MTE-04 Solved Assignment January 2026 | ELEMENTARY ALGEBRA | IGNOU BDP

Q1. Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set' is true, you need to say that this is true because it is a well-defined collection of 3 objects.)

(i)) Eliminating z from x+2y+3z = 2, 3x+2y+3z=6 and 2x+3y=5 gives x + 2y = 2. (75 words)
(ii)) The roots of x³-8x-3=0 are given by x = (8±√64+12)/2 (75 words)
(iii)) √2, 1, 3/5 ∈ Q ∩ Z X R. (75 words)
(iv)) Given any n positive numbers in R, the product of their harmonic mean and their arithmetic mean is 1. (75 words)
(v)) If A and B are two sets such that (A∪B) is empty, then either A = ∅ or B = ∅. (75 words)
(vi)) For any x, y∈ R, |x-y|>|1x|-|y1|. (75 words)
(vii)) The geometrical representation of the set {ix|x∈ R} is a point. (75 words)
(viii)) Any finite set is a subset of Z. (75 words)
(ix)) Every biquadratic equation has at least one real root. (75 words)
(x)) The converse of the statement, ‘Every student of MTE-04 has completed FST-01', is 'Every student of FST-01 has completed MTE-04'. (75 words)

Key Points:

Quadratic formula applies only to degree 2 equations.
Cartesian product `Z x R` contains ordered pairs `(z, r)`.
Product of Arithmetic Mean and Harmonic Mean is not always 1.
The union `A ∪ B` being empty implies both `A` and `B` are empty.

Answer: This response comprehensively addresses ten sub-questions related to Elementary Algebra (MTE-04), evaluating each statement as true or false. Justifications are provided with concise proofs, counterexamples, or explanations, adhering to the specific word limits for each sub-question. The topics covered range from systems of linear equations and polynomial roots to set theory, number systems, inequalities, and mathematical logic, reflecting core concepts taught in the course. Each justification a...

Q2a. Show that 1/√n +……+1/√n > 2√n+1-2√n∈ N.

Key Points:

LHS `1/√n +……+1/√n` simplifies to `√n`.
The inequality to prove is `√n > 2√(n+1) - 2√n`.
Rearrange inequality by adding `2√n`: `3√n > 2√(n+1)`.
Square both positive sides: `(3√n)² > (2√(n+1))²`.

Answer: The question requires proving the inequality `1/√n +……+1/√n > 2√n+1-2√n` for `n ∈ N`. The left-hand side (LHS) `1/√n +……+1/√n` is interpreted as a sum of `n` identical terms, each `1/√n`. Thus, the LHS simplifies to `n * (1/√n) = √n`. The right-hand side (RHS) is `2√(n+1) - 2√n`. We must demonstrate that `√n > 2√(n+1) - 2√n` for all natural numbers `n`. To prove this inequality, we first perform an algebraic rearrangement. Add `2√n` to both sides of the inequality: `√n + 2√n > 2√(n+1)`. This st...

Q2b. Let a, b > 0, a + b = 1, n > 1. Show that (a + 1/a)^(n-1) + (b + 1/b)^(n-1) >= 5^n / 2^(n-1).

Key Points:

Jensen's Inequality applies to convex functions.
The function f(x) = (x + 1/x)^(n-1) is strictly convex for x ∈ (0,1) and n > 1.
Convexity is proven by showing f''(x) > 0 for the domain.
Applying Jensen's inequality yields LHS >= 5^(n-1) / 2^(n-2).

Answer: To address this question from MTE-04, we will use the concept of convexity and Jensen's inequality. The problem requires us to show that (a + 1/a)^(n-1) + (b + 1/b)^(n-1) >= 5^n / 2^(n-1) for a, b > 0, a + b = 1, and n > 1. First, let's define the function f(x) = (x + 1/x)^(n-1). For Jensen's inequality to apply, f(x) must be convex over the interval (0, 1). Let g(x) = x + 1/x. Then g'(x) = 1 - 1/x² and g''(x) = 2/x³. For x ∈ (0, 1), g(x) > 0 and g''(x) > 0. The second derivative of f(x) is f''...

