Not yet checked 8 of 8 questions have not yet been compared with the official paper.
Section A
Q. 1
(a)For the function f(x) graphed in the adjoining figure, find the following limits or explain why they do not exist. (i) lim x o 1 f ( x ) \lim_{x o 1} f(x) lim x o 1 f ( x ) (ii) lim x o 2 f ( x ) \lim_{x o 2} f(x) lim x o 2 f ( x ) (iii) lim x o 3 f ( x ) \lim_{x o 3} f(x) lim x o 3 f ( x ) Also discuss the continuity of f(x) at x = 1 x = 1 x = 1 , x = 2 x = 2 x = 2 and x = 3 x = 3 x = 3 . [10]
(b)Differentiate y = 3 x ( x 2 + 1 ) ( x − 1 ) 2 y = 3\sqrt{\dfrac{x(x^2+1)}{(x-1)^2}} y = 3 ( x − 1 ) 2 x ( x 2 + 1 ) with respect to x . [10]
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Q. 2
(a)For what values of a , m , and b does the function f ( x ) = { 3 , x = 0 − x 3 + 3 x + a , 0 < x < 1 m x + b , 1 ≤ x ≤ 2 f(x) = \begin{cases} 3, & x = 0 \ -x^3 + 3x + a, & 0 < x < 1 \ mx + b, & 1 \leq x \leq 2 \end{cases} f ( x ) = { 3 , x = 0 − x 3 + 3 x + a , 0 < x < 1 m x + b , 1 ≤ x ≤ 2 satisfy the hypotheses of mean value theorem on the interval [0,2] ? [10]
(b)A box with rectangular base, whose length is twice its width, is to have a closed top. The area of the material in the box is to be 192 in². What should the dimensions of the box be in order to have the largest possible volume? [10]
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Q. 3
(a)Show that ∫ 0 π 2 sin x sin x + cos x d x = π 4 \displaystyle\int_0^{\frac{\pi}{2}} \dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\,dx = \dfrac{\pi}{4} ∫ 0 2 π sin x + cos x sin x d x = 4 π . [10]
(b)Evaluate I = I=∫04∫04−x∫04−x−ydzdydx . [10]
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Q. 4
(a)A plot of land lies between a straight fence and a curved stream at distance x from one end of the fence, the width y meters of the plot was measured as follows: x 0 10 20 30 40 50 60 70 80 y 0 32 44 58 63 50 30 32 0 Find the approximate area of the plot by using trapezoidal rule. [10]
(b)Solve the differential equation (x+1)dxdy−ny=ex(x+1)n+1 . [10]
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Q. 5
(a)Assume that the half life of the radium in a piece of lead is 1500 years. How much radium will remain in the lead after 2500 years? [10]
(b)A particle of mass m is moving under the action of the forces F1=−mω2x , F2=mF0t , F3=−2mμdtdx . Assuming that damping is small, set up and solve the equation of motion. [10]
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Section B
Q. 6
(a)Find radius and interval of convergence of the series 1∑∞(2n)!n!xn . [10]
(b)Prove that ( sin x + i cos x ) n = cos n ( π 2 − x ) + i sin n ( π 2 − x ) , n ∈ Z (\sin x + i\cos x)^n = \cos n\left(\frac{\pi}{2} - x\right) + i\sin n\left(\frac{\pi}{2} - x\right), \quad n \in \mathbb{Z} ( sin x + i cos x ) n = cos n ( 2 π − x ) + i sin n ( 2 π − x ) , n ∈ Z [10]
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Q. 7
(a)Construct the analytic function whose real part is e − x [ ( x 2 − y 2 ) cos y + 2 x y sin y ] e^{-x}[(x^2 - y^2)\cos y + 2xy\sin y] e − x [( x 2 − y 2 ) cos y + 2 x y sin y ] . [10]
(b)Compute ∫Cz(z2−4)(z+4)z2+2dz , where C is the curve shown in the figure below. [10]
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Q. 8
(a)Find the tangent and normal to the curve x2−xy+y2=7 at the point (−1,2) . [10]
(b)Find the curvature and torsion of the circular helix r ⃗ = ( a cos u , a sin u , b u ) \vec{r} = (a\cos u,\ a\sin u,\ bu) r = ( a cos u , a sin u , b u ) . [10]
The 2021 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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