MAT2131 — Repeated / Same / Similar Previous-Term Questions¶
1. Cartesian & Polar Coordinates¶
Q1(a) [3]: Find the relation between Cartesian coordinates and Polar coordinates. [171 Term]
Q1(a) [3]: Find the relation between Cartesian coordinate and polar coordinate. [191 Term]
Q1(a) [4]: Find the relation between Cartesian coordinates and Polar coordinates. [201 Term]
Q1(a) [6]: Find the relation between Cartesian coordinates and polar coordinates. Transform to polar coordinates the equation (x3=y2(2a-x)). [211 Term]
2. Condition for Pair of Straight Lines¶
Q1(c) [5]: Find the condition that the general equation (ax2+2hxy+by2+2gx+2fy+c=0) may represents a pair of straight lines. [161 Term]
Q1(a): Find the condition that the general equation of the second degree (ax2+2hxy+by2+2gx+2fy+c=0) may represents a pair of straight lines. [181 Term]
3. Homogeneous Quadratic → Pair of Straight Lines Through Origin¶
Q1(c) [4]: Prove that a homogeneous quadratic equation (ax2+2hxy+by2=0) always represents a pair of straight lines which passing through the origin. [171 Term]
Q1(c) [3]: Prove that a homogeneous quadratic equation (ax2+2hxy+by2=0) always represents a pair of straight lines which passing through the origin. [211 Term]
4. Find Parameter so Equation Represents Pair of Straight Lines¶
Q3(a) [3]: Find the value of (\lambda) so that the equation (\lambda x2+4xy+y2-4x-2y-3=0) may represent a pair of straight lines. [171 Term]
Q2(a): Find the value of k so that the equation (6x2+2kxy+12y2+22x+31y+20=0) may represent a pair of straight lines. [181 Term]
Q3(c) [5]: For what value of (\lambda) the equation (2xy+4x-6y+\lambda=0) represent two intersecting lines? If (\lambda=17), then what type of conic represents by the given equation? [201 Term]
5. Pair of Parallel Lines¶
Q2(b) [3]: Prove that the equation (x2+6xy+9y2+4x+12y-5=0) represent a pair of parallel lines. [171 Term]
Q2(b) [3+5]: Prove that the equation (x2+6xy+9y2+4x+12y-5=0) represent a pair of parallel lines. [211 Term]
6. Straight Lines Equidistant from Origin¶
Q2(a) [5]: Prove that the straight lines represented by the equation (ax2+2hxy+by2+2gx+2fy+c=0) will be equidistance from the origin, if (f4-g4=c(bf2-ag2)). [171 Term]
Q2(b) [5]: Prove that the straight lines represented by the equation (ax2+2hxy+by2+2gx+2fy+c=0) will be equidistance from the origin, if (f4-g4=c(bf2-ag2)). [201 Term]
7. Each Pair Bisects the Angle Between the Other Pair¶
Q3(b) [3]: If the pair of straight lines (x2-2mxy-y2=0) and (x2-2nxy-y2=0) be such that each pair bisects the angle between the other pair, prove that (mn+1=0). [171 Term]
Q2(a) [3]: If the pair of straight lines (x2-2axy-y2=0) and (x2-2bxy-y2=0) be such that each pair bisects the angle between the other pair, prove that (ab=-1). [201 Term]
8. Equation of Angle Bisectors¶
Q1(c) [6]: Show that the necessary condition of bisectors of the angles between the lines represented by (ax2+2hxy+by2=0) is (\frac{x2-y2}{a-b}=\frac{xy}{h}). [201 Term]
Q2(a) [3]: Show that the necessary condition of bisector of the angles between the lines represented by (ax2+2hxy+by2=0) is (\frac{x2-y2}{a-b}=\frac{xy}{h}). [211 Term]
9. Translation / Shifting of Axes¶
Q1(b) [5]: Remove first degree terms from the equation (3x2+4y2-12x+4y+13=0). [161 Term]
Q1(b): By transforming to parallel axes through a properly chosen point ((h,k)), prove that the equation (12x2-10xy+2y2+11x-5y+2=0) can be reduced to one containing only the terms of the second degree. [181 Term]
