Q1 50M Compulsory prove Linear algebra and calculus
(a) If $A=\begin{bmatrix} 1 & -1 & 1 \\ 2 & -1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$, then show that $A^2 = A^{-1}$ (without finding $A^{-1}$). (10 marks)
(b) Find the matrix associated with the linear operator on $V_3(R)$ defined by $T(a, b, c) = (a+b, a-b, 2c)$ with respect to the ordered basis $B = \{(0, 1, 1), (1, 0, 1), (1, 1, 0)\}$. (10 marks)
(c) Given: $$\Delta(x)=\begin{vmatrix} f(x+\alpha) & f(x+2\alpha) & f(x+3\alpha) \\ f(\alpha) & f(2\alpha) & f(3\alpha) \\ f'(\alpha) & f'(2\alpha) & f'(3\alpha) \end{vmatrix}$$ where $f$ is a real valued differentiable function and $\alpha$ is a constant. Find $\displaystyle\lim_{x \to 0} \frac{\Delta(x)}{x}$. (10 marks)
(d) Show that between any two roots of $e^x \cos x = 1$, there exists at least one root of $e^x \sin x - 1 = 0$. (10 marks)
(e) Find the equation of the cylinder whose generators are parallel to the line $x = -\frac{y}{2} = \frac{z}{3}$ and whose guiding curve is $x^2 + 2y^2 = 1$, $z = 0$. (10 marks)
Answer approach & key points
Prove the required results across all five sub-parts with rigorous mathematical reasoning. For (a), compute A² and show A³=I implying A²=A⁻¹ without explicit inversion; for (b), construct the change-of-basis matrix and apply T to basis vectors; for (c), use Taylor expansion and determinant properties; for (d), apply Rolle's theorem to the appropriate auxiliary function; for (e), use the standard cylinder equation with given generator direction. Allocate approximately 15% time to (a), 20% to (b), 20% to (c), 25% to (d), and 20% to (e), reflecting the analytical depth required for the calculus and analysis components.
- For (a): Compute A² by matrix multiplication, then verify A³ = I₃ (identity), hence A² = A⁻¹ without computing inverse explicitly
- For (b): Apply T to each basis vector of B, express results as linear combinations of B, and assemble coefficients as columns of the matrix representation
- For (c): Expand f(x+kα) = f(kα) + xf'(kα) + O(x²), substitute into determinant, identify leading term as x times a 3×3 determinant involving f and f' values
- For (d): Define g(x) = e⁻ˣ - cos x, note roots of g correspond to roots of eˣcos x = 1, apply Rolle's theorem to g between consecutive roots
- For (e): Use direction ratios (1, -2, 3) from generator line, write cylinder as locus of points at fixed distance from axis with given guiding curve x² + 2y² = 1, z = 0
- For (c): The limit equals the determinant with rows [f'(α), f'(2α), f'(3α)], [f(α), f(2α), f(3α)], [f'(α), f'(2α), f'(3α)] which simplifies appropriately
- For (d): Show g'(x) = -e⁻ˣ + sin x = 0 implies eˣsin x = 1, completing the proof via intermediate value property
- For (e): Final equation eliminates parameter to give 13(x² + 2y²) - (x - 2y + 3z)² = 13 or equivalent standard form
Q2 50M prove Analytical geometry and multivariable calculus
(a) Show that the planes, which cut the cone $ax^2 + by^2 + cz^2 = 0$ in perpendicular generators, touch the cone $\frac{x^2}{b+c} + \frac{y^2}{c+a} + \frac{z^2}{a+b} = 0$. (20 marks)
(b) Given that $f(x,y) = |x^2 - y^2|$. Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$. Hence show that $f_{xy}(0,0) = f_{yx}(0,0)$. (15 marks)
(c) Show that $S = \{(x, 2y, 3x) : x, y$ are real numbers$\}$ is a subspace of $R^3(R)$. Find two bases of $S$. Also find the dimension of $S$. (15 marks)
Answer approach & key points
Prove the three mathematical statements systematically, allocating approximately 40% of effort to part (a) given its 20 marks, and 30% each to parts (b) and (c). Begin each part with clear statement of what is to be proved, develop the proof through logical steps with proper mathematical notation, and conclude with explicit verification of the required result. For (a), establish the condition for perpendicular generators first; for (b), carefully handle the absolute value through case analysis; for (c), verify all three subspace axioms before finding bases.
