is allowed.
• 1.84147 2.03562 2.19146 2.29290 2.32521 2.27807 2.14935

Questions

1. Q. 210 marks

   (a) The greatest resultant that two forces can have is of magnitude 𝑃 and the least is of magnitude 𝑄.Show that, when they act at an angle 𝛼, their resultant is of 𝛼 𝛼 magnitude √𝑃2 𝑐𝑜𝑠 2 + 𝑄2 𝑠𝑖𝑛2 . 2 2 (b) A sphere of weight 𝑊 and radius 𝑎 is suspended by a string of length 𝑙 from a (10) point 𝑃 and a weight 𝑤 is also suspended from 𝑃 by a string sufficiently long for the weight to hang below the sphere. Show that the inclination of the first string to the vertical is 𝑤𝑎 sin−1 . (𝑊 + 𝑤)(𝑎 + 𝑙 ) .

2. Q. 310 marks

   (a) Show that the law of force towards the pole, of a particle describing the curve 𝑟𝑛 = 𝑎𝑛 cos 𝑛𝜃 is given by (𝑛 + 1)ℎ2 𝑎2𝑛 𝑓= . 𝑟2𝑛+3 The maximum velocity that a particle executing simple harmonic motion of (10) (b) amplitude 𝑎 attains, is 𝑣. If it is disturbed in such a way that its maximum velocity becomes 𝑛𝑣. Find the change in the amplitude and the time-period of motion.

3. Q. 410 marks

   (a) Define ordinary and singular points of the differential equation 𝑎2 (𝑥 )𝑦 ′′ + 𝑎1 (𝑥)𝑦 ′ + 𝑎0 (𝑥)𝑦 = 0. When a singular point is said to be regular and irregular? Find regular and irregular singular points of the differential equation (𝑥 2 − 4)2 𝑦 ′′ + (𝑥 − 2)𝑦 ′ + 𝑦 = 0. (b) Show that (10) 2 sin 𝑥 𝐽3 ⁄2 = √ [ − cos 𝑥]. 𝜋𝑥 𝑥

4. Q. 510 marks

   (a) Solve the equation by using method of undetermined coefficients 𝑦 ′′ − 𝑦 ′ + 𝑦 = 2 cos 3𝑥. (b) Use the method of Frobenius to find two linear independent series solutions in (10) powers of 𝑥 of the DE. 𝑥 2 𝑦 ′′ − (𝑥 2 + 𝑥 )𝑦 ′ + 𝑦 = 0. APPLIED MATHEMATICS

5. Q. 610 marks

   (a) Classify general second order partial differential equation (PDE) into elliptic, parabolic and hyperbolic form. Discuss the nature of the PDE (1 − 𝑥 2 )𝑢𝑥𝑥 − 2𝑥𝑦𝑢 𝑥𝑦 + (1 − 𝑦 2 )𝑢𝑦𝑦 = 0 at each (𝑥, 𝑦) ∈ 𝑅 2 . (b) Use the method of separation of variables to find the solution 𝑢(𝑥, 𝑡): [0, 𝑇] × (10) [0, 𝐿] → 𝑅 to the initial/boundary value problem 𝑢 𝑡 (𝑥, 𝑡) = 𝑢 𝑥𝑥 (𝑥, 𝑡) 𝑓𝑜𝑟 0 < 𝑡 ≤ 𝑇 𝑎𝑛𝑑 0 ≤ 𝑥 ≤L, 𝑢 (𝑥, 0) = 𝑓(𝑥 ), 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 𝐿, 𝑢 (0, 𝑡) = 𝑢 (𝐿, 𝑡) = 0, 𝑓𝑜𝑟 0 < 𝑡 ≤ 𝑇, where 𝑓: [0, 𝐿] → 𝑅 is a known function.

6. Q. 710 marks

   (a) Use Simpson’s 3/8 rule to estimate the integral 3 ∫(𝑥 3 − 2𝑥 2 + 7𝑥 − 5)𝑑𝑥 . 1 By comparing your answer with exact value, find the error. (b) Solve the system of equations by Jacobi iterative method. (10) 10𝑥 + 3𝑦 + 𝑧 = 19, 3𝑥 + 10𝑦 + 2𝑧 = 29, 𝑥 + 2𝑦 + 10𝑧 = 35

7. Q. 810 marks

   (a) In the following table values of 𝑦 = 𝑥 + sin 𝑥 2 are tabulated x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 f (x) 1.84147 2.03562 2.19146 2.29290 2.32521 2.27807 2.14935 Construct a difference table and estimate 𝑓(1.04) and 𝑓(1.57). (b) 𝜋 ⁄2 (10) Use trapezoidal and Simpson’s 1/3 rules to approximate ∫0 𝑠𝑖𝑛2 (𝑥 )𝑑𝑥. Find a maximum bound for the error in each case. Compare your approximations with the actual result.

Standard FPSC instructions

Printed on this and every CSS paper.

• Part-II is to be attempted on the separate Answer Book.
• All the parts (if any) of each question must be attempted at one place instead of at different places.
• Write Q. No. in the Answer Book in accordance with Q. No. in the question paper.
• No page/space be left blank between the answers. All the blank pages of the Answer Book must be crossed.
• Extra attempt of any question or any part of the attempted question will not be considered.