MTE-01 Solved Assignment January 2026 | CALCULUS | IGNOU BDP

Q: Answer the following.

Answer available — download the full PDF for complete details.

Q: Answer the following.

Answer available — download the full PDF for complete details.

Q: Answer the following.

Calculus is fundamental for analyzing rates of change and accumulation, essential in economic applications like revenue optimization and for understanding the geometric properties of functions through curve tracing. This problem leverages differentiation to find maximum revenue and integration to derive the revenue function, along with analytical methods for describing the shape of a given curve. Part (a) focuses on applying differential calculus concepts to an economic scenario, specifically f

Q: Answer the following.

Answer available — download the full PDF for complete details.

Q: Answer the following.

This response provides a comprehensive solution to the given Calculus problems from the MTE-01 course. It covers the differentiation of exponential and inverse trigonometric functions, the application of Leibnitz's theorem for higher-order derivatives, and the analysis of continuity for a piecewise-defined function. Each sub-question is addressed with step-by-step calculations and clear explanations, adhering to the specified formatting and word count requirements for an IGNOU examination contex

Q1. 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∈ R : x² – 3x + 2 = 0} is an infinite set. (100 words)
(b)) The greatest integer function is continuous on R. (100 words)
(c)) d/dx integral from 3 to x of e^t dt = x*e^x - ln 3. (100 words)
(d)) Every integrable function is monotonic. (100 words)
(e)) a * b = a + b defines a binary operation on Q, the set of rational numbers. (100 words)

Key Points:

A set formed by solving a polynomial equation has a finite number of solutions, not infinite.
The greatest integer function exhibits jump discontinuities at every integer, preventing overall continuity.
The Fundamental Theorem of Calculus, Part 1, states that d/dx ∫[a to x] f(t) dt = f(x).
Integrability does not imply monotonicity; a continuous, non-monotonic function like x² is integrable.

Answer: Here, we evaluate the truthfulness of five given statements related to calculus and basic set theory, providing detailed reasons in the form of proofs or counter-examples as required by the MTE-01 course. Each sub-question's answer adheres to the specified word limits and formatting guidelines.

Q2. Answer the following.

(a)) Find the domain of the function f given by f(x) = (2-x) / sqrt(x^2 + 1) (250 words)
(b)) The set R of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on R as: a*b = (a+b)/2 for all a, b in R. Is (*) associative in R? Is (.) distributive (*) in R? Check. (250 words)

Key Points:

Function domain includes all valid input values for real output.
Square root argument must be non-negative (≥ 0); denominator cannot be zero.
x² + 1 is always positive, making sqrt(x² + 1) always defined and non-zero.
Associativity requires (a*b)*c = a*(b*c) for all elements.

Answer: This response addresses two fundamental concepts in calculus and abstract algebra: finding the domain of a real-valued function and checking algebraic properties (associativity and distributivity) for a defined operation. For function domains, the key is to identify restrictions arising from square roots and denominators. For algebraic properties, direct application of definitions and careful step-by-step verification, including the use of counterexamples where appropriate, is crucial.

Q3. Answer the following.

(a)) If |z - 1 + 2i| = 4, show that the point z + i describes a circle. Also draw this circle. (250 words)
(b)) Express (x-1) / (x^3 - x^2 - 2x) as a sum of partial fractions. (250 words)

Key Points:

The equation |z - z₀| = r represents a circle centered at z₀ with radius r.
Complex number transformations like w = z + k shift the center of geometric loci.
Partial fraction decomposition requires complete factorization of the denominator.
For distinct linear factors, partial fractions are A/linear_factor + B/linear_factor + ...

Answer:

Q4. Answer the following.

(a)) Find the least value of a^2 sec^2 x + b^2 cosec^2 x, where a > 0, b > 0. (250 words)
(b)) Evaluate: integral from 0 to 1 of (x^2 cot^-1(x^3))/(1+x^6) dx (150 words)
(c)) For any two sets S and T, show that: S U T = (S-T) U (S intersection T) U (T-S). Depict this situation in the Venn diagram. (100 words)

Key Points:

Least value of `a^2 sec^2 x + b^2 cosec^2 x` is `(a+b)^2` using AM-GM inequality.
AM-GM inequality `X+Y ≥ 2√(XY)` is applied to `a^2 tan^2 x + b^2 cot^2 x`.
Definite integral `∫[0,1] (x^2 cot^-1(x^3))/(1+x^6) dx` evaluates to `π^2 / 32`.
Integral evaluation involves two substitutions: `u = x^3` and `v = cot^-1(u)`.

Answer: This response comprehensively addresses three distinct problems from the MTE-01 Calculus course: finding the least value of an expression using algebraic inequalities, evaluating a definite integral involving inverse trigonometric functions, and demonstrating a fundamental identity in set theory using both formal proof and a Venn diagram. Each sub-question is tackled using principles and techniques commonly covered in the MTE-01 curriculum, emphasizing clear, step-by-step solutions and conceptua...

Q5. Answer the following.

(a)) Let f and g be two functions defined on R by: f(x) = x^3 - x^2 - 8x + 12 and g(x) = f(x)/(x+3) when x != -3 and alpha when x = -3. i) Find the value of alpha for which f is continuous at x = -3. ii) Find all the roots of f(x) = 0. (250 words)
(b)) Find the area between the curve y^2(4-x) = x^2 and its asymptote parallel to y-axis. (250 words)

Key Points:

Continuity at a point requires lim(x→a) f(x) = f(a).
Synthetic division is used to factor polynomials when a root is known or suspected.
The roots of f(x)=0 are found by setting each factor of f(x) to zero.
Vertical asymptotes occur where the denominator is zero and the numerator is non-zero.

