Download JNTUA B.Tech 1-1 R19 2020 January 19A54101 Algebra And Calculus Question Paper

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Code: 19A54101

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B.Tech I Year I Semester (R19) Regular Examinations January 2020

ALGEBRA & CALCULUS

(Common to all branches)

PART - A

(Compulsory Question)

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Time: 3 hours

Max. Marks: 70

1 Answer the following: (10 X 02 = 20 Marks)

(a) Find the rank of the matrix A = 8 1 3 6 1 0 3 2 2 1 -8 -1 -3 4 .

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(b) If ? is an Eigen value of a matrix A then prove that ?^m is an Eigen value of A^m. (m being a positive integer)

(c) Discuss the application of Rolle's theorem to the function f (x)= secx in [0,2p].

(d) State Maclaurin's theorem with Lagrange's form of remainder.

(e) Evaluate ?z/?x and ?z/?y if z = log(x² + y²).

(f) If u = x² - 2y² ,v = 2x² – y² find J (d(u,v))/(d(x,y))

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(g) Evaluate ?₀¹ ?₀^x e^y dydx .

(h) Evaluate ?₀² ?₀² ?₀² (x² + y² + z²)dzdydx .

(i) Show that G(n) = 2?₀⁸ e^-x²x^2n-1dx, (n > 0).

(j) Express the integral ?₀^p/2 v(cot?) d? in terms of beta function.

PART - B

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(Answer all five units, 5 X 10 = 50 Marks)

UNIT - I

2 (a) Reduce the matrix A = -2 -1 -3 -1 1 1 2 3 -1 1 0 1 1 0 1 1 -1 to Echelon form and hence find its rank.

(b) Find the Eigen values and Eigen vectors of the matrix A = 2 0 1 0 2 0 1 0 2 .

OR

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3 (a) Test for consistency the following equations and solve them if consistent :

5x+3y+7z = 4,

3x + 26y+2z=9,

7x+2y+10z = 5.

(b) Verify Cayley-Hamilton theorem for the matrix A = 1 3 7 4 2 3 1 2 1 and hence find its inverse.

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Code: 19A54101

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UNIT - II

4 (a) Verify Rolle's theorem for f (x) = x^2m-1(a-x)²ⁿ in (0,a).

(b) Verify Taylor's theorem for f(x) = (1 - x)^3/2 with Lagrange's form of remainder up to two terms in the interval [0,1].

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OR

5 (a) Verify Lagrange's Mean value theorem for f(x)=(x-1)(x-2)(x - 3) in [0,4].

(b) Verify Cauchy's mean value theorem for f(x) = sinx and g(x) = cos x in the interval [a, b].

UNIT - III

6 (a) If u = x + y + z, uv = y + z, uvw=z, Find ?(x,y,z)/?(u,v,w)

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(b) Discuss the maxima and minima of f (x, y) = x³ y² (1 - x - y).

OR

7 (a) Determine whether the following functions are functionally dependent or not. If functionally dependent, find the functional relation between them: u = x² + y² + 2xy + 2x + 2y, v = e^x+y.

(b) Find the maximum and minimum distances of the point (3, 4, 12) from the sphere x² + y² + z² =1.

UNIT - IV

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8 (a) Change the order of integration and hence evaluate I = ?₀^a?_y^a (y / (x²+y²)) dx dy.

(b) Compute the volume of the sphere x² + y² + z² = a², using spherical coordinates.

OR

9 (a) Evaluate the double integral ?₀⁸ ?₀⁸ e^{-(x² + y²)} dx dy, using polar coordinates.

(b) Evaluate ?₀^a?₀^v(a²-x²)?₀^{v(a²-x²-y²)} (1 / (x+y+z)) dz dy dx.

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UNIT - V

10 (a) Prove that B(m,n) = B(n,m).

(b) Show that ?₀¹ [x^m (log x)ⁿ] dx = ((-1)ⁿ n!) / ((m + 1)ⁿ⁺¹), where n is a positive integer and m>-1. Hence evaluate ?₀¹ [x²(log x)²] dx.

OR

11 (a) Express the following integrals in terms of gamma functions: (i) ?₀⁸ x⁴ e^-x dx. (ii) ?₀¹ v(1-x²) dx.

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(b) Express ?₀¹ x^m (1-xⁿ) dx in terms of Gamma function and hence evaluate ?₀⁸ (x dx)/(v(1-x⁴)).

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