Q1(b) [5]: Origin shifted at ((4,-6)). Find transformation of equation (3x2+5y2+10xy+10=0). [191 Term]
10. Rotation of Axes¶
Q1(c) [6]: If the direction of axes is turned through an angle (45^\circ) and the origin remains unchanged then find the transformation equation of (x2-2xy+y2+2x-4y+3=0). [191 Term]
Q1(b) [4]: If the Axes are turned through (45^\circ), find the transformed form of the equation (3x2+3y2+2xy=2). [201 Term]
Q1(b) [5]: Determine the equation of the parabola (x2-2xy+y2+2x-4y+3=0) after rotating of axes through (45^\circ). [211 Term]
11. Nature of Conic¶
Q3(a): Test the nature of the conic given by the equation (3x2-8xy-3y2+10x-13y+8=0). [181 Term]
Q3(b) [3]: If (ax2+2hxy+by2+2gx+2fy+c=0) represents the non-degenerate conic, then write down the natures of conic. [201 Term]
Q3(c) [5]: For what value of (\lambda) the equation (2xy+4x-6y+\lambda=0) represent two intersecting lines? If (\lambda=17), then what type of conic represents by the given equation? [201 Term]
Q2(c) [5]: Test the nature and find the center of the conic represented by (2x2-3xy+y2-5x-4y+6=0). [211 Term]
12. Shortest Distance Between Two Straight Lines¶
Q2(a) [8]: Find the length and equation of shortest distance (SD) between the lines (\frac{x-1}{4}=\frac{y-2}{3}=\frac{z-1}{-5}) and (\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-4}{-4}). [161 Term]
Q3(c) [4]: Find the shortest distance between the straight lines through the points (P(6,2,2)) and (Q(-4,0,-1)) in the directions ((4,-2,2)) and ((3,-2,-2)) respectively. [171 Term]
Q2(b) [3+5]: Find the length and the equation of shortest distance between the lines (\frac{x-1}{4}=\frac{y-2}{3}=\frac{z-1}{-5}) and (\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-4}{-4}). [181 Term]
Q5(b) [6]: Find the magnitude and the equation of shortest distance between the lines (\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-4}{4}) and (\frac{x-2}{2}=\frac{y-4}{4}=\frac{z-5}{5}). [191 Term]
Q2(c) [6]: Find the length of the shortest distance between the lines (\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}) and (\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}). State whether the lines are coplanar or not. [201 Term]
13. Coplanarity of Two Lines¶
Q2(c): Show that the lines (\frac{x+3}{2}=\frac{y+15}{3}=\frac{z-7}{-3}) and (\frac{x+1}{4}=\frac{y+1}{5}=\frac{z+1}{-1}) are coplanar. [181 Term]
Q4(a) [3]: Obtain the conditions of coplanarity of two straight lines in vector form. [191 Term]
Q2(c) [6]: Find the length of the shortest distance between the lines (\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}) and (\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}). State whether the lines are coplanar or not. [201 Term]
14. Plane Perpendicular to a Plane and Containing Line of Intersection¶
Q2(b) [6]: Obtain the equation of the plane which is perpendicular to the plane (x+2y-3z+1=0) and which contains the line of intersection of the planes (3x+y+z-5=0) and (2x-3y+z+11=0). [161 Term]
Q2(c) [6]: Find the equation of the plane which is perpendicular to the plane (5x+3y+6z+8=0) and which contains the line of intersection of planes (x+2y+3z-4=0) and (2x+y-z+5=0). [171 Term]
15. Position Vector, (\overrightarrow{PQ}) and Magnitude¶
Q4(a) [1+3]: What is position vector? The position vectors of points P and Q are given by (\vec r_1=2\hat i+3\hat j-\hat k) and (\vec r_2=4\hat i-3\hat j+2\hat k). Determine (\overrightarrow{PQ}) in terms of (\hat i,\hat j,\hat k) and its magnitude. [171 Term]
