Mathematics Honours CBCS ( 2017-18 ) Paper: C-1 of Vidyasagar University
Previous Year Question Paper
Mathematics Honours CBCS Question Papers for 2017-18 provide a valuable resource for undergraduate students aiming to excel in their academic journey. These question papers, structured under the CBCS framework, are ideal for understanding exam patterns and mastering core mathematical concepts. Explore resources like the Vidyasagar University official portal for more details. By reviewing the University of Calcutta Mathematics Hons CBCS question paper, students can enhance their study strategies, identify key topics, and improve their problem-solving skills. This paper serves as a guide to help students gauge their readiness and target areas for improvement. For more information about the University of Calcutta and its programs, visit their official website at University of Calcutta. Prepare effectively with this essential resource, and ensure you’re well-equipped for your exams!
Vidyasagar University
MATHEMATICS
[Honours]
(CBCS)
[First Semester]
Paper : C1T
Full Marks : 60
Time : 3 hours
The figures in the right hand margin indicate.
UNIT-I: Calculus-I
-
Answer any three questions: \( \pmb{[2\times 3]} \)
- What do you mean by asymptote? Does asymptote exist for every curve?
- Your answer: Find the value of \( \lim\limits_{x \to 1} x^{\frac{1}{x – 1} }\)
- Define the point of inflection of a curve.
- Find the envelope of the straight line \( \frac{x}{a}+\frac{y}{b}=1 \), where the parameters \( a \) and \( b \) are connected by the relation \( ab = c^2 \).
- Write the Leibnitz theorem for successive derivatives up to the 4th order.
-
Answer any one question \( \pmb{[10\times 1]} \):
- If \( s \) be the length of the arc \( 3ay^{2}=x(x-a)^{2} \), measured from the origin to any point \( (x, y) \), show that \( 3s^{2} = 4x^2 + 3y^2 \). \( \pmb{[5] } \)
- Show that the curve \( y = 3x^5 – 40x^3 + 3x – 20 \) is concave upwards for \( -2 \lt x \lt 0 \) and \( 2 \lt x \lt \infty \), but convex upwards for \( -\infty \lt x \lt -2 \) and \( 0 \lt x \lt 2 \). Also, show that \( x = -2, 0, 2 \) are its points of inflection. \( \pmb{[2+2+1] } \)
-
- Trace the curve \( x^{2}y^{2}=a(y^{2}-x^{2}) \). \( \pmb{[5] } \)
- If \( \alpha, \beta \) be the roots of the equation \( ax^2 + bx + c = 0 \), show that \( \lim\limits_{x \to a} \frac{1-\cos \left(ax^2 + bx + c \right)}{\left(x-\alpha \right)^{2}}=\frac{1}{2}a^{2}\left(\alpha^{2}-\beta^{2} \right) \). \( \pmb{[5] } \)
UNIT-II: Calculus-II
-
Answer any two questions: \( \pmb{[2\times 2] } \)
- If \( I_{n}=\int^{\frac{\pi}{4}}_{0} \tan^n x \, dx\), where \( n \) is a positive integer greater than 1, then prove that \( I_{n}=\frac{1}{n-1}-I_{n-2}\).
- The volume of the solid generated by the revolution of the curve \( y = \frac{1}{x} \), bounded by \( y = 0, x = 2, x=b~(~0\lt b \lt 2 ) \) about the x-axis is 3. Find the value of \( b \).
- What is the formula for finding the area of the curve \( y = \psi(t), x = \phi(t) \), where \( t \) is the parameter?
-
Answer any two questions: \( \pmb{[5\times 2] } \)
- If \( I_{m,n}=\int^{\frac{\pi}{2}}_{0} \cos^m x \sin nx \, dx\) then show that \( I_{m,n}= \frac{1}{m+n}+\frac{1}{m+n}I_{m-1,n-1}\). And also deduce that \( I_{m,n}= \frac{1}{2^{m+1}}\left[2+\frac{2^2}{2}+\frac{2^3}{3}+…+\frac{2^m}{m} \right]\).
- Find the area of the surface generated by revolving the curve \( x = \cos t, y = 2 + \sin t \), where \( 0 \leq t \leq 2\pi \), about the x-axis.
- Find the arc length parameter along the curve \(C:\”{r}(t)=\left(1+2t \right)\hat{i}+\left(1+3t \right)\hat{j}+6\left(1-t \right)\hat{k} \) from the point \(t=0\).
UNIT-III: Geometry
-
Answer any three questions: \( \pmb{[2\times 3] } \)
- Determine the type of the conic \( 8x^2 + 10xy + 3y^2 + 22x + 14y + 15 = 0 \).
- Show the plane \( z -1=0 \) which intersects the ellipsoid \( \frac{x^2}{48}+\frac{y^2}{12}+\frac{z^2}{4}=1 \) is an ellipse. Determine its semi-axes.
