IIT JAM Mathematics Test Series 2023 : Real Analysis – Functions of Several Variables

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic Functions of Several Variables of the Chapter Real Analysis. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Share with Your Friends.

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IIT JAM Mathematics Test Series 2023 : Real Analysis – Functions of Several Variables


Q1. A function f(x,y) on R2 has the following partial derivatives

∂f(x,y)/∂x = x2  , ∂f(x,y)/∂y = y2

Then

(a) f has directional derivatives in all directions everywhere
(b) f has derivative at all points
(c) f has directional derivative only along the direction (1,1) everywhere
(d) f does not have directional derivatives in any direction everywhere

Q2. A function f:R2 → R is defined by f(x,y) = xy. Let v = (1,2) and a = (a1, a2) be two elements of R2. The directional derivative of  f  in the direction of v at a is

(a) a1 + 2a2
(b) a2 + 2a1
(c) a2/2 + a1
(d) a1/2 + a2

Q3. Let f:R2 → R2 be given by f(r,θ) = (rcosθ , rsinθ). Then for which of the subset U of R2 given below, f restricted to U admits an inverse? 

(a) U = R2
(b) U = {(x,y)∈R2 : x > 0 , y > 0}
(c) U = {(x,y)∈R2 : x2 + y2 < 1}
(d) U = {(x,y)∈R2 : x < -1 , y < -1}

Q4. Consider the function f(x,y) = x2/y2 , (x,y) ∈ [1/2,3/2]×[1/2,3/2]. Then derivative of the function at (1,1) along the direction (1,1)

(a) 0
(b) 1
(c) 2
(d) -2

Q5. Let f(x,y) = xy/(x2 + y2) ,(x,y) ≠ (0,0) ; f(0,0) = 0 . Then

(a) f is continuous at (0,0) and the partial derivatives fx’fy exist at every point of R2
(b) f is discontinuous at (0,0) and fx’f exist at every point on R2
(c) f is discontinuous at (0,0) and fx’f exist only at (0,0)
(d) None of these

Q6. Let f:R2 → R and g:R2 → R be defined by f(x,y) = |x| + |y| and g(x,y) = |xy| . Then

(a) f is differentiable at (0,0), but g is not differentiable at (0,0)
(b) f is differentiable at (0,0), but f is not differentiable at (0,0)
(c) Both f and g are differentiable at (0,0)
(d) Both f and g are continuous at (0,0)

Q7. Consider the map f:R2 → R2 defined by f(x,y) = (3x – 2y + x2, 4x+5y+y2). Then

(a) f us continuous at (0,0)
(b) f is continuous at (0,0) and all directional derivatives exists at (0,0)
(c) f is differentiable at (0,0) but the derivative Df(0,0) is not invertible
(d) f is differentiable at (0,0) but the derivative Df(0,0) is invertible

Q8. The map L:R2 → R2 given by L(x,y) = (x-y) is

(a) Differentiable everywhere on R2
(b) Differentiable only at (0,0)
(c) DL(0,0) = L
(d) DL(x,y) = L for all (x,y) ∈ R2

Q9. Let f:[π, 2π] → R2 be the function f(t) = (cost, sint) . Which of the following are necessarily correct ?

(a) There exists to ∈ [π, 2π] such that f'(to) = 1/π [f(2π) – f(π)]
(b) There does not exist any to ∈ [π, 2π] such that f'(to) = 1/π [f(2π) – f(π)]
(c) There exists to ∈ [π, 2π] such that || f(2π) – f(π) || ≤ || f'(to) ||
(d) f'(t) =  (-sint, cost) for all t ∈ [2π, π]

Q10. Let f:Rn → Rn be the function defined by f(x) = x||x||2 for x ∈ Rn . Which of the following statements are correct ?

(a) (Df)(0) = 0
(b) (Df)(x) = 0 for all x ∈ Rn
(c) f is one-one
(d) f has an inverse

Q11. Let f:A∪E → R2 be differentiable, where

A = {(x,y) ∈ R2 : 1/2 < x2 < y2 < 1} and
E = {(x,y) ∈ R2 : (x-2)2 + (y-2)2 < 1/2} .

Let Df be the derivative of the function  f . Which of the following are necessarily correct ?

(a) If (Df)(x,y) = 0 for all (x,y) ∈ A∪E, then f is constant
(b) If (Df)(x,y) = 0 for all (x,y) ∈ A, then f is constant on A
(c) If (Df)(x,y) = 0 for all (x,y) ∈ E, then f is constant on E
(d) If (Df)(x,y) = 0 for all (x,y) ∈ A∪E, then for some (xo, yo), (x1, x2) ∈ R2,
f(x,y) = (xo, yo) for all (x,y) ∈ A and
f(x,y) = (x1, y1) for all (x,y) ∈ E

Q12. L:Rn → R be the function L(x) = <x, y>, where <•,•> is some inner product on Rn and y is a fixed vector in Rn. Further denote by DL, the derivative of L. Which of the following are necessarily correct ?

(a) DL(u) = DL(v) for all u,v ∈ Rn
(b) DL(0,0,…,0) = L
(c) DL(x) = ||x||2 for all x ∈ Rn
(d) DL(1,1,…,1) = 0

Q13. Let A = {(x,y) ∈ R2 : x+y ≠ -1} define f:A → R2 by f(x,y) = [x/(1+x+y) , y/(1+x+y)]. Then

(a) The Jacobian matrix of f does not vanish on A
(b) f is infinitely differentiable on A
(c) f is injective on A
(d) f(A) = R2

Q14. Let f:Rn → R be the map f(x1,…,xn) = a1x1 + …+anxn , where a = (a1,…an) is a fixed non-zero vector. Let Df(0) denote the derivative of f at 0. Which of the following are correct ?

(a) (Df)(0) is a linear map from Rn to R
(b) [(Df)(0)](a) = ||a||2
(c) [(Df)(0)](a) = 0
(d) [(Df)(0)](b) = a1b1+…+anbn for b = (b1,…,bn)

Q15. Let f(x,y) = √(|xy|) . Then

(a) fx and fy do not exist at (0,0)
(b) fx(0,0) = 1
(c) fy(0,0) = 0
(d) f is differentiable at (0,0)

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Answer Key

01.(a), (b)06.(b), (d)11.(b), (c), (d)
02.(b)07.(b), (d)12.(b)
03.(a)08.(a), (c), (d)13.(a), (b), (c)
04.(a)09.(b), (c), (d)14.(a), (b), (d)
05.(c)10.(a), (c), (d)15.(c)

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