All 8 questions from UPSC Civil Services Mains Mathematics 2022 Paper I (400 marks total). Every stem reproduced in full, with directive-word analysis, marks, word limits, and answer-approach pointers.
8Questions
400Total marks
2022Year
Paper IPaper
Topics covered
Differential equations, mechanics and vector calculus (2)Linear algebra, calculus and 3D geometry (1)Linear equations, optimization and 3D geometry (1)Vector spaces, multiple integration and 3D geometry (1)Linear transformations, curve tracing and 3D geometry (1)Mechanics, differential equations and vector calculus (1)Vector calculus, Laplace transforms and mechanics (1)
A
Q1
50MCompulsoryproveLinear algebra, calculus and 3D geometry
(a) Prove that any set of n linearly independent vectors in a vector space V of dimension n constitutes a basis for V. 10
(b) Let T : ℝ² → ℝ³ be a linear transformation such that T(1, 0) = (1, 2, 3) and T(1, 1) = (−3, 2, 8). Find T(2, 4). 10
(c) Evaluate lim (x → ∞) (eˣ + x)^(1/x). 10
(d) Examine the convergence of ∫₀² dx/(2x − x²). 10
(e) A variable plane passes through a fixed point (a, b, c) and meets the axes at points A, B and C respectively. Find the locus of the centre of the sphere passing through the points O, A, B and C, O being the origin. 10
हिंदी में पढ़ें
(a) सिद्ध कीजिए कि n विमीय सदिश समष्टि V के लिए n रैखिकतः स्वतंत्र सदिशों का कोई भी समुच्चय V के लिए एक आधार बनता है। 10
(b) माना T : ℝ² → ℝ³ एक रैखिक रूपांतरण, ऐसा है कि T(1, 0) = (1, 2, 3) तथा T(1, 1) = (−3, 2, 8) है। T(2, 4) को ज्ञात कीजिए। 10
(c) lim (x → ∞) (eˣ + x)^(1/x) का मान निकालिए। 10
(d) ∫₀² dx/(2x − x²) की अभिसारिता का परीक्षण कीजिए। 10
(e) एक चर समतल एक स्थिर बिंदु (a, b, c) से गुजरता है तथा अक्षों को क्रमशः A, B व C बिंदुओं पर मिलता है । बिंदुओं O, A, B तथा C से गुजरते हुए गोले के केंद्र का बिंदुपथ ज्ञात कीजिए, जहाँ O मूल-बिंदु है । 10
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, rigorous derivations with all steps justified and verified.
State definition of basis (independent + spanning)
Prove the set spans V using dimension argument
Reference the theorem: independent set in n-dim space is basis
Express (2, 4) as linear combination of (1, 0) and (1, 1)
Apply linearity: T(ax + by) = aT(x) + bT(y)
Compute final vector in R3
Take logarithm to handle the exponent
Apply L'Hopital's rule to the log-limit
Exponentiate the result to find the final limit
Identify singularities at both x = 0 and x = 2
Split integral or use comparison test near singularity
Evaluate or bound the integral to determine convergence
50MsolveLinear equations, optimization and 3D geometry
(a) Find all solutions to the following system of equations by row-reduced method : x₁ + 2x₂ − x₃ = 2, 2x₁ + 3x₂ + 5x₃ = 5, − x₁ − 3x₂ + 8x₃ = − 1. 15
(b) A wire of length l is cut into two parts which are bent in the form of a square and a circle respectively. Using Lagrange's method of undetermined multipliers, find the least value of the sum of the areas so formed. 15
(c) If P, Q, R; P', Q', R' are feet of the six normals drawn from a point to the ellipsoid x²/a² + y²/b² + z²/c² = 1, and the plane PQR is represented by lx + my + nz = p, show that the plane P'Q'R' is given by x/a²l + y/b²m + z/c²n + 1/p = 0. 20
हिंदी में पढ़ें
(a) निम्नलिखित समीकरण निकाय के सभी हलों को पंक्ति-समानित विधि से ज्ञात कीजिए : x₁ + 2x₂ − x₃ = 2, 2x₁ + 3x₂ + 5x₃ = 5, − x₁ − 3x₂ + 8x₃ = − 1. 15
(b) एक l लम्बाई के तार को दो भागों में काटकर क्रमशः एक वर्ग तथा एक वृत्त के रूप में मोड़ा गया है । लग्रांज की अनिर्धारित गुणक विधि का प्रयोग करके, इस तरह से प्राप्त किए गए क्षेत्रफलों के योगफल का न्यूनतम मान ज्ञात कीजिए । 15
(c) यदि P, Q, R; P', Q', R', एक बिंदु से दीर्घवृत्तज x²/a² + y²/b² + z²/c² = 1 पर छः (सिक्स) अभिलंब पाद हैं तथा lx + my + nz = p से समतल PQR निरूपित है, दर्शाइए कि x/a²l + y/b²m + z/c²n + 1/p = 0, समतल P'Q'R' को निरूपित करता है । 20
Answer approach & key points
(a) calculate: given > formula > substitution > result with units > interpretation | (b) calculate: given > formula > substitution > result with units > interpretation | (c) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete, stepwise derivations with all justifications and checks.
