Questions asked by Swati V.

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• Find the equation of the conic of which one focus lies at (2,1), one directrix is x+y=0 and it passes through (1,4)?Also identify the conic.
  
• Find the equation of the conic of which one focus lies at (2,1), one directrix is x+y=0 and it passes through (1,4)?Also identify the conic.Also reduced conic obtained to standard form and draw a rough skech of the conic obtained.
  
• A rotating liquid forms a surface in the form of paraboloid.The surface is 2m deep at the centre and 10m across.Obtain an equation of the surface?
  
• Identify an axis of revolution and generating conic of the surface #4x^2+25y^2+4z^2=100#. Does this conic also generate #x^2/4+y^2/25+z^2/4=1# ?Give reasons for your answer.'
  
• Prove that product of the distance from any point on a hyperbola to its asymptotes is a constant?
  
• Can any conic have its focus lying on the corresponding directrix?Give reasons for your answer
  
• Under what conditions on #alpha# do the spheres #x^2+y^2+z^2+alphax-y=0# and #x^2+y^2+z^2+x+2z+1=0# intersect each other at an angle of #45^circ#?
  
• Find the distance of the centre of the circle x^2+y^2+z^2+x-2y+2z=3,2x+y+2z=1 from the plane ax+by+cz=d, where a,b,c,d are constants.?
  
• Check whether or not the conicoid represented by #5x^2+4y^2-4yz+2xz+2x-4y-8z+2=0# is central or not. If it is, transform the equation by shifting the origin to the center. Else, change any one coefficient to make the equation that of a central conicoid.?
  
• Trace the surface #x^2/4+y^2/9-z^2/4=1#. Also, describe its sections by the planes x=±2,algebraically and geometrically.?
  
• Find the equation of right circular cylinder whose curve is the circle in #(x+1/2)^2+(y-1)^2+(z+1)^2=21/4# has as center the point #(-1/2,1,-1)#?
  
• Find the equation of those tangent planes to the sphere #x^2+y^2+z^2+2x-4y+6z-7=0# which intersect in the line #6x-3y-23=0,3z+2=0#?
  
• 'If x/1=y/1=z/-1 represents one of the three mutually perpendicular generators of the cone 3xy+8xz-5yz=0,find the equations of other two.?
  
• Prove that the paraboloids x^2/a1^2+y^2/b1^2=2z/c1,x^2/a2^2+y^2/b2^2=2z/c2,x^2/a3^2+y^2/b3^2=2z/c3 have a common tangent plane,if |a1^2 a2^2 a3^2 | |b1^2 b2^2 b3^2 |=0? |c1 c2 c3 |
  
• Under a rotation of axes, a parabola can become a hyperbola.please give a short proof or a counterexample?
  
• Which of the following statement are true? Give reasons for your answers, either with a short proof or counterexample. 1.Given any two conics C1 and C2, atleast one element of R×R will belong to C1∩C2. 2.1/√2,1/√3,1/√5form direction cosines of a line.
  
• Give reasons for your answers, either with a short proof or counterexample.? 1.#2x+3y=7z# represents a line in three dimensional space. 2.#x=y=z-1# does not intersect cone #x^2+y^2+z^2+2(yz+zx+xy)=0#.
  
• Give reasons for your answers, either with a short proof or counterexample.? ax^2+by^2+cz^2=d represents a sphere with radius √a^2+b^2+c^2-d, where a,b,c,d are positive real numbers.
  
• Prove that an element of an integral domain is a unit iff it generates the domain.?
  
• Give an example, with justification, to show that: 1.the union of integral domains need not be an integral domain; 2.all finite fields are not isomorphic.?
  
• Check whether or not #{z∈C|z^5=1}# is a group with respect to addition.?
  
• Write the Cayley tables for addition and multiplication in Z7.?
  
• Let R be a ring with identity , S an integral domain and f:R--->S a non-trivial ring homomorphism.Show that the ideal,Ker f, is a prime ideal of R.?
  
• Which of the following statements are true/false?Give reasons for your answers. 1.If σ is an even permutation,then σ^2=1.
  
• Give an example of a function which is one to one but not onto,with reason.?
  
• Which of the following statement are true/false? Give reasons for your answers. 1.Every cylinder has a circular base. 2. Every conicoid has a unique centre.
  
• Differentiate cos^-1(2x^2-1) with respect to sin^-1√1-x^2?
  
