BEY-018 Solved Assignment January 2026 | LINEAR ALGEBRA AND CALCULUS | IGNOU B.SC.

Q: Answer the following questions.

Answer available — download the full PDF for complete details.

Q: Answer the following questions.

Answer available — download the full PDF for complete details.

Q: Find the first order partial derivatives

Answer available — download the full PDF for complete details.

QQ.1. Answer the following questions.

(a)) If A, B are symmetric matrices of the same order, then what will be the type of matrix (AB – BA)? Give reasons in support of your answer. (50 words)
(b)) If the matrix A is both symmetric and skew-symmetric, then determine A. (50 words)
(c)) If A is a square matrix, such that A² =A, then find (I + A)³ - 7A. (50 words)
(d)) If A = [[1 0 0], [0 1 -2], [1 0 4]] ; I = [[1 0 0], [0 1 0], [0 0 1]], A⁻¹ = 1/6 (A² + aA + bI). Find a & b. (100 words)

Key Points:

Skew-symmetric matrix: Mᵀ = -M.
Symmetric matrix: Mᵀ = M.
Zero matrix is the only matrix that is both symmetric and skew-symmetric.
Idempotent matrix property: If A² = A, then (I+A)³ = I+7A.

Answer: This question assesses fundamental concepts and applications in Linear Algebra, specifically properties of matrices and the Cayley-Hamilton Theorem. It covers identifying matrix types based on transpose properties, understanding idempotent matrices, and deriving a matrix inverse using its characteristic equation. **(a) Type of matrix (AB – BA)** If A and B are symmetric matrices, then Aᵀ = A and Bᵀ = B. Let M = (AB - BA). To determine its type, we compute its transpose: Mᵀ = (AB - BA)ᵀ. Using t...

QQ.2. Answer the following questions.

(a)) If a=i+j+k and b=j-k, find vector c such that a×c=b and a.c =3. (125 words)
(b)) Show that area of a parallelogram whose diagonals are given by a and b is |a x b| / 2. Also find the area of the parallelogram whose diagonals are 2i-j+k and i+3j-k. (125 words)

Key Points:

To find an unknown vector c with conditions involving cross and dot products, represent c as xi+yj+zk and solve the resulting system of linear equations.
The cross product a×c=b yields three scalar equations by equating corresponding components.
The dot product a.c=k provides a fourth scalar equation, completing the system for x, y, z.
The area of a parallelogram with diagonals a and b is given by the formula |a × b| / 2.

Answer:

QQ.3. Answer the following questions.

(a)) Find a vector of magnitude 6, which is perpendicular to both the vectors 2i-j+2k and 4i-j+3k. (50 words)
(b)) Find the angle between the vectors 2i-j+k and 3i+4j-k. (50 words)
(c)) If a+b+c=0, show that a×b=b×c=c×a. (50 words)
(d)) If A,B,C,D are the points with position vectors i+j-k, 2i-j+3k, 3i-2j+k, respectively. Find the projection of AB along CD. (50 words)
(e)) Using vectors, find the area of the triangle ABC with vertices A(1,2,3), B(2,-1,4) and C(4,5,-1). (50 words)

Key Points:

Vector perpendicular to two others is their cross product, scaled to desired magnitude.
Angle between vectors uses dot product: `cosθ = (a ⋅ b) / (|a||b|)`.
If `a+b+c=0`, then `a×b=b×c=c×a` due to cross product properties.
Projection of vector `p` along `q` is `(p ⋅ q) / |q|`.

Answer:

QQ.4. Answer the following questions.

(a)) Prove that determinant of [[b+c, a, a], [c, c+a, a], [b, a, a+b]] = 4abc (125 words)
(b)) Prove that determinant of [[1, bc, bc(b+c)], [1, ca, ca(c+a)], [1, ab, ab(a+b)]] is independent of a, b, c. (125 words)

Key Points:

Determinant properties: Elementary row/column operations are crucial for simplification.
For (a), operations R1→R1-R2 and R2→R2-R3 simplify the determinant.
Result of (a): The determinant is generally `ab²+b²c+ac²+bc²`, not `4abc` universally.
Conditional Identity (a): `ab²+b²c+ac²+bc² = 4abc` is true only if `a=b=c`.