- Part (a): Derive condition for perpendicular generators of cone ax² + by² + cz² = 0 using direction cosines and orthogonality condition l₁l₂ + m₁m₂ + n₁n₂ = 0
- Part (a): Show that tangent plane condition leads to the reciprocal cone x²/(b+c) + y²/(c+a) + z²/(a+b) = 0 using the determinant condition for tangency
- Part (b): Analyze f(x,y) = |x² - y²| in four quadrants/regions to compute partial derivatives fx and fy near origin
- Part (b): Calculate mixed partial derivatives f_xy(0,0) and f_yx(0,0) using limit definition, showing both equal zero despite |x²-y²| not being C²
- Part (c): Verify S is subspace of R³ by checking: (i) non-empty/contains zero, (ii) closed under addition, (iii) closed under scalar multiplication
- Part (c): Express S as span{(1,0,3), (0,2,0)} = span{(1,0,3), (0,1,0)}, verify linear independence, conclude dim(S) = 2 with two distinct bases
Q3 50M solve Calculus, Linear Algebra and Analytical Geometry
(a)(i) If $u = x^2 + y^2$, $v = x^2 - y^2$, where $x = r\cos\theta$, $y = r\sin\theta$, then find $\frac{\partial(u,v)}{\partial(r,\theta)}$. (7 marks)
(a)(ii) If $\int\limits_{0}^{x} f(t)\,dt = x + \int\limits_{x}^{1} tf(t)\,dt$, then find the value of $f(1)$. (5 marks)
(a)(iii) Express $\int\limits_{a}^{b} (x-a)^m (b-x)^n\,dx$ in terms of Beta function. (8 marks)
(b) A sphere of constant radius $r$ passes through the origin $O$ and cuts the axes at the points $A, B$ and $C$. Find, the locus of the foot of the perpendicular drawn from $O$ to the plane $ABC$. (15 marks)
(c)(i) Prove that the eigen vectors, corresponding to two distinct eigen values of a real symmetric matrix, are orthogonal. (8 marks)
(c)(ii) For two square matrices A and B of order 2, show that trace (AB) = trace (BA). Hence show that AB - BA ≠ I₂, where I₂ is an identity matrix of order 2. (7 marks)
Answer approach & key points
Solve all six sub-parts systematically, allocating approximately 35% time to part (b) (15 marks) as the highest-weighted component, 30% to part (a) (20 marks across three items), and 35% to part (c) (15 marks across two items). Begin each sub-part with clear statement of given conditions, show complete working with proper mathematical notation, and conclude with boxed final answers. For (a)(iii) and (c)(i), explicitly state theorems being applied (Beta function definition, spectral theorem for symmetric matrices).
- For (a)(i): Correct application of chain rule for Jacobians, computing ∂(u,v)/∂(x,y) and ∂(x,y)/∂(r,θ) separately, then multiplying to get final result 8r³sinθcosθ or equivalent simplified form
- For (a)(ii): Differentiation under integral sign using Leibniz rule, establishing f(x) = 1 - xf(x), solving to get f(x) = 1/(1+x), hence f(1) = 1/2
- For (a)(iii): Substitution x = a + (b-a)t to transform limits to 0 and 1, identifying parameters p = m+1, q = n+1, final answer as (b-a)^(m+n+1) B(m+1, n+1)
- For (b): Setting up sphere equation x²+y²+z²-2ux-2vy-2wz=0 with center (u,v,w), using |OA|=|OB|=|OC|=2r condition, finding plane ABC as x/u + y/v + z/w = 2, deriving foot of perpendicular coordinates and eliminating parameters to get locus x⁻² + y⁻² + z⁻² = r⁻²
- For (c)(i): Using definition of eigenvectors AX=λX, AY=μY with λ≠μ, exploiting symmetry A=Aᵀ to show λXᵀY = μXᵀY, hence XᵀY=0 proving orthogonality
- For (c)(ii): Direct computation of trace(AB) and trace(BA) showing equality via ΣΣaᵢⱼbⱼᵢ, then using trace(AB-BA)=0 while trace(I₂)=2 to establish contradiction
Q4 50M solve Linear Algebra, Calculus and Three Dimensional Geometry
