BSc Mathematics SEM II 2021 2022 2022 CALCULUS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Tick mark the correct option ( 2 mks each)
   • 1)The geometric series is convergent if
   • A) always B)|r| >1
   • 2)The function f(x) 10-12x-3x?-x? is
   • A) always increasing B) always decreasingC) both A and B D) neither increasing nor
   • 3) critical points of f(x)=5x?+4x are
   • 4) The derivative of the inverse function of f(x)=x?+3x-7 at x=1 is
   • 5)The sum of the series given below if exist then it is
   • 6) If the series given Za, is convergent then
   • 7)If the convergent then is convergent is not convergent is not convergent
   • 8) The series is said to be conditionally convergent if it is
   • A)convergent but not absolutely convergent
   • B)convergent and absolutely convergent
   • C) absolutely convergent but not convergent
   • 9)The sequence a, =1/n? then
   • A)a, and Za, both converge B) a, and Xa, both diverge C)a, is convergent and La, does not converge. D)a, and both undefined
   • 10)Every bounded sequence in R has
   • A) unbounded subsequence
   • B) convergent subsequence
   • C) divergent subsequence
   • D)no subsequence
   • 11) Ifa function is continuous then
   • A)f is bounded and attains its bounds
   • B)f is bounded and doesn't attain its bounds
   • C)f is not bounded and attains its bounds
   • D)f is neither bounded nor attain its bounds
   • 12)The polynomial has \
   • A) No zero in R B) two zero in R
   • C) three zero in R D) four zero in R are continuous such that g(a)<f(a)<0, g(b)>f(b)>0 then
   • A)f(x).g(x) is not equal to zero for any x in [a,b] is to zero for some x in [a,b]
   • C)f(x).g(x) is equal to one for some x in [a,b] is equal to 2 for some x
   • 14)The series
   • A) oscillates between-1 and 0
   • B) oscillates between-| and 2
   • C) oscillates between-2and 0
   • D) oscillates between | and 0
   • 15)The series is convergent if
   • 16) Ifa function from I to R where is an open interval , is differentiable at point p in I
   • A)f is a nonconstant function on R
   • B)f is a constant function onR
   • C)f is continuous on R
   • D)f is discontinuous onR
   • 17) If y= xe* then nth derivative y,=
   • 18)If a function from R to Ris differentiable even function then is odd function B)f is even function
   • C)f is odd function D)f is zero function
   • 19) The function on R defined as f(x) =1
   • A)continuous and unbounded
   • B)continuous and bounded
   • D)discontinuous and bounded
   • 20) Divide 100 into two parts such that sum of their square is minimum then parts
   • 21) ---gives us existence of atleast one point between x=b at which derivative of the
   • C)Rolle's theorem
   • D)Bolzano,s theorem
   • 25)n" derivative of F Q.2) Attempt any three of the following
   • 1)Prove that the geometric series is convergent if |r|<1
   • 2)Express the number 5.232323.as a ratio of two integers
   • 3) Prove that If the series convergent
   • 4)State The Limit Comparison test for convergence of the series and investigate convergence of
2. Q3 Attempt any three of the following (5 MKS each)
   • 1)Show that has multiple zeros in [-4,4]
   • 2) Prove that If a function f:[a,b]—R is continuous then it is bounded and attains it's bounds
   • 3)Find the derivative of the inverse function of y=xe
   • 4) Prove that If a function f:I—R where I is an open interval is differentiable at p|I then f is
3. Q4 Attempt any three of the following (5 MKS each)
   • 1)Prove that if a real valued function f is derivable in (a,b) and for any x then the function is one one in (a,b)
   • 2) State and prove Lagrange's Mean Value theorem for real valued function
   • 4) Obtain the expansion of upto the first four terms 7 Q.5) Attempt any one of the following (5 MKS each)
   • 1) Discuss the convergence of alternating harmonic series given by
   • 2) Check whether the following function is differentiable at 0 or not?
   • 3) Verify Rolle's theorem for as f(x)=x?

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