{ "schema": "mathmagic/1", "id": "batch-001", "title": "Mathematical Physics · Launch 100", "description": "100 original single-correct practice questions for IIT JAM / CUET-PG preparation. Not official past-paper questions or a complete syllabus mock.", "tests": [ { "id": "test-1", "title": "Test 1", "description": "25 mixed theory and numerical questions · untimed practice" }, { "id": "test-2", "title": "Test 2", "description": "25 mixed theory and numerical questions · untimed practice" }, { "id": "test-3", "title": "Test 3", "description": "25 mixed theory and numerical questions · untimed practice" }, { "id": "test-4", "title": "Test 4", "description": "25 mixed theory and numerical questions · untimed practice" } ], "questions": [ { "id": "mm1-001", "topic": 1, "test": "test-1", "type": "theory", "difficulty": "moderate", "question": { "en": "In $x=A\\sin(\\omega t)+Bt^2$, $x$ is a length. What are the dimensions of $A\\omega/B$?", "hi": "यदि $x=A\\sin(\\omega t)+Bt^2$ में $x$ लंबाई है, तो $A\\omega/B$ की विमा क्या है?" }, "options": [ "$LT$", "$T$", "$T^{-1}$", "$L$" ], "answer": 1, "solution": { "steps": [ "Sine का argument dimensionless होना चाहिए: $[\\omega]=T^{-1}$.", "दोनों terms length हैं, इसलिए $[A]=L$ और $[B]=LT^{-2}$.", "Ratio की dimension $L T^{-1}/(L T^{-2})=T$ है." ], "trap": "$B$ को velocity समझना गलत है; वह $t^2$ से multiply हो raha है." }, "note": { "topic": 1, "formula": "01.014" } }, { "id": "mm1-002", "topic": 1, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "A drag force has the form $F=k\\rho^a v^b r^c$, where $k$ is dimensionless. If $b=2$, what are $(a,c)$?", "hi": "ड्रैग बल $F=k\\rho^a v^b r^c$ में $k$ विमाहीन है। यदि $b=2$, तो $(a,c)$ क्या हैं?" }, "options": [ "$(1,2)$", "$(1,1)$", "$(2,1)$", "$(0,2)$" ], "answer": 0, "solution": { "steps": [ "Density, speed और radius की dimensions respectively $ML^{-3},LT^{-1},L$ हैं.", "RHS की dimension $M^aL^{-3a+b+c}T^{-b}$ है; force की $MLT^{-2}$.", "Mass से $a=1$; length से $-3+2+c=1$, इसलिए $c=2$." ], "trap": "Dimensional analysis numerical constant $k$ का value नहीं batata." }, "note": { "topic": 1, "formula": "01.005" } }, { "id": "mm1-003", "topic": 1, "test": "test-3", "type": "theory", "difficulty": "moderate", "question": { "en": "Which conclusion follows from dimensional consistency alone?", "hi": "केवल विमीय संगति से कौन-सा निष्कर्ष निकलता है?" }, "options": [ "The equation is physically correct", "Every numerical factor is correct", "The functional form is unique", "Every additive term has the same dimensions" ], "answer": 3, "solution": { "steps": [ "Addition/subtraction सिर्फ same-dimensional quantities के बीच valid है.", "लेकिन $s=ut+at^2$ और $s=ut+\\tfrac12at^2$ दोनों dimensionally consistent हैं.", "इसलिए consistency necessary check है, physical correctness का sufficient proof नहीं." ], "trap": "Dimensionless constants और boundary conditions dimensional check से fix नहीं होते." }, "note": { "topic": 1, "formula": "01.005" } }, { "id": "mm1-004", "topic": 3, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "For $R=V/I$, independent standard relative uncertainties in $V$ and $I$ are 3% and 4%. Find the standard relative uncertainty in $R$.", "hi": "$R=V/I$ में $V,I$ की स्वतंत्र मानक सापेक्ष अनिश्चितताएँ 3% और 4% हैं। $R$ की मानक सापेक्ष अनिश्चितता कितनी है?" }, "options": [ "1%", "25%", "5%", "7%" ], "answer": 2, "solution": { "steps": [ "यहाँ independent standard uncertainties di हैं, maximum-error bounds नहीं.", "Quotient के liye first-order propagation: $(\\sigma_R/R)^2=(\\sigma_V/V)^2+(\\sigma_I/I)^2$.", "इसलिए $\\sigma_R/R=\\sqrt{3^2+4^2}\\%=5\\%$." ], "trap": "7% worst-case bound होगा; standard uncertainty के liye quadrature use होती है.", "shortcut": "3–4–5 triangle yaad हो तो root turant मिलता है." }, "note": { "topic": 3, "formula": "03.017" } }, { "id": "mm1-005", "topic": 3, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "For $g=4\\pi^2L/T^2$, maximum relative errors in $L,T$ are 1% and 2%. Find the first-order maximum relative error in $g$.", "hi": "$g=4\\pi^2L/T^2$ में $L,T$ की अधिकतम सापेक्ष त्रुटियाँ 1% और 2% हैं। $g$ की प्रथम-क्रम अधिकतम सापेक्ष त्रुटि कितनी है?" }, "options": [ "1%", "5%", "$\\sqrt{17}$%", "3%" ], "answer": 1, "solution": { "steps": [ "Maximum-error estimate में absolute exponent contributions add करते हैं.", "$\\vert \\Delta g\\vert /g\\le \\vert \\Delta L\\vert /L+2\\vert \\Delta T\\vert /T$.", "Values substitute करें: $1\\%+2(2\\%)=5\\%$." ], "trap": "Quotient में error contributions subtract नहीं होते; worst case में magnitudes add होते हैं." }, "note": { "topic": 3, "formula": "03.013" } }, { "id": "mm1-006", "topic": 3, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "Independent measurements give $x=10\\pm0.2$, $y=5\\pm0.1$ (standard uncertainties). Find the standard uncertainty of $z=x-y$.", "hi": "स्वतंत्र माप $x=10\\pm0.2$, $y=5\\pm0.1$ हैं (मानक अनिश्चितताएँ)। $z=x-y$ की मानक अनिश्चितता क्या है?" }, "options": [ "$\\sqrt{0.05}\\approx0.224$", "0.1", "0.3", "0.05" ], "answer": 0, "solution": { "steps": [ "Derivatives $\\partial z/\\partial x=1$ और $\\partial z/\\partial y=-1$ हैं.", "Independent propagation में derivatives square होते हैं: $\\sigma_z^2=0.2^2+(-1)^2(0.1)^2=0.05$.", "इसलिए $z=5$ और $\\sigma_z=\\sqrt{0.05}\\approx0.224$, same units as $x$." ], "trap": "Subtraction से uncertainties cancel नहीं होती जब measurements independent hon." }, "note": { "topic": 3, "formula": "03.014" } }, { "id": "mm1-007", "topic": 3, "test": "test-3", "type": "theory", "difficulty": "challenging", "question": { "en": "Two equal-uncertainty measurements $x,y$ have correlation coefficient $+1$. What is the standard uncertainty of $x-y$ if both standard uncertainties are $\\sigma$?", "hi": "समान मानक अनिश्चितता $\\sigma$ वाले $x,y$ का सहसंबंध गुणांक $+1$ है। $x-y$ की मानक अनिश्चितता क्या है?" }, "options": [ "$\\sqrt2\\sigma$", "$2\\sigma$", "$\\sigma$", "0" ], "answer": 3, "solution": { "steps": [ "Correlated difference के liye $\\operatorname{Var}(x-y)=\\sigma_x^2+\\sigma_y^2-2\\rho\\sigma_x\\sigma_y$.", "Equal uncertainties और $\\rho=1$ रखने पर variance $2\\sigma^2-2\\sigma^2=0$.", "Perfectly shared fluctuation difference में cancel होती है; standard uncertainty zero है." ], "trap": "यह cancellation perfect correlation और equal uncertainty पर depend करती है; generally zero नहीं होती." }, "note": { "topic": 3, "formula": "03.016" } }, { "id": "mm1-008", "topic": 3, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "A gauge reads 2.38 mm and has a positive zero error of 0.04 mm. What is the corrected reading?", "hi": "गेज की रीडिंग 2.38 mm और धनात्मक शून्य त्रुटि 0.04 mm है। संशोधित रीडिंग क्या है?" }, "options": [ "2.38 mm", "2.30 mm", "2.34 mm", "2.42 mm" ], "answer": 2, "solution": { "steps": [ "Positive zero error का matlab instrument true zero पर भी extra positive reading देता है.", "Correction rule: true reading = observed reading − zero error.", "$2.38-0.04=2.34$ mm." ], "trap": "Zero correction का sign, zero error के opposite होता है." }, "note": { "topic": 3, "formula": "03.001" } }, { "id": "mm1-009", "topic": 2, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "Four independent readings each have standard deviation $s$. How many readings are needed to halve the standard error relative to these four readings?", "hi": "चार स्वतंत्र रीडिंग में प्रत्येक का मानक विचलन $s$ है। इन चार की तुलना में mean की standard error आधी करने के लिए कुल कितनी रीडिंग चाहिए?" }, "options": [ "32", "16", "8", "4" ], "answer": 1, "solution": { "steps": [ "Independent equal-variance observations के mean की standard error $s/\\sqrt n$ है.", "Four readings पर error $s/2$; target इसका half, यानी $s/4$.", "$s/\\sqrt n=s/4$ से $n=16$ total readings चाहिए." ], "trap": "16 total हैं; already 4 li हैं तो additional 12 चाहिए.", "shortcut": "Error half करने के liye sample size four times करें." }, "note": { "topic": 2, "formula": "02.006" } }, { "id": "mm1-010", "topic": 2, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "For the population data $1,2,3$, what are population variance and unbiased sample variance, respectively?", "hi": "डेटा $1,2,3$ के लिए population variance और unbiased sample variance क्रमशः क्या हैं?" }, "options": [ "$2/3,1$", "$1,2/3$", "$2,1$", "$1/3,1/2$" ], "answer": 0, "solution": { "steps": [ "Mean $\\bar x=2$ है; squared deviations $1,0,1$ का sum 2 है.", "Population variance denominator $N=3$: $\\sigma^2=2/3$.", "Unbiased sample estimator denominator $N-1=2$: $s^2=2/2=1$." ], "trap": "Population और sample variance के denominators अलग हैं; units दोनों के squared units हैं." }, "note": { "topic": 2, "formula": "02.003" } }, { "id": "mm1-011", "topic": 9, "test": "test-3", "type": "numerical", "difficulty": "moderate", "question": { "en": "For $\\mathbf a=(1,2,2)$ and $\\mathbf