(a)(i) Reduce the following matrix to a row-reduced echelon form and hence also, find its rank:
A = [1 3 2 4 1
0 0 2 2 0
2 6 2 6 2
3 9 1 10 6] (10 marks)
(a)(ii) Find the eigen values and the corresponding eigen vectors of the matrix
A = (0 -i
i 0), over the complex-number field. (10 marks)
(b) Show that the entire area of the Astroid : x^(2/3) + y^(2/3) = a^(2/3) is (3/8)πa². (15 marks)
(c) Find equation of the plane containing the lines
$$\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7},$$
$$\frac{x-2}{1} = \frac{y-4}{3} = \frac{z-6}{5}.$$
Also find the point of intersection of the given lines. (15 marks)
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 40% time to part (a) covering matrix reduction and eigenvalue computation (20 marks), 30% to part (b) deriving the astroid area using parametric integration (15 marks), and 30% to part (c) finding the plane equation and intersection point of skew lines (15 marks). Begin with clear statement of given data, proceed with systematic computational steps showing all row operations/integrations/vector calculations, and conclude with boxed final answers for each sub-part.
- For (a)(i): Correct application of elementary row operations to reduce 4×5 matrix to row-reduced echelon form; identification of pivot positions and correct rank determination (rank = 3)
- For (a)(ii): Setting up and solving characteristic equation det(A - λI) = 0; obtaining eigenvalues λ = ±1; finding normalized eigenvectors (1, i) and (1, -i) or equivalent scalar multiples
- For (b): Parametric representation x = a cos³θ, y = a sin³θ; correct area integral setup using symmetry and Jacobian; evaluation yielding (3/8)πa²
- For (c): Verification that lines are coplanar via condition [a₂-a₁, b₁, b₂] = 0; calculation of plane normal via cross product of direction vectors; point of intersection by solving parametric equations simultaneously
- Correct handling of complex numbers in (a)(ii) and proper geometric interpretation of astroid as hypocycloid with four cusps
Q5 50M Compulsory solve Differential equations, mechanics, vector calculus
Solve the differential equation: d²y/dx² + 2y = x²e^(3x) + e^x cos 2x (10 marks)
Solve the initial value problem: d²y/dx² + 4y = e^(-2x) sin 2x; y(0) = y'(0) = 0 using Laplace transform method. (10 marks)
Two rods LM and MN are joined rigidly at the point M such that (LM)² + (MN)² = (LN)² and they are hanged freely in equilibrium from a fixed point L. Let ω be the weight per unit length of both the rods which are uniform. Determine the angle, which the rod LM makes with the vertical direction, in terms of lengths of the rods. (10 marks)
If a planet, which revolves around the Sun in a circular orbit, is suddenly stopped in its orbit, then find the time in which it would fall into the Sun. Also, find the ratio of its falling time to the period of revolution of the planet. (10 marks)
Show that ∇²[∇·(r⃗/r)] = 2/r⁴, where r⃗ = xî + yĵ + zk̂. (10 marks)
Answer approach & key points
Solve demands complete analytical solutions with rigorous mathematical derivation. Structure as: (1) Differential equation using undetermined coefficients/operator method, (2) Laplace transform with partial fractions and inversion, (3) Static equilibrium with virtual work or moment balance for LMN triangle configuration, (4) Central force motion converting circular to radial free-fall using Kepler's laws, (5) Vector calculus with proper coordinate treatment of r/|r|. Each part requires distinct mathematical machinery.
