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B.Sc. Part-I (Semester-I) Examination
MATHEMATICS
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(Differential & Integral Calculus)
Paper—II
Time : Three Hours] [Maximum Marks : 60
Note :— (1) Question No. 1 is compulsory. Attempt once.
(2) Attempt ONE question from each unit.
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- Choose the correct alternatives (1 mark each) : 10
- The value of lim x?0 sin x / x is:
- 0
- 1
- 8
- None of these
- If y = e2x then yn is:
- 2n ex
- 2n e2x
- 2n ex
- None of these
- The series : x - x3/3! + x5/5! - ... is the expansion of function :
- sin x
- sinh x
- cos x
- cosh x
- |x - x0| < d represents :
- x0 - d < x < x0 + d
- x0 + d < x < x0 - d
- x0 - d = x = x0 + d
- x0 - d < x < x0 + d
- If f be differentiable on (a, b) and f'(x) = 0, ? x ? [a, b], then f(x) is :
- Monotonic increasing in [a, b]
- Monotonic decreasing in [a, b]
- Constant in [a, b]
- None of these
- For f(x) = x2 in [1, 3] then the value of ‘C’ by Lagrange’s mean value theorem is :
- 6/5
- 2
- 0
- 1
- ?ab f(x) dx = - ?ba f(x) dx
- True
- False
- The functions f and g be
- continuous in [a, b]
- derivable in (a, b) and
- g'(x) ? 0 for all x ? (a, b).
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- Rolle’s
- Lagrange's
- Cauchy’s
- Leibnitz
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- The function f(x) has the removable discontinuity if :
- f(x+) = f(x-)
- f(x+) = f(x-) = f(x)
- f(x+), f(x-) do not exist
- None of these
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- d/dx cosh x is
- sinh x
- -sinh x
- h sinh x
- -h sinh x
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- The value of lim x?0 sin x / x is:
UNIT—I
-
- If limx?x0 f(x) = l and limx?x0 g(x) = m, then prove that : limx?x0 [f(x)+g(x)] = limx?x0 f(x) + limx?x0 g(x) = l + m. 4
- Prove that the function defined by f(x) = x2 is continuous for all x ? R. 3
- Using definition of limit, prove that :
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- Define limit of a function and show that the limit of a function if it exist is unique. 1+3
- Prove that limx?2 x2 = 4; by using e-d definition. 3
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- If f(x) = x2 sin(1/x), x ? 0 and f(x) = 0, x = 0 then show that f(x) has a simple discontinuity at x = 0.
UNIT—II
-
- Prove that if f(x) is differentiable at x = x0, then it is continuous at x = x0. Is converse of this statement true ? Justify. 5
- Evaluate : limx?0 (cos x)cot2x 4
- If y = A sin mx + B cos mx, then prove that y2 + m2y = 0. 3
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- If y = sin(m sin-1x), then show that :
- (1 - x2)y2 - xy1 + m2y = 0
- (1 - x2)yn+2 - (2n + 1)xyn+1 - (n2 - m2)yn = 0.
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- If y = sin(m sin-1x), then show that :
-
- If y = 1/(ax + b), then prove that yn = 2n(-1)n n! / (ax+b)n+1. 5
- Evaluate : limx?0 (x - sinx) / x3 4
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UNIT—ITI
-
- State and prove Lagrange’s mean value theorem. 5
- Verify Cauchy mean value theorem for the functions : f(x) = ex and g(x) = e-x in [a, b]. 4
- Expand sin x in powers of (x - p/2), upto first four terms. 3
-
- State and prove Cauchy’s mean value theorem. 5
- Expand 3x2 + 4x2 + 5x - 3 about the point x = 1 by Taylor’s theorem. 4
- Verify the Rolle’s theorem for the function : f(x) = ex sin x in [0,p]. 3
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UNIT—IV
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- If u= f(x, y, z) is a homogeneous function of degree n, then show that : x ?u/?x + y ?u/?y + z ?u/?z = nu 4
- If u=ex (xcosy-ysiny), then find the value uxx + uyy. 3
-
- If u= f(x, y) be homogeneous function of degree n then prove that :
- ?u/?x, ?u/?y are homogeneous functions of degree ‘n - 1’ in x, y and
- x2 ?2u/?x2 + 2xy ?2u/?x?y + y2 ?2u/?y2 = n(n-1)u. 4
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- If u=3(ax +by+cz) - (x2+y2+2z2) and a2 + b2 + c2 = 1, then show that : ?2u/?x2 + ?2u/?y2 + ?2u/?z2 = 0 3
- If u=log(x3 + y3) / (x - y), x ? y, then prove that :
- x ?u/?x + y ?u/?y = 2
- x2 ?2u/?x2 + 2xy ?2u/?x?y + y2 ?2u/?y2 = -3. 3
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- If u= f(x, y) be homogeneous function of degree n then prove that :
UNIT—V
-
- Prove that : ?sinmx cosnx dx = (sinm+1x cosn-1x) / (m+n) + (n-1)/(m+n) ?sinmx cosn-2x dx 4
- Evaluate : ?(x2 +2x+3) / v(x2 +x+1) dx 4
- Show that 8a is the length of an arc of the cycloid x = a(t - sin t), y = a(1 - cos t); 0<t<2p 4
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- Prove that : ?tannx dx = (tann-1x) / (n-1) - ?tann-2x dx. Hence evaluate ?tan4x dx . 4
- Find the area bounded by the x-axis, the curve y = c cosh(x/c) and the ordinates x = 0, x = a. 3
- Show that length of the curve y = log sec x between the points, where x = 0 and x = p/3 is log (2+v3). 3
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