Q1 50M Compulsory solve Linear algebra, calculus and 3D geometry
(a) Let V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3) and V₃ = (1, 1, 9, -5) be three vectors of the space ℝ⁴. Does (3, -1, 0, -1) ∈ span {V₁, V₂, V₃} ? Justify your answer. (10 marks)
(b) Find the rank and nullity of the linear transformation : T : ℝ³ → ℝ³ given by T(x, y, z) = (x + z, x + y + 2z, 2x + y + 3z) (10 marks)
(c) Find the values of p and q for which limₓ→₀ [x(1 + p cos x) - q sin x]/x³ exists and equals 1. (10 marks)
(d) Examine the convergence of the integral ∫₀¹ (log x)/(1+x) dx (10 marks)
(e) A variable plane which is at a constant distance 3p from the origin O cuts the axes in the points A, B, C respectively. Show that the locus of the centroid of the tetrahedron OABC is 9(1/x² + 1/y² + 1/z²) = 16/p². (10 marks)
हिंदी में पढ़ें
(a) मान लीजिए V₁ = (2, -1, 3, 2), V₂ = (-1, 1, 1, -3), V₃ = (1, 1, 9, -5) समष्टि ℝ⁴ के तीन सदिश हैं । क्या (3, -1, 0, -1) ∈ विस्तृति {V₁, V₂, V₃} ? अपने उत्तर को तर्कसहित सिद्ध कीजिए । (10 अंक)
(b) T(x, y, z) = (x + z, x + y + 2z, 2x + y + 3z) द्वारा दिए गए रैखिक रूपांतरण : T : ℝ³ → ℝ³ की कोटि तथा शून्यता ज्ञात कीजिए । (10 अंक)
(c) p तथा q के वो मान निकालिए जिसके लिए limₓ→₀ [x(1 + p cos x) - q sin x]/x³ का अस्तित्व है एवं 1 के बराबर है । (10 अंक)
(d) समाकल ∫₀¹ (log x)/(1+x) dx की अभिसारिता का परीक्षण कीजिए । (10 अंक)
(e) एक चर समतल, जो कि मूल-बिंदु O से अचर दूरी 3p पर है, अक्षों को क्रमशः बिंदुओं A, B, C पर काटता है । दर्शाइए कि चतुष्फलक OABC के केंद्रक का बिंदुपथ 9(1/x² + 1/y² + 1/z²) = 16/p² है । (10 अंक)
Answer approach & key points
(a) justify: claim > 3-4 reasons > evidence > conclusion | (b) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation | (d) examine: intro > how/why with reasoning > evidence > conclusion | (e) derive: given > assumptions > stepwise derivation > result > check Full marks: Complete stepwise derivation with named theorems and verification.
- Set up alpha V1 + beta V2 + gamma V3 = w
- Four equations in three unknowns
- Show inconsistency
- Conclude not in span
- Matrix of T
- Row reduction to rank 2
- Kernel spanned by (1,1,-1)
- Rank-nullity: nullity 1
- Taylor series of cos x and sin x
- Coefficient of x vanishes: q = 1 + p
- Coefficient of x^3 equals 1
- p = -5/2, q = -3/2
- Singularity only at 0
- Compare with -log x
- Integral of -log x is 1
- Conclusion: converges
- Intercept plane
- Distance condition 1/a^2+1/b^2+1/c^2 = 1/(9p^2)
- Centroid (a/4,b/4,c/4)
- Substitute and simplify
Q2 50M solve Linear algebra, multivariable calculus and 3D geometry
(a) If the matrix of a linear transformation T : IR³→IR³ relative to the basis (1, 0, 0), (0, 1, 0), (0, 0, 1) is
1 & 1 & 2
-1 & 2 & 1
0 & 1 & 3 ,
then find the matrix of T relative to the basis (1, 1, 1), (0, 1, 1), (0, 0, 1). (15 marks)