Q3a. Using the discriminant, give the nature of the roots of 7x³ +x²-35x = 5. Also solve the equation.

Key Points:

Rearrange cubic equation to standard form: 7x³+x²-35x-5=0.
Identify rational root x = -1/7 by testing values (e.g., using Rational Root Theorem).
Factor the cubic equation into (7x+1)(x²-5) = 0.
Discriminant D = b² - 4ac used for quadratic factor x² - 5 = 0.

Answer: The given equation is 7x³ + x² - 35x = 5. To analyze its roots using the discriminant, we first need to rearrange it into the standard polynomial form and attempt to reduce it to a quadratic equation, as the discriminant is primarily defined for quadratic equations in elementary algebra. Rearranging the equation, we get 7x³ + x² - 35x - 5 = 0. This is a cubic equation. For such equations, we often look for rational roots using the Rational Root Theorem. Let's test simple rational values for x =...

Q3b. Find the cubic equation whose roots are the cubes of the roots of x³ + ax² + bx + c = 0, a, b, c∈ R.

Key Points:

Vieta's formulas relate polynomial coefficients to elementary symmetric sums of roots.
The sum of cubes (α³+β³+γ³) is found using (Σα)((Σα)²-3Σαβ)+3αβγ.
The product of the new roots (α³β³γ³) is simply the cube of the original roots' product (αβγ)³.
The sum of products taken two at a time (Σα³β³) is derived by finding the sum of cubes of roots of an auxiliary polynomial with roots αβ, βγ, γα.

Answer: Let α, β, γ be the roots of the given equation x³ + ax² + bx + c = 0. According to Vieta's formulas (MTE-04 Block 2, Unit 5), the elementary symmetric sums are: Σα = α + β + γ = -a Σαβ = αβ + βγ + γα = b αβγ = -c We need to find a new cubic equation, say y³ + Ay² + By + C = 0, whose roots are α³, β³, γ³. The coefficients of this new equation are determined by: A = -(α³ + β³ + γ³) B = α³β³ + β³γ³ + γ³α³ C = -(α³β³γ³) 1. **For Coefficient A**: We use the identity α³ + β³ + γ³ = (α + β + γ)((α +...

Q3c. Obtain the resolvent cubics, by Descartes' method and by Ferrari's method, of the equation x⁴ + 4x³ +8=0. Are the cubics the same? Further, use either method to obtain the roots of this equation.

Key Points:

Descartes' method reduces quartic to y⁴ + Py² + Qy + R = 0 via x = y - a/4 substitution.
Descartes' resolvent cubic for x⁴ + 4x³ + 8 = 0 is Z³ - 12Z² + 16Z - 64 = 0 (Z=k²).
Ferrari's method transforms quartic directly by forming (x²+ax/2+y)² and setting RHS discriminant to zero.
Ferrari's resolvent cubic for x⁴ + 4x³ + 8 = 0 is y³ - 8y - 16 = 0.

Answer: To obtain the resolvent cubics for the equation x⁴ + 4x³ + 8 = 0, we will apply both Descartes' method and Ferrari's method. Both techniques transform the quartic equation into a cubic equation, whose roots are then used to factorize the original quartic. **Descartes' Method** Descartes' method typically applies to depressed quartics of the form y⁴ + Py² + Qy + R = 0. First, we eliminate the x³ term from our given equation, x⁴ + 4x³ + 8 = 0, by substituting x = y - (coefficient of x³)/4. Here,...

Q4a. If A and B are the set of even integers and set of odd integers, respectively, find A∪B and (A∪B).

Key Points:

A is the set of even integers.
B is the set of odd integers.
Union (A∪B) contains elements present in A or B.
A∪B for even and odd integers is the set of all integers (Z).