• Prove that #In=sec^n-2xtanx/n-1+n-2/n-1 In-2#,where #In=∫sec^n dx#. Hence deduce #I4#?
  
• Integrate the ∫1/x^2-6x+11 dx from -5 to 5?
  
• Please Integrate the ∫(x^2+1)/(2x+1)(x-1)(x+1) dx?
  
• If # (1+x^2)y_(n+1)+(2nx-m)y_n+n(n-1)y_(n-1)=0 #?
  
• If xsiny=sin(p+y),p∈R,show that sinpdy/dx+sin^2y=0?
  
• Prove that ∫√cos^n x dx/√cos^n x +√sin^n x from 0 to π/2=π/4.?
  
• Find the intervals in #R# over which #∫(t+1)^3 e^t dt# from #-1# to #x# is decreasing?
  
• Reduce the conic x^2+6xy+y^2-8=0 to standard form.Hence verify the given conic?
  
• Find the inverse of matrix B={(-1,-3,0),(2,4,0)(-1,-1,2)} by finding the adjoint as well as using Cayley-Hamiltion theorem?
  
• Let #T:P_2→P_1# be defined by #T(a+bx+cx^2)=b+2c+(a-b)x#. Check that #T# is a linear transformation. Find the matrix of the transformation with respect to the ordered bases #B_1={x^2,x^2+x,x^2+x+1}# and #B_2={1,x}#. Find the kernel of #T#.?
  
• Find the inverse of the matrix #{[-1,2,1],[0,1,1],[1,0,2]}# using row reduction?
  
• Check whether the following system of equations has a solution. 4x+2y+8z+6z=3 2x+2y+2z+2w=1 x+3z+2w=3?
  
• Which of the following are binary operations on #S={x∈R|x>0}#? Justify your answer. (i)The operations #∇# is defined by #x∇y=|ln(xy)|# where #lnx# is a natural logarithm. (ii) The operations #∆# is defined by #x∆y=x^2+y^3#.
  
• Check whether the matrices A and B are diagonalisable?Diagonalise those matrices which are diagonalisable. (i) #A={[-2,-5,-1],[3,6,1],[-2,-3,1]}# (ii) #B={[-1,-3,0],[2,4,0],[-1,-1,2]}#.
  
• If #G# is a group such that #o(G)=2m#, where #m in NN#, then #G# has a subgroup of order #m#. This statement is true/false? Please give reasons for your answers.
  
• If #G=<x># is of order 25, then #x^α# generates #G#, where #α# is a factor of 25. This statement is true/false? Please give reasons for your answer.
  
• Give an example with justification,of a function #f:R_1 rarr R_2#, where #(R_1,+,*)# and #(R_2,+,*)# are rings and such that #f:(R_1,+) rarr (R_2,+)# is a group homomorphism but #f# is not a ring homomorphism?
  
• Let #p# be a prime.Show that #S={m+nsqrt(-p) | m,n in ZZ}# is a subring of #CC#..Further,check whether or not #S# is an ideal of #CC#?
  
• Give an example,with justification, of each of the following:(i) A zero divisor in ZZ_5, (ii)An element of C [0,1] which is not a zero divisor, (iii)A subring of an integral domain which is not an integral domain.?
  
• Which of the following statements are true/false?Justify your answer. (i)R² has infinitely many non-zero, proper vector subspaces.(ii)Every system of homogeneous linear equations has a non zero solution.
  
• The function f,defined by f(x)=x-1/3-x, has the same set as domain and as range. This statement is true/false?Please give reasons for your answer.
  
• Prove that if G≠{e} and G has no proper non-trivial subgroup, then G is finite and o(G) is a prime number?
  
• Prove that R^n/R^m≃R^(n-m) as groups,where n,m∈N,n≥m?
  
• If #f(x)=(4x^2-7x-2)/(x-2),x!=2#,find #aδ>0# such that #|f(x)-9|<1/100# for #0<|x-2|<δ#. Hence show that #lim_(xto2)f(x)=9#?
  
• Prove that for every #x>0,x/(1+x^2)<tan^(-1)(x)<x#?
  
• Which of the following statements are true or false?Give reasons for your answers.(i) the greatest integer function is continuous on RR.(ii)The domain of the function f,given by f(x)=√2-x/x, is ]0,1[.'
  
