Page 1 : Functions of several variables, , [functions of more variables, directional derivative, partial derivative, derivative as a linear transformation, inverse, and implicit function theorems], , Marks per question: 3 and __— Penalty for wrong answer: -1, , xy? ., Ql. For the function f(x, y) defined f(x,y) =} x?+y* if Gy) +o, , 0, if @y)=0, , at the origin, , directional derivative, (A) exists and f(x) is continuous, (B) exists and f(x) is not continuous, (C) does not exists and f (x) is continuous, , (D) does not exists and f(x) is not continuous, , Q2. Let f:IR? > R? be defined by f(x,y) = fe + ae * ney ee ares, Then, , (A) f is not continuous at (0,0), , (B) f is continuous at (0,0) but not differentiable at (0,0), , (C) f is differentiable only at (0,0), , (D) f is differentiable everywhere, , Q3. Let f,g:IR? > R be defined by f(x,y) = x* + y?; g(x,y) =x*+y? —10x7y., Then at (0,0), , (A) f has a local minimum but not g, , (B) g has a local minimum but not f, , (C) both f and g havea local minimum, , (D) neither f nor g has a local minimum, , : 1-cos(x?+y?) : ob at, Q4. The values of etna: ete and ea (ysin r + xsin 2 are, (A) 1,0 (8) >,0 (C) 0,1 (D) 0,>, , , , , , , , NET/GATE-MATHEMATICS WhatsApp: 8918629317

Page 2 : Q6., , qQ7., , Qs., , a9., correct? (Here Df denotes the derivatives of f), , Qio., , Given a equation F(x, y) = x? — y” = 0. Then Implicit function theorem can apply, (A) at (0,0) but not at (1,1) (B) at (1,1) but not at (0,0), (C) neither at (0,0) nor at (1,1) | (D) both at (0,0) and (1,1), , The function f (x, y) = x3 + y? — 63(x+ y) + 12xy, (A) has four stationary points (B) is minimum at (—7, —7), , (C) is maximum at (3,3) (D) is maximum at (5, —1), , The matrix of derivative Df of f: IR? > R? given by f(x,y) = (x + y, xy) is, , my) eG) G7) (5), , Consider the map f: R* > R2defined by f(x,y) = (7x + x*,3x + 4y + y*). Then,, (A) f is discontinuous at (0,0)., , (B) f is discontinuous at (0,0) but all directional derivatives exist at (0,0)., , (C) f is differentiable at (0,0) but the derivatives Df (0,0) is not invertible., , (D) f is differentiable at (0,0) and the derivative Df (0,0) is invertible., , Let f(r, ®) = (rcos®,rsin®)for (r, 8) € IR? with r # 0. Which of the following is, , (A) The linear transformation Df (r, 8) is zero for any (r,®) € R? withr + 0, (B) f is one-one on { (7,0) € R?:r #0}, (C) For any (r, 6) € R? with r + 0, f is one-one on some neighbourhood of (r, 8), , (D) Df (r, 8) = r7I, for any (r, 6) € R? withr # 0; Izis identity matrix of order 2, , 2xy, , ae Ae ,y) #0, , The function f(x,y) =} (*+y7)" if %Y) is continuous at (0,0) if, 0 , if @y)=0, , (A)n=1 (B)n = 2 (C)n => (D)n =>, , , , NET/GATE-MATHEMATICS WhatsApp: 8918629317

Page 3 : xy?, = 0, Q11. For the function f(x,y) =} x?+y? i Oy) +, 0, if @y)=0, (A) the conditions of Schwarz’s theorem are satisfied., (B) the conditions of Young’s theorem are satisfied., , (C) both the conditions of Schwarz’s theorem and Young’s are satisfied., , (D) fey = fyxat (0,0), , Q12. Let f:IR? > IR? be the function defined by f(x,y) = (x3 + xy?,x?y + y*) for all, (x, y) € R. Which of the following statement is correct?, , (A) Df (0,0) # 0 (B) Df (x,y) = 0 for all (x,y) ER, , (C) f is one-one (D) f is not differentiable, , Q13. Consider IR? with the usual metric and the functions f:[0,21) > IR? and g: [0,27] >, R? defined by f(t) = (cost, sint),0 < t < 2mand g(t) = (cost, sint),0 < t <2. Then, , (A) f is uniformly continuous but g, (B) g is uniformly continuous but not f, (C) both f and g are uniformly continuous, , (D) neither f nor g is uniformly continuous, , Q14. The Taylor’s expansion of f(x, y) = e*In(1 + y) at (0,0) is, , 2, , 2, A) fay) =ytxy-S +e (8) f@y) =y-xy-2, y Re, (fy) =y Fx Se (D) f@y)=x+y—-St+e, Q15. Using implicit function theorem, there exists a single variable function g defined by, , the equation F(x, y) = x° + y° — 16x%y — 1 = 0 ina neighbourhood of x = 1 such that, g(1) = 2. Then derivative function, g’(x) is, , 5x4-48x? 5x4-48x?, (A) e Sx Y ( ) Ce x VY, 5y*-16x3 5Sy*+16x3, 5x4448x? —5x*+48x2, (oe (=, 5Sy*-16x3 5y*-16x3, , , , , , , , NET/GATE-MATHEMATICS WhatsApp: 8918629317