IGNOU BEY 18 SOLVED ASSIGNMENT

Ques 1.

If A, B are symmetric matrices of the same order, then what will be the type of matrix (AB-BA)? Give reasons in support of your answer.

Ques 2.

If the matrix A is both symmetric and skew-symmetric, then determine A.

Ques 3.

If A is a square matrix, such that A² =A, then find (I + A)³ - 7A.

Ques 4.

Ques 5.

, and  find vector  such that and

Ques 6.

Show that area of a parallelogram whose diagonals are given by and  is Also find the area of the parallelogram whose diagonals are  and 

Ques 7.

Find a vector of magnitude 6, which is perpendicular to both the vectors  and

Ques 8.

Find the angle between the vectors and .

Ques 9.

If , show that   x  =  x  =  x .

Ques 10.

If A,B,C,D are the points with position vectors , respectively. Find the projection of  along .

Ques 11.

Using vectors, find the area of the triangle ABC with vertices A(1,2,3), B(2, -1, 4) and C(4,5,-1).

Ques 12.

Prove that

Ques 13.

Prove that

   is independent of a, b, c.

Ques 14.

Show that the conical tent of given capacity will require the least amount of canvas if its height is √2 times its base radius.

Ques 15.

An open storage bin with square base and vertical sides is to be constructed from a given amount of material. Determine its dimensions if its volume is to be maximum neglecting the thickness of material and waste in constructing it.

Ques 16.

Find the height of a right cylinder with greatest lateral surface area that may be inscribed in a given sphere of radius R.

Ques 17.

Given a point on the axis of the parabola y² = 2px at a distance a from the vertex, find the abscissa of the point of the curve closest to it.

Ques 18.

Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied.

i. f(x) = sin² x on the interval [0, π].

ii. f (x) = x² + 4 on [-2, 2].

iii. f(x) = sinx + cos x on  

iv. f(x) = x³- 2x on [0, 1].

Ques 19.

Explain why Lagrange's mean value theorem is not applicable to the following functions in the respective intervals: f(x) = 13x +11, x∈ [1,3].

Ques 20.

Verify means values theorem for the function f(x) = 4x³-4x in the interval [a,b], where a=0, and b=3.

Ques 21.

Find 'c' of Cauchy's mean value theorem for the function f(x) = 2.ln(x) and g(x) = x²-1 in the interval [2,3].

Ques 22.

Find the first order partial derivatives

Ques 23.

Ques 24.

Ques 25.

Find  and  for the function x² sin(y³)+xe^3z-cos(z²)=3y-6z+8 

Ques 26.

Find the orthogonal trajectories of the family of circles x² + (y - c)² = c², where c is a parameter.

Ques 27.

In a certain isolated population p (t) the rate of population growth  is equal to , where k and ɛ are both positive constants. If p (0) = 1, then find the limiting population as t→ ∞.

Ques 28.

Use the method of Laplace transforms to find the solution of the initial value problem y" + 9y = 6 cos 3t, y(0) = 2, y'(0) = 0

Ques 29.

Determine the solution of the undamped (forced vibrations) system mü + Ku = F_o cos wt, u(0) = 0, u(0) = 1 When, w ≠  

Ques 30.

Reduce the equation

xy" + y' + xy = 0, x > 0

into Bessel's equation and hence write its general solution in terms of Bessel's functions.

Ques 31.

Prove that

Ques 32.

If A, B are symmetric matrices of the same order, then what will be the type of matrix (AB-BA)? Give reasons in support of your answer.

Ques 33.

If the matrix A is both symmetric and skew-symmetric, then determine A.

Ques 34.

If A is a square matrix, such that A² =A, then find (I + A)³ - 7A.

Ques 35.

Ques 36.

, and  find vector  such that and

Ques 37.

Show that area of a parallelogram whose diagonals are given by and  is Also find the area of the parallelogram whose diagonals are  and 

Ques 38.

Find a vector of magnitude 6, which is perpendicular to both the vectors  and

Ques 39.

Find the angle between the vectors and .

Ques 40.

If , show that   x  =  x  =  x .

Ques 41.

If A,B,C,D are the points with position vectors , respectively. Find the projection of  along .

Ques 42.

Using vectors, find the area of the triangle ABC with vertices A(1,2,3), B(2, -1, 4) and C(4,5,-1).

Ques 43.

Prove that

Ques 44.

Prove that

   is independent of a, b, c.

Ques 45.

Show that the conical tent of given capacity will require the least amount of canvas if its height is √2 times its base radius.

Ques 46.

An open storage bin with square base and vertical sides is to be constructed from a given amount of material. Determine its dimensions if its volume is to be maximum neglecting the thickness of material and waste in constructing it.

Ques 47.

Find the height of a right cylinder with greatest lateral surface area that may be inscribed in a given sphere of radius R.

Ques 48.

Given a point on the axis of the parabola y² = 2px at a distance a from the vertex, find the abscissa of the point of the curve closest to it.

Ques 49.

Can Rolle's theorem be applied to each of the following functions? Find 'c' in case it can be applied.

i. f(x) = sin² x on the interval [0, π].

ii. f (x) = x² + 4 on [-2, 2].

iii. f(x) = sinx + cos x on  

iv. f(x) = x³- 2x on [0, 1].

Ques 50.

Explain why Lagrange's mean value theorem is not applicable to the following functions in the respective intervals: f(x) = 13x +11, x∈ [1,3].

Ques 51.

Verify means values theorem for the function f(x) = 4x³-4x in the interval [a,b], where a=0, and b=3.

Ques 52.

Find 'c' of Cauchy's mean value theorem for the function f(x) = 2.ln(x) and g(x) = x²-1 in the interval [2,3].

Ques 53.

Find the first order partial derivatives

Ques 54.

Ques 55.

Ques 56.

Find  and  for the function x² sin(y³)+xe^3z-cos(z²)=3y-6z+8 

Ques 57.

Find the orthogonal trajectories of the family of circles x² + (y - c)² = c², where c is a parameter.

Ques 58.

In a certain isolated population p (t) the rate of population growth  is equal to , where k and ɛ are both positive constants. If p (0) = 1, then find the limiting population as t→ ∞.

Ques 59.

Use the method of Laplace transforms to find the solution of the initial value problem y" + 9y = 6 cos 3t, y(0) = 2, y'(0) = 0

Ques 60.

Determine the solution of the undamped (forced vibrations) system mü + Ku = F_o cos wt, u(0) = 0, u(0) = 1 When, w ≠  

Ques 61.

Reduce the equation

xy" + y' + xy = 0, x > 0

into Bessel's equation and hence write its general solution in terms of Bessel's functions.

Ques 62.

Prove that

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