Q1 50M Compulsory solve Linear algebra and calculus
(a) Let H be a subspace of R⁴ spanned by the vectors v₁ = (1, –2, 5, –3), v₂ = (2, 3, 1, –4), v₃ = (3, 8, –3, –5). Then find a basis and dimension of H, and extend the basis of H to a basis of R⁴. (10 marks)
(b) Let T : R³ → R³ be a linear operator and B = {v₁, v₂, v₃} be a basis of R³ over R. Suppose that Tv₁ = (1, 1, 0), Tv₂ = (1, 0, –1), Tv₃ = (2, 1, –1). Find a basis for the range space and null space of T. (10 marks)
(c) Discuss the continuity of the function
f(x) = { 1/(1–e^(–1/x)), x ≠ 0
{ 0, x = 0
for all values of x. (10 marks)
(d) Expand ln(x) in powers of (x–1) by Taylor's theorem and hence find the value of ln(1·1) correct up to four decimal places. (10 marks)
(e) Find the equation of the right circular cylinder which passes through the circle x² + y² + z² = 9, x – y + z = 3. (10 marks)
हिंदी में पढ़ें
(a) माना H, R⁴ की एक उपसमष्टि है, जो कि सदिशों v₁ = (1, –2, 5, –3), v₂ = (2, 3, 1, –4), v₃ = (3, 8, –3, –5) द्वारा जनित है। तब H का एक आधार एवं विमा ज्ञात कीजिए तथा H के इस आधार को R⁴ के एक आधार तक विस्तृत कीजिए। (10 अंक)
(b) माना T : R³ → R³ एक रैखिक संकारक है तथा R पर R³ का एक आधार B = {v₁, v₂, v₃} है। माना कि Tv₁ = (1, 1, 0), Tv₂ = (1, 0, –1), Tv₃ = (2, 1, –1) है। T की परिसर समष्टि तथा शून्य समष्टि के लिए एक आधार ज्ञात कीजिए। (10 अंक)
(c) x के सभी मानों के लिए फलन
f(x) = { 1/(1–e^(–1/x)), x ≠ 0
{ 0, x = 0
के सांतत्य की चर्चा कीजिए। (10 अंक)
(d) टेलर प्रमेय द्वारा ln(x) का (x–1) की घात में प्रसार कीजिए तथा ln(1·1) का दशमलव के चार स्थानों तक सही मान ज्ञात कीजिए। (10 अंक)
(e) वृत्त x² + y² + z² = 9, x – y + z = 3 से होकर जाने वाले लम्ब वृत्तीय बेलन का समीकरण ज्ञात कीजिए। (10 अंक)
Answer approach & key points
Solve each sub-part systematically with approximately equal time allocation (2 marks per minute). For (a), use row reduction to find basis and extend; for (b), construct matrix representation to find range and null spaces; for (c), analyze left and right limits at x=0 and behavior at other points; for (d), apply Taylor expansion about x=1 with Lagrange remainder; for (e), use the standard cylinder formula with given circle as directrix. Present solutions clearly with proper mathematical notation.
- For (a): Correctly row reduce the matrix formed by v₁, v₂, v₃ to identify linearly independent vectors, state dim(H)=2, and find two appropriate vectors to extend basis to R⁴
- For (b): Construct the matrix of T relative to standard basis, correctly identify rank(T)=2 for range space basis, and solve Tx=0 to find null space basis with dimension 1
- For (c): Evaluate lim(x→0⁺) f(x) = 1, lim(x→0⁻) f(x) = 0, conclude discontinuity at x=0 (jump discontinuity), and verify continuity elsewhere
- For (d): Derive Taylor series ln(x) = Σ(-1)^(n+1)(x-1)^n/n for 0<x≤2, compute ln(1.1) ≈ 0.0953 using first 4-5 terms with error estimation
- For (e): Identify center (0,0,0), radius √3, axis direction (1,-1,1), and derive cylinder equation (x+y)² + (y+z)² + (z+x)² = 18 or equivalent standard form
Q2 50M solve Linear operators and multivariable calculus
(a) Consider a linear operator T on R³ over R defined by T(x, y, z) = (2x, 4x – y, 2x + 3y – z). Is T invertible? If yes, justify your answer and find T⁻¹. (15 marks)
(b) If u = (x+y)/(1-xy) and v = tan⁻¹x + tan⁻¹y, then find ∂(u, v)/∂(x, y). Are u and v functionally related? If yes, find the relationship. (15 marks)
(c) Find the image of the line x = 3-6t, y = 2t, z = 3+2t in the plane 3x+4y-5z+26 = 0. (20 marks)
हिंदी में पढ़ें