Answer: A is the set of even integers, and B is the set of odd integers. The union A∪B includes all elements that are either even or odd. Since every integer is inherently either even or odd, the union of these two sets encompasses all integers. Therefore, A∪B = Z, the set of all integers. Assuming the second "(A∪B)" in the question was a typo for A∩B (intersection), which seeks elements common to both sets. An integer cannot be simultaneously both even and odd. Thus, the intersection of A and B is an...

Q4b. Find A×B, and the number of elements in it, where A = {3n+2|1≤ n ≤ 10} ⊆ Z, and B = {n ∈ Z | 1≤ n ≤15} ∩ {m∈Z | 2|m} . Given any two sets C and D, under what conditions on them will C×D and D×C have the same number of elements? Give reasons for your answer.

(i)) Find A×B, and the number of elements in it, where A = {3n+2|1≤ n ≤ 10} ⊆ Z, and B = {n ∈ Z | 1≤ n ≤15} ∩ {m∈Z | 2|m} . (100 words)
(ii)) Given any two sets C and D, under what conditions on them will C×D and D×C have the same number of elements? Give reasons for your answer. (75 words)

Key Points:

Set A elements: {5, 8, 11, 14, 17, 20, 23, 26, 29, 32}; |A| = 10.
Set B elements: {2, 4, 6, 8, 10, 12, 14}; |B| = 7.
Cartesian product A×B is a set of ordered pairs (a,b) where a∈A, b∈B.
Number of elements in A×B is |A| × |B| = 10 × 7 = 70.

Answer: To find the Cartesian product A×B and its number of elements, we first determine the elements of sets A and B based on their definitions. Set A consists of integers formed by 3n+2 for n from 1 to 10. Set B contains even integers between 1 and 15, inclusive. Once the cardinalities of A and B are known, the number of elements in A×B is simply their product. For the second part, the condition for C×D and D×C to have the same number of elements relies on the fundamental property of set cardinalitie...

Q4c. Express the following situation in a Venn diagram: In a survey of 60 women, it is found that 25 have studied upto Class 12 only, 10 have studied till Class 10 only, 26 got scholarships, 9 of those studying till Class 12 got scholarships, 8 of those studying till Class 10 got scholarships, and 11 had completed their BA degree.

Key Points:

Universal set (U) represents 60 women.
L10, L12, BA are mutually exclusive education levels.
Scholarships (S) set overlaps with L10, L12, BA.
L10 regions: (L10 ∩ S) = 8; (L10 \ S) = 2.

Answer: The situation is represented by a Venn diagram within a universal set of 60 women. Let L10 be women who studied till Class 10 only, L12 for Class 12 only, and BA for those with a BA degree. These three education levels are mutually exclusive sets, as per the phrasing in set theory (MTE-04, Unit 1, Section 1.4). A fourth set, S (Scholarships), overlaps these. The Venn diagram features three distinct ovals (L10, L12, BA) for education levels. A larger circle, 'S' (Scholarships), intersects each o...

Q5. In the context of your IGNOU studies, give the following:

(i)) an example of an implication; (40 words)
(ii)) the converse of your statement in (i) above; (40 words)
(iii)) the contrapositive of your statement in (i) above; (40 words)
(iv)) a statement using ∀; (40 words)
(v)) a statement using ∃. (40 words)

Key Points:

An implication (P ⇒ Q) is a conditional statement 'If P, then Q'.
The converse of (P ⇒ Q) is (Q ⇒ P), swapping hypothesis and conclusion.
The contrapositive of (P ⇒ Q) is (¬Q ⇒ ¬P), negating and swapping.
An implication and its contrapositive are logically equivalent statements.

Answer: This response addresses key logical constructs fundamental to Elementary Algebra (MTE-04), specifically focusing on implications, their related statements, and quantifier usage. Understanding these concepts is crucial for constructing and analyzing mathematical proofs and statements. Each sub-question provides a concrete example relevant to the course content.

Q6a. Give the following:

(i)) a 2×4 matrix; (30 words)
(ii)) the transpose of the matrix in (i) above; (30 words)
(iii)) a system of linear equations represented by AX=B, where A is the matrix in (ii) above. (30 words)

Key Points:

A 2×4 matrix has 2 rows and 4 columns.
Matrix transpose (Aᵀ) swaps rows and columns of the original matrix.
If A is mxn, its transpose Aᵀ is nxm.
A system of linear equations can be represented as AX=B.