b=(2,1,-2)$, find the angle between them.", "hi": "सदिश $\\mathbf a=(1,2,2)$ और $\\mathbf b=(2,1,-2)$ के बीच कोण ज्ञात करें।" }, "options": [ "$0^\\circ$", "$60^\\circ$", "$180^\\circ$", "$90^\\circ$" ], "answer": 3, "solution": { "steps": [ "Dot product $\\mathbf a\\cdot\\mathbf b=2+2-4=0$.", "दोनों vectors nonzero हैं, इसलिए $\\cos\\theta=(\\mathbf a\\cdot\\mathbf b)/(\\vert \\mathbf a\\vert \\vert \\mathbf b\\vert )=0$.", "Angle का standard range $0\\le\\theta\\le\\pi$ है, so $\\theta=90^\\circ$." ], "trap": "Negative z components को ignore करने से orthogonality miss हो जाएगी." }, "note": { "topic": 9, "formula": "09.004" } }, { "id": "mm1-012", "topic": 9, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "Find the vector projection of $(2,1,2)$ onto $(1,0,1)$.", "hi": "$(2,1,2)$ का $(1,0,1)$ पर सदिश प्रक्षेप क्या है?" }, "options": [ "$(4,0,4)$", "$(1,0,1)$", "$(2,0,2)$", "$(\\sqrt2,0,\\sqrt2)$" ], "answer": 2, "solution": { "steps": [ "Vector projection formula $(\\mathbf a\\cdot\\mathbf b)\\mathbf b/(\\mathbf b\\cdot\\mathbf b)$ use करें.", "यहाँ $\\mathbf a\\cdot\\mathbf b=4$ और $\\mathbf b\\cdot\\mathbf b=2$.", "Projection $=2(1,0,1)=(2,0,2)$; remaining vector $(0,1,0)$, $\\mathbf b$ के perpendicular है." ], "trap": "Scalar projection $2\\sqrt2$ एक number है; question vector projection माँगता है." }, "note": { "topic": 9, "formula": "09.007" } }, { "id": "mm1-013", "topic": 9, "test": "test-1", "type": "theory", "difficulty": "challenging", "question": { "en": "For nonzero real vectors, when does $\\vert \\mathbf a\\cdot\\mathbf b\\vert =\\vert \\mathbf a\\vert \\vert \\mathbf b\\vert $ hold?", "hi": "अशून्य वास्तविक सदिशों के लिए $\\vert \\mathbf a\\cdot\\mathbf b\\vert =\\vert \\mathbf a\\vert \\vert \\mathbf b\\vert $ कब होता है?" }, "options": [ "For every pair of vectors", "The vectors are parallel or antiparallel", "The vectors are perpendicular", "Only when the vectors are equal" ], "answer": 1, "solution": { "steps": [ "Dot product identity से LHS $=\\vert \\mathbf a\\vert \\vert \\mathbf b\\vert \\vert \\cos\\theta\\vert $.", "Equality तभी जब $\\vert \\cos\\theta\\vert =1$, यानी $\\theta=0$ या $\\pi$.", "इसलिए vectors linearly dependent, parallel या antiparallel हो सकते हैं; magnitudes equal होना ज़रूरी नहीं." ], "trap": "Absolute value की wajah से antiparallel case भी include होता है." }, "note": { "topic": 9, "formula": "09.005" } }, { "id": "mm1-014", "topic": 10, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "If $\\vert \\mathbf a\\vert =3$, $\\vert \\mathbf b\\vert =4$ and their angle is $30^\\circ$, find the area of the triangle formed by them.", "hi": "$\\vert \\mathbf a\\vert =3$, $\\vert \\mathbf b\\vert =4$ और बीच का कोण $30^\\circ$ है। इनसे बने त्रिभुज का क्षेत्रफल क्या है?" }, "options": [ "3", "6", "12", "$3\\sqrt3$" ], "answer": 0, "solution": { "steps": [ "Cross-product magnitude parallelogram का area देता है: $ab\\sin\\theta$.", "यहाँ parallelogram area $3\\times4\\times\\tfrac12=6$.", "Triangle isi का half है, so area $=3$ square units." ], "trap": "$\\vert \\mathbf a\\times\\mathbf b\\vert $ triangle नहीं, parallelogram का area है." }, "note": { "topic": 10, "formula": "10.010" } }, { "id": "mm1-015", "topic": 10, "test": "test-3", "type": "numerical", "difficulty": "moderate", "question": { "en": "A force $(0,3,0)$ N acts at position $(2,0,0)$ m. Find the torque about the origin.", "hi": "स्थिति $(2,0,0)$ m पर बल $(0,3,0)$ N लगता है। मूल बिंदु के सापेक्ष बलाघूर्ण क्या है?" }, "options": [ "$(0,0,-6)$ N m", "$(6,0,0)$ N m", "0", "$(0,0,6)$ N m" ], "answer": 3, "solution": { "steps": [ "Torque की order fixed है: $\\boldsymbol\\tau=\\mathbf r\\times\\mathbf F$.", "$2\\hat i\\times3\\hat j=6\\hat k$, क्योंकि $\\hat i\\times\\hat j=\\hat k$.", "इसलिए $\\boldsymbol\\tau=(0,0,6)$ N m; right-hand rule positive z देता है." ], "trap": "$\\mathbf F\\times\\mathbf r$ लेने से sign reverse हो जाएगा." }, "note": { "topic": 10, "formula": "10.013" } }, { "id": "mm1-016", "topic": 11, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "Find the volume of the parallelepiped with edges $(1,0,1),(0,2,0),(1,0,-1)$.", "hi": "किनारे $(1,0,1),(0,2,0),(1,0,-1)$ वाले समांतरषट्फलक का आयतन कितना है?" }, "options": [ "2", "0", "4", "$-4$" ], "answer": 2, "solution": { "steps": [ "Volume scalar triple product का absolute value है.", "Second cross third $=(0,2,0)\\times(1,0,-1)=(-2,0,-2)$.", "First vector से dot $=-4$; volume $\\vert -4\\vert =4$." ], "trap": "Signed triple product orientation dikhata है; geometric volume negative नहीं हो सकता." }, "note": { "topic": 11, "formula": "11.004" } }, { "id": "mm1-017", "topic": 11, "test": "test-1", "type": "theory", "difficulty": "challenging", "question": { "en": "For arbitrary vectors, which expression equals $\\mathbf a\\times(\\mathbf b\\times\\mathbf c)$?", "hi": "मनमाने सदिशों के लिए $\\mathbf a\\times(\\mathbf b\\times\\mathbf c)$ किसके बराबर है?" }, "options": [ "$\\mathbf a(\\mathbf b\\cdot\\mathbf c)$", "$\\mathbf b(\\mathbf a\\cdot\\mathbf c)-\\mathbf c(\\mathbf a\\cdot\\mathbf b)$", "$\\mathbf c(\\mathbf a\\cdot\\mathbf b)-\\mathbf b(\\mathbf a\\cdot\\mathbf c)$", "$(\\mathbf a\\times\\mathbf b)\\times\\mathbf c$" ], "answer": 1, "solution": { "steps": [ "Vector triple-product identity BAC–CAB use होती है.", "Inner pair के first vector $\\mathbf b$ को $\\mathbf a\\cdot\\mathbf c$ से multiply करें.", "फिर $\\mathbf c(\\mathbf a\\cdot\\mathbf b)$ subtract करें; result $\\mathbf b,\\mathbf c$ के plane में होता है." ], "trap": "Cross product associative नहीं है; brackets change करना valid नहीं." }, "note": { "topic": 11, "formula": "11.007" } }, { "id": "mm1-018", "topic": 7, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "Two nonzero vectors satisfy $\\vert \\mathbf a+\\mathbf b\\vert =\\vert \\mathbf a-\\mathbf b\\vert $. Find their angle.", "hi": "दो अशून्य सदिशों के लिए $\\vert \\mathbf a+\\mathbf b\\vert =\\vert \\mathbf a-\\mathbf b\\vert $ है। उनका कोण क्या है?" }, "options": [ "$90^\\circ$", "$0^\\circ$", "$45^\\circ$", "$180^\\circ$" ], "answer": 0, "solution": { "steps": [ "दोनों sides square करें: $a^2+b^2+2\\mathbf a\\cdot\\mathbf b=a^2+b^2-2\\mathbf a\\cdot\\mathbf b$.", "Common terms cancel करने पर $4\\mathbf a\\cdot\\mathbf b=0$.", "Nonzero vectors का dot zero हो तो angle $90^\\circ$ होता है." ], "trap": "Equal magnitudes of sum/difference का matlab original vectors equal नहीं है." }, "note": { "topic": 7, "formula": "07.021" } }, { "id": "mm1-019", "topic": 12, "test": "test-3", "type": "theory", "difficulty": "challenging", "question": { "en": "Under spatial inversion $\\mathbf r\\to-\\mathbf r$, which quantity remains unchanged?", "hi": "स्थानिक उत्क्रमण $\\mathbf r\\to-\\mathbf r$ में कौन-सी राशि अपरिवर्तित रहती है?" }, "options": [ "Position $\\mathbf r$", "Linear momentum $\\mathbf p$", "Force $\\mathbf F$", "Angular momentum $\\mathbf r\\times\\mathbf p$" ], "answer": 3, "solution": { "steps": [ "Position और momentum polar vectors हैं, inversion पर दोनों sign बदलते हैं.", "Angular momentum $\\mathbf L=\\mathbf r\\times\\mathbf p$ transforms as $(-\\mathbf r)\\times(-\\mathbf p)$.", "दो minus cancel होते हैं, इसलिए $\\mathbf L$ unchanged; यह axial vector है." ], "trap": "Axial और polar vectors proper rotations में similar, inversion में different behave करते हैं." }, "note": { "topic": 12 } }, { "id": "mm1-020", "topic": 8, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "A unit vector makes equal acute angles with all three positive Cartesian axes. What is its x component?", "hi": "एक इकाई सदिश तीनों धनात्मक Cartesian अक्षों से समान न्यून कोण बनाता है। उसका x घटक क्या है?" }, "options": [ "$1/\\sqrt2$", "1", "$1/\\sqrt3$", "$1/3$" ], "answer": 2, "solution": { "steps": [ "Direction cosines $l,m,n$ equal और acute-angle condition से positive हैं.", "Unit-vector relation $l^2+m^2+n^2=1$ से $3l^2=1$.", "Positive root लें: $l=1/\\sqrt3$, जो x component भी है." ], "trap": "Components का sum 1 नहीं, squares का sum 1 होता है." }, "note": { "topic": 8, "formula": "08.015" } }, { "id": "mm1-021", "topic": 13, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "Find the principal argument of $z=-1+i$.", "hi": "$z=-1+i$ का principal argument ज्ञात करें।" }, "options": [ "$-3\\pi/4$", "$3\\pi/4$", "$-\\pi/4$", "$\\pi/4$" ], "answer": 1, "solution": { "steps": [ "Real part negative और imaginary part positive है, so point second quadrant में है.", "Reference angle $\\tan^{-1}\\vert 1/(-1)\\vert =\\pi/4$.", "Principal argument $\\pi-\\pi/4=3\\pi/4$, within $(-\\pi,\\pi]$." ], "trap": "सिर्फ $\\tan^{-1}(y/x)$ लेने से quadrant information kho सकती है." }, "note": { "topic": 13, "formula": "13.015" } }, { "id": "mm1-022", "topic": 13, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "Evaluate $(1+i)^8$.", "hi": "$(1+i)^8$ का मान ज्ञात