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Roll No. Total No. of Pages : 03
Total No. of Questions : 09
Bachelor of Science - Honours (Mathematics) (Sem.-1)
CALCULAS-I
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Subject Code : UC-BSHM-101-19
M.Code : 77312
Time : 3 Hrs. Max. Marks : 60
INSTRUCTIONS TO CANDIDATES :
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- SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
- SECTION -B & C have FOUR questions each.
- Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
- Select atleast TWO questions from SECTION - B & C.
SECTION-A
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- Solve the following :
{x: |2x+5| > 1; |x—4| < 2}. - a) Find the L.u.b. and g.l.b., if they exist for the set. A = {x + 1/x : x > 0}.
b) Define the greatest integer function. Also write its domain and range. - c) Prove that d/dx(sinh x)=cosh x.
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d) Differentiate cos-1(2x2-1) with respect to x if 0 < x < 1. - e) Discuss the applicability of Rolle’s Theorem for the function f(x) = | x | in the interval [-3, 3].
- f) Evaluate limx?1 log x / (x-vx).
- g) Show that y = x + a is the only asymptote of the curve x2 (x — y) + ay2 = 0.
- h) Find the nth derivative of 1/((x+2)(x+3)).
- i) Using e — d definition, prove that f(x) =3x +2 is continuous at x = 2.
- j) State Cauchy’s Mean Value theorem.
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SECTION-B
- State and prove Archimedean property of real numbers.
- Express the function f(x) = 1/(x-1) as a composite of two ‘simpler’ functions, and state necessary conditions on their domains.
- Prove that the function f(x) = 1/x is continuous in (0, 1) but is not uniformly continuous.
- Find all the asymptotes of the following curve : x3 + 2x2y - xy2 - 2y3 + 3x2 - 7xy + 2y2 - 3x+2y-1=0
- If f(x)=| x+a ab ac |
&right. | ab x+b bc | , find f’ (x).--- Content provided by FirstRanker.com ---
&right. | ac bc x+c | , find f’ (x). - If sin y = x sin (a + y), prove that dy/dx = sin2(a+y) / sina
- If y=log (x+v(x2 +a2) , show that (x2 +a2)y2 +xy1=0.
- Find the derivative of xxx.
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SECTION-C
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- Find the interval of concave upwards for the curve y = (cos x + sin x) e-x in (0, 2p).
- Show that the curve y = log2(x) / x has a point of inflexion at (e2, 4/e2).
- a) Find the values of a and b, so that the limx?0 (x(1-acosx)+bsinx) / x3 exists and is equal to 1/3.
b) Use Lagrange’s Mean Value theorem to prove that x / v(1-x2) < sin-1 x < x for 0 < x < 1. - a) Find the nth derivative of sin x sin 2x sin 3x.
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b) If y=(sin-1 x)2, find yn (0) - a) If f(x)=tanx, then prove that
nC0fn(0) + nC2fn-2(0) + nC4fn-4(0) + ... = sec(np/2).
b) Use Maclaurin’s Theorem (with Lagrange’s form of remainder) to expand sin x.
NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.
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This download link is referred from the post: PTU B.Sc (Honours) 2020 March Previous Question Papers
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