All 8 questions from UPSC Civil Services Mains Mathematics 2021 Paper I (400 marks total). Every stem reproduced in full, with directive-word analysis, marks, word limits, and answer-approach pointers.
8Questions
400Total marks
2021Year
Paper IPaper
Topics covered
Linear algebra and calculus (1)Analytical geometry and multivariable calculus (1)Calculus, Linear Algebra and Analytical Geometry (1)Linear Algebra, Calculus and Three Dimensional Geometry (1)Differential equations, mechanics, vector calculus (1)Catenary, differential equations, line integrals (1)Vector calculus, differential equations, particle dynamics (1)Orthogonal trajectories, differential equations, particle dynamics, Stokes theorem (1)
A
Q1
50MCompulsoryproveLinear algebra and calculus
(a) If A= 1 & -1 & 1
2 & -1 & 0
1 & 0 & 0 , then show that A² = A⁻¹ (without finding A⁻¹). (10 marks)
(b) Find the matrix associated with the linear operator on V₃(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: Δ(x)= f(x+α) & f(x+2α) & f(x+3α)
f(α) & f(2α) & f(3α)
f'(α) & f'(2α) & f'(3α) where f is a real valued differentiable function and α is a constant. Find displaystylelimₓ → 0 (Δ(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 = -y/2 = z/3 and whose guiding curve is x² + 2y² = 1, z = 0. (10 marks)
हिंदी में पढ़ें
(a) यदि A= 1 & -1 & 1
2 & -1 & 0
1 & 0 & 0 है, तो A⁻¹ को ज्ञात किए बिना दर्शाइए कि A² = A⁻¹। (10 अंक)
(b) क्रमित आधारक B = (0, 1, 1), (1, 0, 1), (1, 1, 0) के सापेक्ष V₃(R) पर परिभाषित रैखिक संकारक : T(a, b, c) = (a+b, a-b, 2c) से संबंधित आव्यूह ज्ञात कीजिए। (10 अंक)
(c) दिया गया है : Δ(x)= f(x+α) & f(x+2α) & f(x+3α)
f(α) & f(2α) & f(3α)
f'(α) & f'(2α) & f'(3α) जहाँ f एक वास्तविक-मान अवकलनीय फलन है तथा α एक अचर है। displaystylelimₓ → 0 (Δ(x))/x को ज्ञात कीजिए। (10 अंक)
(d) दर्शाइए कि e^x cos x = 1 के किन्हीं दो मूलों के बीच में e^x sin x - 1 = 0 का कम से कम एक मूल विद्यमान है। (10 अंक)
(e) उस बेलन का समीकरण ज्ञात कीजिए जिसके जनक, रेखा : x = -y/2 = z/3 के समानांतर हैं तथा जिसका निर्देशक-वक्र x² + 2y² = 1, z = 0 है। (10 अंक)
Answer approach & key points
(1(a)) justify: claim > 3-4 reasons > evidence > conclusion | (1(b)) calculate: given > formula > substitution > result with units > interpretation | (1(c)) calculate: given > formula > substitution > result with units > interpretation | (1(d)) justify: claim > 3-4 reasons > evidence > conclusion | (1(e)) derive: given > assumptions > stepwise derivation > result > check Full marks: All parts fully solved with correct methods and no errors.