करें।" }, "options": [ "16", "$-16$", "$16i$", "$-16i$" ], "answer": 0, "solution": { "steps": [ "Polar form $1+i=\\sqrt2 e^{i\\pi/4}$ है.", "Power लगाएँ: $(\\sqrt2)^8e^{i2\\pi}=16e^{i2\\pi}$.", "$e^{i2\\pi}=1$, इसलिए answer 16 real positive है." ], "trap": "Modulus भी power होता है; सिर्फ phase को 8 से multiply करना incomplete है.", "shortcut": "$(1+i)^2=2i$; square twice और करने से 16 मिलता है." }, "note": { "topic": 13, "formula": "13.033" } }, { "id": "mm1-023", "topic": 13, "test": "test-3", "type": "numerical", "difficulty": "challenging", "question": { "en": "Two coherent phasors are $3e^{i0}$ and $4e^{i\\pi/2}$. Find the modulus of their sum.", "hi": "दो coherent phasors $3e^{i0}$ और $4e^{i\\pi/2}$ हैं। इनके योग का modulus क्या है?" }, "options": [ "7", "1", "$\\sqrt7$", "5" ], "answer": 3, "solution": { "steps": [ "पहला phasor real axis पर 3, दूसरा imaginary axis पर $4i$ है.", "Sum $z=3+4i$; amplitudes directly add नहीं होती क्योंकि phases different हैं.", "Resultant modulus $\\vert z\\vert =\\sqrt{3^2+4^2}=5$." ], "trap": "7 सिर्फ इन-phase phasors के liye मिलता." }, "note": { "topic": 13, "formula": "13.030" } }, { "id": "mm1-024", "topic": 13, "test": "test-4", "type": "theory", "difficulty": "challenging", "question": { "en": "Which set contains all cube roots of unity?", "hi": "निम्न में से कौन-सा समुच्चय 1 के सभी घनमूल देता है?" }, "options": [ "$\\{1,e^{\\pi i/3},e^{2\\pi i/3}\\}$", "$\\{1,1,1\\}$", "$\\{1,e^{2\\pi i/3},e^{4\\pi i/3}\\}$", "$\\{1,-1,i\\}$" ], "answer": 2, "solution": { "steps": [ "Unity का general argument $2\\pi k$ है.", "Cube root के phases $2\\pi k/3$ लें, with $k=0,1,2$; उसके baad roots repeat होते हैं.", "तीनों roots unit circle पर equally spaced हैं और cube करने पर 1 देते हैं." ], "trap": "Principal root सिर्फ 1 है, लेकिन all roots में तीन distinct values चाहिए." }, "note": { "topic": 13, "formula": "13.036" } }, { "id": "mm1-025", "topic": 13, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "Simplify $(1+i)/(1-i)$.", "hi": "$(1+i)/(1-i)$ को सरल करें।" }, "options": [ "$1+i$", "$i$", "1", "$-i$" ], "answer": 1, "solution": { "steps": [ "Denominator के conjugate $1+i$ से numerator और denominator multiply करें.", "Numerator $(1+i)^2=2i$ और denominator $(1-i)(1+i)=2$.", "Ratio $2i/2=i$." ], "trap": "Denominator का modulus-square 2 है; terms को individually divide नहीं कर सकते." }, "note": { "topic": 13, "formula": "13.008" }, "svg": "1 + i1iReIm0" }, { "id": "mm1-026", "topic": 14, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "Find $\\sum_{n=0}^{\\infty}(-1/3)^n$.", "hi": "$\\sum_{n=0}^{\\infty}(-1/3)^n$ का योग ज्ञात करें।" }, "options": [ "$3/4$", "$3/2$", "$1/4$", "$4/3$" ], "answer": 0, "solution": { "steps": [ "Geometric series का first term 1 और common ratio $r=-1/3$ है.", "$\\vert r\\vert <1$, इसलिए infinite sum formula valid है.", "$S=1/(1-r)=1/(1+1/3)=3/4$." ], "trap": "Negative ratio को denominator में subtract करते waqt sign ध्यान से लें." }, "note": { "topic": 14, "formula": "14.017" } }, { "id": "mm1-027", "topic": 14, "test": "test-3", "type": "theory", "difficulty": "challenging", "question": { "en": "The series $\\sum_{n=1}^{\\infty}(-1)^{n+1}/n$ is:", "hi": "श्रेणी $\\sum_{n=1}^{\\infty}(-1)^{n+1}/n$ किस प्रकार की है?" }, "options": [ "Absolutely convergent", "Divergent because terms alternate", "Finite after some terms", "Conditionally convergent" ], "answer": 3, "solution": { "steps": [ "Magnitudes $1/n$ monotonically zero की तरफ जाते हैं, so alternating-series test convergence देता है.", "Absolute values की series $\\sum1/n$ harmonic series है, जो diverge करती है.", "Original convergent और absolute series divergent हो तो conditional convergence होती है." ], "trap": "Terms का zero होना alone sufficient नहीं; यहाँ alternating test use hua है." }, "note": { "topic": 14, "formula": "14.042" } }, { "id": "mm1-028", "topic": 14, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "How many terms of $1-1/2+1/3-\\cdots$ guarantee an absolute truncation error below 0.01 by the alternating-series bound?", "hi": "$1-1/2+1/3-\\cdots$ में alternating-series bound से त्रुटि 0.01 से कम सुनिश्चित करने हेतु न्यूनतम कितने पद चाहिए?" }, "options": [ "10", "101", "100", "99" ], "answer": 2, "solution": { "steps": [ "$N$ terms के baad remainder bound $\\vert R_N\\vert \\le1/(N+1)$ है.", "Strictly below 0.01 guarantee के liye $1/(N+1)<1/100$ चाहिए.", "$N+1>100$, so minimum integer $N=100$." ], "trap": "99 terms का bound exactly 0.01 है, below 0.01 नहीं." }, "note": { "topic": 14, "formula": "14.042" } }, { "id": "mm1-029", "topic": 14, "test": "test-1", "type": "theory", "difficulty": "moderate", "question": { "en": "For what real $x$ does $\\sum_{n=0}^{\\infty}(x/2)^n$ converge?", "hi": "वास्तविक $x$ के किन मानों के लिए $\\sum_{n=0}^{\\infty}(x/2)^n$ अभिसरित होती है?" }, "options": [ "All real $x$", "$-20$, find $y'(1)$.", "hi": "$x>0$ के लिए $y=x^x$ है। $y'(1)$ क्या है?" }, "options": [ "$e$", "2", "1", "0" ], "answer": 2, "solution": { "steps": [ "Variable base और exponent के liye logarithmic differentiation use करें: $\\ln y=x\\ln x$.", "Differentiate: $y'/y=\\ln x+1$, इसलिए $y'=x^x(\\ln x+1)$.", "$x=1$ पर $y'=1(0+1)=1$." ], "trap": "Ordinary power rule $nx^{n-1}$ constant exponent के liye है." }, "note": { "topic": 15, "formula": "15.028" } }, { "id": "mm1-033", "topic": 15, "test": "test-1", "type": "theory", "difficulty": "moderate", "question": { "en": "Which statement about $f(x)=\\vert x\\vert $ at $x=0$ is correct?", "hi": "$x=0$ पर $f(x)=\\vert x\\vert $ के लिए कौन-सा कथन सही है?" }, "options": [ "Derivative equals zero", "Continuous but not differentiable", "Differentiable but not continuous", "Both discontinuous and nondifferentiable" ], "answer": 1, "solution": { "steps": [ "Left और right limits दोनों 0 हैं, और $f(0)=0$, so function continuous है.", "Left slope $-1$ और right slope $+1$ है.", "One-sided derivatives unequal हैं, इसलिए derivative at zero exist नहीं करता." ], "trap": "Continuity differentiability guarantee नहीं करती; corner इसका standard example है." }, "note": { "topic": 15, "formula": "15.019" } }, { "id": "mm1-034", "topic": 15, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $f(x)=xe^{-x}$ on $x\\ge0$, find the location of its global maximum.", "hi": "$x\\ge0$ पर $f(x)=xe^{-x}$ का वैश्विक अधिकतम किस $x$ पर है?" }, "options": [ "1", "0", "2", "There is no finite maximum" ], "answer": 0, "solution": { "steps": [ "Product rule से $f'(x)=e^{-x}(1-x)$.", "Exponential positive है, so derivative $x<1$ पर positive और $x>1$ पर negative है.", "$f(0)=0$ और $f\\to0$ as $x\\to\\infty$; इसलिए global maximum at $x=1$." ], "trap": "सिर्फ stationary point nikalna enough नहीं; sign change और domain boundaries भी check kiye." }, "note": { "topic": 15, "formula": "15.034" } }, { "id": "mm1-035", "topic": 16, "test": "test-3", "type": "numerical", "difficulty": "moderate", "question": { "en": "Evaluate $\\lim_{x\\to0}(1-\\cos x)/x^2$.", "hi": "$\\lim_{x\\to0}(1-\\cos x)/x^2$ का मान क्या है?" }, "options": [ "0", "1", "Infinity", "$1/2$" ], "answer": 3, "solution": { "steps": [ "Small x पर $\\cos x=1-x^2/2+O(x^4)$.", "इसलिए numerator $1-\\cos x=x^2/2+O(x^4)$.", "$x^2$ से divide करके limit $1/2$ मिलती है." ], "trap": "Direct substitution से $0/0$ आता है; इसका matlab answer zero नहीं." }, "note": { "topic": 16, "formula": "16.024" } }, { "id": "mm1-036", "topic": 16, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "For $\\vert x\\vert \\ll1$, what is $(1+x)^{-1/2}$ through order $x^2$?", "hi": "$\\vert x\\vert \\ll1$ में $(1+x)^{-1/2}$ का $x^2$ तक विस्तार क्या है?" }, "options": [ "$1+x/2-3x^2/8$", "$1-x+ x^2$", "$1-x/2+3x^2/8$", "$1-x/2+x^2/8$" ], "answer": 2, "solution": { "steps": [ "Binomial expansion $(1+x)^a=1+ax+a(a-1)x^2/2+O(x^3)$ है.", "$a=-1/2$ put करने पर linear coefficient $-1/2$ मिलता है.", "Quadratic coefficient $(-1/2)(-3/2)/2=3/8$, इसलिए required series." ], "trap": "Quadratic coefficient में factorial $2!