- Part (a): Correct complementary function y_c = A cos(√2 x) + B sin(√2 x) and particular integral using operator method or undetermined coefficients for both x²e^(3x) and e^x cos 2x terms
- Part (b): Proper Laplace transform application with ℒ{e^(-2x) sin 2x} = 2/((s+2)²+4), correct partial fraction decomposition, and inverse transform satisfying y(0) = y'(0) = 0
- Part (c): Recognition that angle at M is 90°, use of virtual work principle or moment equilibrium about L, expressing tan θ = (MN)²/(LM·LN) or equivalent in terms of given lengths
- Part (d): Conversion of circular orbit to degenerate ellipse with semi-major axis a = R/2, application of Kepler's third law giving fall time T/4√2, ratio 1/(4√2) or √2/8
- Part (e): Correct computation of ∇·(r⃗/r) = 2/r, then ∇²(2/r) = 0 for r ≠ 0, with proper handling of singularity or verification via direct Cartesian differentiation
Q6 50M solve Catenary, differential equations, line integrals
A heavy string, which is not of uniform density, is hung up from two points. Let T₁, T₂, T₃ be the tensions at the intermediate points A, B, C of the catenary respectively where its inclinations to the horizontal are in arithmetic progression with common difference β. Let ω₁ and ω₂ be the weights of the parts AB and BC of the string respectively. Prove that (i) Harmonic mean of T₁, T₂ and T₃ = 3T₂/(1 + 2cos β) (ii) T₁/T₃ = ω₁/ω₂ (20 marks)
Solve the equation: d²y/dx² + (tan x - 3cos x)dy/dx + 2y cos²x = cos⁴x completely by demonstrating all the steps involved. (15 marks)
Evaluate ∫_C F⃗ · dr⃗, where C is an arbitrary closed curve in the xy-plane and F⃗ = (-yî + xĵ)/(x² + y²). (15 marks)
Answer approach & key points
Solve requires complete working with all intermediate steps demonstrated. Structure as: Part I (20 marks) - establish catenary tension relations using ψ = θ - β, θ, θ + β, derive T = T₀ sec ψ, prove harmonic mean identity and tension-weight ratio using vertical equilibrium; Part II (15 marks) - reduce second-order ODE via substitution t = sin x to standard form, find complementary function and particular integral; Part III (15 marks) - identify singularity at origin, apply Green's theorem excluding origin with limiting circle, evaluate residue. Conclude with boxed final answers for each part.
- Catenary: Correct tension formula T = T₀ sec ψ with inclinations ψ₁ = θ - β, ψ₂ = θ, ψ₃ = θ + β in AP
- Catenary: Derivation of T₁ + T₃ = 2T₂ cos β from horizontal equilibrium and harmonic mean manipulation
- Catenary: Vertical equilibrium giving ω₁ = T₂ sin β - T₁ sin β and ω₂ = T₃ sin β - T₂ sin β leading to T₁/T₃ = ω₁/ω₂
- ODE: Substitution t = sin x reducing equation to d²y/dt² - 3dy/dt + 2y = t² with CF = Aeᵗ + Be²ᵗ and PI = ½t² + 3/2t + 7/4
- Line integral: Recognition that ∂Q/∂x - ∂P/∂y = 0 except at origin, Green's theorem with indentation, limit evaluation giving 2π for curves enclosing origin and 0 otherwise
- Complete final answers: (i) H = 3T₂/(1+2cos β), (ii) T₁/T₃ = ω₁/ω₂; y = Aeˢⁱⁿˣ + Be²ˢⁱⁿˣ + ½sin²x + 3/2 sin x + 7/4; integral = 2πn where n = winding number about origin
Q7 50M prove Vector calculus, differential equations, particle dynamics
(a) Verify Gauss divergence theorem for $\vec{F} = 2x^2y\hat{i} - y^2\hat{j} + 4xz^2\hat{k}$ taken over the region in the first octant bounded by $y^2 + z^2 = 9$ and $x = 2$. (20 marks)
(b) Find all possible solutions of the differential equation: $y^2 \log y = xy\dfrac{dy}{dx} + \left(\dfrac{dy}{dx}\right)^2$. (15 marks)
(c) A heavy particle hangs by an inextensible string of length $a$ from a fixed point and is then projected horizontally with a velocity $\sqrt{2gh}$. If $\dfrac{5a}{2} > h > a$, then prove that the circular motion ceases when the particle has reached the height $\dfrac{1}{3}(a + 2h)$ from the point of projection. Also, prove that the greatest height ever reached by the particle above the point of projection is $\dfrac{(4a-h)(a+2h)^2}{27a^2}$. (15 marks)
Answer approach & key points
Begin with a clear statement of intent to verify, solve, and prove across all three parts. For part (a), set up the volume integral of divergence and surface integrals over the cylindrical region in the first octant, allocating approximately 35-40% of effort due to its 20 marks. For part (b), identify the Clairaut form and solve by substitution p = dy/dx, spending about 25-30% of time. For part (c), apply energy conservation and circular motion dynamics to derive the height conditions, using ~30-35% of effort. Conclude each part with boxed final answers and explicit verification statements.