Q4(a) [3]: The position vectors of points P and Q are given by (\vec r_1=2\hat i+3\hat j-\hat k), (\vec r_2=4\hat i-3\hat j+2\hat k). Determine (\overrightarrow{PQ}) in terms of (\hat i,\hat j,\hat k) and finds its magnitude. [201 Term]
Q4(b) [3]: If (\overrightarrow{OA}=2\hat i+3\hat j-4\hat k) and (\overrightarrow{OB}=4\hat i-3\hat j+2\hat k), then find (|\overrightarrow{AB}|). [211 Term]
16. Vector Equation of Line Through Two Points¶
Q4(b) [3]: Find the vector equation of the line passing through the points ((3,1,1)) and ((2,2,-1)). [161 Term]
Q4(d) [3]: Find the vector equation of the line passing through the points ((3,1,1)) and ((2,2,-1)). [171 Term]
17. Angle Made by a Vector with Coordinate Axes¶
Q4(a): Find the angles which the vector (\vec P=3\hat i-6\hat j+2\hat k) makes with the coordinate axes. [181 Term]
Q4(c) [4]: Find the angle which the vector (\vec A=3\hat i-6\hat j+2\hat k) makes with the coordinate axes. [201 Term]
18. Vector Projection¶
Q4(b) [6]: Find the projection of (\vec a) along (\vec b) and (\vec b) along (\vec a) where (\vec a=2\hat i+6\hat j-3\hat k) and (\vec b=\hat i+4\hat j+8\hat k). [191 Term]
Q4(b) [3]: Find the projection of the vector (4\hat i-3\hat j+\hat k) on the line passing through the points ((2,3,-1)), ((-2,-4,3)). [201 Term]
19. Vector Triple Product¶
Q4(a) [5]: Prove that (\vec a\times(\vec b\times\vec c)+\vec b\times(\vec c\times\vec a)+\vec c\times(\vec a\times\vec b)=0). [161 Term]
Q3(c): If a, b, c are three vectors, prove that (\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c). [181 Term]
Q4(c) [5]: If (\vec a=4\hat i-\hat j-3\hat k), (\vec b=2\hat i+5\hat j), and (\vec c=\hat i+5\hat j-2\hat k), then verify the vector triple product formula. [191 Term]
20. Directional Derivative¶
Q4(c) [6]: Find the directional derivative of (g=x2yz2+4y^2z) at ((-1,-2,-1)) in the direction (5\hat i-2\hat j-3\hat k). [161 Term]
Q3(a) [4]: Find the directional derivative of (\phi=x2yz2+4y^2z) at ((-1,-2,-1)) in the direction (5\hat i-2\hat j-3\hat k). [191 Term]
Q4(c): Find the directional derivative of (\phi=4e^{2x-y+z}) at ((1,1,-1)) in a direction toward the point ((-3,5,6)). [181 Term]
Q5(b): Find the directional derivative of (\phi=4e^{2x-y+z}) at ((1,1,-1)) in a direction toward the point ((-3,5,6)). [211 Term]
Q5(b) [4]: Find the directional derivative of (\phi=x2yz+4xz2) at ((1,-2,-1)) in the direction (2\hat i-\hat j-2\hat k). [201 Term]
21. (\vec V=\vec\omega\times\vec r) and Curl¶
Q5(a) [4]: If (\vec V=\vec\omega\times\vec r), prove that (\vec\omega=\frac12\operatorname{curl}\vec V), where (\vec\omega) is a constant vector. [171 Term]
Q3(c) [5]: Suppose (\vec v=\vec w\times\vec r), show that (\vec w=\frac12\operatorname{curl}\vec v), where (\vec w) is a constant vector. [191 Term]
Q6(a) [5]: What is curl of a vector? If (\vec V=\vec\omega\times\vec r), prove that (\vec\omega=\frac12\operatorname{curl}\vec V), where (\vec\omega) is a constant vector. [201 Term]
Q6(c): If (\vec v=\vec\omega\times\vec r), prove (\vec\omega=\frac12\operatorname{curl}\vec v), where (\vec\omega) is a constant vector. [211 Term]
22. Divergence and Curl¶
Q4(b): Define gradient, divergence and curl. [181 Term]
Q5(a): Define divergence and curl of a vector. [211 Term]
Q5(c) [5]: Find the divergence and curl of the vector field (\vec F=x^2z\hat i-2yz^3\hat j+xy^2z\hat k) at the point ((1,2,-1)). [171 Term]