(b) Evaluate the triple integral which gives the volume of the solid enclosed between the two paraboloids Z = 5(x² + y²) and Z = 6 – 7x² – y². (15 marks)
(c)(i) Show that the equation 2x² + 3y² – 8x + 6y – 12z + 11 = 0 represents an elliptic paraboloid. Also find its principal axis and principal planes. (10 marks)
(c)(ii) The plane x/a+y/b+z/c=1 meets the coordinate axes in A, B, C respectively. Prove that the equation of the cone generated by the lines drawn from the origin O to meet the circle ABC is
yz(b/c+c/b)+zx(c/a+a/c)+xy(b/a+a/b)=0. (10 marks)
हिंदी में पढ़ें
(a) यदि आधार (1, 0, 0), (0, 1, 0), (0, 0, 1) के सापेक्ष रैखिक रूपांतरण T : IR³→IR³ का आव्यूह
1 & 1 & 2
-1 & 2 & 1
0 & 1 & 3
है, तब आधार (1, 1, 1), (0, 1, 1), (0, 0, 1) के सापेक्ष T का आव्यूह ज्ञात कीजिए। (15 अंक)
(b) दो परवलयजों Z = 5(x² + y²) और Z = 6 – 7x² – y² के बीच घिरे ठोस के आयतन को दर्शाने वाले त्रिशः समाकल का मान निकालिए। (15 अंक)
(c)(i) दर्शाइए कि समीकरण 2x² + 3y² – 8x + 6y – 12z + 11 = 0 एक दीर्घवृत्तीय परवलयज प्रदर्शित करता है। साथ ही मुख्य अक्ष और मुख्य समतलों को भी ज्ञात कीजिए। (10 अंक)
(c)(ii) समतल x/a+y/b+z/c=1, निर्देशांक अक्षों को क्रमशः A, B, C में मिलता है। सिद्ध कीजिए कि मूल बिंदु O से वृत्त ABC को मिलाने वाली रेखाओं द्वारा जनित शंकु का समीकरण
yz(b/c+c/b)+zx(c/a+a/c)+xy(b/a+a/b)=0
है। (10 अंक)
Answer approach & key points
(a) calculate: given > formula > substitution > result with units > interpretation | (b) calculate: given > formula > substitution > result with units > interpretation | (c(i)) explain: definition/context > points in order > small example > short close | (c(ii)) justify: claim > 3-4 reasons > evidence > conclusion Full marks: All parts fully solved with correct methods and verification
- Change-of-basis matrix P
- Inverse of P
- M = P^-1 A P
- Result [4 3 2; -2 0 -1; 2 1 2]
- Intersection curve 2x^2+y^2=1
- Limits in z
- Elliptic change of variables
- V = 3 pi sqrt2 / 2
- Complete the squares
- Standard form z = X^2/6 + Y^2/4
- Axis x=2, y=-1
- Planes x=2 and y=-1
- Circle as plane section of a sphere through O, A, B, C
- Ray meets plane at lambda(x,y,z), lambda = 1/D
- Eliminate lambda
- Expand and cancel to the given equation
Q3 50M solve Linear algebra, multivariable calculus, 3D geometry
Let A = 1 & 0 & 0
1 & 0 & 1
0 & 1 & 0
(i) Verify the Cayley-Hamilton theorem for the matrix A.
(ii) Show that Aⁿ = Aⁿ⁻² + A² - I for n ≥ 3, where I is the identity matrix of order 3. Hence, find A⁴⁰. 10+10
(b) Justify whether (0, 0) is an extreme point for the function f(x, y) = 2x⁴ - 3x^2y + y². 15
(c) Find the equation of the sphere through the circle
x² + y² + z² - 4x - 6y + 2z - 16 = 0; 3x + y + 3z - 4 = 0
in the following two cases.
(i) the point (1, 0, -3) lies on the sphere.
