Download Previous year question paper maths and more Cheat Sheet Computer Science in PDF only on Docsity!
Bihar Engineering University, Patna
B.Tech 1st^ Semester Exam-
Course: B. Tech. Time: 03 Hours
Code: 105102 Subject: Mathematics-1(Calculus & Linear Algebra) Full Marks: 70
Instructions:-
(i) The marks are indicated in the right-hand margin.
(ii) There are NINE questions in this paper.
(iii) Attempt FIVE questions in all.
(iv) Question No. l is compulsory.
Q.1 Choose the correct answer of the following (any seven Question only):
(a) Lagrange's mean value theorem can be proved for a function f(x) by applying Roll's mean value
theorem
(a) φ(x) = f(x) + kx^2
(b) φ(x) = f(x) – kx^2
(c) φ(x) = f(x) + kx
(d) φ(x) = {f(x)}^2 + kx^2
(b) The function f(x) = x(x + 3) 𝑒−
𝑥 (^2) satisfy all conditions of Roll's mean value theorem in the interval [-
3, 0]. Then the value of c is:
(a) 0.
(b) 1.
(c) 2.
(d) -2.
(c) if
is an eig e nvector of
. What is the associated eigenvalue?
(i) 4/
(ii) 5.
(iii) -2.
(iv) None of the above
(d) if f is continuous on [-2.354, 2.354] then
(i) 𝑓(𝑐𝑜𝑠𝑥)𝑑𝑥
−2.354 = 0
(ii) 𝑓 𝑐𝑜𝑠𝑥 𝑑𝑥
−2.354 = 2^ 𝑓(𝑐𝑜𝑠𝑥)𝑑𝑥
0
(iii) 2.
(iv) none of the above
(e)let f(x) = 𝑥
3 (^2) , x ϵ R then
(i) f is uniformly continuous
(iii) f is continuous, but not differentiable at x = 0
(iii) f is differentiable, and derivative of f is continuous
(iv) f is differentiable, but derivative of x is discontinuous at x = 0
(f) if A(2) = 2i - j + 2k, A(3) = 4i – 2j + 3k. then 𝐴. 𝑑𝐴 𝑑𝑡 𝑑𝑡^ 𝑖𝑠
3 2
(i) 5
(ii) 10
(iii) 15
(iv) 20
(g) if ∇ x 𝐹 0. Then it is called.
(i) solenoidal
(ii) Rotational
(iii) irrotational
(iv) None
(h) if 3x + 2y + z = 0, x + 4y + z = 0, 2x + y + 4Z = 0 be a system of equations, then
(i) It is inconsistent
2 𝑛 − 2 2 3.4.5…(2𝑛−1)2𝑛 𝑥
2 𝑛
Q. 4 (a) Obtain the fourth-degree Taylor's polynomial approximation to
f(x) = e2x^ about x = 0. Find the maximum error when 0 ≤ x ≤ 0.
(b) It is given the Rolle's theorem holds the function f(x) x^3 + bx^2 + cx, 1 ≤ x ≤ 2 at the Point x = 4/3.
Find the value of b and c.
Q. 5 (a) Evaluate log 𝑥 + 1 𝑥
𝑑𝑥 1+𝑥^2
∞ 0
(b) Discuss the convergence of the sequence whose n-th term is an =
− 1 𝑛 𝑛 + 1.
Q. 6 if f (x) = log (1 + x), x > 0 using <Madaurin's theorem, show that for 0 < θ < 1,
log (1 + x) = x – 𝑥^2 2 +^
𝑥^3 3 1+ 𝜃𝑥 3
Deduce that log (1+x) = < x – 𝑥^2 2 +^
𝑥^3 3 𝑓𝑜𝑟^ 𝑥^ > 0
(b) Using Taylor's theorem, express the polynomial 2x^3 + 7x^2 + x - 6 in powers of (x - l).
Q. 7 (a) Find the values of a,b,c if A =
is orthogonal?
(b) Verify Cayley-Hamilton theorem for the matrix A =
and find its inverse. Also express A^5 –
4A^4 - 7A^3 + 11A^2 – A – 10I as a linear polynomial in A. https://www.akubihar.com
Q. 8 (a) Expand long x in power of (x - 1) by Taylor's theorem.
(b) By using Beta function evaluate 𝑥^5 1 + 𝑥^3 10 𝑑𝑥. 1 0
Q. 9 (a) Find the Fourier series to represent the function defined as
𝜋 2 ,^ −^ 𝜋^ <^ 𝑥^ < 0 𝜋 2 −^ 𝑥, 0 <^ 𝑥^ <^ 𝜋
(b) Evaluate: (i) div F and Curl F, where F= grad (x^3 + y^3 + z^3 – 3xyz).
(ii) If F = (x + y + z) 𝑖 + 𝑗 - (x + y)𝑘 show that F Curl F = 0.
