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Download SGBAU BSc 2019 Summer 1st Sem Mathematics Algebra n Trignometry Question Paper

Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 1st Sem Mathematics Algebra n Trignometry Previous Question Paper

This post was last modified on 10 February 2020

This download link is referred from the post: SGBAU BSc Last 10 Years 2010-2020 Question Papers || Sant Gadge Baba Amravati university


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B.Sc. (Part—I) (Semester—I) Examination

MATHEMATICS

(Algebra & Trigonometry)

Paper—I

Time : Three Hours] [Maximum Marks : 60

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Note :— (1) Question ONE is compulsory. Attempt once.

(2) Attempt ONE question from each Unit.

1. Choose the correct alternative :— 10

(1) Which one of the following statements is true :—

(a) cosh(x + iy) = coshx-cosy + isinhx-siny

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(b) cosh(x + iy) = cosx cosy + isinx siny

(c) cosh (x + iy) = coshx + cosy - isinhx-siny

(d) cosh(x +iy) = coshx siny + isinhx cosy

(2) What is the value of sinh-1x :

(a) log[x+√(x2+1) ] (b) log[x+√(x2—1) ]

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(c) log[x+√(1—x2) ] (d) None of these

(3) The value of 4tan-1(1/5) - tan-1(1/70) + tan-1(1/99) is

(a) π/2 (b) π/4

(c) √3 (d) 1

(4) Sum of the series x - x2/2 + x3/3 +(-1)n+1xn/n + ..... ; —1 < x < 1 is denoted by ___ .

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(a) log(1 + x) (b) sinhx

(c) coshx (d) ex

(5) If q=2+2i—j + 4k then the norm of q is .

(a) -5 (b) 5

(c) 1/5 (d) None of these

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(6) The inverse of unit quaternion is its

(a) Purely imaginary (b) Purely real

(c) Complex conjugate (d) None of these

(7) If α + iβ be the root of quadratic polynomial f(x) = 0 then its another root is

(a) α—iβ (b) α

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(c) β (d) None of these

(8) If α, β, γ are the roots of the equation px3+qx2 + rx + s =0 then Σα is

(a) q/3 (b) -q/p

(c) r/p (d) s/p

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(9) If A and B are the non-singular matrices of order n then

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(a) (AB)' = AB (b) (AB)-1 =AB

(c) (AB)-1 =B-1A-1 (b) None of these

(10) ‘Every square matrix satisfies its own characteristics equation’ is the statement of

(a) Lagrange’s MVT (b) De-Moivre’s theorem

(c) Cayley-Hamilton theorem (d) Cauchy’s MVT

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UNIT—I

2. (a) Prove that (1+sinθ+icosθ)/(1+sinθ-icosθ) = sint +icosθ.

Hence prove that [1+sin(π/5)+icos(π/5)]n + [1+sin(π/5)-icos(π/5)]n =0.

(b) If sin(α +iβ) =x +iy then prove that cosh2β - sin2α = x2+y2 and cosα sinhβ =1.

3. (p) Prove that one of the value of :

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(cosθ + isinθ)n is (cosnθ + isinnθ); when n is negative integer. 5

(q) Separate real and imaginary parts of tan (x +iy). 5

UNIT—II

4. (a) Find the Sum of the series :

C=1+ecosx.cos(sinx) + e2cosx.cos(2sin x)+ e3cosx.cos(3sinx)+.... 5

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(b) Prove that 4tan-1(1/5) - tan-1(1/70) + tan-1(1/99) = π/4 5

5. (p) Find the sum of the series sinh x + (1/2)sinh 2x+(1/3)sinh 3x+ ..... 5

(q) If -π/4 < x < π/4 then prove that

x = tan-1(tan x) - (1/3)tan3x + (1/5)tan5x + ..... 5

UNIT—III

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6. (a) Prove that for p, q∈H, N(pq) = N(p) N(q) and N(q*) = N(q). 5

(b) For the quaternion q = cos(π/3)+isin(π/3) and the input vector v = i, compute the output vector w under the action of the operators Lq and Lq* . 5

7. (p) Show that the quaternian product need not be commutative. 5

(q) For any p, q∈H, show that pq = qp if and only if p and q are parallel. 5

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8. (a) Find the roots of the equation, 8x4+ 18x3—27x2 — 27x = 0, if these roots are in geometric progression. 5

(b) State Descartes rule of sign. Find the nature of the roots of the equation 2x4 -x2 +4x3 -5 =0. 5

9. (a) Prove that in an equation with real coefficients complex roots occur in pairs. 5

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(b) Solve the equation x4 -2x3 — 22x2 + 62x — 15 = 0; given that 2+√3 is one of the root. 5

UNIT—V

10. (a) Show that if λ is the eigen value of a nonsingular matrix A then λ-1 is the eigen value of A-1 5

(b) Find the eigen values and the corresponding eigen vector for smallest eigen value of the matrix A= | 8 -8 -2 | | 4 -3 -2 | . 5 | 3 -4 1 |

11. (p) Show that the eigen values of any square matrix A and A' are same. 5

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(q) Reduce to canonical form and find the rank of the matrix A= | 1 1 -1 1 | | 1 -1 2 -1 | . 5 | 3 1 0 1 |

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This download link is referred from the post: SGBAU BSc Last 10 Years 2010-2020 Question Papers || Sant Gadge Baba Amravati university

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