Q5(b): Determine the divergence and curl of the vector field (\vec V=(2xz^3+6y)\hat i+(6x-2yz)\hat j+(3x^2z^2-y^2)\hat k) at point ((-1,-2,1)). [181 Term]
23. Solenoidal Vector Field¶
Q6(a) [3]: Determine the constant a so that (\vec V=(3x^2+y)\hat i+(2y-z)\hat j+(x+az)\hat k) is solenoidal. [171 Term]
Q6(b) [5]: When a vector is called solenoidal? Show that (\vec A=(2x2+8xy2z)\hat i+(3x^3y-3xy)\hat j-(4y^2z^2+2x^3z)\hat k) is not solenoidal but (\vec B=xyz^2\vec A) is solenoidal. [201 Term]
Q6(b): Show that if (\vec A=(2x2+8xy2z)\hat i+(3x^3y-3xy)\hat j-(4y^2z^2+2x^3z)\hat k) is not solenoidal but (\vec B=xyz^2\vec A) is solenoidal. [211 Term]
24. Irrotational Vector Field¶
Q5(b) [4]: If (\vec A=x^3z\hat i-2xyz\hat j+2yz^3\hat k), find curl A at ((1,-1,1)). Test whether this vector is irrotational or not. [161 Term]
Q4(d): Show that (\vec r=(6xy+z^3)\hat i-(3x^2-z)\hat j+(3xz^2-y)\hat k) is irrotational. [181 Term]
Q3(b) [5]: Check whether (\vec F=(2xy+z^3)\hat i+x^2\hat j+3xz^2\hat k) is irrotational or solenoidal. [191 Term]
Q6(a): When a vector field is called irrotational? Find constants a, b, c so that (\vec V=(x+2y+az)\hat i+(bx-3y-z)\hat j+(4x+cy+2z)\hat k) is irrotational. [211 Term]
25. Conservative Vector Field / Scalar Potential¶
Q5(a) [6]: If (\vec F=(2xy+z^3)\hat i+(x^2+2y)\hat j+(3xz^2-2)\hat k), show that F is conservative and find a scalar (\phi) such that (\vec F=\nabla\phi). [161 Term]
Q7(c): Show that (\vec F=(2xy+z^3)\hat i+x^2\hat j+3xz^2\hat k) is a conservative force field; also find the total work done in moving an object in this field from ((1,-2,1)) to ((3,1,4)). [211 Term]
26. (\nabla^2(1/r)=0)¶
Q6(a) [1+3]: What is vector differential operator? Prove that (\nabla^2\left(\frac1r\right)=0), where (\vec r=x\hat i+y\hat j+z\hat k). [181 Term]
Q5(c): What is differential operator? Prove that (\nabla^2\left(\frac1r\right)=0), where (\vec r=x\hat i+y\hat j+z\hat k). [211 Term]
27. Velocity and Acceleration of a Particle¶
Q5(a): Find the velocity and acceleration of a particle which moves along the curve (x=2\sin3t), (y=2\cos3t), (z=8t) at any time (t>0); find the magnitude of the velocity and acceleration. [181 Term]
Q5(d): A particle moves along the curve (x=2t^2), (y=t^2-4t), (z=3t-5), where t is the time. Find the components of its velocity and acceleration at time (t=1) in the direction (\hat i-3\hat j+2\hat k). [201 Term]
Q4(d): A particle moves along the curve (x=2t^2), (y=t^2-4t), (z=3t-5), where t is time. Find the components of its velocity and acceleration at time (t=1) in the direction (\hat i-3\hat j+2\hat k). [211 Term]
Q6(d): Define ordinary derivatives of vector. A particle moves along a curve whose parametric equations are (x=e^{-t}), (y=2\cos3t), (z=2\sin3t), where t is the time. Determine its velocity and acceleration at any time. [211 Term]
28. Green's Theorem — Statement / Proof¶
Q6(c) [2]: State Green's theorem in the plane. [161 Term]
Q7(a) [6]: State and prove Green's theorem in the plane. [171 Term]
Q7(a): State Green's theorem, Stokes theorem and the Gauss divergence theorem. [181 Term]
Q7(a): State and prove Green's theorem in the plane. [201 Term]
Q7(a): State and prove Green's theorem. [211 Term]
29. Green's Theorem — Apply / Verify¶
Q7(a) [4]: Apply Green's theorem to evaluate (\oint_C[(2x2-y2)dx+(x2+y2)dy]), where C is the boundary of the area enclosed by the x-axis and the upper-half of the circle (x2+y2=a^2). [161 Term]