Compute A² correctly
Show A³ = I
Conclude A² = A⁻¹
Compute T of each basis vector
Express each image in basis B
Write the columns as [T]_B = [0 1 −1; 2 1 1; −1 0 1]
Observe Δ(0) = 0 since two rows coincide
Identify the limit as Δ′(0)
Show Δ′(0) has two equal rows so the limit is 0
Define φ(x) = e⁻ˣ − cos x with φ(a) = φ(b) = 0
Apply Rolle's theorem
Derive eᶜ sin c − 1 = 0
Direction ratios (1, −2, 3)
Eliminate parameters to get (x − z/3)² + 2(y + 2z/3)² = 1
50MproveAnalytical geometry and multivariable calculus
(a) Show that the planes, which cut the cone ax² + by² + cz² = 0 in perpendicular generators, touch the cone (x²)/(b+c) + (y²)/(c+a) + (z²)/(a+b) = 0. (20 marks)
(b) Given that f(x,y) = |x² - y²|. 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 R³(R). Find two bases of S. Also find the dimension of S. (15 marks)
हिंदी में पढ़ें
(a) दर्शाइए कि वे समतल, जो कि शंकु ax² + by² + cz² = 0 को लंब जनकों में काटते हैं, शंकु (x²)/(b+c) + (y²)/(c+a) + (z²)/(a+b) = 0 को स्पर्श करते हैं। (20 अंक)
(b) दिया गया है : f(x,y) = |x² - y²|, तब f_xy(0,0) तथा f_yx(0,0) ज्ञात कीजिए। अतः दर्शाइए कि f_xy(0,0) = f_yx(0,0)। (15 अंक)
(c) दर्शाइए कि S = (x, 2y, 3x) : x, y वास्तविक संख्याएँ हैं R³(R) का एक उपसमष्टि है। S के दो आधार ज्ञात कीजिए। S की विमा भी ज्ञात कीजिए। (15 अंक)
Answer approach & key points
(a) justify: claim > 3-4 reasons > evidence > conclusion | (b) calculate: given > formula > substitution > result with units > interpretation | (c) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete derivations with all steps shown, correct results, and clear justification.
Derive the condition (b+c)l² + (c+a)m² + (a+b)n² = 0 for perpendicular generators
Write the tangent plane condition for the second cone
Show the two conditions coincide
Compute f_x(0,k) = 0 from the limit definition, including the sign of h² − k²
Compute f_xy(0,0) = 0
Compute f_yx(0,0) = 0 and conclude equality
Verify closure under addition and scalar multiplication and the zero vector
50MsolveCalculus, Linear Algebra and Analytical Geometry
(a)(i) If u = x² + y², v = x² - y², where x = rcosθ, y = rsinθ, then find (∂(u,v))/(∂(r,θ)). (7 marks)
(a)(ii) If ∫limits₀^x f(t) dt = x + ∫limitsₓ¹ tf(t) dt, then find the value of f(1). (5 marks)
(a)(iii) Express ∫limitsₐ^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)
हिंदी में पढ़ें
(a)(i) यदि u = x² + y², v = x² - y², जहाँ पर x = rcosθ, y = rsinθ है, तब (∂(u,v))/(∂(r,θ)) ज्ञात कीजिए। (7 अंक)
(a)(ii) यदि ∫limits₀^x f(t) dt = x + ∫limitsₓ¹ tf(t) dt है, तो f(1) का मान ज्ञात कीजिए। (5 अंक)
(a)(iii) ∫limitsₐ^b (x-a)^m (b-x)ⁿ dx को बीटा-फलन के रूप में व्यक्त कीजिए। (8 अंक)
(b) अचर त्रिज्या r का एक गोला मूल-बिंदु O से गुजरता है तथा अक्षों को A, B, C बिंदुओं पर काटता है। O से समतल ABC पर खींचे गए लंब-पाद का बिंदुपथ ज्ञात कीजिए। (15 अंक)
(c)(i) सिद्ध कीजिए कि एक वास्तविक सममित आव्यूह के दो भिन्न अभिलक्षणिक मानों के संगत अभिलक्षणिक सदिश, लंबिक हैं। (8 अंक)
(c)(ii) दो वर्ग आव्यूह A तथा B जिनकी कोटि, 2 है के लिए दर्शाइए कि अनुरेख (AB) = अनुरेख (BA)। अतैव दर्शाइए कि AB - BA ≠ I₂ जहाँ I₂ एक 2-कोटि का तत्समक आव्यूह है। (7 अंक)
Answer approach & key points
(a(i)) calculate: given > formula > substitution > result with units > interpretation | (a(ii)) calculate: given > formula > substitution > result with units > interpretation | (a(iii)) derive: given > assumptions > stepwise derivation > result > check | (b) derive: given > assumptions > stepwise derivation > result > check | (c(i)) justify: claim > 3-4 reasons > evidence > conclusion | (c(ii)) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete stepwise derivations with all justifications, correct final results, and verification steps.