$ को miss करना common error है." }, "note": { "topic": 16, "formula": "16.040" } }, { "id": "mm1-037", "topic": 16, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "Using the quadratic Taylor polynomial at zero, estimate $e^{0.1}$.", "hi": "शून्य पर द्विघात Taylor बहुपद से $e^{0.1}$ का अनुमान लगाएँ।" }, "options": [ "1.005", "1.105", "1.1", "1.11" ], "answer": 1, "solution": { "steps": [ "Exponential का Taylor polynomial $P_2(x)=1+x+x^2/2$ है.", "At $x=0.1$, quadratic contribution $0.01/2=0.005$.", "Sum $1+0.1+0.005=1.105$; omitted terms $O(0.1^3)$ हैं." ], "trap": "Quadratic approximation exact exponential value नहीं; next positive term about 0.000167 है." }, "note": { "topic": 16, "formula": "16.016" } }, { "id": "mm1-038", "topic": 15, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "If $y=\\ln(\\sin x)$ for $00$, so real logarithm valid है.", "At $\\pi/4$, sine और cosine equal हैं; derivative 1 है." ], "trap": "Outer log derivative के साथ inner sine derivative भी multiply करना है." }, "note": { "topic": 15, "formula": "15.025" } }, { "id": "mm1-039", "topic": 15, "test": "test-3", "type": "numerical", "difficulty": "challenging", "question": { "en": "A curve obeys $x^2+xy+y^2=3$. Find $dy/dx$ at $(1,1)$.", "hi": "वक्र $x^2+xy+y^2=3$ पर $(1,1)$ में $dy/dx$ ज्ञात करें।" }, "options": [ "1", "$-2$", "0", "$-1$" ], "answer": 3, "solution": { "steps": [ "Implicit differentiation में $y=y(x)$ रखें: $2x+y+xy'+2yy'=0$.", "Collect terms: $(x+2y)y'=-(2x+y)$.", "At $(1,1)$, $y'=-3/3=-1$." ], "trap": "$xy$ के derivative में product rule से $y+xy'$ दोनों terms आते हैं." }, "note": { "topic": 15 } }, { "id": "mm1-040", "topic": 16, "test": "test-4", "type": "theory", "difficulty": "challenging", "question": { "en": "For $\\sin x\\approx x$, what is the leading absolute error order as $x\\to0$?", "hi": "$\\sin x\\approx x$ में $x\\to0$ पर absolute error का leading order क्या है?" }, "options": [ "$O(\\vert x\\vert )$", "$O(x^4)$", "$O(\\vert x\\vert ^3)$", "$O(x^2)$" ], "answer": 2, "solution": { "steps": [ "Taylor expansion $\\sin x=x-x^3/6+O(x^5)$ है.", "Linear approximation subtract करने पर error $-x^3/6+O(x^5)$ बचता है.", "Magnitude leading order $\\vert x\\vert ^3/6$, so absolute error $O(\\vert x\\vert ^3)$ है." ], "trap": "Relative error about $x^2/6$ है, लेकिन question absolute error का है." }, "note": { "topic": 16, "formula": "16.046" } }, { "id": "mm1-041", "topic": 17, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "Evaluate $\\int_0^1 x^2\\,dx$.", "hi": "$\\int_0^1 x^2\\,dx$ का मान ज्ञात करें।" }, "options": [ "3", "$1/3$", "$1/2$", "1" ], "answer": 1, "solution": { "steps": [ "Power integration rule $\\int x^n dx=x^{n+1}/(n+1)$ applies for $n\\ne-1$.", "Antiderivative $x^3/3$ है.", "Bounds लगाएँ: $[x^3/3]_0^1=1/3$." ], "trap": "Derivative rule को reverse करते waqt new exponent से divide करना होता है." }, "note": { "topic": 17, "formula": "17.004" } }, { "id": "mm1-042", "topic": 18, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "Evaluate $\\int_0^1 xe^{x^2}\\,dx$.", "hi": "$\\int_0^1 xe^{x^2}\\,dx$ का मान क्या है?" }, "options": [ "$(e-1)/2$", "$e-1$", "$e/2$", "$(e^2-1)/2$" ], "answer": 0, "solution": { "steps": [ "Exponent का derivative factor available है; substitute $u=x^2$, so $du=2x\\,dx$.", "Bounds $x=0,1$ से $u=0,1$ होते हैं.", "Integral $\\tfrac12\\int_0^1e^u du=(e-1)/2$." ], "trap": "Substitution में factor 1/2 और transformed bounds दोनों maintain करें." }, "note": { "topic": 18, "formula": "18.001" } }, { "id": "mm1-043", "topic": 18, "test": "test-3", "type": "numerical", "difficulty": "moderate", "question": { "en": "Evaluate $\\int_0^{\\pi}x\\sin x\\,dx$.", "hi": "$\\int_0^{\\pi}x\\sin x\\,dx$ ज्ञात करें।" }, "options": [ "0", "$2\\pi$", "2", "$\\pi$" ], "answer": 3, "solution": { "steps": [ "Integration by parts में $u=x$, $dv=\\sin x dx$ लें; then $du=dx$, $v=-\\cos x$.", "Antiderivative $-x\\cos x+\\sin x$ है.", "At $\\pi$ value $\\pi$ और at zero 0; answer $\\pi$." ], "trap": "Endpoint $\\cos\\pi=-1$ का sign answer decide करता है." }, "note": { "topic": 18, "formula": "18.003" } }, { "id": "mm1-044", "topic": 17, "test": "test-4", "type": "theory", "difficulty": "moderate", "question": { "en": "If $f(x)$ is odd and integrable on $[-a,a]$, what is $\\int_{-a}^{a}f(x)dx$?", "hi": "यदि $f(x)$ विषम और $[-a,a]$ पर समाकलनीय है, तो $\\int_{-a}^{a}f(x)dx$ क्या है?" }, "options": [ "$\\int_0^a f(x)dx$", "Always positive", "0", "$2\\int_0^a f(x)dx$" ], "answer": 2, "solution": { "steps": [ "Odd symmetry का matlab $f(-x)=-f(x)$.", "Negative half के substitution $u=-x$ से उसका contribution positive half के opposite मिलता है.", "Symmetric limits पर दोनों cancel, so integral zero है." ], "trap": "Integrability ज़रूरी है; singular nonintegrable functions को blindly odd-symmetry rule मत लगाएँ." }, "note": { "topic": 17 } }, { "id": "mm1-045", "topic": 18, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $a>0$, evaluate $\\int_0^\\infty e^{-ax}\\,dx$.", "hi": "$a>0$ के लिए $\\int_0^\\infty e^{-ax}\\,dx$ का मान क्या है?" }, "options": [ "Diverges", "$1/a$", "$a$", "0" ], "answer": 1, "solution": { "steps": [ "पहले finite upper limit $R$ लें: integral $=[-e^{-ax}/a]_0^R=(1-e^{-aR})/a$.", "$a>0$ होने से $R\\to\\infty$ पर exponential zero होता है.", "Limit $1/a$ है; convergence positivity condition पर depend करती है." ], "trap": "Upper limit term zero है, लेकिन lower limit term को भूलने से wrong answer 0 मिलेगा." }, "note": { "topic": 18, "formula": "18.015" } }, { "id": "mm1-046", "topic": 18, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "Evaluate $\\int_0^\\infty xe^{-2x}\\,dx$.", "hi": "$\\int_0^\\infty xe^{-2x}\\,dx$ का मान ज्ञात करें।" }, "options": [ "$1/4$", "$1/2$", "1", "2" ], "answer": 0, "solution": { "steps": [ "Parts लें: $u=x$, $dv=e^{-2x}dx$, giving $v=-e^{-2x}/2$.", "Boundary term $[-xe^{-2x}/2]_0^\\infty=0$, क्योंकि exponential polynomial से faster decay करता है.", "Remaining $\\tfrac12\\int_0^\\infty e^{-2x}dx=\\tfrac12\\times\\tfrac12=1/4$." ], "trap": "Decay constant 2 दोनों integrations में factor देता है.", "shortcut": "General formula $\\int_0^\\infty xe^{-ax}dx=1/a^2$ for $a>0$." }, "note": { "topic": 18, "formula": "18.017" } }, { "id": "mm1-047", "topic": 18, "test": "test-3", "type": "numerical", "difficulty": "challenging", "question": { "en": "Evaluate $\\int_0^1\\ln x\\,dx$ as an improper integral.", "hi": "अनुचित समाकल $\\int_0^1\\ln x\\,dx$ का मान ज्ञात करें।" }, "options": [ "0", "1", "Diverges to $-\\infty$", "$-1$" ], "answer": 3, "solution": { "steps": [ "Lower endpoint singular है, इसलिए $\\epsilon>0$ से start करके limit लें.", "Parts से antiderivative $x\\ln x-x$ मिलता है.", "At 1 value $-1$; as $\\epsilon\\to0^+$, $\\epsilon\\ln\\epsilon\\to0$ और $\\epsilon\\to0$. इसलिए integral $-1$." ], "trap": "Integrand unbounded होने का matlab integral automatically divergent नहीं." }, "note": { "topic": 18, "formula": "18.010" } }, { "id": "mm1-048", "topic": 17, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "If $F(x)=\\int_0^{x^2}\\cos t\\,dt$, find $F'(x)$.", "hi": "यदि $F(x)=\\int_0^{x^2}\\cos t\\,dt$, तो $F'(x)$ क्या है?" }, "options": [ "$2x\\sin(x^2)$", "$\\cos x$", "$2x\\cos(x^2)$", "$\\cos(x^2)$" ], "answer": 2, "solution": { "steps": [ "Fundamental theorem upper limit के integrand को evaluate करता है.", "Upper limit $g(x)=x^2$ variable है, so chain rule factor $g'(x)=2x$ lagega.", "Thus $F'(x)=\\cos(x^2)\\cdot2x$." ], "trap": "Upper limit का derivative miss करना common trap है." }, "note": { "topic": 17 } }, { "id": "mm1-049", "topic": 18, "test": "test-1", "type": "theory", "difficulty": "challenging", "question": { "en": "For which real $p$ does $\\int_1^\\infty x^{-p}dx$ converge?", "hi": "किन वास्तविक $p$ के लिए $\\int_1^\\infty x^{-p}dx$ अभिसरित होता है?" }, "options": [ "$p<1$", "$p>1$", "$p\\ge1$", "$p>0$" ], "answer": 1, "solution": { "steps": [ "For $p\\ne1$, finite integral $[x^{1-p}/(1-p)]_1^R$ है.", "$R\\to\\infty$ पर upper power zero तभी जब $1-p<0$, यानी $p>1$.", "At $p=1$, integral $\\ln R$ diverge करता है; endpoint exclude है." ], "trap": "Integrand zero की तरफ जाए, फिर भी integral diverge हो सकता है; $1/x$ example है." }, "note": { "topic": 18, "formula": "18.013" } }, { "id": "mm1-050", "topic": 18, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "Evaluate $\\int_{-\\infty}^{\\infty}e^{-2x^2}dx$.", "hi": "$\\int_{-\\infty}^{\\infty}e^{-2x^2}dx$ का मान ज्ञात करें।" }, "options": [ "$\\sqrt{\\pi/2}$", "$\\sqrt\\pi/2$", "$\\sqrt{2\\pi}$", "$\\pi/2$" ], "answer": 0, "solution": { "steps": [ "Standard Gaussian result $\\int_{-\\infty}^{\\infty}e^{-u^2}du=\\sqrt\\pi$ use करें.", "$u=\\sqrt2x$ लें, so $dx=du/\\sqrt2$; infinite limits unchanged हैं.", "Answer $\\sqrt\\pi/\\sqrt2=\\sqrt{\\pi/2}$." ], "trap": "Coefficient a के liye scaling $1/\\sqrt a$ होती है, $1/a$ नहीं." }, "note": { "topic": 18, "formula": "18.022" } }, { "id": "mm1-051", "topic": 19, "test": "test-3", "type": "numerical", "difficulty": "moderate", "question": { "en": "For $f(x,y)=x^2y+\\sin y$, find $\\partial f/\\partial y$ at $(2,0)$.", "hi": "$f(x,y)=x^2y+\\sin y$ के लिए $(2,0)$ पर $\\partial f/\\partial y$ क्या है?" }, "options": [ "4", "1", "0", "5" ], "answer": 3, "solution": { "steps": [ "Partial derivative wrt y lete waqt x constant treat होता है.", "$f_y=x^2+\\cos y$.", "At $(2,0)$, $4+1=5$." ], "trap": "Point को derivative से पहले substitute करने पर dependence kho सकती है." }, "note": { "topic": 19, "formula": "19.002" } }, { "id": "mm1-052", "topic": 19, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "If $z=x^2+y^2$, $x=t$, $y=t^2$, find $dz/dt$ at $t=1$.", "hi": "$z=x^2+y^2$, $x=t$, $y=t^2$ में $t=1$ पर $dz/dt$ कितना है?" }, "options": [ "2", "8", "6", "4" ], "answer": 2, "solution": { "steps": [ "Multivariable chain rule $dz/dt=z_x x'+z_y y'$ है.", "$z_x=2x$, $z_y=2y$, $x'=1$, $y'=2t$.", "At $t=1$, $x=y=1$, इसलिए $2(1)+2(1)(2)=6$." ], "trap": "सिर्फ explicit x dependence लेने से y के through variation miss हो जाएगा.", "shortcut": "Direct substitution $z=t^2+t^4$ भी same derivative देता है." }, "note": { "topic": 19, "formula": "19.019" } }, { "id": "mm1-053", "topic": 19, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $f(x,y)=x^2+2xy+3y^2$, what is the nature of the stationary point at the origin?", "hi": "$f(x,y)=x^2+2xy+3y^2$ में मूल बिंदु के stationary point की प्रकृति क्या है?" }, "options": [ "Not a stationary point", "Strict minimum", "Strict maximum", "Saddle point" ], "answer": 1, "solution": { "steps": [ "Gradient $(2x+2y,2x+6y)$ origin पर zero है.", "Hessian $\\begin{pmatrix}2&2\\\\2&6\\end{pmatrix}$ का first leading minor 2 और determinant $12-4=8$ positive है.", "Hessian positive definite है, so origin strict minimum है; equivalently $f=(x+y)^2+2y^2$." ], "trap": "Positive determinant alone enough नहीं; $f_{xx}$ का sign भी check करें." }, "note": { "topic": 19, "formula": "19.011" } }, { "id": "mm1-054", "topic": 20, "test": "test-2", "type": "numerical", "difficulty": "moderate", "question": { "en": "For plane polar coordinates $x=r\\cos\\theta,y=r\\sin\\theta$ with $r\\ge0$, what is $dx\\,dy$?", "hi": "समतलीय ध्रुवीय निर्देशांक $x=r\\cos\\theta,y=r\\sin\\theta$ में $r\\ge0$ के लिए $dx\\,dy$ क्या है?" }, "options": [ "$r\\,dr\\,d\\theta$", "$dr\\,d\\theta$", "$r^2\\,dr\\,d\\theta$", "$r\\sin\\theta\\,dr\\,d\\theta$" ], "answer": 0, "solution": { "steps": [ "Area element coordinate transformation के absolute Jacobian से आता है.", "$J=\\det\\begin{pmatrix}\\cos\\theta&-r\\sin\\theta\\\\\\sin\\theta&r\\cos\\theta\\end{pmatrix}=r$.", "इसलिए $dxdy=r\\,drd\\theta$ for $r\\ge0$." ], "trap": "Radial strip की tangential width $r\\,d\\theta$ है; Jacobian factor drop मत करें." }, "note": { "topic": 20, "formula": "20.038" } }, { "id": "mm1-055", "topic": 20, "test": "test-3", "type": "theory", "difficulty": "moderate", "question": { "en": "With spherical polar angle $\\theta$ measured from positive z, which is the volume element?", "hi": "धनात्मक z अक्ष से मापे गए spherical polar angle $\\theta$ के लिए आयतन अवयव क्या है?" }, "options": [ "$r\\,dr\\,d\\theta\\,d\\phi$", "$r^2\\cos\\theta\\,dr\\,d\\theta\\,d\\phi$", "$dr\\,d\\theta\\,d\\phi$", "$r^2\\sin\\theta\\,dr\\,d\\theta\\,d\\phi$" ], "answer": 3, "solution": { "steps": [ "Spherical coordinates के orthogonal scale factors $1,r,r\\sin\\theta$ हैं.", "Volume element इन तीन local lengths का product है.", "इसलिए $dV=r^2\\sin\\theta\\,drd\\theta d\\phi$; convention $0\\le\\theta\\le\\pi$ में sine nonnegative है." ], "trap": "Latitude angle use करने पर cosine aa सकता है; यहाँ polar angle from z specified है." }, "note": { "topic": 20, "formula": "20.045" } }, { "id": "mm1-056", "topic": 21, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "Find $\\nabla\\phi$ at $(1,2,1)$ for $\\phi=x^2y+z^2$.", "hi": "$\\phi=x^2y+z^2$ के लिए $(1,2,1)$ पर $\\nabla\\phi$ ज्ञात करें।" }, "options": [ "$(4,2,1)$", "$(1,4,2)$", "$(4,1,2)$", "$(2,2,2)$" ], "answer": 2, "solution": { "steps": [ "Cartesian gradient $(\\partial_x\\phi,\\partial_y\\phi,\\partial_z\\phi)$ है.", "Derivatives respectively $2xy,x^2,2z$ मिलते हैं.", "At $(1,2,1)$ gradient $(4,1,2)$ है; यह scalar field का vector derivative है." ], "trap": "Gradient vector होता है, derivatives का scalar sum नहीं." }, "note": { "topic": 21, "formula": "21.004" } }, { "id": "mm1-057", "topic": 21, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $\\phi=x^2+y^2$, find the directional derivative at $(1,2)$ along $(3,4)$.", "hi": "$\\phi=x^2+y^2$ के लिए $(1,2)$ पर दिशा $(3,4)$ में directional derivative क्या है?" }, "options": [ "5", "$22/5$", "22", "$11/5$" ], "answer": 1, "solution": { "steps": [ "Gradient at point $=(2,4)$.", "Direction को normalize करें: $\\hat u=(3,4)/5$.", "Directional derivative $\\nabla\\phi\\cdot\\hat u=(6+16)/5=22/5$." ], "trap": "Unnormalised direction से dot करने पर 22 मिलेगा, जो per-unit-distance derivative नहीं." }, "note": { "topic": 21, "formula": "21.012" } }, { "id": "mm1-058", "topic": 21, "test": "test-2", "type": "theory", "difficulty": "moderate", "question": { "en": "At a point where $\\nabla f\\ne0$, which direction gives the largest positive directional derivative?", "hi": "जहाँ $\\nabla f\\ne0$, वहाँ अधिकतम धनात्मक directional derivative किस दिशा में मिलता है?" }, "options": [ "Along $\\nabla f$", "Opposite to $\\nabla f$", "Perpendicular to $\\nabla f$", "Every direction equally" ], "answer": 0, "solution": { "steps": [ "Unit direction $\\hat u$ में derivative $D_{\\hat u}f=\\nabla f\\cdot\\hat u=\\vert \\nabla f\\vert \\cos\\theta$.", "Largest positive value तब आता है जब $\\cos\\theta=1$.", "इसलिए gradient की direction में maximum $\\vert \\nabla f\\vert $ मिलता है." ], "trap": "Opposite direction maximum decrease देती है, maximum positive derivative नहीं." }, "note": { "topic": 21, "formula": "21.016" } }, { "id": "mm1-059", "topic": 19, "test": "test-3", "type": "numerical", "difficulty": "challenging", "question": { "en": "Find the maximum of $xy$ subject to $x^2+y^2=2$ for real $x,y$.", "hi": "वास्तविक $x,y$ के लिए प्रतिबंध $x^2+y^2=2$ के अंतर्गत $xy$ का अधिकतम क्या है?" }, "options": [ "2", "$\\sqrt2$", "0", "1" ], "answer": 3, "solution": { "steps": [ "Nonnegative square $(x-y)^2\\ge0$ से $2xy\\le x^2+y^2=2$.", "Thus $xy\\le1$.", "Equality $x=y$ पर possible है; constraint से $(x,y)=(1,1)$ या $(-1,-1)$, दोनों पर product 1." ], "trap": "Constraint के बिना product unbounded है; constrained extrema अलग problem है." }, "note": { "topic": 19 } }, { "id": "mm1-060", "topic": 20, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "In plane polar coordinates, find $\\nabla f$ for $f=r^2\\sin\\theta$ at $r=2,\\theta=\\pi/2$.", "hi": "ध्रुवीय निर्देशांक में $f=r^2\\sin\\theta$ के लिए $r=2,\\theta=\\pi/2$ पर $\\nabla f$ क्या है?" }, "options": [ "$2\\hat e_r$", "$4\\hat e_r+4\\hat e_\\theta$", "$4\\hat e_r$", "$4\\hat e_\\theta$" ], "answer": 2, "solution": { "steps": [ "Polar gradient $\\nabla f=\\hat e_r\\partial_r f+\\hat e_\\theta(1/r)\\partial_\\theta f$.", "Derivatives से $2r\\sin\\theta\\hat e_r+r\\cos\\theta\\hat e_\\theta$ मिलता है.", "At given point sine 1, cosine 0; answer $4\\hat e_r$." ], "trap": "Angular component में scale factor $1/r$ zaroor आता है." }, "note": { "topic": 20, "formula": "20.049" } }, { "id": "mm1-061", "topic": 22, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "For $\\mathbf F=(x^2,y^2,z^2)$, find $\\nabla\\cdot\\mathbf F$ at $(1,2,3)$.", "hi": "$\\mathbf F=(x^2,y^2,z^2)$ का $(1,2,3)$ पर divergence क्या है?" }, "options": [ "36", "12", "6", "14" ], "answer": 1, "solution": { "steps": [ "Divergence matching component derivatives का sum है.", "$\\nabla\\cdot\\mathbf F=2x+2y+2z$.", "At $(1,2,3)$ result $2+4+6=12$ है, एक scalar." ], "trap": "Vector magnitude या components का sum divergence नहीं है; differentiation पहले करें." }, "note": { "topic": 22, "formula": "22.002" } }, { "id": "mm1-062", "topic": 22, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $\\mathbf F=(-y,x,0)$, find $\\nabla\\times\\mathbf F$.", "hi": "$\\mathbf F=(-y,x,0)$ का curl ज्ञात करें।" }, "options": [ "$(0,0,2)$", "$(0,0,0)$", "$(0,0,-2)$", "$(1,1,0)$" ], "answer": 0, "solution": { "steps": [ "Curl का z component $\\partial_xF_y-\\partial_yF_x$ है.", "यहाँ $\\partial_x(x)=1$ और $\\partial_y(-y)=-1$, so z component $1-(-1)=2$.", "बाकी components zero हैं, इसलिए curl $(0,0,2)$." ], "trap": "Field का divergence zero है, लेकिन curl nonzero हो सकता है." }, "note": { "topic": 22, "formula": "22.014" }, "svg": "F = (−y, x, 0)" }, { "id": "mm1-063", "topic": 22, "test": "test-3", "type": "theory", "difficulty": "moderate", "question": { "en": "For a twice continuously differentiable scalar field $\\phi$, which identity always holds?", "hi": "दो बार सतत अवकलनीय scalar field $\\phi$ के लिए कौन-सी सर्वसमिका सदैव सही है?" }, "options": [ "$\\nabla\\cdot(\\nabla\\phi)=0$", "$\\nabla\\phi=0$", "$\\nabla^2\\phi=1$", "$\\nabla\\times(\\nabla\\phi)=0$" ], "answer": 3, "solution": { "steps": [ "Curl of gradient के components mixed second derivatives के differences हैं.", "$C^2$ smoothness से mixed derivatives commute: $\\partial_x\\partial_y\\phi=\\partial_y\\partial_x\\phi$.", "हर component cancel होता है, इसलिए curl zero; Laplacian generally zero नहीं है." ], "trap": "Smoothness condition और field का domain matter करते हैं; scalar का gradient automatically constant नहीं." }, "note": { "topic": 22, "formula": "22.036" } }, { "id": "mm1-064", "topic": 22, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "In three dimensions and for $r>0$, find $\\nabla^2(1/r)$.", "hi": "तीन आयामों में $r>0$ के लिए $\\nabla^2(1/r)$ क्या है?" }, "options": [ "$-1/r^2$", "$-4\\pi$", "0", "$1/r^3$" ], "answer": 2, "solution": { "steps": [ "Radial scalar के liye $\\nabla^2 f=r^{-2}\\frac{d}{dr}(r^2f')$.", "$f=1/r$ पर $f'=-1/r^2$, so $r^2f'=-1$.", "Constant का derivative zero, इसलिए Laplacian zero for $r>0$." ], "trap": "Origin excluded है; distributional identity में $-4\\pi\\delta^{(3)}(\\mathbf r)$ आता है." }, "note": { "topic": 22, "formula": "22.023" } }, { "id": "mm1-065", "topic": 24, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "Find the outward flux of $\\mathbf F=(x,y,z)$ through a sphere of radius $a$.", "hi": "त्रिज्या $a$ के गोले से $\\mathbf F=(x,y,z)$ का बाहर की ओर flux क्या है?" }, "options": [ "0", "$4\\pi a^3$", "$4\\pi a^2$", "$4\\pi a^3/3$" ], "answer": 1, "solution": { "steps": [ "Divergence $\\nabla\\cdot\\mathbf F=3$ है.", "Divergence theorem से closed-surface flux $\\iiint_V3dV$.", "Sphere का volume $4\\pi a^3/3$, इसलिए flux $3(4\\pi a^3/3)=4\\pi a^3$." ], "trap": "Field magnitude surface पर a है; area से multiply करने पर भी $a^3$ hi आएगा." }, "note": { "topic": 24, "formula": "24.029" } }, { "id": "mm1-066", "topic": 24, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $\\mathbf F=(-y/2,x/2,0)$, find circulation around the unit circle counterclockwise as viewed from positive z.", "hi": "$\\mathbf F=(-y/2,x/2,0)$ का unit circle पर positive z से देखने पर वामावर्त circulation कितना है?" }, "options": [ "$\\pi$", "$2\\pi$", "0", "$-\\pi$" ], "answer": 0, "solution": { "steps": [ "Stokes theorem में specified counterclockwise orientation का normal $+\\hat k$ है.", "Curl $\\nabla\\times\\mathbf F=\\hat k$.", "Unit disk पर curl flux $1\\times\\pi=\\pi$, जो circulation है." ], "trap": "Clockwise traversal लेने पर answer $-\\pi$ होगा; orientation ignore मत करें." }, "note": { "topic": 24, "formula": "24.027" } }, { "id": "mm1-067", "topic": 23, "test": "test-3", "type": "numerical", "difficulty": "challenging", "question": { "en": "Evaluate $\\int_C(y\\,dx+x\\,dy)$ from $(0,0)$ to $(1,2)$ along any smooth path.", "hi": "$(0,0)$ से $(1,2)$ तक किसी smooth path पर $\\int_C(y\\,dx+x\\,dy)$ क्या है?" }, "options": [ "3", "1", "Path dependent", "2" ], "answer": 3, "solution": { "steps": [ "Differential $d(xy)=y\\,dx+x\\,dy$ identify करें.", "Integrand exact differential है, so line integral सिर्फ endpoints पर depend karega.", "Value $[xy]_{(0,0)}^{(1,2)}=2-0=2$." ], "trap": "Exact differential पहचानने के baad path parametrise करना ज़रूरी नहीं." }, "note": { "topic": 23, "formula": "23.033" } }, { "id": "mm1-068", "topic": 23, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "Evaluate $\\iint_D(x^2+y^2)dA$ over the unit disk $D$.", "hi": "Unit disk $D$ पर $\\iint_D(x^2+y^2)dA$ का मान क्या है?" }, "options": [ "$2\\pi$", "$1/4$", "$\\pi/2$", "$\\pi$" ], "answer": 2, "solution": { "steps": [ "Disk और integrand radial हैं, so polar coordinates choose करें.", "$x^2+y^2=r^2$, और $dA=r\\,drd\\theta$; limits $0\\le r\\le1$, $0\\le\\theta\\le2\\pi$.", "Integral $2\\pi\\int_0^1r^3dr=2\\pi/4=\\pi/2$." ], "trap": "Area Jacobian r को drop करने से wrong power मिलेगी." }, "note": { "topic": 23, "formula": "23.010" } }, { "id": "mm1-069", "topic": 22, "test": "test-1", "type": "theory", "difficulty": "challenging", "question": { "en": "A smooth curl-free vector field is guaranteed to be conservative on which kind of open connected domain?", "hi": "एक smooth curl-free vector field किस प्रकार के open connected domain पर निश्चित रूप से conservative होता है?" }, "options": [ "Only a spherical surface", "A simply connected domain", "Any domain, including all domains with holes", "Only a bounded domain" ], "answer": 1, "solution": { "steps": [ "Conservative field के liye potential चाहिए जिससे field gradient हो.", "Curl zero local condition है; global path independence के liye topology भी important है.", "Simply connected open domain में smooth curl-free field conservative होता है." ], "trap": "Punctured plane का circulation field origin के bahar curl-free होते hue भी global potential नहीं de सकता." }, "note": { "topic": 22, "formula": "22.039" } }, { "id": "mm1-070", "topic": 22, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $\\mathbf F=\\mathbf r/r^3$ and $r>0$, what is its divergence?", "hi": "$r>0$ में $\\mathbf F=\\mathbf r/r^3$ का divergence क्या है?" }, "options": [ "0", "$3/r^3$", "$-2/r^3$", "$4\\pi$" ], "answer": 0, "solution": { "steps": [ "Radial vector field $\\mathbf F=f(r)\\hat e_r$ यहाँ $f=1/r^2$ है.", "Spherical divergence $r^{-2}\\frac{d}{dr}[r^2f(r)]$ use करें.", "Bracket $r^2/r^2=1$, derivative zero, इसलिए divergence zero away from origin." ], "trap": "Closed sphere से flux $4\\pi$ है, फिर भी divergence r>0 पर zero; singular origin essential है." }, "note": { "topic": 22 } }, { "id": "mm1-071", "topic": 26, "test": "test-3", "type": "numerical", "difficulty": "moderate", "question": { "en": "For $A=\\begin{pmatrix}2&1\\\\1&2\\end{pmatrix}$, find the eigenvalues.", "hi": "$A=\\begin{pmatrix}2&1\\\\1&2\\end{pmatrix}$ के eigenvalues ज्ञात करें।" }, "options": [ "2 and 2", "$-1$ and 3", "0 and 4", "1 and 3" ], "answer": 3, "solution": { "steps": [ "Characteristic equation $\\det(A-\\lambda I)=(2-\\lambda)^2-1=0$.", "Isse $2-\\lambda=\\pm1$, इसलिए $\\lambda=1,3$.", "Trace check $1+3=4$ और determinant check $1\\times3=3$ दोनों match करते हैं." ], "trap": "Diagonal entries hi eigenvalues सिर्फ triangular matrices में directly मिलते हैं." }, "note": { "topic": 26, "formula": "26.020" } }, { "id": "mm1-072", "topic": 26, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "Which vector is an eigenvector of $\\begin{pmatrix}2&1\\\\1&2\\end{pmatrix}$ with eigenvalue 3?", "hi": "$\\begin{pmatrix}2&1\\\\1&2\\end{pmatrix}$ के eigenvalue 3 से संबंधित eigenvector कौन-सा है?" }, "options": [ "$(1,0)^T$", "$(0,0)^T$", "$(1,1)^T$", "$(1,-1)^T$" ], "answer": 2, "solution": { "steps": [ "Eigenvector equation $(A-3I)v=0$ solve करें.", "Matrix $\\begin{pmatrix}-1&1\\\\1&-1\\end{pmatrix}$ से $v_2=v_1$.", "Any nonzero multiple of $(1,1)^T$ valid है; zero vector eigenvector नहीं होता." ], "trap": "$(1,-1)^T$ isi matrix का eigenvalue 1 वाला vector है." }, "note": { "topic": 26, "formula": "26.017" } }, { "id": "mm1-073", "topic": 25, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "For a $3\\times3$ matrix with determinant 2, find $\\det(3A)$.", "hi": "एक $3\\times3$ matrix के लिए $\\det A=2$ है। $\\det(3A)$ क्या है?" }, "options": [ "162", "54", "6", "18" ], "answer": 1, "solution": { "steps": [ "Matrix को scalar 3 से multiply करने पर हर row 3 times होती है.", "Determinant हर row scaling से multiply होता है; three rows give $3^3$.", "Thus $\\det(3A)=27\\times2=54$." ], "trap": "General rule n×n matrix के liye $\\det(cA)=c^n\\det A$ है." }, "note": { "topic": 25 } }, { "id": "mm1-074", "topic": 25, "test": "test-2", "type": "theory", "difficulty": "moderate", "question": { "en": "A square matrix has two identical rows. Which statement must be true?", "hi": "एक square matrix की दो पंक्तियाँ समान हैं। कौन-सा कथन आवश्यक रूप से सही है?" }, "options": [ "Its determinant is zero", "Its trace is zero", "Every entry is zero", "All eigenvalues are zero" ], "answer": 0, "solution": { "steps": [ "Identical rows linearly dependent हैं.", "Dependent rows के determinant का value zero होता है; alternatively row swap से matrix unchanged but determinant sign changes.", "इसलिए determinant must be zero, so matrix singular है." ], "trap": "Singular matrix में at least one eigenvalue zero होता है; all eigenvalues zero ज़रूरी नहीं." }, "note": { "topic": 25, "formula": "25.018" } }, { "id": "mm1-075", "topic": 26, "test": "test-3", "type": "theory", "difficulty": "challenging", "question": { "en": "Which statement is always true for a finite-dimensional Hermitian matrix?", "hi": "Finite-dimensional Hermitian matrix के लिए कौन-सा कथन सदैव सही है?" }, "options": [ "All entries are real", "All eigenvalues are positive", "Its determinant is necessarily 1", "All eigenvalues are real" ], "answer": 3, "solution": { "steps": [ "Hermitian condition $A^\\dagger=A$ है, real-entry condition नहीं.", "If $Av=\\lambda v$, scalar $v^\\dagger Av$ अपने conjugate के equal है, so real है.", "$\\lambda=(v^\\dagger Av)/(v^\\dagger v)$ real होता है; positivity separately require होती है." ], "trap": "Hermitian matrix complex off-diagonal entries rakh सकती है, provided conjugate symmetry हो." }, "note": { "topic": 26, "formula": "26.056" } }, { "id": "mm1-076", "topic": 26, "test": "test-4", "type": "theory", "difficulty": "challenging", "question": { "en": "Eigenvectors of a real symmetric matrix belonging to distinct eigenvalues are:", "hi": "Real