- Find the equation of the sphere through the circle \( x^2 + y^2 + z^2 = 25 \), \( x + 2y -z+ 2 = 0 \) and the point \( (1, 1, 1) \).
- Find the equation of the cylinder where generators are parallel to the line \(x=-\frac{y}{2}=\frac{Z}{3} \) and whose guiding curve is the ellipse \(x^2 +y^2 =1, z=3 \).
- Find the equation of the right circular cone where axis is \(\frac{x}{1}=\frac{y}{0}=\frac{Z}{-2} \) and radius is equal to \(7 \).
-
Answer any one question: \( \pmb{[5\times 1] } \)
- Find the polar equation of the tangent to the circle \( r = 2d \cos \theta \) at the point whose vectorial angle is \( \theta_0 \).
- Find the equations of the generating lines of the hyperboloid of one sheet \(\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{16}=1 \) which passes through the point \((2,3,-4)\).
-
Answer any one question: \( \pmb{[10\times 1] } \)
- Find the locus of a luminous point, if the ellipsoid \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \) casts a circular shadow on the plane \(z=0\). \( \pmb{[5] } \)
- Reduce the equation \(4x^2 -4xy +y^2-8x-6y+5=0 \) to canonical form. \( \pmb{[5] } \)
- Prove that the locus of the foot of the perpendicular from the a focus of the conic \(\frac{l}{r}=1-e\cos \theta \) on a tangent to it, is given by \(r^2 (1-e^2)-2ler \cos \theta – l^2=0 \). \( \pmb{[5] } \)
- Prove that the axes of sections of the conicoid \(ax^2 +by^2+cz^2=1 \) which pass through the line \(\frac{x}{l}=\frac{y}{m}=\frac{z}{n} \) lie on the cone \(\frac{(b-c)}{x}\left(mz-ny \right)+\frac{(c-a)}{y}\left(nx-lz \right)+\frac{(a-b)}{z}\left(ly-mx \right)=0 \). \( \pmb{[5] } \)
UNIT-IV: Differential Equations
-
Answer any two questions: \( \pmb{[2\times 2] } \)
- Find the integrating factor of the following differential equation \( \frac{dy}{dx}+ \frac{xy}{1-y^2} -y\sqrt{x} =0 \).
- Reduce the equation \( \sin y \frac{dy}{dx} = \cos x (2 \cos y – \sin^2 x) \) into a linear equation.
- Show that the equation \( M(x, y) \, dx + N(x, y) \, dy = 0 \) will be exact if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).
-
Answer any one question: \( \pmb{[5\times 1] } \)
- Show that the substitution \( z = ax + by + c \) changes \( y^{\prime} = f(ax + by + c) \) into an equation with separable variables, and apply this method to solve the equation \( y^{\prime} = \sin^2 (x-y+1) \).
- Reduce the equation \(\left( 2x^2 -1\right)\left( \frac{dy}{dx}\right)^2+\left( x^2 + y^2 +2xy+2\right)\frac{dy}{dx}+2y^2+1=0 \) to clairaut’s form by the substitution \(x+y=u \) and \(xy-1=v \), hence solve the equation.
- Show that if the substitution \(y_1 \) and \(y_2 \) be the solutions of the equation \(\frac{dy}{dx}+Py=Q \) where \(P\) and \(Q\) are functions of \(x\) alone, and \(y_2 =y_1 z\) then \( z=1+ae^{-\int \frac{Q}{y_1} dx}\).
- Show that the solution \(\frac{dy}{dx}+Py=Q \) can also be written in the form \(y=\frac{P}{Q}-e^{\int P \, dx}\left[c+\int e^{\int P \, dx} \, \left(\frac{P}{Q} \right) \right] \).
FAQs
-
Where can Mathematics Honours CBCS Question Papers for 2017-18 be found?
These papers are available on the Vidyasagar University website and academic repositories. -
Why are Mathematics Honours CBCS question papers important?
They provide a comprehensive understanding of exam patterns and essential topics. -
Are 2017-18 question papers aligned with the current CBCS syllabus?
Yes, these papers follow the CBCS structure, offering relevant preparation material. -
How to use Mathematics Honours CBCS Question Papers effectively?
Practicing them regularly helps identify key topics and improve problem-solving skills. -
What are the main topics covered in the 2017-18 Mathematics question papers?
Topics include Algebra, Calculus, Differential Equations, Linear Programming, and more. -
Can Mathematics Honours CBCS Question Papers be downloaded?
Yes, they can be downloaded from university portals and academic websites. -
Are solutions for Mathematics CBCS Question Papers available?
Solved papers are offered by online platforms and educational publishers. -
How many years of question papers should be reviewed?
Reviewing at least five years of papers, including 2017-18, is recommended. -
Do question patterns differ each year under CBCS?
While the syllabus remains consistent, question patterns may vary slightly. -
What resources can be used to find old question papers?
University archives, online academic platforms, and libraries are valuable resources.
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