Express u = r² and v = r² cos 2θ
Compute the four partial derivatives
Evaluate the determinant −4r³ sin 2θ
Differentiate both sides using the fundamental theorem
Obtain f(x) = 1 − x f(x)
State f(1) = 1/2
Substitute x = a + (b−a)t
Change limits to 0 and 1
Result (b−a)^(m+n+1) B(m+1, n+1)
Equation of the sphere and intercepts 2α, 2β, 2γ
Foot of the perpendicular and expressions for α, β, γ
50MsolveLinear 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
(x+1)/3 = (y+3)/5 = (z+5)/7,
(x-2)/1 = (y-4)/3 = (z-6)/5.
Also find the point of intersection of the given lines. (15 marks)
हिंदी में पढ़ें
(a)(i) निम्नलिखित आव्यूह का पंक्ति-समानीत सोपानक रूप में समान्यन कीजिए एवं अतैव इसकी कोटि भी ज्ञात कीजिए।
A = [1 3 2 4 1
0 0 2 2 0
2 6 2 6 2
3 9 1 10 6] (10 अंक)
(a)(ii) सम्मिश्र संख्या क्षेत्र पर आव्यूह A = (0 -i
i 0) के अभिलक्षणिक मान तथा संगत अभिलक्षणिक सदिशों को ज्ञात कीजिए। (10 अंक)
(b) दर्शाइए कि ऐस्ट्रॉइड : x^(2/3) + y^(2/3) = a^(2/3) का पूरा क्षेत्रफल (3/8)πa² है। (15 अंक)
(c) रेखाओं
(x+1)/3 = (y+3)/5 = (z+5)/7,
(x-2)/1 = (y-4)/3 = (z-6)/5
को अंतर्विष्ट करने वाले समतल का समीकरण ज्ञात कीजिए। दी गई रेखाओं के प्रतिच्छेद बिंदु को भी ज्ञात कीजिए। (15 अंक)
Answer approach & key points
(a(i)) calculate: given > formula > substitution > result with units > interpretation | (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: Complete working with all steps shown, correct results, verification included
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)
हिंदी में पढ़ें
अवकल समीकरण: d²y/dx² + 2y = x²e^(3x) + e^x cos 2x को हल कीजिए। (10)
लाप्लास रूपान्तर विधि का उपयोग करते हुए प्रारम्भिक मान समस्या: d²y/dx² + 4y = e^(-2x) sin 2x; y(0) = y'(0) = 0 को हल कीजिए। (10)
दो छड़ें LM व MN बिन्दु M पर दृढ़ता से इस प्रकार जुड़ी हैं कि (LM)² + (MN)² = (LN)² तथा वे स्वतन्त्र रूप से साम्यावस्था में स्थिर बिन्दु L पर टंगी हैं। माना कि दोनों एकसमान छड़ों का प्रति एकांक लम्बाई, भार ω है। छड़ LM का उद्वधर दिशा के साथ बने कोण को छड़ों की लम्बाई के रूप में ज्ञात कीजिए। (10)
यदि एक ग्रह, जो सूर्य के परितः वृत्तीय कक्षा में परिभ्रमण करता है, अचानक अपनी कक्षा में रोक दिया जाता है, तो वह समय, जिसमे वह सूर्य में गिर जाएगा, ज्ञात कीजिए। इसके गिरने के समय का ग्रह के परिभ्रमण आवर्तकाल से अनुपात भी ज्ञात कीजिए। (10)
दर्शाइए कि ∇²[∇·(r⃗/r)] = 2/r⁴, जहाँ r⃗ = xî + yĵ + zk̂ है। (10)
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) derive: given > assumptions > stepwise derivation > result > check | (c) derive: given > assumptions > stepwise derivation > result > check | (d) derive: given > assumptions > stepwise derivation > result > check | (e) derive: given > assumptions > stepwise derivation > result > check Full marks: Complete stepwise derivations with all intermediate steps shown, correct application of named theorems/methods, and verification where appropriate.