Form the augmented matrix of the system
Perform row operations to reach RREF
State the general solution in parametric form (infinitely many solutions, rank 2)
50MsolveVector spaces, multiple integration and 3D geometry
(a) Let the set P = {(x, y, z)ᵀ : x − y − z = 0 and 2x − y + z = 0} be the collection of vectors of a vector space R³(R). Then (i) prove that P is a subspace of R³. (ii) find a basis and dimension of P. 10+10
(b) Use double integration to calculate the area common to the circle x² + y² = 4 and the parabola y² = 3x. 15
(c) Find the equation of the sphere of smallest possible radius which touches the straight lines : (x-3)/3 = (y-8)/(-1) = (z-3)/1 and (x+3)/(-3) = (y+7)/2 = (z-6)/4. 15
हिंदी में पढ़ें
(a) माना समुच्चय P = {(x, y, z)ᵀ : x − y − z = 0 तथा 2x − y + z = 0} सदिश समष्टि R³(R) के सदिशों का एक समूह है । तब (i) सिद्ध कीजिए कि P, R³ की एक उपसमष्टि है । (ii) P का एक आधार तथा विमा ज्ञात कीजिए । 10+10
(b) द्विश: समाकलन का उपयोग करके, वृत्त x² + y² = 4 तथा परवलय y² = 3x के उभयनिष्ठ क्षेत्रफल का परिकलन कीजिए । 15
(c) लघुतम संभाव्य त्रिज्या के गोले का समीकरण ज्ञात कीजिए जो सरल रेखाओं : (x-3)/3 = (y-8)/(-1) = (z-3)/1 तथा (x+3)/(-3) = (y+7)/2 = (z-6)/4 को स्पर्श करता है । 15
Answer approach & key points
(a(i)) justify: claim > 3-4 reasons > evidence > conclusion | (a(ii)) calculate: given > formula > substitution > result with units > interpretation | (b) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Rigorous proofs, correct calculations, clear steps, and verification.
Verify P is non-empty (zero vector)
Verify closure under vector addition
Verify closure under scalar multiplication
State subspace criteria used
Solve the system of linear equations
Express general solution in parametric form
Identify linearly independent vectors for basis
State dimension of P
Find intersection points of curves
Set up double integral with correct limits
Evaluate the integral step-by-step
State final area (4π+√3)/3 sq units
Identify direction vectors of lines
Find shortest distance between skew lines
Determine center of the sphere
Write final sphere x²+(y−1/2)²+(z−9/2)²=135/2 (r=3√30/2)
50MtraceLinear transformations, curve tracing and 3D geometry
(a) Find a linear map T : R² → R² which rotates each vector of R² by an angle θ. Also, prove that for θ = (π)/2, T has no eigenvalue in R. 15
(b) Trace the curve y²x² = x² – a², where a is a real constant. 20
(c) If the plane ux + vy + wz = 0 cuts the cone ax² + by² + cz² = 0 in perpendicular generators, then prove that (b + c) u² + (c + a) v² + (a + b) w² = 0. 15
हिंदी में पढ़ें
(a) एक रैखिक प्रतिचित्र T : R² → R² ज्ञात कीजिए जो कि R² के प्रत्येक सदिश को θ कोण से घुमा देता है । यह भी सिद्ध कीजिए कि θ = (π)/2 के लिए, T का कोई भी अभिलक्षणिक मान (आइगेनमान) R में नहीं है । 15
(b) वक्र y²x² = x² – a² का अनुरेख (ट्रेस) कीजिए, जहाँ a एक वास्तविक अचर है । 20
(c) यदि समतल ux + vy + wz = 0, शंकु ax² + by² + cz² = 0 को लंब जनकों में काटता है, तो सिद्ध कीजिए कि (b + c) u² + (c + a) v² + (a + b) w² = 0. 15
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) trace: start point > the stages in sequence > end point > what changed | (c) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete derivations with all steps, correct results, and clear justification