- For (a): Correct computation of div F = 4xy - 2y + 8xz and proper setup of volume integral over 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ √(9-y²) in first octant
- For (a): Evaluation of surface integrals over all five faces (x=0, x=2, y=0, z=0, and cylindrical surface y²+z²=9) with correct normal vectors
- For (b): Recognition of equation as Clairaut's form y = xp + f(p) where p = dy/dx, leading to substitution and factorization
- For (b): Complete solution including general solution y = cx + c²/log y and singular solution obtained by eliminating c
- For (c): Application of energy conservation ½mv² = ½m(2gh) - mga(1-cos θ) and tension condition T = 0 for circular motion cessation
- For (c): Derivation that circular motion ceases at height (a+2h)/3 using the condition 5a/2 > h > a
- For (c): Calculation of projectile motion after string slackens to reach greatest height (4a-h)(a+2h)²/(27a²)
Q8 50M solve Orthogonal trajectories, differential equations, particle dynamics, Stokes theorem
(a)(i) Find the orthogonal trajectories of the family of confocal conics $$\frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1; \quad a > b > 0 \text{ are constants and } \lambda \text{ is a parameter.}$$ Show that the given family of curves is self orthogonal. (10 marks)
(a)(ii) Find the general solution of the differential equation: $$x^2\frac{d^2y}{dx^2} - 2x(1+x)\frac{dy}{dx} + 2(1+x)y = 0.$$ Hence, solve the differential equation: $x^2\dfrac{d^2y}{dx^2} - 2x(1+x)\dfrac{dy}{dx} + 2(1+x)y = x^3$ by the method of variation of parameters. (10 marks)
(b) Describe the motion and path of a particle of mass $m$ which is projected in a vertical plane through a point of projection with velocity $u$ in a direction making an angle $\theta$ with the horizontal direction. Further, if particles are projected from that point in the same vertical plane with velocity $4\sqrt{g}$, then determine the locus of vertices of their paths. (15 marks)
(c) Using Stokes' theorem, evaluate $\displaystyle\iint_S (\nabla \times \vec{F})\cdot \hat{n}dS$, where $\vec{F} = (x^2+y-4)\hat{i} + 3xy\hat{j} + (2xy+z^2)\hat{k}$ and $S$ is the surface of the paraboloid $z = 4-(x^2+y^2)$ above the $xy$-plane. Here, $\hat{n}$ is the unit outward normal vector on $S$. (15 marks)
Answer approach & key points
Solve this multi-part numerical problem by allocating approximately 40% time to part (a) covering orthogonal trajectories and differential equations, 30% to part (b) on projectile dynamics and locus derivation, and 30% to part (c) applying Stokes' theorem. Begin with clear problem identification for each sub-part, show complete derivations with intermediate steps, and conclude with boxed final answers for each component.
- For (a)(i): Derive differential equation of confocal conics family, find orthogonal trajectories by replacing dy/dx with -dx/dy, and verify self-orthogonality by showing the family coincides with its orthogonal trajectories
- For (a)(ii): Transform the homogeneous equation using y = vx substitution or identify one solution by inspection, find second linearly independent solution, then apply variation of parameters for the particular integral
- For (b): Derive parametric equations of projectile motion x = ut cos θ, y = ut sin θ - ½gt², eliminate t to get trajectory equation y = x tan θ - (gx²)/(2u²cos²θ), find vertex coordinates, and eliminate θ to obtain locus
- For (c): Verify Stokes' theorem applicability, compute curl of F, identify boundary curve C as circle x²+y²=4 at z=0, parametrize C as r(t) = (2cos t, 2sin t, 0), evaluate line integral ∮F·dr directly
- For (c) alternative: Compute surface integral of curl F over paraboloid with proper normal orientation, verify consistency with line integral result