BMTC-131 English Medium Solved Assignment 2026

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BMTC-131 English Medium Solved Assignment 2026 Available

Assignment Code: BMTC-131/TMA/2026

1. Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.

a) The set \{S \in R / (x ^ 2) – 3x + 2 = 0\} is an infinite set.

b) The greatest interger function is continuous on R.

c) d/dx [integrate ln(t) dt from 3 to x’] = x * e ^ x – ln

d) Every integrable function is monotonic.

e) 1 oplus b = sqrt(a + b) defines a binary operation on Q, the set of rational numbers.

2. a) Find the domain of the function f given by f(x) = sqrt((2 – x)/(x ^ 2 + 1))

b) The set R of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on R as:

Is () associative in R? Is (.) distributive () in R? Check. ab = (a + b)/2 , forall a,b in R

3. a) |z – 1 + 2i| = 4 show that the point z+i describes a circle. Also draw this circle.

b) Express x-1 x^ 3 – x ^ 2 – 2x a sum of partial fractions.

a) Find the least value of a ^ 2 * sec^2 x + b ^ 2 * co * sec^2 x where a > 0, b > 0

b) Evaluate:

integrate (x ^ 2 * arccot(x ^ 3))/(1 + x ^ 6) dx

c) For any two sets S and T, show that:

S cup T= (S – T) cup(S cap T) cup(T-S).

Depict this situation in the Venn diagram.

5. a) Let f and g be two functions defined on R by:

(5)

f(x) = x ^ 3 – x ^ 2 – 8x + 12 f(x), when x≠-3 and g(x)=x+3′ a, when x=-3

i) Find the value of a for which f is continuous at x = – 3

ii) Find all the roots of f(x) = 0

b) Find the area between the curve y ^ 2 * (4 – x) = x ^ 3 and its asymptote parallel to y-axis.

6. a) If the revenue function is given by d/dx (R) = 15 + 2x – x ^ 2 x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

b) Trace the curve y ^ 2 * (x + 1) = x ^ 2 * (3 – x) clearly stating all the properties used for tracing it.

7. a) Find the length of the cycloid x = c(theta – sin theta) y = alpha(l – cos theta) and show that the line

b) Find the condition for the curves, a * x ^ 2 + b * y ^ 2 = 1 a’ * x ^ 2 + b’ * y ^ 2 = 1 intersecting orthogonally.

8.a) If y = e ^ m * arcsin(x) then show that (1 – x ^ 2) * y_{2} – x*y_{1} – m ^ 2 * y = 0 Hence using Leibnitz’s formula, find the value of (1 – x ^ 2) y n+2 -(2b+1)xy n+1 .

b) Find the largest subset of R on which the function f / R -> R defined as: (4) is continuous. f(x)= 2x,x>5\\ x+5, 1 <= x <= 5 \\ |x|&,x<1

9. a) Solve the equation: x ^ 4 + 15x ^ 3 + 70x ^ 2 + 120x + 64 = 0 given that its roots are in G.P.

b) Evaluate: lim x -> 0 (tan x – sin x)/(x ^ 3)

a) If Ima = [x³(log x)” dx, show that:

(m + 1) I m,n =x.^ m+1 (log x)^ n -n I m ,n-1. Hence find the value of integrate x ^ 4 * (log(x)) ^ 3 dx

b) Verigy Lagrange’s mean value theorem for the function f defined by f(x) = 2x ^ 2 – 7x – 10 over [2, 5].