Matrix T = [[cosθ, -sinθ], [sinθ, cosθ]]
Characteristic polynomial det(T-λI) = 0
Substitute θ=π/2 to get λ²+1=0
Conclude roots are imaginary, not in R
Domain analysis: x² ≥ a² (x ≥ a or x ≤ -a)
Symmetry about x and y axes
Asymptotes: y = ±1 (horizontal), no vertical asymptote
Sketch showing branches in quadrants I, II, III, IV
Substitute plane equation into cone equation
Derive condition for perpendicular lines
Apply condition to get (b+c)u² + (c+a)v² + (a+b)w² = 0
50MCompulsoryproveDifferential equations, mechanics and vector calculus
(a) Show that the general solution of the differential equation dy/dx + Py = Q can be written in the form y = Q/P − e^(−∫P dx) {C + ∫ e^(∫P dx) d(Q/P)}, where P, Q are non-zero functions of x and C, an arbitrary constant. 10
(b) Show that the orthogonal trajectories of the system of parabolas : x² = 4a(y + a) belong to the same system. 10
(c) A body of weight w rests on a rough inclined plane of inclination θ, the coefficient of friction, μ, being greater than tan θ. Find the work done in slowly dragging the body a distance 'b' up the plane and then dragging it back to the starting point, the applied force being in each case parallel to the plane. 10
(d) A projectile is fired from a point O with velocity √2gh and hits a tangent at the point P(x, y) in the plane, the axes OX and OY being horizontal and vertically downward lines through the point O, respectively. Show that if the two possible directions of projection be at right angles, then x² = 2hy and then one of the possible directions of projection bisects the angle POX. 10
(e) Show that A⃗ = (6xy + z³)î + (3x² - z)ĵ + (3xz² - y)k̂ is irrotational. Also find φ such that A⃗ = ∇φ. 10
हिंदी में पढ़ें
(a) दर्शाइए कि अवकल समीकरण dy/dx + Py = Q का व्यापक हल y = Q/P − e^(−∫P dx) {C + ∫ e^(∫P dx) d(Q/P)} के रूप में लिखा जा सकता है, जहाँ P, Q, x के शून्येतर फलन हैं तथा C एक स्वेच्छ अचर है। 10
(b) दर्शाइए कि परवलयों के निकाय : x² = 4a(y + a) के लंबकोणीय संहेडी, उसी निकाय में स्थित होते हैं। 10
(c) w भार का एक पिंड, θ कोण से झुके हुए एक रूक्ष समतल पर स्थित है, घर्षण गुणांक μ, tan θ से अधिक है। पिंड को समतल पर ऊपर की तरफ 'b' दूरी तक धीरे-धीरे खींचने तथा वापस आरंभिक बिंदु तक खींचने में किए गए कार्य को ज्ञात कीजिए, जहाँ लगाया गया बल प्रत्येक दशा में समतल के समांतर है। 10
(d) एक प्रक्षेप्य √2gh वेग के साथ बिंदु O से प्रक्षेपित किया गया तथा समतल के बिंदु P(x, y) पर स्पर्शरेखा से टकराता है जहाँ अक्ष OX तथा OY क्रमशः बिंदु O से क्षैतिज तथा अधोमुखी उर्ध्वाधर रेखाएँ हैं। यदि प्रक्षेपण की दो संभव दिशाएँ समकोण पर हों, तो दर्शाइए कि x² = 2hy तथा प्रक्षेपण की संभव दिशाओं में से एक, कोण POX को द्विभाजित करती है। 10
(e) दर्शाइए कि A⃗ = (6xy + z³)î + (3x² - z)ĵ + (3xz² - y)k̂ अघूर्णी है। φ को भी ज्ञात कीजिए जबकि A⃗ = ∇φ। 10
Answer approach & key points
(a) derive: given > assumptions > stepwise derivation > result > check | (b) derive: given > assumptions > stepwise derivation > result > check | (c) calculate: given > formula > substitution > result with units > interpretation | (d) derive: given > assumptions > stepwise derivation > result > check | (e) derive: given > assumptions > stepwise derivation > result > check Full marks: All parts fully derived with clear steps and verification.