Complementary function C₁cos(√2 x) + C₂sin(√2 x)
Particular integral for x²e³ˣ with a = 1/11, b = −12/121, c = 50/1331
Particular integral eˣ(4 sin 2x − cos 2x)/17 and the general solution
Transform with zero initial conditions to get Y = 2/[(s²+4)(s²+4s+8)]
50MsolveCatenary, 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)
हिंदी में पढ़ें
एक भारी डोरी, जिसका घनत्व एक समान नहीं है, दो बिन्दुओं से टंगी हुई है। माना कि T₁, T₂, T₃ क्रमशः कैटिनरी के बीच के बिन्दुओं A, B, C पर तनाव हैं, जिन पर इसके क्षैतिज के साथ आनति कोण, सार्व अंतर β के साथ समांतर श्रेढ़ी में हैं। माना कि डोरी के AB तथा BC भागों के भार क्रमशः ω₁ तथा ω₂ हैं। सिद्ध कीजिए (i) T₁, T₂ तथा T₃ का हरात्मक माध्य = 3T₂/(1 + 2cos β) (ii) T₁/T₃ = ω₁/ω₂ (20)
सभी अंतरस्थ (शामिल) चरणों को दर्शाते हुए समीकरण: d²y/dx² + (tan x - 3cos x)dy/dx + 2y cos²x = cos⁴x को पूर्ण रूप से हल कीजिए। (15)
∫_C F⃗ · dr⃗ का मान निकालिए, जहाँ C, xy-समतल में एक सैच्छिक संयुक्त वक्र है तथा F⃗ = (-yî + xĵ)/(x² + y²) है। (15)
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Complete, rigorous derivations with all steps shown and verified.
Constant horizontal tension gives T = H sec ψ
Prove the harmonic mean equals 3T₂/(1 + 2cos β) using cos(θ−β) + cos(θ+β) = 2cos θ cos β
Weights as differences of vertical tension components giving ω₁/ω₂ = cos(θ+β)/cos(θ−β) = T₁/T₃
Substitution t = sin x with correct transformation of y′ and y″
Reduce to Y″ − 3Y′ + 2Y = 1 − t²
Complementary function and particular integral giving the general solution
Show F·dr = dθ
State the value 2π times the winding number about the origin
Treat the cases: origin outside (0), inside once anticlockwise (2π), through the origin (undefined)
(a) Verify Gauss divergence theorem for F⃗ = 2x^2yî - y^2ĵ + 4xz^2k̂ taken over the region in the first octant bounded by y² + z² = 9 and x = 2. (20 marks)
(b) Find all possible solutions of the differential equation: y² log y = xydy/dx + (dy/dx)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 √2gh. If 5a/2 > h > a, then prove that the circular motion ceases when the particle has reached the height 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 ((4a-h)(a+2h)²)/(27a²). (15 marks)
हिंदी में पढ़ें
(a) प्रथम अष्टांश में y² + z² = 9 तथा x = 2 द्वारा परिबद्ध क्षेत्र पर F⃗ = 2x^2yî - y^2ĵ + 4xz^2k̂ के लिए गॉस अपसरण प्रमेय को सत्यापित कीजिए। (20 अंक)
(b) अवकल समीकरण: y² log y = xydy/dx + (dy/dx)² के सभी संभव हल ज्ञात कीजिए। (15 अंक)
(c) एक भारी कण a लम्बाई की अवितान्य डोरी से एक स्थिर बिंदु से टंगा है तथा √2gh वेग से क्षैतिज दिशा में प्रक्षेपित किया जाता है। यदि 5a/2 > h > a है, तो सिद्ध कीजिए कि प्रक्षेपण बिंदु से 1/3(a + 2h) ऊँचाई पहुँचने पर कण की वृत्तीय गति समाप्त हो जाती है। यह भी सिद्ध कीजिए कि उस कण द्वारा प्रक्षेपण बिंदु से ऊपर प्राप्य अधिकतम ऊँचाई ((4a-h)(a+2h)²)/(27a²) है। (15 अंक)
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) calculate: given > formula > substitution > result with units > interpretation | (c) derive: given > assumptions > stepwise derivation > result > check Full marks: Complete, correct derivations with all steps shown and results verified.
