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PPSC · PMS Punjab 2022

Mathematics, Paper I

100 marks · 3 hours · 5 questions
This paper Mathematics · all yearsQ. 3 · Find the area of the…Q. 4 · Discuss the motion of a…Q. 5 · Solve the differential equation ;…Q. 6 · The nth term of a…Q. 7 · Prove that the Necessary and…With this paper← 20212022 →
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Q. 3
  1. (a)Find the area of the region bounded by the curve y = 4 x − y=4x−x2y = 4x - x^2 , the x-axis, and the lines x=1x=1 and x=3x=3 . [10]
  2. (b)Using rectangular rule for n = 4 n = 4 n = 4 , approximate the value of the definite integral ∫0111+x2dx\int_{0}^{1} \frac{1}{1+x^2} dx [10]
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Q. 4
  1. (a)Discuss the motion of a particle moving in a straight line if it starts from rest at a distance a from a point 0 and moves with an acceleration equal to μ\mu times its distance from 0. [10]
  2. (b)A culture initially has P0P_0 number of bacteria. At t = 1 t = 1 t = 1 hour the number of bacteria is measured to be P0P_0 . If the rate of growth is proportional to the number of bacteria P(t)P(t) present at time t, determine the time necessary for the number of bacteria to triple. [10]
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Q. 5
  1. (a)Solve the differential equation dydx=(−2x+y)2−7\frac{dy}{dx} = (-2x+y)^2 - 7 ; y(0)=0y(0)=0 . [10]
  2. (b)A particle of mass m oscillates in a line with natural period T = T=2πωT = \frac{2\pi}{\omega} . If an applied force F cos ⁡ p t F \cos pt F cos pt now acts in the line so that the particle is instantaneously at rest at zero time at a distance d from the centre of oscillation, prove that the displacement of the particle from the centre at subsequent time t is d cos ⁡ ω t + F m ( ω 2 − p 2 ) ( cos ⁡ p t − cos ⁡ ω t ) d \cos \omega t + \frac{F}{m(\omega^2 - p^2)} (\cos pt - \cos \omega t) d cos ω t + m ( ω 2 − p 2 ) F ​ ( cos pt − cos ω t ) . [10]
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Q. 6
  1. (a)The nth term of a sequence is n1/nn^{1/n} . Determine whether sequence converges or diverges. [10]
  2. (b)Determine the convergence or divergence of the series 1 1 ⋅ 3 + 1 ⋅ 2 1 ⋅ 3 ⋅ 5 + 1 ⋅ 2 ⋅ 3 1 ⋅ 3 ⋅ 5 ⋅ 7 + .. \frac{1}{1\cdot 3} + \frac{1\cdot 2}{1\cdot 3 \cdot 5} + \frac{1\cdot 2\cdot 3}{1\cdot 3\cdot 5 \cdot 7} + ... 1 ⋅ 3 1 ​ + 1 ⋅ 3 ⋅ 5 1 ⋅ 2 ​ + 1 ⋅ 3 ⋅ 5 ⋅ 7 1 ⋅ 2 ⋅ 3 ​ + ... by applying any appropriate test. [10]
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Q. 7
  1. (a)Prove that the Necessary and sufficient condition for a function W = f ( Z ) = U ( x , y ) + i V ( x , y ) W = f(Z) = U(x, y) + i V(x, y) W = f ( Z ) = U ( x , y ) + iV ( x , y ) to be an analytic function is that the four partial derivatives Ux,Uy,Vx,VyU_x, U_y, V_x, V_y exist, are continuous and satisfy the Cauchy Riemann equations at each point of D i.e., ∂U∂x=∂V∂y\frac{\partial U}{\partial x} = \frac{\partial V}{\partial y} and ∂U∂y=−∂V∂x\frac{\partial U}{\partial y} = -\frac{\partial V}{\partial x} . [10]
  2. (b)Prove that the function f ( z ) = ∣ x y ∣ f(z) = \sqrt{|xy|} f ( z ) = ∣ x y ∣ ​ is not analytic at the origin although the Cauchy Riemann equations are satisfied at the origin. [10]
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The 2022 PMS Punjab Mathematics paper set by the PPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.

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