(ii) the given circle is a great circle of the sphere. 15
हिंदी में पढ़ें
दिया गया है A= 1 & 0 & 0
1 & 0 & 1
0 & 1 & 0
(i) आव्यूह A के लिये कैले-हैमिल्टन प्रमेय को सत्यापित कीजिए।
(ii) दर्शाइए कि n ≥ 3 के लिये Aⁿ = Aⁿ⁻² + A² – I; जहाँ I कोटि 3 का तत्समक आव्यूह है।
अतः A⁴⁰ ज्ञात कीजिए। 10+10
(b) तर्क सहित दर्शाइये कि (0, 0), फलन f(x, y) = 2x⁴ - 3x^2y + y² का चरम-बिन्दु है अथवा नहीं। 15
(c) वृत्त x² + y² + z² - 4x - 6y + 2z - 16 = 0; 3x + y + 3z - 4 = 0 से होकर गुजरने वाले गोले का समीकरण निम्न दो स्थितियों में ज्ञात कीजिए।
(i) बिन्दु (1, 0, -3) गोले पर हो।
(ii) दिया गया वृत्त गोले का एक बृहत् वृत्त हो। 15
Answer approach & key points
(a(i)) derive: given > assumptions > stepwise derivation > result > check | (a(ii)) calculate: given > formula > substitution > result with units > interpretation | (b) justify: claim > 3-4 reasons > evidence > conclusion | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: All steps shown with correct theorems and verification.
- Characteristic polynomial lambda^3 - lambda^2 - lambda + 1
- A^2 and A^3
- A^3 - A^2 - A + I = 0 shown
- A^3 - A = A^2 - I and AB = B
- Induction
- A^(2m) = A^2 + (m-1)B
- A^40 = [1 0 0; 20 1 0; 20 0 1]
- Critical point at origin
- Hessian singular so test fails
- Path y = 0 gives positive
- Path y = 3x^2/2 gives negative; not extreme
- Family S + lambda L = 0
- Case (i): lambda = -8/5
- Case (ii): centre on plane, lambda = 4/19
- Final equations
Q4 50M solve Matrix rank, curve tracing, 3D geometry
(a) Find the rank of the matrix
A = 1 & 2 & -1 & 0
-1 & 3 & 0 & -4
2 & 1 & 3 & -2
1 & 1 & 1 & -1
by reducing it to row-reduced echelon form. 15
(b) Trace the curve y²(x² - 1) = 2x - 1. 20
(c) Prove that the locus of a line which meets the lines
y = mx, z = c; y = -mx, z = -c and the circle x² + y² = a², z = 0 is
c²m²(cy - mzx)² + c²(yz - cmx)² = a²m²(z² - c²)². 15
हिंदी में पढ़ें
(a) आव्यूह A = 1 & 2 & -1 & 0
-1 & 3 & 0 & -4
2 & 1 & 3 & -2
1 & 1 & 1 & -1
का पंक्ति समानीत सोपानक रूप में समानयन करके उसकी कोटि ज्ञात कीजिए। 15
(b) वक्र y²(x² - 1) = 2x - 1 को अनुरेखित कीजिए। 20
(c) सिद्ध कीजिए कि रेखाओं y = mx, z = c; y = -mx, z = -c और
वृत्त x² + y² = a², z = 0 से मिलने वाली रेखा का बिन्दु-पथ
c²m²(cy - mzx)² + c²(yz - cmx)² = a²m²(z² - c²)² है। 15
Answer approach & key points
(a) calculate: given > formula > substitution > result with units > interpretation | (b) trace: start point > the stages in sequence > end point > what changed | (c) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete working, correct results, clear presentation, and verification steps.
- Row operations on A
- Reach RREF [1 0 0 1; 0 1 0 -1; 0 0 1 -1; 0 0 0 0]
- Count non-zero rows
- rank = 3
- Symmetry about x-axis
- Intercepts (1/2,0), (0,+-1)
- Asymptotes x = +-1 and y = 0
- Region -1<x<=1/2 and x>1; two branches
- Points P and Q on the two lines
- Line PQ parametrised, solve for S and D
- Point at z = 0 on the circle
- Substitute and simplify to the given equation