Download Original PDF
Get the official Barkatullah University print version scanned document.
🤝 Help Your Juniors!
Have previous year question papers that aren't on our website? Help the next batch of students by sending them to us! With your consent, we will proudly feature your name as a Top Contributor on our platform.
Submit Papers 📩Roll No...............
Total No. of Questions: 16
[Total No. of Printed Pages: 07]
C-244
11321
B.Sc. (First Year) (NEP Examination, 2024)
(REG/PVT.) (Major, Minor)
MATHEMATICS
BA/BSc (MJMI1341, I212)
Calculus and Differential Equations
(REG/PVT.) (Major, Minor)
MATHEMATICS
BA/BSc (MJMI1341, I212)
Calculus and Differential Equations
Time: 3 Hours
[Maximum Marks: 70]
नोट : किन्हीं दो प्रश्नों के उत्तर दीजिए । सभी प्रश्नों के अंक समान हैं ।
Note: Attempt any two questions. All questions carry equal marks.
1.
माधव की जीवनी के बारे में संक्षिप्त टिप्पणी लिखिए ।
Write a short note about biography of Madhava.
2.
यदि y = a cos(log x) + b sin(log x), तब दर्शाइए कि x²y₂ + xy₁ + y = 0.
If y = a cos(log x) + b sin(log x), then show that x²y₂ + xy₁ + y = 0.
x²y₂ + xy₁ + y = 0.
3.
परवलय y² = 4ax के बिन्दु (x, y) पर वक्रता त्रिज्या ज्ञात कीजिए ।
Find out the radius of curvature at the point (x, y) of the parabola y² = 4ax.
y² = 4ax.
4.
∫sin⁷ x dx को हल कीजिए ।
Solve ∫sin⁷ x dx.
∫sin⁷ x dx
खण्ड 'अ'
Section A
(अति लघु उत्तरीय प्रश्न)
(Very Short Answer Type Questions)
2×3=6
नोट : प्रत्येक खण्ड में दिये गये निर्देशों के अनुसार प्रश्नों के उत्तर दीजिए ।
Note: Attempt the questions as per instructions given in each Section.
5.
अवकल समीकरण को हल कीजिए :
Solve the differential equation:
(1 + x²)² dy/dx + 2xy = cos x.
खण्ड 'ब'
Section B
(लघु उत्तरीय प्रश्न)
(Short Answer Type Questions)
4×9=36
नोट : किन्हीं चार प्रश्नों के उत्तर दीजिए । सभी प्रश्नों के अंक समान हैं ।
Note: Attempt any four questions. All questions carry equal marks.
6.
टेलर प्रमेय द्वारा tan⁻¹ x का प्रसार
(x - π/4)
की घातों में कीजिए ।Expand tan⁻¹ x in the powers of
(x - π/4)
by Taylor's theorem.7.
वक्र x³ + 2x²y - xy² - 2y³ + 4y² + 2xy + y - 1 = 0. सभी अनन्तस्पर्शियाँ ज्ञात कीजिए ।
Find the all asymptotes of the curve:
x³ + 2x²y - xy² - 2y³ + 4y² + 2xy + y - 1 = 0.
8.
वक्र y²(2a - x) = x³ का अनुरेखण कीजिए ।
Trace the curve y²(2a - x) = x³.
y²(2a - x) = x³
9.
Evaluate
∫π/20 dx / (4 + 5 sin x)
का मान ज्ञात कीजिए ।Evaluate
∫π/20 dx / (4 + 5 sin x)
10.
Evaluate 
का मान ज्ञात कीजिए ।
Evaluate
11.
Solve (1 + y²)dx = (tan⁻¹ y - x)dy को हल कीजिए ।
Solve (1 + y²)dx = (tan⁻¹ y - x)dy.
(1 + y²)dx = (tan⁻¹ y - x)dy
12.
Solve
d²y/dx² - 3 dy/dx - 4y = 0
को हल कीजिए ।Solve
d²y/dx² - 3 dy/dx - 4y = 0
खण्ड 'स'
Section C
(दीर्घ उत्तरीय प्रश्न)
(Long Answer Type Questions)
2×14=28
नोट : किन्हीं दो प्रश्नों के उत्तर दीजिए । सभी प्रश्नों के अंक समान हैं ।
Note: Attempt any two questions. All questions carry equal marks.
13.
यदि u = sin⁻¹ ((x1/3 + y1/3) / (x1/2 + y1/2))1/2, हो, तो सिद्ध कीजिए कि :
x² (∂²u/∂x²) + 2xy (∂²u/∂x∂y) + y² (∂²u/∂y²) = (tan u) / 144 (13 + tan² u).
If u = sin⁻¹ ((x1/3 + y1/3) / (x1/2 + y1/2))1/2, then prove that:
x² (∂²u/∂x²) + 2xy (∂²u/∂x∂y) + y² (∂²u/∂y²) = (tan u) / 144 (13 + tan² u).
14.
Evaluate
∫∫ e(2x+3y) dxdy
over the triangle bounded by x = 0, y = 0 and x + y = 1.Evaluate
∫∫ e(2x+3y) dxdy
over the triangle bounded by x = 0, y = 0 and x + y = 1.15.
Solve
x² (d²y/dx²) - 3x (dy/dx) + 4y = 2x²
को हल कीजिए ।Solve
x² (d²y/dx²) - 3x (dy/dx) + 4y = 2x²
16.
प्रचाल-विचरण की विधि से अवकल समीकरण
d²y/dx² + 4y = 4 tan 2x
को हल कीजिए ।Solve the differential equation
d²y/dx² + 4y = 4 tan 2x
by the method of variation of parameters.