QP-IFSM-23-MATHEMATHICS-PAPER-I-051223

INDIAN FOREST SERVICE (NAINJEXAM- 2023 MATHEMATICS Paper -I BJKE-U-MTH Time Allowed : Three Hours Maximum Marks : 200 Question Paper Specific Instructions Please read each of the following instructions carefully before attempting questions: There are EIGHT questions in all, out of which FIVE are to be attempted. Questions no. 1 and 5 are compulsory. Out of the remaining SIX questions, THREE are to be attempted selecting at least ONE question from each of the two Sections A and B. Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a question shall be counted even if attempted partly. Any page or portion of the page left blank in the Question-cum-Answer Booklet must be clearly struck off. All questions carry equal marks. The number of marks carried by a question/part is indicated against it. Unless otherwise mentioned, symbols and notations have their usual standard meanings. Assume suitable data, if necessary, and indicate the same clearly. Answers must be written in ENGLISH only. BJKE-U-MTH 1 Ql. (a) (b) () (6) (9) Q2. (a) (b) BJKE-U-MTH SECTION A Let V be a vector space of the dimension n over a field F. Then show that is isomorphic 10 . Let 1 : 13 _ R® bea linear map defined by T(x, y, 2) = (x, 2, - 2$ - 2) and let 100) = - 13 + 2. Then find f(T). Test the convergence of improper integral b dx ? sin( 2]; where x = 312 + 58; y= 4r - 28, 2172 - 392; then x find 2. or 68 If the equation ax? + 2hxy + by? + 2gx + 2fy + c = 0, represents two intersecting straight lines, then show that the square of the distance of c(a +b) - (12 + 2) ab — h? the point of intersection from the origin is If 51 = {(, , z)|x + 2y + > = 0} and ‎ = {(x, y, z)|x + y ¬ 2 = 0} are subspaces of R®, then (i) ` 110 8 basis 0 51 82. (ii) ` determine dim (51 + Sg). (1) describe 51 Sg and 5] + Sg geometrically. 55942570 Let the function : R? -> R be defined by 2 >, 83) = (0, 0) flx,y)= † ±“ +y 0 » if(x, y)=(0,0) 2 (c) Q3. (a) (b) (c) BJKE-U-MTH Then show that Gi) Dy, f(0, 0) and Dg f(0, 0) exist. Gi) f(x, y) is continuous at (0, 0) by e-5 method. (iii) fx, y) is not differentiable at (0, 0). where D; and De denote partial derivatives with respect to x and y respectively. 44447=15 (i) Find the equation of the plane which passes through the point (2, 1, —1) and is orthogonal to each of the planes x — y + z = 1, 3x + 4y —2z=0. 8 Gi) Find the equation of the sphere for which the circle x24 y24 224 Ty—-224+2=0, 2x + 8y+ 42 =8 isa great circle. 7 Find the maximum and minimum distances from the origin to the curve 5x2 4 6xy + By” —8=0, using Lagrange’s Multiplier method. 10 Let T: R°—> R® bea linear transformation defined by T(x, y, z) = (2x, 4x — y, 2x + 8y — z). Prove that T is invertible and find pol. 15 (i) Find the coordinates of the vertex, focus and the length of the latus rectum of the principal sections of the paraboloid given by the ; x? y? 22 equation ae ae = 8 a b c Gi) Find the nature of the quadric surface given by the equation 2x? + By + 822 — 4x + 20y — 6z = 5. Also find its associated characteristics, principal axes and principal planes. 7 3 . (a) (b) (c) BJKE-U-MTH Prove that the straight lines -1 2410 x-4 y+3 241 = = d = ai . intersect and find the equation of the plane containing them. Also find their point of intersection. 10 Show that the matrix 7 -6 6 A=/2 0 4 ¶ ‎ 18 is diagonalizable and find a spectral decomposition of the matrix A. 15 Find the volume bounded by the cylinder ±2 + 3 =4and planesy+z=4 and z= 0. 15 Q5. (a) (b) (c) (d) (e) BJKE-U-MTH SECTION B Find the solution of the differential equation 2x 2y( 4 = tan (xy?) — 2xy”. What will be the definite value of the arbitrary constant, appearing in the solution, on coordinate axes ? Identify that one solution of the equation xy” +(x-l)y’'-y=0 which is of the form ce*™* and then find the other solution by method of reduction of order. Four bars are joined together to form a rhombus. The bars are uniform and each bar is of weight W. A rhombus is suspended vertically from one of the joints and a spherical ball of weight S is balanced inside the rhombus so as to keep the system intact. If 20 is the angle at a fixed joint in the state of equilibrium, then find the ratio of weight of the rhombus to that of the spherical ball in terms of the radius of the sphere, the length of a bar and the angle ’. In a central orbit, the central force is given as wur(3 + 2a%u2). If a particle is projected at a distance a with velocity [Pe in a direction a making an angle tan"! ‘ with the radius, then show that equation of its path can be written as r=a tan 0. > > If v.E =0, v.H =0, vxE --& and ‘Vx H = “, then show oO eH ee that V?H = —— and V2E = °~ ot ot 5 Q6. (a) (b) (c) Q7. (a) (b) BJKE-U-MTH Solve the differential equation y (xy + 2x7?) dx + x (xy — xy?) dy = 0. Verify that the obtained solution satisfies the given differential equation. A particle slides down the smooth curve, y = a sinh (): the axis of x being horizontal and the axis of y downwards, starting from rest at the point where the tangent is inclined at « to the horizon. Show that the particle will leave the curve when it has fallen through a vertical distance a sec a. (i) Show that the area bounded by a simple closed C is given by 5 { (x dy — y dx). Hence obtain the area of an ellipse. Cc (ii) Evaluate { (e* dx + 2y dy — dz) by using Stokes’ theorem, where r I’ is the curve x2 4 y2= a2, z= hi A uniform ladder of 10 m length and of 10 kg weight rests with its foot on the rough ground and its upper end against a smooth wall, the inclination to the vertical being a. A force P is applied horizontally to the ladder at a point distant 3m from the foot, so as to make the foot approach the wall. Prove that the force P must exceed “(u + 5 tan a), where wu is the coefficient of friction at the foot. Find the complete solution of 2 s yay OY yy - los xsin (log x) +1 dx? dx x 10 15 10 15 (c) Q8. (a) (b) (c) BJKE-U-MTH @ Given that u =x +y+z,v=x2 + y* +2 and w = xy + yz + 2x. Show that the vectors grad u, grad v and grad w are coplanar. 8 > 3 Gi) For the curve given by r = [2s t?, 5 find the curvature and torsion at t = 1. a > > (i) Given that the vectors f and g are irrotational. Show that the > Oo vector f x g_ is solenoidal. > A A A Gi) Show that the vector q = (6xy + z3)i + (8x2—2)j + (3x22 — y)k EN is irrotational and find a scalar function @ such that q = grad 9. 3+7=10 Find singular solution of the differential equation y2 (8) —axy(8E tan? + y? soc? 2? tan? B=0 directly and from its complete primitive. Determine tac-locus. Show that the envelope of family of curves, which is represented by the given equation, is y=+x tan B. 15 An elliptic lamina is completely immersed in water with its plane vertical. Its minor axis is horizontal and is at a depth h. Determine the centre of pressure. 15