A Firstranker's choice
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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+v(x2+1) ] (b) log[x+v(x2—1) ]
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(c) log[x+v(1—x2) ] (d) None of these
(3) The value of 4tan-1(1/5) - tan-1(1/70) + tan-1(1/99) is
(a) p/2 (b) p/4
(c) v3 (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 a + iß be the root of quadratic polynomial f(x) = 0 then its another root is
(a) a—iß (b) a
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(c) ß (d) None of these
(8) If a, ß, ? are the roots of the equation px3+qx2 + rx + s =0 then Sa is
(a) q/3 (b) -q/p
(c) r/p (d) s/p
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(Contd.)
Firstranker's choice
(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+sin?+icos?)/(1+sin?-icos?) = sint +icos?.
Hence prove that [1+sin(p/5)+icos(p/5)]n + [1+sin(p/5)-icos(p/5)]n =0.
(b) If sin(a +iß) =x +iy then prove that cosh2ß - sin2a = x2+y2 and cosa sinhß =1.
3. (p) Prove that one of the value of :
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(cos? + isin?)n is (cosn? + isinn?); 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) = p/4 5
5. (p) Find the sum of the series sinh x + (1/2)sinh 2x+(1/3)sinh 3x+ ..... 5
(q) If -p/4 < x < p/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(p/3)+isin(p/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
www.FirstRanker.com (Contd.)
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A Firstranker's choice
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+v3 is one of the root. 5
UNIT—V
10. (a) Show that if ? is the eigen value of a nonsingular matrix A then ?-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 |
Firstranker's choice www.FirstRanker.com
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