Q7(b) [5]: Apply Green's theorem to evaluate (\oint_C{(3x-8y^2)dx+(4y-6xy)dy}), where C is the boundary enclosed by (x=0), (y=0) and (x+y=1). [171 Term]
Q7(b) [6]: Compute (\oint_C x2y,dx+y3,dy) by Green's theorem where C is the closed path formed by (y=x) and (y3=x2) from ((0,0)) to ((1,1)). [191 Term]
Q7(c): Verify Green's theorem in the plane for (\oint_C(3x2-8y2)dx+(4y-6xy)dy), where C is the boundary of the region defined by (y=\sqrt{x}) and (y=x^2). [201 Term]
Q7(b): Verify Green's theorem in the plane for (\oint_C(xy+y2)dx+x2dy), where C is the closed curve of the region bounded by (y=x) and (y=x^2). [211 Term]
30. Parametric Line Integral — (x=t2, y=2t, z=t3)¶
Q6(c) [7]: If (\phi=2xyz^2), (\vec F=xy\hat i-2\hat j+x^2\hat k) and C is the curve (x=t^2), (y=2t), (z=t^3) from (t=0) to (t=1), evaluate the line integrals (i) (\int_C\phi,d\vec r); (ii) (\int_C\vec F\times d\vec r). [171 Term]
Q6(a): If (\psi=2xyz^2), (\vec B=xy\hat i-z\hat j+x^2\hat k) and C is the curve (x=t^2), (y=2t), (z=t^3) from (t=0) to (t=1), evaluate the line integrals (i) (\int_C\psi,d\vec r); (ii) (\int_C\vec B\times d\vec r). [191 Term]
Q7(b) [4]: If (\phi=2xyz^2), (\vec F=xy\hat i-z\hat j+x^2\hat k) and C is the curve (x=t^2), (y=2t), (z=t^3) from (t=0) to (t=1), evaluate the line integrals (i) (\int_C\phi,d\vec r); (ii) (\int_C\vec F\times d\vec r). [201 Term]
31. (x,dy-y,dx) Line Integral¶
Q6(a) [5]: Find the line integral (I=\oint(x,dy-y,dx)) round the square ((0,0)), ((1,0)), ((1,1)) and ((0,1)), ((0,0)). [161 Term]
Q6(b) [4]: Find the line integral (I=\int(xdy-ydx)) from ((0,0)) to ((2,2)) over the (i) straight line (y=x); (ii) square ((0,0)) to ((1,0)) to ((1,1)) to ((0,0)). [191 Term]
32. Line Integral Along Different Paths¶
Q6(b) [7]: If (\vec F=y\hat i+(x-2xz)\hat j-xy\hat k), evaluate (\int_C\vec F\cdot d\vec r) from ((0,0,0)) to ((2,4,8)) along the path C: (i) (x=t), (y=t), (z=0); (ii) the straight line joining ((0,0,0)) to ((2,2,2)). [161 Term]
Q5(d): If (\vec A=(3x^2+6y)\hat i-14yz\hat j+20xz^2\hat k), evaluate (\int_C\vec A\cdot d\vec r) from ((0,0,0)) to ((1,1,1)) along the following paths C: (i) (x=t), (y=t^2), (z=t^3); (ii) the straight lines from ((0,0,0)) to ((1,0,0)) then ((1,1,0)). [211 Term]
33. Path Independence / Work Done¶
Q6(c): Show that the line integral (\int[(x2-yz)dx+(y2-zx)dy+(z^2-xy)dz]) is independent of the path of integration between the points ((1,1,1)) and ((2,3,4)) and find its value. [181 Term]
Q7(c): Show that (\vec F=(2xy+z^3)\hat i+x^2\hat j+3xz^2\hat k) is a conservative force field; also find the total work done in moving an object in this field from ((1,-2,1)) to ((3,1,4)). [211 Term]
34. Divergence / Volume Integral¶
Q7(b): Evaluate the volume integral (\iiint\operatorname{div}\vec F,dv) for the vector field (\vec F=(2xy+z)\hat i+y^3\hat j-(e^x+3y)\hat k) taken over the region bounded by (2x+2y+z=6), (x=0), (y=0), (z=0). [181 Term]
Q7(c) [5]: Evaluate (\iiint\nabla\cdot\vec F,dv) where (\vec F=xy\hat i+z^2\hat j+2xz\hat k) bounded by the region (x=0), (y=0), (z=0) and the plane (x+y+z=1). [191 Term]
35. Cylindrical / Spherical / Curvilinear Coordinates¶
Q7(c) [2]: Give the representation of a point (F(x,y,z)) in cylindrical and spherical coordinates. [161 Term]
Q7(d) [4]: Prove that a cylindrical coordinate system is orthogonal. [161 Term]
Q7(c) [3]: Explain a curvilinear coordinate system. [171 Term]
Q7(d): Write down curvilinear coordinates in polar, cylindrical and spherical coordinates. [181 Term]