Tiêu đề Mathematises II Question Paper.pdf ✅

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Prepared by Uni Bytes / www.unibytes.xyz Don't use our content without permission. Tribhuvan University Faculty of Humanities and Social Science Semester: II Subject: Mathematics II 2021 Batch Group B Attempt any SIX questions 2. Evaluate the limit, limn→θ xcosθ−θcosx x−θ 3. Find the derivative y = e 3x−1 by definition method. 4. Verify Rolle's theorem for the function f(x) = sinx, x ∈ [0, π] and find the point on the curve where the tangent is parallel to the x axis. 5. 5Find maximum and minimum value of the function f(x) = x 3 + 6x 2 + 9x − 2. Also find the point of inflection if any. 6. Evaluate the integral a) ∫ 2x+5 √x 2+5x dx b) ∫ e 2x log x dx 7. Evaluate ∫ √1 + x 3 2 0 dx by using simpson’s 1 3 rule by taking n = 4. 8. Solve the differential equation: 2 dy dx = 2y x + y 2 x 2 Group C Attempt any TWO questions. 9. Define pivot element, pivot column and consistency in the system of equations. Using simplex method, Maximize F = 5x - 3y subject to 3x + 2y ≤= 6 , - x + 3y ≥= - 4, x ≥= 0 and y ≥= 0 10. a) Verify Lagrange's mean value theorem for the function f(x) = √x − 1 , x ∈ [1,3] . b) Solve the differential equation xy dy dx = x 2 + y 2 . 11. Compute the approximate value of integrate ∫ 1 1+x 2 dx by using composite trapezoidal rule with three points and compare the result with the actual value. Determine the error formula and numerically verify an upper bound on it. Don’t Forget to Follow Uni Bytes
Prepared by Uni Bytes / www.unibytes.xyz Don't use our content without permission. Tribhuvan University Faculty of Humanities and Social Science Semester: II Subject: Mathematics II 2020 Batch Group B Attempt any SIX questions 2. A function f(x) is defined as f(x) = { 2x + 3 for − 3/2 ≤ x < 0 3 − 2x for 0 ≤ x > 3/2 −3 − 2x for x > 3/2 show that f(x) is continuous at x = 0 and discontinuous at x = 3/2. 3. Find the dy dx when x = a(t + sint), y = a(1 − cost). 4. State L 'Hospital's Rule. Use it to evaluate:limn→0 xe x−In(1+x) x 2 5. State Roll's Theorem. Verify the Rolle's Theorem for the function f(x) = x 2 − 9 in the interval -3 ≤ x ≤ 3. 6. Evaluate: a) ∫ dx x 2−25 2 0 b) ∫ 2x+5 √x 2+5x dx 7. Solve: sin x dy dx + (cosx)y = −sinxcosx. 8. Solve the following system of equations by Gauss-Seidel method - 4x + y - z = -8 3x + 6y + 2z = 1 x – y + 3z = 2 Group C Attempt any TWO questions. 9. Using simplex method, find the optimal solution of Z = 7x1 + 5x2 Subject to x1 + 2x2 ≤ 6 4x1 + 3x2 ≤ 6 x1 ≥ 0, x2 ≥ 0 10. What do you mean by stationary points and inflection points? Using derivatives, find two numbers whose sum is 20 and sum of whose squares is minimum. 11. Using Simpson's 1 3 rule evaluate, ∫ 1 1+x dx 1 0 with 3 points of intervals. Find the error of approximation. How many points are to be considered to make the approximation value within 10-5? Don’t Forget to Follow Uni Bytes
Prepared by Uni Bytes / www.unibytes.xyz Don't use our content without permission. Tribhuvan University Faculty of Humanities and Social Science Semester: II Subject: Mathematics 2019 Batch Don’t Forget to Follow Uni Bytes