Answer: Determinants are fundamental concepts in linear algebra, covered extensively in courses like BEY-018. They provide crucial information about matrices, such as invertibility and area/volume scaling factors. Proving determinant identities often involves strategic application of elementary row or column operations to simplify the matrix before expansion, as these operations change the determinant in predictable ways or keep it invariant. This approach helps in systematically reducing the matrix to ...

QQ.5. Answer the following questions.

(a)) Show that the conical tent of given capacity will require the least amount of canvas if its height is √2 times its base radius. (50 words)
(b)) An open storage bin with square base and vertical sides is to be constructed from a given amount of material. Determine its dimensions if its volume is to be maximum neglecting the thickness of material and waste in constructing it. (50 words)

Key Points:

Optimization uses derivatives to find maximum or minimum values.
Define primary function and constraint function from problem statement.
Express the primary function in terms of a single variable.
Set the first derivative of the single-variable function to zero.

Answer: Since no course PDF was provided, this answer assumes BEY-018 covers calculus-based optimization techniques, using standard formulas for volume and surface area. The solutions involve setting the first derivative of the primary function (area or volume) with respect to one variable to zero, after incorporating the constraint. **(a) Conical Tent Optimization** To minimize canvas A = πr√(r² + h²) for given volume V = (1/3)πr²h. Substitute h = 3V/(πr²) into A. Minimizing A²(r) by setting its deriv...

QQ.6. Answer the following questions.

(a)) Explain why Lagrange's mean value theorem is not applicable to the following functions in the respective intervals: f(x) = 13x +11, x∈[1,3]. (75 words)
(b)) Verify means valueS theorem for the function f(x) = 4x³-4x in the interval [a,b], where a=0, and b=3. (75 words)
(c)) Find 'c' of Cauchy's mean value theorem for the function f(x) = 2.ln(x) and g(x) = x²-1 in the interval [2,3]. (100 words)
(e.i)) Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied. f (x) = sin² x on the interval [0, π]. (15 words)
(e.ii)) Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied. f (x) = x² + 4 on [– 2, 2]. (15 words)
(e.iii)) Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied. f (x) = sin x + cosx on [0, π/2]. (15 words)
(e.iv)) Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied. f(x) = x³ - 2x on [0, 1]. (15 words)

Key Points:

LMVT requires continuity on [a,b] and differentiability on (a,b).
CMVT extends LMVT for two functions, including g'(x) ≠ 0 condition.
Rolle's Theorem is a special case of LMVT where f(a) = f(b) implies f'(c)=0.
Polynomials are always continuous and differentiable everywhere.

Answer: The following answers comprehensively address the applicability and verification of Lagrange's, Cauchy's, and Rolle's Mean Value Theorems for the given functions and intervals, adhering to the principles taught in BEY-018 Linear Algebra and Calculus. Each solution checks the necessary conditions and performs the required calculations to find 'c' where applicable.

QQ.7. Find the first order partial derivatives

(a)) p²(r+1)/t³ + pre^(2p+3r+4t) (75 words)
(b)) g(s,t,v)=t ln(s+2t)-ln(3v)(s³+t²-4v) (75 words)
(c)) Find ∂z/∂x and ∂z/∂y for the function x² sin(y³)+ xe³- cos(z²)=3y-6z+8 (100 words)

Key Points:

Partial derivatives treat all variables other than the one being differentiated as constants.
The Product Rule d/dx(uv) = u'v + uv' is essential when variables are multiplied.
The Chain Rule d/dx f(g(x)) = f'(g(x)) * g'(x) applies to composite functions.
Implicit differentiation is used when variables are implicitly related (e.g., z is a function of x and y).

Answer:

QQ.8. Answer the following questions.