Compute div F = 4xy − 2y + 8xz and the volume integral 180
Compute the curved-surface integral 108 with correct normal and parametrisation
Compute the plane x = 2 integral 72, show other faces give 0, and equate 180 = 180
Substitute u = log y to reach Clairaut's form u = x u′ + u′²
General solution y = e^(Cx + C²)
Singular solution y = e^(−x²/4)
Use energy and the radial equation with T = 0 to get cos θ = −2(h−a)/(3a) and height (a+2h)/3
Projectile motion from that point with speed² = 2g(h−a)/3
(a)(i) Find the orthogonal trajectories of the family of confocal conics (x²)/(a²+λ) + (y²)/(b²+λ) = 1; a > b > 0 are constants and λ 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: x²(d^2y)/(dx²) - 2x(1+x)dy/dx + 2(1+x)y = 0. Hence, solve the differential equation: x²(d^2y)/(dx²) - 2x(1+x)dy/dx + 2(1+x)y = x³ 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 θ with the horizontal direction. Further, if particles are projected from that point in the same vertical plane with velocity $4√g$, then determine the locus of vertices of their paths. (15 marks)
(c) Using Stokes' theorem, evaluate displaystyle∬_S (∇ × F⃗)· n̂dS, where F⃗ = (x²+y-4)î + 3xyĵ + (2xy+z²)k̂ and S is the surface of the paraboloid z = 4-(x²+y²) above the xy-plane. Here, n̂ is the unit outward normal vector on S. (15 marks)
हिंदी में पढ़ें
(a)(i) संनाभि शंकु कुल (x²)/(a²+λ) + (y²)/(b²+λ) = 1; a > b > 0 अचर हैं तथा λ एक प्राचल है, के लम्बकोणीय संघेदी ज्ञात कीजिए। दर्शाइए कि दिया गया वक्र-कुल स्वलंबिक है। (10 अंक)
(a)(ii) अवकल समीकरण: x²(d^2y)/(dx²) - 2x(1+x)dy/dx + 2(1+x)y = 0 का व्यापक हल ज्ञात कीजिए। अतः अवकल समीकरण: x²(d^2y)/(dx²) - 2x(1+x)dy/dx + 2(1+x)y = x³ को प्राचल विचरण विधि द्वारा हल कीजिए। (10 अंक)
(b) द्रव्यमान m का एक कण, जो कि प्रक्षेपण बिन्दु से वेग u के साथ क्षैतिज दिशा के साथ θ कोण बनाने वाली दिशा में प्रक्षेपण बिन्दु से गुजरने वाले उद्धवाधर समतल में प्रक्षेपित किया जाता है, उसकी गति तथा पथ का वर्णन कीजिए। यदि कणों को उसी बिन्दु से उसी उद्धवाधर समतल में वेग $4√g$ के साथ प्रक्षेपित किया जाता है, तो उनके पथों के शीर्षों के बिन्दुपथ को भी निर्धारित कीजिए। (15 अंक)
(c) स्टोक्स प्रमेय का उपयोग करते हुए displaystyle∬_S (∇ × F⃗)· n̂dS का मान निकालिए, जहाँ पर F⃗ = (x²+y-4)î + 3xyĵ + (2xy+z²)k̂ तथा S, परवलयज z = 4-(x²+y²) का xy-समतल से ऊपर का पृष्ठ है। यहाँ n̂, S पर एकक बहिर्मुखी अभिलम्ब सदिश है। (15 अंक)
Answer approach & key points
(a(i)) derive: given > assumptions > stepwise derivation > result > check | (a(ii)) derive: given > assumptions > stepwise derivation > result > check | (b) describe: define > structure or process in order > labelled diagram > significance | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Complete derivations with all steps shown, correct results, and verification
Eliminate λ to get xy p² + (x² − y² − c²)p − xy = 0
Replace p by −1/p and show the equation is unchanged
Conclude the trajectories are the same confocal family
Complementary function C₁x + C₂xe²ˣ with y₁ = x identified
Wronskian 2x²e²ˣ with the equation in standard form (R = x)
Particular integral −x²/2 and general solution
Equations of motion and parabolic path
Vertex coordinates X = u² sin 2θ/(2g), Y = u² sin²θ/(2g)
Locus X²/64 + (Y − 4)²/16 = 1
Boundary C is the circle x² + y² = 4, z = 0, anticlockwise
Reduce F·dr to (x² + y − 4)dx + 3xy dy on C and parametrise