Identify the equation as linear first-order ODE
Compute the integrating factor e^∫P dx
Integrate to find the general solution
Rearrange to match the required form
Differentiate the given equation to find dy/dx
Replace dy/dx with -dx/dy for orthogonal trajectories
Solve the resulting differential equation
Show the new equation is identical, so trajectories are members of x² = 4a(y + a) with another value of a
Resolve forces parallel and perpendicular to the plane
Calculate the applied force for upward motion
Calculate the applied force for downward motion
Total work W = 2μwb cosθ
Write the trajectory equation for the projectile
Use the condition that the two angles are at right angles
50MproveMechanics, differential equations and vector calculus
(a) A cable of weight w per unit length and length 2l hangs from two points P and Q in the same horizontal line. Show that the span of the cable is 2l(1 - (2h²)/(3l²)), where h is the sag in the middle of the tightly stretched position. 20
(b) Solve the following differential equation by using the method of variation of parameters : (x² - 1)(d^2y)/(dx²) - 2xdy/dx + 2y = (x² - 1)², given that y = x is one solution of the reduced equation. 15
(c) Verify Green's theorem in the plane for ∮_C (3x² - 8y²) dx + (4y - 6xy) dy, where C is the boundary curve of the region defined by x = 0, y = 0, x + y = 1. 15
हिंदी में पढ़ें
(a) 2l लम्बाई का एक तार (केबल) जिसका भार w प्रति इकाई (यूनिट) लम्बाई है, एक क्षैतिज रेखा के दो बिन्दुओं P तथा Q से लटकी हुई है। दर्शाइए कि तार की विस्तृति (स्पैन) 2l(1 - (2h²)/(3l²)) है, जहाँ h तार के कसकर खींची हुई स्थिति में मध्य का झोल है। 20
(b) प्राचल-विचरण विधि का उपयोग करके, निम्नलिखित अवकल समीकरण : (x² - 1)(d^2y)/(dx²) - 2xdy/dx + 2y = (x² - 1)² को हल कीजिए, जहाँ समानीत समीकरण का एक हल y = x दिया गया है। 15
(c) समतल में ग्रीन के प्रमेय को ∮_C (3x² - 8y²) dx + (4y - 6xy) dy के लिए सत्यापित कीजिए, जहाँ C, x = 0, y = 0, x + y = 1 द्वारा परिभाषित क्षेत्र का सीमा वक्र 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 derivations with all steps, correct final results, and clear diagrams.
State catenary equation y = c cosh(x/c)
Relate length 2l to span 2a and sag h
Apply Taylor expansion for cosh(x/c)
Solve S² − 2lS + 8h²/3 = 0 and expand to S ≈ 2l(1 − 2h²/(3l²))
Find second solution y2 using reduction of order
Compute Wronskian W(y1, y2)
Set up and solve integrals for u1, u2
General solution y = c₁x + c₂(x² + 1) + x⁴/6 − x²/2
50MverifyVector calculus, Laplace transforms and mechanics
(a) Verify Stokes' theorem for F⃗ = xî + z²ĵ + y²k̂ over the plane surface : x + y + z = 1 lying in the first octant. 20
(b) Solve the following initial value problem by using Laplace's transformation (d²y)/(dt²) - 3dy/dt + 2y = h(t), where h(t) = 2 for 0 < t < 4 and h(t) = 0 for t > 4, y(0) = 0, y′(0) = 0 15
(c) Suppose a cylinder of any cross-section is balanced on another fixed cylinder, the contact of curved surfaces being rough and the common tangent line horizontal. Let ρ and ρ′ be the radii of curvature of the two cylinders at the point of contact and h be the height of centre of gravity of the upper cylinder above the point of contact. Show that the upper cylinder is balanced in stable equilibrium if h < (ρρ′)/(ρ+ρ′). 15
हिंदी में पढ़ें
(a) स्टोक्स प्रमेय को F⃗ = xî + z²ĵ + y²k̂ के लिए प्रथम अष्टांशक में स्थित समतल पृष्ठ : x + y + z = 1 पर सत्यापित कीजिए। 20
(b) लाप्लास रूपांतरण का उपयोग करके निम्नलिखित प्रारंभिक मान समस्या : (d²y)/(dt²) - 3dy/dt + 2y = h(t), जहाँ h(t) = 2 (0 < t < 4) तथा h(t) = 0 (t > 4), y(0) = 0, y′(0) = 0 को हल कीजिए । 15
(c) माना किसी भी अनुप्रस्थ-काट का एक बेलन दूसरे स्थिर बेलन पर संतुलित है, जहाँ वक्रीय पृष्ठों का संपर्श रूक्ष है तथा उभयनिष्ठ स्पर्श-रेखा क्षैतिज है । माना दोनों बेलनों के स्पर्श बिंदु पर उनकी वक्रता त्रिज्याएं ρ तथा ρ′ हैं और संपर्श बिंदु से ऊपरी बेलन के गुरुत्व केंद्र की ऊँचाई h है । दर्शाइए कि स्थायी साम्य में ऊपरी बेलन संतुलित है यदि h < (ρρ′)/(ρ+ρ′) । 15
Answer approach & key points
Framework: UPSC Mathematics Paper 1. (a) derive: given > assumptions > stepwise derivation > result > check | (b) calculate: given > formula > substitution > result with units > interpretation | (c) justify: claim > 3-4 reasons > evidence > conclusion Full marks: Complete, rigorous derivations with all steps shown and correct final results.