symmetric matrix के भिन्न eigenvalues से संबंधित eigenvectors कैसे होते हैं?" }, "options": [ "Always identical in length", "Necessarily coordinate basis vectors", "Orthogonal", "Always parallel" ], "answer": 2, "solution": { "steps": [ "Let $Av=\\lambda v$, $Aw=\\mu w$ with $\\lambda\\ne\\mu$.", "Symmetry से $v^TAw=(Av)^Tw$, so $\\mu v^Tw=\\lambda v^Tw$.", "Difference $(\\mu-\\lambda)v^Tw=0$ से $v^Tw=0$, इसलिए orthogonal." ], "trap": "Distinct eigenvalues condition important है; degenerate eigenspace में arbitrary vectors automatically orthogonal नहीं." }, "note": { "topic": 26, "formula": "26.057" } }, { "id": "mm1-077", "topic": 25, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "The system $x+y=2$, $2x+2y=4$ has how many real solutions?", "hi": "समीकरण $x+y=2$, $2x+2y=4$ के वास्तविक हलों की संख्या कितनी है?" }, "options": [ "Exactly two", "Infinitely many", "Exactly one", "None" ], "answer": 1, "solution": { "steps": [ "Second equation first का exactly 2 times है; independent constraint सिर्फ एक है.", "Two unknowns में एक free parameter रहेगा: $x=t$, $y=2-t$.", "Every real t solution देता है, so infinitely many solutions हैं." ], "trap": "Coefficient determinant zero से सिर्फ uniqueness fail होती है; consistency separately check karni होती है." }, "note": { "topic": 25, "formula": "25.027" } }, { "id": "mm1-078", "topic": 26, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "Find the rank of $\\begin{pmatrix}1&2&3\\\\2&4&6\\\\0&1&1\\end{pmatrix}$.", "hi": "$\\begin{pmatrix}1&2&3\\\\2&4&6\\\\0&1&1\\end{pmatrix}$ की rank क्या है?" }, "options": [ "2", "1", "3", "0" ], "answer": 0, "solution": { "steps": [ "Second row first का double है, so at most two independent rows हैं.", "First और third rows proportional नहीं, so at least two independent rows हैं.", "इसलिए rank exactly 2; row reduction भी एक zero row देता है." ], "trap": "Zero determinant rank को 3 से kam batata है, exact rank 1 या 2 के liye independence check चाहिए." }, "note": { "topic": 26, "formula": "26.014" } }, { "id": "mm1-079", "topic": 26, "test": "test-3", "type": "theory", "difficulty": "challenging", "question": { "en": "For $A=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}$ over the real numbers, which statement is correct?", "hi": "वास्तविक संख्याओं पर $A=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}$ के लिए कौन-सा कथन सही है?" }, "options": [ "It has two distinct eigenvalues", "It is a real symmetric matrix", "It has no eigenvectors", "It is not diagonalizable" ], "answer": 3, "solution": { "steps": [ "Triangular matrix के eigenvalues दोनों 1 हैं.", "$A-I=\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}$ का nullspace सिर्फ $(1,0)^T$ की span है.", "Algebraic multiplicity 2 but geometric multiplicity 1; two independent eigenvectors नहीं, so diagonalizable नहीं." ], "trap": "Repeated eigenvalue alone non-diagonalizable नहीं banata; eigenspace dimension decisive है." }, "note": { "topic": 26, "formula": "26.030" } }, { "id": "mm1-080", "topic": 26, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "For a real rotation matrix $R=\\begin{pmatrix}\\cos\\theta&-\\sin\\theta\\\\\\sin\\theta&\\cos\\theta\\end{pmatrix}$, find $R^{-1}$.", "hi": "Real rotation matrix $R=\\begin{pmatrix}\\cos\\theta&-\\sin\\theta\\\\\\sin\\theta&\\cos\\theta\\end{pmatrix}$ का inverse क्या है?" }, "options": [ "$-R$", "$R^2$", "$R^T$", "$R$ for every angle" ], "answer": 2, "solution": { "steps": [ "Rotation matrix के columns orthonormal हैं.", "इसलिए $R^TR=I$, जो inverse की defining relation है.", "इसलिए $R^{-1}=R^T=R(-\\theta)$, opposite angle की rotation." ], "trap": "Inverse को entrywise reciprocal लेना matrix algebra में valid नहीं." }, "note": { "topic": 26, "formula": "26.043" } }, { "id": "mm1-081", "topic": 27, "test": "test-1", "type": "numerical", "difficulty": "moderate", "question": { "en": "Solve $dy/dx=2y$, $y(0)=3$.", "hi": "$dy/dx=2y$, $y(0)=3$ का हल क्या है?" }, "options": [ "$y=3e^{-2x}$", "$y=3e^{2x}$", "$y=3+2x$", "$y=2e^{3x}$" ], "answer": 1, "solution": { "steps": [ "Separable equation: $dy/y=2dx$.", "Integrate करके $\\ln\\vert y\\vert =2x+C$, so $y=Ke^{2x}$.", "Initial condition $y(0)=3$ से $K=3$; इसलिए $y=3e^{2x}$." ], "trap": "Initial derivative 6 है; $3+2x$ satisfy नहीं करता." }, "note": { "topic": 27, "formula": "27.012" } }, { "id": "mm1-082", "topic": 27, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "Solve $y'+y=e^{-x}$ with $y(0)=0$.", "hi": "$y'+y=e^{-x}$ और $y(0)=0$ का हल क्या है?" }, "options": [ "$y=xe^{-x}$", "$y=e^{-x}$", "$y=xe^x$", "$y=1-e^{-x}$" ], "answer": 0, "solution": { "steps": [ "Linear ODE का integrating factor $e^{\\int1dx}=e^x$ है.", "Multiply करने पर $(e^xy)'=1$, so $e^xy=x+C$.", "Initial condition से $C=0$, इसलिए $y=xe^{-x}$." ], "trap": "RHS homogeneous solution जैसी है; trial constant times $e^{-x}$ alone kaam नहीं karega." }, "note": { "topic": 27, "formula": "27.009" } }, { "id": "mm1-083", "topic": 27, "test": "test-3", "type": "theory", "difficulty": "moderate", "question": { "en": "For $y'+P(x)y=Q(x)$, what is an integrating factor?", "hi": "$y'+P(x)y=Q(x)$ के लिए integrating factor क्या है?" }, "options": [ "$e^{\\int Q(x)dx}$", "$\\int P(x)dx$", "$P(x)Q(x)$", "$e^{\\int P(x)dx}$" ], "answer": 3, "solution": { "steps": [ "ऐसा factor $\\mu$ चाहिए जिससे LHS $(\\mu y)'$ ban जाए.", "Product rule demands $\\mu'=P\\mu$, यानी $d\\mu/\\mu=Pdx$.", "Integrate करके $\\mu=e^{\\int Pdx}$; nonzero constant factor irrelevant है." ], "trap": "Equation को पहले coefficient of y' equal 1 form में lao." }, "note": { "topic": 27, "formula": "27.007" } }, { "id": "mm1-084", "topic": 28, "test": "test-4", "type": "numerical", "difficulty": "moderate", "question": { "en": "Solve $y''+4y=0$ with $y(0)=1$, $y'(0)=0$.", "hi": "$y''+4y=0$, $y(0)=1$, $y'(0)=0$ का हल क्या है?" }, "options": [ "$y=\\cos4x$", "$y=e^{-2x}$", "$y=\\cos2x$", "$y=\\sin2x$" ], "answer": 2, "solution": { "steps": [ "Characteristic equation $m^2+4=0$ से roots $m=\\pm2i$.", "General real solution $A\\cos2x+B\\sin2x$ है.", "$y(0)=1$ से $A=1$; $y'(0)=2B=0$ से $B=0$." ], "trap": "Angular frequency 2 है, coefficient 4 नहीं." }, "note": { "topic": 28, "formula": "28.027" } }, { "id": "mm1-085", "topic": 28, "test": "test-1", "type": "theory", "difficulty": "challenging", "question": { "en": "What is the general solution of $y''-2y'+y=0$?", "hi": "$y''-2y'+y=0$ का सामान्य हल क्या है?" }, "options": [ "$C_1e^{2x}+C_2$", "$(C_1+C_2x)e^x$", "$C_1e^x+C_2e^{-x}$", "$C_1\\cos x+C_2\\sin x$" ], "answer": 1, "solution": { "steps": [ "Characteristic polynomial $m^2-2m+1=(m-1)^2$ है.", "Repeated root 1 के liye independent solutions $e^x$ और $xe^x$ चाहिए.", "General solution इनका linear combination $(C_1+C_2x)e^x$ है." ], "trap": "Same exponential दो बार likhne से दो independent constants नहीं मिलते." }, "note": { "topic": 28, "formula": "28.017" } }, { "id": "mm1-086", "topic": 28, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $y''+y=\\cos x$, which is a particular solution?", "hi": "$y''+y=\\cos x$ का एक particular solution कौन-सा है?" }, "options": [ "$\\tfrac{x}{2}\\sin x$", "$\\cos x$", "$\\sin x$", "$\\tfrac12\\cos x$" ], "answer": 0, "solution": { "steps": [ "Forcing frequency natural frequency 1 के equal है, so resonance trial में x factor चाहिए.", "Try $y_p=Ax\\sin x$. Then $y_p''+y_p=2A\\cos x$.", "RHS compare करने पर $2A=1$, so $A=1/2$." ], "trap": "$\\cos x$ और $\\sin x$ homogeneous solutions हैं, unka constant multiple forcing produce नहीं karega." }, "note": { "topic": 28, "formula": "28.044" } }, { "id": "mm1-087", "topic": 29, "test": "test-3", "type": "theory", "difficulty": "moderate", "question": { "en": "For an odd integrable function on $[-\\pi,\\pi]$, which Fourier coefficients vanish?", "hi": "$[-\\pi,\\pi]$ पर विषम समाकलनीय फलन के कौन-से Fourier coefficients शून्य होते हैं?" }, "options": [ "All $b_n$ only", "All coefficients", "Only $a_0$", "$a_0$ and all $a_n$" ], "answer": 3, "solution": { "steps": [ "Odd function का symmetric integral zero है, so constant coefficient zero.", "Cosine even है; odd f times cosine odd hi रहता है, so every cosine coefficient zero.", "Sine odd है; f times sine even हो सकता है, इसलिए sine coefficients generally remain." ], "trap": "Odd symmetry का matlab sine series होता है; sine coefficients vanish नहीं करते." }, "note": { "topic": 29, "formula": "29.036" } }, { "id": "mm1-088", "topic": 29, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $f(x)=x$ on $(-\\pi,\\pi)$, with $f\\sim\\sum b_n\\sin nx$, find $b_1$.", "hi": "$(-\\pi,\\pi)$ में $f(x)=x$ और $f\\sim\\sum b_n\\sin nx$ के लिए $b_1$ क्या है?" }, "options": [ "$\\pi$", "$-2$", "2", "1" ], "answer": 2, "solution": { "steps": [ "Coefficient $b_1=\\pi^{-1}\\int_{-\\pi}^{\\pi}x\\sin x\\,dx$.", "Integrand even है, so $b_1=(2/\\pi)\\int_0^\\pi