(a) माना R के ऊपर R³ पर एक रैखिक संकारक T, T(x, y, z) = (2x, 4x – y, 2x + 3y – z) द्वारा परिभाषित है। क्या T व्युत्क्रमणीय है? यदि हाँ, तो अपने उत्तर का तर्क प्रस्तुत कीजिए तथा T⁻¹ ज्ञात कीजिए। (15 अंक)
(b) यदि u = (x+y)/(1-xy) तथा v = tan⁻¹x + tan⁻¹y है, तब ∂(u, v)/∂(x, y) ज्ञात कीजिए। क्या u तथा v फलनतः सम्बन्धित हैं? यदि हाँ, तो सम्बन्ध ज्ञात कीजिए। (15 अंक)
(c) रेखा x = 3-6t, y = 2t, z = 3+2t का समतल 3x+4y-5z+26 = 0 में प्रतिबिम्ब ज्ञात कीजिए। (20 अंक)
Answer approach & key points
Solve all three parts systematically, allocating approximately 30% time to part (a) on linear operator invertibility, 30% to part (b) on Jacobian and functional dependence, and 40% to part (c) on finding image of a line in a plane. Begin each part with clear statement of the mathematical approach, show complete working with logical flow, and conclude with verified final answers. For part (c), explicitly verify that the given line intersects the plane before finding the image line.
- For (a): Construct matrix representation of T, compute determinant to establish invertibility (det = 2 ≠ 0), and find inverse using adjugate or row reduction: T⁻¹(x,y,z) = (x/2, 2x-y, 7x-3y-z)/2
- For (b): Compute Jacobian ∂(u,v)/∂(x,y) = 0, establishing functional dependence; derive relationship u = tan(v) using tan⁻¹ addition formula
- For (c): Verify line intersects plane at point P(3,0,3), find direction vector of image line using reflection formula or two-point method with foot of perpendicular
- For (c): Correctly apply image line formula: find reflected point or use symmetric point method to determine image line equation
- All parts: Explicit verification of results (TT⁻¹ = I for (a), direct substitution for (b), checking image line lies in plane for (c))
Q3 50M solve Linear algebra, optimization and 3D geometry
(a) Let V = M₂ₓ₂(ℝ) denote a vector space over the field of real numbers. Find the matrix of the linear mapping φ: V → V given by φ(v) = $\begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$v with respect to standard basis of M₂ₓ₂(ℝ), and hence find the rank of φ. Is φ invertible? Justify your answer. (15 marks)
(b) Find the volume of the greatest cylinder which can be inscribed in a cone of height h and semi-vertical angle α. (20 marks)
(c) Find the vertex of the cone 4x² - y² + 2z² + 2xy - 3yz + 12x - 11y + 6z + 4 = 0. (15 marks)
हिंदी में पढ़ें
(a) माना V = M₂ₓ₂(ℝ) वास्तविक संख्याओं के क्षेत्र पर एक सदिश समष्टि दर्शाता है। M₂ₓ₂(ℝ) के मानक आधार के सन्दर्भ में φ(v) = $\begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$v द्वारा दिए गए रैखिक प्रतिचित्रण φ: V → V का आव्यूह ज्ञात कीजिए और तब φ की कोटि (रैंक) ज्ञात कीजिए। क्या φ व्युत्क्रमणीय है? अपने उत्तर का तर्क प्रस्तुत कीजिए। (15 अंक)
(b) ऊँचाई h तथा अर्ध-शीर्ष कोण α वाले एक शंकु के अंतर्गत सबसे बड़े बेलन का आयतन ज्ञात कीजिए। (20 अंक)
(c) शंकु 4x² - y² + 2z² + 2xy - 3yz + 12x - 11y + 6z + 4 = 0 का शीर्ष ज्ञात कीजिए। (15 अंक)
Answer approach & key points
Solve all three parts systematically, allocating approximately 30% time to part (a) on linear mapping matrix representation, 40% to part (b) on optimization using calculus for the inscribed cylinder, and 30% to part (c) on finding the cone vertex through partial derivatives. Begin with clear identification of basis elements for (a), set up coordinate geometry for (b) with proper diagrammatic visualization, and use the condition for singular points for (c). Present each part with distinct sub-headings and conclude with boxed final answers.
