Download SGBAU BSc 2019 Summer 1st Sem Mathematics Differential Integral Calculus Question Paper

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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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1. Choose the correct alternatives (1 mark each) : 10
   1. The value of lim x→0 sin x / x is:
      1. 0
      2. 1
      3. ∞
      4. None of these

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   2. If y = e^2x then y_n is:
      1. 2ⁿ e^x
      2. 2ⁿ e^2x
      3. 2ⁿ e^x
      4. None of these

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   3. The series : x - x³/3! + x⁵/5! - ... is the expansion of function :
      1. sin x
      2. sinh x
      3. cos x
      4. cosh x

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   4. |x - x₀| < δ represents :
      1. x₀ - δ < x < x₀ + δ
      2. x₀ + δ < x < x₀ - δ
      3. x₀ - δ ≤ x ≤ x₀ + δ
      4. x₀ - δ < x < x₀ + δ

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   5. If f be differentiable on (a, b) and f'(x) = 0, ∀ x ∈ [a, b], then f(x) is :
      1. Monotonic increasing in [a, b]
      2. Monotonic decreasing in [a, b]
      3. Constant in [a, b]
      4. None of these

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   6. For f(x) = x² in [1, 3] then the value of ‘C’ by Lagrange’s mean value theorem is :
      1. 6/5
      2. 2
      3. 0
      4. 1

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   7. ∫_a^b f(x) dx = - ∫_b^a f(x) dx
      1. True
      2. False
   8. The functions f and g be
      1. continuous in [a, b]
      2. derivable in (a, b) and

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      3. g'(x) ≠ 0 for all x ∈ (a, b).
      These are the hypothesis of mean value theorem by :
      1. Rolle’s
      2. Lagrange's
      3. Cauchy’s
      4. Leibnitz

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   9. The function f(x) has the removable discontinuity if :
      1. f(x⁺) = f(x^-)
      2. f(x⁺) = f(x^-) = f(x)
      3. f(x⁺), f(x^-) do not exist
      4. None of these

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   10. d/dx cosh x is
       1. sinh x
       2. -sinh x
       3. h sinh x
       4. -h sinh x

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UNIT—I

2. 1. If lim_x→x0 f(x) = l and lim_x→x0 g(x) = m, then prove that : lim_x→x0 [f(x)+g(x)] = lim_x→x0 f(x) + lim_x→x0 g(x) = l + m. 4
   2. Prove that the function defined by f(x) = x² is continuous for all x ∈ R. 3

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   3. Using definition of limit, prove that :
3. 1. Define limit of a function and show that the limit of a function if it exist is unique. 1+3
   2. Prove that lim_x→2 x² = 4; by using ε-δ definition. 3

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4. 1. 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

5. 1. 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

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   2. Evaluate : lim_x→0 (cos x)^cot2x 4
   3. If y = A sin mx + B cos mx, then prove that y₂ + m²y = 0. 3
6. 1. If y = sin(m sin^-1x), then show that :
      1. (1 - x²)y₂ - xy₁ + m²y = 0
      2. (1 - x²)y_n+2 - (2n + 1)xy_n+1 - (n² - m²)y_n = 0.

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7. 1. If y = 1/(ax + b), then prove that y_n = 2ⁿ(-1)ⁿ n! / (ax+b)ⁿ⁺¹. 5
   2. Evaluate : lim_x→0 (x - sinx) / x³ 4

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UNIT—ITI

7. 1. State and prove Lagrange’s mean value theorem. 5
   2. Verify Cauchy mean value theorem for the functions : f(x) = e^x and g(x) = e^-x in [a, b]. 4
   3. Expand sin x in powers of (x - π/2), upto first four terms. 3

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8. 1. State and prove Cauchy’s mean value theorem. 5
   2. Expand 3x² + 4x² + 5x - 3 about the point x = 1 by Taylor’s theorem. 4
   3. Verify the Rolle’s theorem for the function : f(x) = e^x sin x in [0,π]. 3

UNIT—IV

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9. 1. 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
   2. If u=e^x (xcosy-ysiny), then find the value u_xx + u_yy. 3
10. 1. If u= f(x, y) be homogeneous function of degree n then prove that :
       1. ∂u/∂x, ∂u/∂y are homogeneous functions of degree ‘n - 1’ in x, y and
       2. x² ∂²u/∂x² + 2xy ∂²u/∂x∂y + y² ∂²u/∂y² = n(n-1)u. 4

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    2. 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
    3. If u=log(x³ + y³) / (x - y), x ≠ y, then prove that :
       1. x ∂u/∂x + y ∂u/∂y = 2
       2. x² ∂²u/∂x² + 2xy ∂²u/∂x∂y + y² ∂²u/∂y² = -3. 3

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UNIT—V

10. 1. Prove that : ∫sin^mx cosⁿx dx = (sin^m+1x cos^n-1x) / (m+n) + (n-1)/(m+n) ∫sin^mx cos^n-2x dx 4
    2. Evaluate : ∫(x² +2x+3) / √(x² +x+1) dx 4
    3. Show that 8a is the length of an arc of the cycloid x = a(t - sin t), y = a(1 - cos t); 0<t<2π 4

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11. 1. Prove that : ∫tanⁿx dx = (tan^n-1x) / (n-1) - ∫tan^n-2x dx. Hence evaluate ∫tan⁴x dx . 4
    2. Find the area bounded by the x-axis, the curve y = c cosh(x/c) and the ordinates x = 0, x = a. 3
    3. Show that length of the curve y = log sec x between the points, where x = 0 and x = π/3 is log (2+√3). 3

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