• Find the values of a,b,c such that lim_"x→0"ae^(x)-bcosx+ce^(-x)/xsinx=3/2?
  
• Show that ∫dx from 0 to 1 ∫(x^2-y^2)/(x^2+y^2)dy from 0 to 1=∫dy from 0 to 1 ∫(x^2-y^2)/(x^2+y^2)dx from 0 to 1?
  
• Find the slopes of the tangents to the curves of intersections of planes x=0,y=2 and the surface #z=x^3+e^(yx)# at point (0,2,1)?
  
• Find the point on the ellipse x^2/4+y^2=1,that is nearest to the origin?
  
• Check whether the function #f:RR^2->RR# defined by #f(x,y)=2x^4-3x^2y+y^2# has an extrema at #(0,0)#?
  
• Check whether there exists a continuously differentiable function #g# defined by #f(x,y)=0# in the neighborhood of #x=3#, such that #g(3)=1/3#. Find #g'(3)#, if it exists?
  
• If possible, find a function f such that #grad f = (4x^3+9x^2y^2, 6x^3y+6y^5)#?
  
• Write down the sample spaces for the following experiments (i)A coin is tossed and at the same time a die is rolled.(ii)The order in which a mouse,a frog,and a rabbit arrive at a lake is observed.?
  
• Find the probability of drawing an ace or a spade from a deck of 52 cards in a single draw?
  
• In Bengal, 30% of the population has a certain blood type.What is the probability that exactly four out of a randomly selected group of 10 Bengalis will have that blood type?
  
• Records show that the probability is 0.00006 that a car will have a flat tire while driving through a certain tunnel.Find the probability that at least 2 of 10,000 cars passing through this channel will have flat tires?
  
• Let the function f:RR²→RR be defined as f(x,y)={xy(x²-y²)/x²+y², (x,y)≠(0,0) {0, (x,y)=(0,0) Show that (i)f_x(0,y)=y,for all y (ii)f_x(x,0)=x,for all x. Hence verify that f_xy(0,0)≠f_yx(0,0).?
  
• Write the Cayley tables for addition and multiplication in #ZZ_7#?
  
• D/dx[∫sin(t²)dt from x² to 0]=-sin(x²).This statement is true or false?Please give reasons for your answers.
  
• Taking an example,write the reaction of glycoside formation in monosaccharides.What is the significance of this reaction in carbohydrate chemistry?
  
• How does the presence of double bonds affect the melting point in fatty acids?Explain with the help of two examples.
  
• Describe the bonds and forces that stabilise the tertiary globular structure of proteins.?
  
• Explain the mechanism by which enzymes are able to lower the activation energy during a reaction.?
  
• Calculate the approximate value of #int_0^6x^3 dx# by taking 6 subintervals of equal length and applying Simpson's rule?
  
• Find the volume of the solid obtained by revolving the curve x=acos^(3)θ,y=asin^(3)θ about the y-axis?
  
• Evaluate the integral by converting to polar coordinates# \int_{0}^{sqrt3} \int_{y}^{sqrt(4-y^2)} (dxdy)/(4+x^(2)+y^(2))#.?
  
• Show that the line integral #int_{-1,2}^{3,1}(y^(2)+2xy)dx+(x^(2)+2xy)dy# is independent of path and evaluate it?
  
• Evaluate #int(xsqrt(a^2-x^2))/sqrt(a^2+x^2) dx#?
  
• Trace the curve x[y^(2)+4]=8 stating all the points used for doing so?
  
• Find the domain of the function f,defined by f(x)=√x^(3) (9-x)?
  
• Find the length of the curve #y=ln(e^(x)-1/e^(x)+1)# from #x=1# to #x=2#?
  
• Solve the following differential equations: (i)#(2x+y+3)dy/dx=x+2y+1# and (ii)#[x^(2)+1]dy/dx+2xy=4x^(2)#.?
  
• Evaluate the integral: ∫xln|x+1|dx. ?
  
• D/dx[∫sin(t²)dt from x² to 0]=-sin(x²).This statement is true or false?Please give reasons for your answers.
  
• The cost of fuel is running an engine is proportional to the square of the speed in km/h is ₹48 per hour when the speed is 16 kn/h.Other costs amount to ₹300 per hour.Find the most economical speed?
  
• Integrate the int 1/(x^2-6x+11) dx from -5 to 5?