Define boundary curve C in first octant
Compute line integral ∮ F · dr
Compute surface integral ∬ (∇ × F) · n̂ dS
Show both integrals equal 0
Apply Laplace transform to the differential equation
Represent h(t) using unit step functions
Solve for Y(s) in the s-domain
Inverse transform: y = (e^t − 1)² for t below 4 and e^(2t) − 2e^t + 2e^(t−4) − e^(2t−8) for t above 4
Define the geometry of the two cylinders
Express the height of the center of gravity
Rolling condition φ = ((ρ + ρ′)/ρ)α and second-order change in height of G
50MsolveDifferential equations, mechanics and vector calculus
(a) (i) Find the general and singular solutions of the differential equation : (x² - a²)p² - 2xyp + y² + a² = 0, where p = dy/dx. Also give the geometric relation between the general and singular solutions. 10
(ii) Solve the following differential equation : (3x + 2)²(d²y)/(dx²) + 5(3x + 2)dy/dx - 3y = x² + x + 1 10
(b) A chain of n equal uniform rods is smoothly jointed together and suspended from its one end A₁. A horizontal force P⃗ is applied to the other end Aₙ₊₁ of the chain. Find the inclinations of the rods to the downward vertical line in the equilibrium configuration. 15
(c) Using Gauss' divergence theorem, evaluate ∬_S F⃗·n⃗ dS, where F⃗ = xî - yĵ + (z²-1)k̂ and S is the cylinder formed by the surfaces z = 0, z = 1, x² + y² = 4. 15
हिंदी में पढ़ें
(a) (i) अवकल समीकरण : (x² - a²)p² - 2xyp + y² + a² = 0, जहाँ p = dy/dx, के व्यापक व विचित्र हलों को ज्ञात कीजिए । व्यापक व विचित्र हलों के बीच ज्यामितीय संबंध को भी दीजिए । 10
(ii) निम्नलिखित अवकल समीकरण को हल कीजिए : (3x + 2)²(d²y)/(dx²) + 5(3x + 2)dy/dx - 3y = x² + x + 1 10
(b) n बराबर एकसमान छड़ों की एक श्रृंखला एक-दूसरे के साथ चिकने रूप से जुड़ी हुई है तथा इसके एक सिरे A₁ से लटकी हुई है । एक क्षैतिज बल P⃗ श्रृंखला के दूसरे सिरे Aₙ₊₁ पर लगाया गया है । साम्य विन्यास में अधोमुखी उद्वाधर रेखा से छड़ों के झुकाव ज्ञात कीजिए । 15
(c) गॉस के अपसरण प्रमेय का उपयोग करके ∬_S F⃗·n⃗ dS का मान निकालिए, जहाँ F⃗ = xî - yĵ + (z²-1)k̂ तथा S, पृष्ठों z = 0, z = 1, x² + y² = 4 द्वारा बना हुआ बेलन है । 15
Answer approach & key points
(a(i)) derive: given > assumptions > stepwise derivation > result > check | (a(ii)) derive: given > assumptions > stepwise derivation > result > check | (b) derive: given > assumptions > stepwise derivation > result > check | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Rigorous derivation with all steps, correct final answers, and clear geometric/physical interpretation.
Reduce to Clairaut form y = xp ± a√(p² − 1)
General solution (y − cx)² = a²(c² − 1)
Singular solution y² = x² − a² via c-discriminant or p-discriminant
State geometric relation (envelope)
Apply substitution t = 3x + 2
Solve the resulting constant coefficient ODE
Find complementary function
Find particular integral
Free body diagram of the chain
Equilibrium of forces at each joint
Derive tanθᵢ = 2P/[(2n − 2i + 1)W] by virtual work or joint equilibrium