Answer:

Q6. Answer the following.

(a)) If the revenue function is given by dR/dx = 15 + 2x - x^2, x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0. (250 words)
(b)) Trace the curve y^2(x+1) = x^2(3-x), clearly stating all the properties used for tracing it. (250 words)

Key Points:

Maximum/minimum of a function occurs where its first derivative is zero.
The second derivative test confirms a maximum if `d²R/dx² < 0` at the critical point.
Integrating a marginal function (like `dR/dx`) yields the total function (`R(x)`).
The constant of integration `C` is determined using initial conditions (e.g., `R(0)=0`).

Answer: Calculus is fundamental for analyzing rates of change and accumulation, essential in economic applications like revenue optimization and for understanding the geometric properties of functions through curve tracing. This problem leverages differentiation to find maximum revenue and integration to derive the revenue function, along with analytical methods for describing the shape of a given curve. Part (a) focuses on applying differential calculus concepts to an economic scenario, specifically f...

Q7. Answer the following.

(a)) Find the length of the cycloid x = a(theta - sin theta), y = a(1 - cos theta) and show that the line theta = 2pi/3 divides it in the ratio 1:3. (250 words)
(b)) Find the condition for the curves, ax^2 + by^2 = 1 and a'x^2 + b'y^2 = 1 intersecting orthogonally. (250 words)

Key Points:

Arc length for parametric curves: L = ∫√[(dx/dθ)² + (dy/dθ)²] dθ.
Cycloid derivatives: dx/dθ = a(1-cosθ), dy/dθ = a sinθ.
Trigonometric identity 1-cosθ = 2sin²(θ/2) simplifies arc length calculations.
Total length of one arch of the cycloid is 8a.

Answer:

Q8. Answer the following.

(a)) If y = e^(m sin^-1 x), then show that (1-x^2)y2 - xy1 - m^2y = 0. Hence using Leibnitz's formula, find the value of (1-x^2)y(n+2) - (2n+1)xy(n+1). (300 words)
(b)) Find the largest subset of R on which the function f : R -> R defined as: f(x) = { 2x, x > 5 ; x+5, 1 <= x <= 5 ; x, x < 1 } is continuous. (200 words)

Key Points:

Differentiate y = e^(m sin⁻¹x) using chain rule to find y1 and y2.
Square y1 and rearrange to form (1-x²)y1² = m²y².
Differentiate the squared equation implicitly to derive the second-order differential equation.
Leibnitz's theorem: dⁿ/dxⁿ(uv) = Σ(k=0 to n) nCk u^(n-k) v^(k).

Answer: This response provides a comprehensive solution to the given Calculus problems from the MTE-01 course. It covers the differentiation of exponential and inverse trigonometric functions, the application of Leibnitz's theorem for higher-order derivatives, and the analysis of continuity for a piecewise-defined function. Each sub-question is addressed with step-by-step calculations and clear explanations, adhering to the specified formatting and word count requirements for an IGNOU examination contex...

Q9. Answer the following.

(a)) Solve the equation: x^4 + 15x^3 + 70x^2 + 120x + 64 = 0 given that its roots are in G.P. (250 words)
(b)) Evaluate: lim x->0 (tan x - sin x) / x^3 (250 words)

Key Points:

Vieta's formulas link polynomial coefficients to sums/products of its roots.
For roots in G.P. (a/r³, a/r, ar, ar³), products simplify using powers of 'a'.
L'Hopital's Rule resolves 0/0 or ∞/∞ limits by differentiating numerator and denominator.
Repeated application of L'Hopital's Rule may be needed until a determinate form appears.

Answer: This question assesses fundamental concepts in algebra and calculus, specifically the properties of polynomial roots in a Geometric Progression (G.P.) and the evaluation of limits using advanced techniques. Part (a) requires the application of Vieta's formulas for a quartic polynomial and an understanding of G.P. sequences. Part (b) tests the ability to evaluate indeterminate limits, typically using L'Hopital's Rule or Maclaurin series expansions, which are core topics in MTE-01 Calculus. Both ...

Q10. Answer the following.

(a)) If I(m,n) = integral x^m (log x)^n dx, show that: (m+1)I(m,n) = x^(m+1) (log x)^n - n I(m,n-1). Hence find the value of integral x^4 (log x)^2 dx. (300 words)
(b)) Verify Lagrange's mean value theorem for the function f defined by f(x) = 2x^2 - 7x - 10 over [2, 5]. (200 words)

Key Points:

Reduction formulae simplify complex integrals through repeated integration by parts.
Integration by parts: ∫ u dv = uv - ∫ v du, choose u to simplify upon differentiation.
Lagrange's Mean Value Theorem (LMVT) relates average rate of change to instantaneous rate of change.
LMVT conditions: Function must be continuous on [a,b] and differentiable on (a,b).

Answer: This answer addresses two fundamental concepts in calculus: reduction formulae for integration and verification of Lagrange's Mean Value Theorem. Both topics are crucial for understanding advanced integral calculus techniques and the properties of differentiable functions, respectively, as covered in MTE-01 Calculus. Part (a) focuses on deriving a reduction formula using integration by parts, a powerful technique for integrating products of functions, and then applying it to solve a specific in...

MTE-01 Solved Assignment — January 2026 / July 2026

CALCULUS | IGNOU BA CHELORS DEGREE PROGRAMME (BDP)

MTE-01—January 2026 / July 2026

CALCULUS

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MTE-01 Assignment Questions — January 2026

Q1. 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.

Q2. Answer the following.

Q3. Answer the following.

Q4. Answer the following.

Q5. Answer the following.

Q6. Answer the following.

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Q8. Answer the following.

Q9. Answer the following.

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