- For (a): Correctly identify standard basis {E₁₁, E₁₂, E₂₁, E₂₂} of M₂ₓ₂(ℝ), compute φ(Eᵢⱼ) for each basis element, and construct the 4×4 matrix representation; determine rank via row reduction or determinant, and conclude non-invertibility due to zero determinant
- For (a): Explicitly state that φ is not invertible because det(φ) = 0 (the representing matrix is singular), connecting to the original 2×2 matrix having determinant -7 ≠ 0 but the induced map on 4×4 space being singular
- For (b): Set up coordinate system with cone vertex at origin, axis along z-axis, derive cone equation z = r cot(α), express cylinder volume V = πr²(h - r cot(α)) or equivalent, apply dV/dr = 0 to find optimal r = (2h/3)tan(α), and compute maximum volume V_max = (4πh³/27)tan²(α)
- For (b): Verify second derivative test confirms maximum, and express final volume in terms of given parameters h and α with proper dimensional analysis
- For (c): Apply condition that vertex satisfies ∂F/∂x = ∂F/∂y = ∂F/∂z = 0 for F(x,y,z) = 0, solve the resulting linear system: 8x + 2y + 12 = 0, -2y + 2x - 3z - 11 = 0, 4z - 3y + 6 = 0 to obtain vertex coordinates
- For (c): Verify the solution satisfies original cone equation and confirm the point is indeed a singular point (vertex) by checking all partial derivatives vanish simultaneously
Q4 50M solve Eigenvalues, multiple integration and sphere geometry
(a) Let A = $\begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{pmatrix}$ be a 3×3 matrix. Find the eigenvalues and the corresponding eigenvectors of A. Hence find the eigenvalues and the corresponding eigenvectors of A⁻¹⁵, where A⁻¹⁵ = (A⁻¹)¹⁵. (20 marks)
(b) Using double integration, find the area lying inside the cardioid r = a(1+cos θ) and outside the circle r = a. (15 marks)
(c) Find the equation of the sphere which touches the plane 3x+2y-z+2=0 at the point (1, -2, 1) and cuts orthogonally the sphere x²+y²+z²-4x+6y+4=0. (15 marks)
हिंदी में पढ़ें
(a) माना A = $\begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{pmatrix}$ एक 3×3 आव्यूह है। A का अभिलक्षणिक मान तथा संगत अभिलक्षणिक सदिश ज्ञात कीजिए। अतः A⁻¹⁵ का अभिलक्षणिक मान तथा संगत अभिलक्षणिक सदिश ज्ञात कीजिए, जहाँ A⁻¹⁵ = (A⁻¹)¹⁵ है। (20 अंक)
(b) दिशा: समाकलन का प्रयोग करते हुए हृदयाक्ष (कार्डिओइड) r = a(1+cos θ) के अंदर तथा वृत्त r = a के बाह्य स्थित क्षेत्र का क्षेत्रफल ज्ञात कीजिए। (15 अंक)
(c) उस गोले का समीकरण ज्ञात कीजिए, जो समतल 3x+2y-z+2=0 को बिंदु (1, -2, 1) पर स्पर्श करता है और गोले x²+y²+z²-4x+6y+4=0 को लंबिकतः काटता है। (15 अंक)
Answer approach & key points
Solve all three parts systematically, allocating approximately 40% of time to part (a) due to its 20 marks weightage, and 30% each to parts (b) and (c). Begin with setting up the characteristic equation for eigenvalues in (a), then proceed to polar integration for the cardioid area in (b), and finally apply the orthogonal sphere condition in (c). Present each part with clear headings and show all computational steps explicitly.
- Part (a): Correct characteristic equation det(A-λI)=0 yielding λ³-6λ²-3λ+18=0, eigenvalues λ=6,-1,-1, corresponding eigenvectors for distinct and repeated eigenvalues, and application of eigenvalue power property for A⁻¹⁵
- Part (a): Proper handling of inverse matrix eigenvalues (reciprocals) raised to power 15, giving eigenvalues (1/6)¹⁵, (-1)¹⁵=-1, (-1)¹⁵=-1 with same eigenvectors
- Part (b): Correct identification of intersection points at θ=±π/2, proper setup of double integral ∫∫ r dr dθ with limits r from a to a(1+cos θ) and θ from -π/2 to π/2
- Part (b): Accurate evaluation yielding area = a²(8+π)/2 or equivalent simplified form, with correct polar area element r dr dθ
- Part (c): Application of sphere touching plane condition: center lies on normal line through (1,-2,1), giving center as (1+3t, -2+2t, 1-t)
- Part (c): Use of orthogonal spheres condition 2u₁u₂+2v₁v₂+2w₁w₂=d₁+d₂ for spheres x²+y²+z²+2ux+2vy+2wz+d=0, leading to correct radius and final equation