Answer: This response addresses fundamental matrix operations and their application in representing systems of linear equations, aligning with MTE-04 Elementary Algebra concepts. It provides a 2×4 matrix, demonstrates its transpose, and then formulates a corresponding system of linear equations in the AX=B format, using the transposed matrix as the coefficient matrix.

Q6b. Consider the linear system 2x-3y + 4z = 20-2/3, x+2y-3z+13.5 = 0, -x-2y+5z = 113/6. Give the two reasons for Cramer's Rule being applicable for solving this system. Also use the rule to solve the linear system.

Key Points:

Cramer's Rule requires a square system of linear equations (N equations, N variables).
The determinant of the coefficient matrix (det(A)) must be non-zero.
det(A) = 14 for this system, confirming Cramer's Rule applicability.
Each variable `x_i` is solved using the formula `det(A_i) / det(A)`.

Answer: The given linear system is first converted into standard algebraic form: 1) 2x - 3y + 4z = 58/3 2) x + 2y - 3z = -27/2 3) -x - 2y + 5z = 113/6 This system is expressed in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant vector. Cramer's Rule is applicable to this system for two primary reasons, as taught in MTE-04. Firstly, the system is a 'square system', meaning it has an equal number of equations (3) and variables (3), which is a funda...

Q7a. Find the values of a∈ R for which ai is a solution of z⁴ – 2z³ +7z² -4z+10=0. Also find all the roots of this equation.

Key Points:

Substitute `z = ai` into the polynomial equation.
Separate the resulting expression into real and imaginary parts.
Equate both the real and imaginary parts to zero.
Solve `2a³ - 4a = 0` and `a⁴ - 7a² + 10 = 0` to find `a = ±√2`.

Answer: To find the values of `a ∈ R` for which `ai` is a solution of the equation `z⁴ – 2z³ +7z² -4z+10=0`, we substitute `z = ai` into the polynomial. Since the coefficients of the polynomial are real, if `ai` is a root, then its conjugate, `-ai`, must also be a root, a concept covered in MTE-04 Block 1, Unit 2, Section 2.3. Substituting `z = ai`, we get: `(ai)⁴ – 2(ai)³ +7(ai)² -4(ai)+10 = 0` `a⁴i⁴ – 2a³i³ +7a²i² -4ai+10 = 0` `a⁴(1) – 2a³(-i) +7a²(-1) -4ai+10 = 0` `a⁴ + 2a³i - 7a² - 4ai + 10 = 0` Gr...

Q7b. Find all the 8th roots of 3i-3. Also show any one of them in an Argand diagram.

Key Points:

Convert complex number `z = x + iy` to polar form `r(cosθ + i sinθ)`.
Modulus `r = √(x² + y²)` and Argument `θ = tan⁻¹(y/x)` (adjusted for quadrant).
De Moivre's Theorem finds `n`-th roots: `z_k = r^(1/n) [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]`.
For 8th roots, `k` ranges from 0 to 7, yielding 8 distinct roots.

Answer: To find the 8th roots of a complex number `z = 3i - 3`, we first convert it into its polar form `z = r(cosθ + i sinθ)`. Here, `z = -3 + 3i`. The modulus `r` is calculated as `r = √((-3)² + 3²) = √(9 + 9) = √18 = 3√2`. The argument `θ` is `tan⁻¹(3/-3) = tan⁻¹(-1)`. Since the real part is negative and the imaginary part is positive, `z` lies in the second quadrant. Thus, `θ = π - π/4 = 3π/4` radians. So, `z = √18 (cos(3π/4) + i sin(3π/4))`. According to De Moivre's Theorem for roots, the `n`-t...

