Test 1: Calculus
Instruction: Select the correct option corresponding to questions given below in the form shared at the bottom of the page
Total Marks: 100
1) Define a sequence sn of real numbers by sn=∑ni=1(log(n+i)−logn)2n+i. Then, the value of limn→∞sn is equal to:
(a) 0
(b) (log2)34
(c) (log2)33
(d) Limit does not exist
2) Let
L=limn→∞(1+1n)n. Then, L lies in the interval
(a) (0,1)
(b) (1,2)
(c) (2,3)
(d) (3,4)
3) If √1−x2+√1−y2=a(x−y), then dydx is equal to
(a) 1
(b) √1−y2√1−x2
(c) 0
(d) √1−x2√1−y2
4) If In=dndxn(xnlogx), then In−nIn−1 is equal to:
(a) 0
(b) 1
(c) n!
(d) (n−1)!
5) The maximum distance of the normal from the centre of the ellipse
x2a2+y2b2=1is given by
(a) √a2+b2
(b) (a−b)
(c) 12√a2+b2
(d) 12(a−b)
6) Find the values of a and b in order that limx→0x(1+acosx)−bsinxx3
(a) a=−52, b=−32
(b) a=−58, b=−38
(c) a=−54, b=−34
(d) a=52, b=32
7) If z is a homogeneous function of x, y of order n. Consilder the following statements:
(i): x∂f∂x+y∂f∂y=nf
(ii): x2∂2f∂x2+2xy∂2f∂x∂y+y2∂2f∂y2=n2f
Which of the following options is correct?
(a) (i) is correct but (ii) is incorrect
(b) (ii) is correct but (i) is incorrect
(c) Both (i) and (ii) are correct
(d) Both (i) nor (ii) are incorrect
8) The extreme value of xy(a−x−y) is given by:
(a) (0,0)
(b) (0,a)
(c) (a3,a3)
(d) The given function has no extreme values
9) Let If f′(x)=11+x2 for all x and f(0)=0. Then, f(2) lies in the interval:
(a) (0.4,2)
(b) (3,4)
(b) (4,8)
(b) (0,1)
10) If u=x+y1−xy and v=tan−1x+tan−1y, then,
(a) u=tanv
(b) v=tanu
(c) u=cotv
(d) v=cotu
11) Let f(x,y)={x2tan−1yx−y2tan−1xy,x≠0,y≠00,x=y=0
Then,
(a) f(x,y) is continuous and differentiable at (0,0)
(b) f(x,y) is continuous but not differentiable at (0,0)
(c) f(x,y) is neither continuous nor differentiable at (0,0)
(d) f(x,y) is diffrentiable but not continuous at (0,0)
12) The volume generated by revolving y2=4ax about the latus rectum is given by:
(a) 32πa315
(b) 16πa315
(c) 8πa315
(d) 4πa315
13) The value of ∫x1tdt1+t2+∫1x1dtt(1+t2) is given by:
(a) 1
(b) tanx
(c) tan−1x
(d) 0
14) The maximum and minimum values of f(x,y)=7x2+8xy+y2, where x, y are constrained by the relation x2+y2=1, are given by:
(a) Max= 9, Min=1
(b) Max= 9, Min= -1
(c) Max= 8, Min=2
(d) Max= 8, Min=-1
15) Let f(x)=x if x is rational, and 1−x if x is irrational. Then,
(a) f(x) is continuous at all points
(b) f(x) is discontinuous at all points
(c) f(x) has a removable discontinuity at x=12
(d) f(x) is continuous at x=12
16) The value of
∬(1−x2a2−y2b2)32dxdyover the area of the ellipse x2a2+y2b2=1 is given by:
(a) πab
(b) πab5
(c) 2πab5
(d) 4πab5
17) Let the volume of a right circular cylinder of greatest volume which can be inscribed in a sphere of volume V is given by v. Then, V/v is given by:
(a) 2
(b) √2
(c) 3
(d) √3
18) The value of the integral ∫π20log(sinx)dx is given by:
(a) Integral is divergent
(b) −π2log2
(c) π2log2
(d) 0
19) The percentage error in the volume of a right circular cone, when an error of 1% is made in measuring the height and an error of 0.5% is made in measuring the base radius, is given by:
(a) 2
(b) 0.5
(c) 1
(d) π3
20) The value of ∫∞0log(1+a2x2)1+b2x2dx is given by:
(a) πblog(b+ab)
(b) πalog(b+ab)
(c) 2πblog(b+ab)
(d) π2blog(b+ab)