(a)) Find the orthogonal trajectories of the family of circles x² + (y - c)² = c², where c is a parameter. (125 words)
(b)) In a certain isolated population p (t) the rate of population growth dp/dt is equal to p-(k/ε)p², where k and ε are both positive constants. If p (0) = 1, then find the limiting population as t →∞. (125 words)

Key Points:

Orthogonal trajectories are curves intersecting a given family at right angles.
Steps for orthogonal trajectories: 1) Find the differential equation of the given family. 2) Replace dy/dx with -dx/dy. 3) Solve the new differential equation.
Homogeneous differential equations (dy/dx = f(y/x)) are solved using the substitution y = vx.
Population growth models like dp/dt = p - (k/ε)p² are separable differential equations solved via partial fractions.

Answer: This response comprehensively addresses the provided questions on orthogonal trajectories and population dynamics using differential equations, as typically covered in an IGNOU BEY-018 Linear Algebra and Calculus course. The solutions involve standard techniques for first-order differential equations, including variable separation, homogeneous equations, and limits. Due to the requirement to show step-by-step calculations within strict word limits for each sub-question, the answers are presented...

QQ.9. Answer the following questions.

(a)) Use the method of Laplace transforms to find the solution of the initial value problem y" + 9y = 6 cos 3t, y(0) = 2, y'(0) = 0 (100 words)
(b)) Determine the solution of the undamped (forced vibrations) system mü + Ku = Fo cos wt, u(0) = 0, u̇(0) = 1 When, w ≠ √(K/m) (150 words)

Key Points:

Laplace Transform converts ODEs into algebraic equations in the s-domain.
Initial conditions are directly incorporated when transforming derivatives: L{y"} = s²Y(s) - sy(0) - y'(0).
Inverse Laplace Transform converts solutions from s-domain back to time-domain, e.g., L⁻¹{s/(s² + a²)} = cos at.
Undamped forced vibration solution = Complementary (natural) + Particular (forced) solutions.

Answer: The solution of initial value problems and undamped forced vibration systems involves applying differential equation techniques like Laplace transforms and the method of undetermined coefficients, as covered in BEY-018 Linear Algebra and Calculus. For Laplace transforms, the initial conditions are directly integrated into the transformed equation, simplifying the process of solving higher-order differential equations. In forced vibration systems, the solution comprises both a complementary solut...

QQ.10. Answer the following questions.

(a)) Reduce the equation xy" + y' + xy = 0, x > 0 into Bessel's equation and hence write its general solution in terms of Bessel's functions. (125 words)
(b)) Prove that 4J" + 2Jn(x) = Jn−2(x) + Jn+2(x) (125 words)

Key Points:

Standard form of Bessel's equation: x²y" + xy' + (x² - ν²)y = 0.
Equation xy" + y' + xy = 0 transforms to Bessel's equation of order zero (ν=0).
General solution for integer order n Bessel's equation is y(x) = C₁Jn(x) + C₂Yn(x).
Key Bessel recurrence relation: 2J'n(x) = Jn−1(x) - Jn+1(x).

Answer: This response comprehensively addresses the provided questions concerning Bessel's equation and functions, applying standard mathematical methods pertinent to the BEY-018 Linear Algebra and Calculus course. The first part involves reducing a given differential equation into Bessel's form and stating its general solution, while the second part focuses on proving a key recurrence relation for Bessel functions. **(a) Reduction to Bessel's equation and general solution** The given differential equa...

BEY-018 Solved Assignment — January 2026 / July 2026

LINEAR ALGEBRA AND CALCULUS | IGNOU Bachelor of Science (B.SC.)

BEY-018—January 2026 / July 2026

LINEAR ALGEBRA AND CALCULUS

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BEY-018 Assignment Questions — January 2026

QQ.1. Answer the following questions.

QQ.2. Answer the following questions.

QQ.3. Answer the following questions.

QQ.4. Answer the following questions.

QQ.5. Answer the following questions.

QQ.6. Answer the following questions.

QQ.7. Find the first order partial derivatives

QQ.8. Answer the following questions.

QQ.9. Answer the following questions.

QQ.10. Answer the following questions.

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