x\\sin xdx$.", "Parts integral $\\pi$ देता है, इसलिए $b_1=2$." ], "trap": "Fourier normalization $1/\\pi$ और even-integrand factor 2 दोनों चाहिए." }, "note": { "topic": 29, "formula": "29.022" } }, { "id": "mm1-089", "topic": 29, "test": "test-1", "type": "theory", "difficulty": "challenging", "question": { "en": "A periodic function has one-sided limits 2 and 6 at a jump and satisfies the usual Dirichlet conditions. What value does its Fourier series approach at the jump?", "hi": "किसी periodic फलन के jump पर एकतरफा सीमाएँ 2 और 6 हैं तथा usual Dirichlet conditions पूरी हैं। Fourier series वहाँ किस मान पर जाती है?" }, "options": [ "0", "4", "2", "6" ], "answer": 1, "solution": { "steps": [ "Jump पर Fourier convergence theorem one-sided limits का arithmetic mean देता है.", "Mean $[f(x^-)+f(x^+)]/2=(2+6)/2$.", "इसलिए series 4 पर converge करती है, chahe isolated point पर f की assigned value kuch और हो." ], "trap": "Gibbs overshoot nearby होता है; jump point का limiting value midpoint है." }, "note": { "topic": 29, "formula": "29.053" } }, { "id": "mm1-090", "topic": 30, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "A string fixed at $x=0,L$ has mode $X(x)=\\sin(kx)$. What is the smallest positive allowed $k$?", "hi": "$x=0,L$ पर बँधी डोरी में mode $X(x)=\\sin(kx)$ है। सबसे छोटा धनात्मक अनुमत $k$ क्या है?" }, "options": [ "$\\pi/L$", "$2\\pi/L$", "$\\pi/(2L)$", "0" ], "answer": 0, "solution": { "steps": [ "At x=0 sine automatically zero है.", "At x=L boundary requires $\\sin(kL)=0$, इसलिए $kL=n\\pi$.", "Nontrivial smallest positive mode n=1 है, giving $k=\\pi/L$." ], "trap": "n=0 sine function identically zero देता है, physical nontrivial mode नहीं." }, "note": { "topic": 30, "formula": "30.005" } }, { "id": "mm1-091", "topic": 31, "test": "test-3", "type": "numerical", "difficulty": "moderate", "question": { "en": "A density is $p(x)=Cx$ on $0\\le x\\le2$, and zero elsewhere. Find $C$.", "hi": "$0\\le x\\le2$ पर density $p(x)=Cx$ है और बाहर शून्य है। $C$ ज्ञात करें।" }, "options": [ "1", "$1/4$", "2", "$1/2$" ], "answer": 3, "solution": { "steps": [ "Density normalization $\\int p(x)dx=1$ demand करती है.", "$\\int_0^2Cx\\,dx=C[x^2/2]_0^2=2C$.", "Set $2C=1$ तो get $C=1/2$, nonnegative density के साथ consistent." ], "trap": "Density का endpoint value 1 होना normalization condition नहीं; total area 1 चाहिए." }, "note": { "topic": 31, "formula": "31.008" } }, { "id": "mm1-092", "topic": 31, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $p(x)=2x$ on $0\\le x\\le1$, find $P(X>1/2)$.", "hi": "$0\\le x\\le1$ पर $p(x)=2x$ है। $P(X>1/2)$ क्या है?" }, "options": [ "$1/4$", "1", "$3/4$", "$1/2$" ], "answer": 2, "solution": { "steps": [ "Continuous probability relevant interval पर density का area है.", "$P(X>1/2)=\\int_{1/2}^12x dx=[x^2]_{1/2}^1$.", "Answer $1-1/4=3/4$; endpoint equality probability zero है." ], "trap": "p(1/2)=1 density है, interval probability नहीं." }, "note": { "topic": 31, "formula": "31.009" } }, { "id": "mm1-093", "topic": 31, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $p(x)=2x$ on $[0,1]$, find $E[X]$.", "hi": "$[0,1]$ पर $p(x)=2x$ के लिए $E[X]$ ज्ञात करें।" }, "options": [ "$3/4$", "$2/3$", "$1/2$", "$1/3$" ], "answer": 1, "solution": { "steps": [ "Expectation formula $E[X]=\\int x p(x)dx$ है.", "Substitute density: $\\int_0^1x(2x)dx=2\\int_0^1x^2dx$.", "Value $2/3$ मिलती है; high x पर density larger होने से mean midpoint से बड़ा है." ], "trap": "Density का integral 1 है; expectation के liye extra factor x चाहिए." }, "note": { "topic": 31, "formula": "31.016" } }, { "id": "mm1-094", "topic": 31, "test": "test-2", "type": "numerical", "difficulty": "challenging", "question": { "en": "For $p(x)=2x$ on $[0,1]$, find $\\operatorname{Var}(X)$.", "hi": "$[0,1]$ पर $p(x)=2x$ के लिए $\\operatorname{Var}(X)$ क्या है?" }, "options": [ "$1/18$", "$1/12$", "$1/2$", "$4/9$" ], "answer": 0, "solution": { "steps": [ "First moment $E[X]=2\\int_0^1x^2dx=2/3$.", "Second moment $E[X^2]=2\\int_0^1x^3dx=1/2$.", "Variance $E[X^2]-E[X]^2=1/2-4/9=1/18$." ], "trap": "Second moment variance के equal तब होता है जब mean zero हो." }, "note": { "topic": 31, "formula": "31.024" } }, { "id": "mm1-095", "topic": 31, "test": "test-3", "type": "theory", "difficulty": "moderate", "question": { "en": "For a continuous random variable with a regular density, what is $P(X=a)$ for a single point $a$?", "hi": "Regular density वाले continuous random variable के लिए किसी एक बिंदु $a$ पर $P(X=a)$ क्या है?" }, "options": [ "$p(a)$", "1", "Always $1/2$", "0" ], "answer": 3, "solution": { "steps": [ "Probability density को finite-width interval पर integrate करते हैं.", "Single point का interval width zero है, इसलिए its area zero.", "इसलिए $P(X=a)=0$, even if density $p(a)$ positive हो." ], "trap": "Density probability नहीं; density की units reciprocal-x भी हो सकती हैं." }, "note": { "topic": 31, "formula": "31.005" } }, { "id": "mm1-096", "topic": 31, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "For independent $X,Y$ with variances 2 and 3, find $\\operatorname{Var}(2X-Y)$.", "hi": "स्वतंत्र $X,Y$ की variances 2 और 3 हैं। $\\operatorname{Var}(2X-Y)$ क्या है?" }, "options": [ "7", "1", "11", "5" ], "answer": 2, "solution": { "steps": [ "General formula $\\operatorname{Var}(aX+bY)=a^2\\operatorname{Var}X+b^2\\operatorname{Var}Y+2ab\\operatorname{Cov}(X,Y)$.", "Independence से covariance zero; coefficients 2 और −1 square होंगे.", "Variance $4(2)+1(3)=11$." ], "trap": "Minus sign variance को subtract नहीं करता; coefficients square होते हैं." }, "note": { "topic": 31 } }, { "id": "mm1-097", "topic": 31, "test": "test-1", "type": "numerical", "difficulty": "challenging", "question": { "en": "A normalized Gaussian density is $p(x)=Ce^{-x^2/(2\\sigma^2)}$, with $\\sigma>0$. Find $C$.", "hi": "Normalized Gaussian density $p(x)=Ce^{-x^2/(2\\sigma^2)}$, $\\sigma>0$ में $C$ क्या है?" }, "options": [ "$1/(\\sqrt\\pi\\sigma)$", "$1/(\\sqrt{2\\pi}\\sigma)$", "$1/(2\\pi\\sigma^2)$", "$\\sigma/\\sqrt{2\\pi}$" ], "answer": 1, "solution": { "steps": [ "Gaussian integral $\\int_{-\\infty}^{\\infty}e^{-ax^2}dx=\\sqrt{\\pi/a}$ है.", "यहाँ $a=1/(2\\sigma^2)$, so integral $\\sqrt{2\\pi}\\sigma$.", "Total area $C\\sqrt{2\\pi}\\sigma=1$, इसलिए required inverse factor." ], "trap": "Exponent convention important है: missing factor 2 changes normalization." }, "note": { "topic": 31, "formula": "31.030" } }, { "id": "mm1-098", "topic": 32, "test": "test-2", "type": "theory", "difficulty": "challenging", "question": { "en": "For an isotropic 3D velocity density $f(\\mathbf v)=f(v)$, what is the speed density $g(v)$ for $v\\ge0$?", "hi": "Isotropic 3D velocity density $f(\\mathbf v)=f(v)$ के लिए $v\\ge0$ पर speed density $g(v)$ क्या है?" }, "options": [ "$4\\pi v^2 f(v)$", "$f(v)$", "$2\\pi v f(v)$", "$f(v)/(4\\pi v^2)$" ], "answer": 0, "solution": { "steps": [ "Speed interval $[v,v+dv]$ velocity-space में spherical shell है.", "Shell volume $4\\pi v^2dv$; isotropy से density पूरी shell पर same है.", "Probability $g(v)dv=f(v)4\\pi v^2dv$, इसलिए $g(v)=4\\pi v^2f(v)$." ], "trap": "3D density और one-dimensional speed density की units अलग होती हैं." }, "note": { "topic": 32, "formula": "32.013" } }, { "id": "mm1-099", "topic": 32, "test": "test-3", "type": "numerical", "difficulty": "challenging", "question": { "en": "Particles have speeds 1 and 3 with equal probability. Find the RMS speed.", "hi": "कणों की चाल 1 और 3 समान प्रायिकता से है। RMS चाल क्या है?" }, "options": [ "2", "$\\sqrt2$", "5", "$\\sqrt5$" ], "answer": 3, "solution": { "steps": [ "RMS definition $v_{\\rm rms}=\\sqrt{E[v^2]}$ है.", "Squared speeds का mean $(1^2+3^2)/2=5$.", "Square root लेने पर $\\sqrt5$ मिलता है, जो arithmetic mean 2 से बड़ा है." ], "trap": "$\\sqrt{E[v^2]}$ को $E[v]$ से confuse मत करें." }, "note": { "topic": 32, "formula": "32.012" } }, { "id": "mm1-100", "topic": 31, "test": "test-4", "type": "numerical", "difficulty": "challenging", "question": { "en": "A nonnegative variable has density $p(x)=\\lambda e^{-\\lambda x}$, $\\lambda>0$. Find $P(X>s+t\\mid X>s)$ for $s,t\\ge0$.", "hi": "गैरऋणात्मक चर की density $p(x)=\\lambda e^{-\\lambda x}$, $\\lambda>0$ है। $s,t\\ge0$ पर $P(X>s+t\\mid X>s)$ क्या है?" }, "options": [ "$1-e^{-\\lambda t}$", "$e^{-\\lambda s}$", "$e^{-\\lambda t}$", "$e^{-\\lambda(s+t)}$" ], "answer": 2, "solution": { "steps": [ "Survival probability $P(X>u)=\\int_u^\\infty\\lambda e^{-\\lambda x}dx=e^{-\\lambda u}$.", "Conditional probability numerator $P(X>s+t)$ और denominator $P(X>s)$ है, because s,t nonnegative हैं.", "Ratio $e^{-\\lambda(s+t)}/e^{-\\lambda s}=e^{-\\lambda t}$; यह memoryless property है." ], "trap": "Conditioning को ignore करने पर unconditional tail $e^{-\\lambda(s+t)}$ मिलता है." }, "note": { "topic": 31, "formula": "31.009" } } ] }