Q8a. Using the method of substitution, obtain the solution set in R³, of the following:

(i)) x-π = 5 (30 words)
(ii)) 2x - y + z = 1, x - 2y + z = 3, y = √2 - z (100 words)
(iii)) x - y = 5, x = 7, 2x – 3y = 5 (100 words)

Key Points:

Method of substitution involves expressing one variable to simplify equations.
In R³, unconstrained variables are 'free' and can take any real value.
A unique solution in R³ is an ordered triple (x, y, z).
An inconsistent system leads to a false statement (e.g., 8=5).

Answer: To obtain the solution set in R³ using the method of substitution, we systematically replace variables with their equivalent expressions from other equations. This process continues until a single variable is solved, which is then back-substituted to find the remaining variables. The nature of the solution (unique, infinitely many, or none) depends on the consistency of the system and the number of independent equations relative to variables in R³. **(i) x - π = 5** The given equation directly...

Q8b. Give a real life situation problem, which is mathematically translated into 2x + y + 2z=18, x+3y+3z = 24, 3y = 6. Also, explain how this linear system models your problem.

Key Points:

Linear systems model real-world scenarios involving multiple unknown quantities.
Variables (x, y, z) represent specific unknown values, e.g., price or quantity.
Each equation translates a distinct constraint or condition from the problem statement.
Coefficients often represent per-unit costs, quantities, or specific ratios.

Answer: Consider a bakery selling three types of cookies: chocolate (x), vanilla (y), and almond (z). The variables x, y, and z represent the individual price (in rupees) of one chocolate cookie, one vanilla cookie, and one almond cookie, respectively. This real-life situation can be mathematically translated into the given linear system. The system 2x + y + 2z = 18, x + 3y + 3z = 24, 3y = 6 accurately models this problem. The equation `3y = 6` signifies that three vanilla cookies cost ₹6, directly det...

MTE-04 Solved Assignment — January 2026 / July 2026

ELEMENTARY ALGEBRA | IGNOU BA CHELORS DEGREE PROGRAMME (BDP)

MTE-04—January 2026 / July 2026

ELEMENTARY ALGEBRA

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MTE-04 Assignment Questions — January 2026

Q2a. Show that 1/√n +……+1/√n > 2√n+1-2√n∈ N.

Q2b. Let a, b > 0, a + b = 1, n > 1. Show that (a + 1/a)^(n-1) + (b + 1/b)^(n-1) >= 5^n / 2^(n-1).

Q3a. Using the discriminant, give the nature of the roots of 7x³ +x²-35x = 5. Also solve the equation.

Q3b. Find the cubic equation whose roots are the cubes of the roots of x³ + ax² + bx + c = 0, a, b, c∈ R.

Q3c. Obtain the resolvent cubics, by Descartes' method and by Ferrari's method, of the equation x⁴ + 4x³ +8=0. Are the cubics the same? Further, use either method to obtain the roots of this equation.

Q4a. If A and B are the set of even integers and set of odd integers, respectively, find A∪B and (A∪B).

Q4b. Find A×B, and the number of elements in it, where A = {3n+2|1≤ n ≤ 10} ⊆ Z, and B = {n ∈ Z | 1≤ n ≤15} ∩ {m∈Z | 2|m} . Given any two sets C and D, under what conditions on them will C×D and D×C have the same number of elements? Give reasons for your answer.

Q5. In the context of your IGNOU studies, give the following:

Q6a. Give the following:

Q6b. Consider the linear system 2x-3y + 4z = 20-2/3, x+2y-3z+13.5 = 0, -x-2y+5z = 113/6. Give the two reasons for Cramer's Rule being applicable for solving this system. Also use the rule to solve the linear system.

Q7a. Find the values of a∈ R for which ai is a solution of z⁴ – 2z³ +7z² -4z+10=0. Also find all the roots of this equation.

Q7b. Find all the 8th roots of 3i-3. Also show any one of them in an Argand diagram.

Q8a. Using the method of substitution, obtain the solution set in R³, of the following:

Q8b. Give a real life situation problem, which is mathematically translated into 2x + y + 2z=18, x+3y+3z = 24, 3y = 6. Also, explain how this linear system models your problem.

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