Thursday, November 9, 2017

Sheet-6

Progressions

General Term of Arithmetic Progression

1. If pth term, qth term, rth term of an arithmetic progression are in arithmetic progression, show that p, q, r are in arithmetic progression.

2. If mth and nth terms of an arithmetic progression be equal to n and m respectively, find its pth term, where p equals mn.

General Term of Geometric Progression

3. Natural numbers p, q, r, s are in arithmetic progression. If P, Q, R, S be respectively the pth, qth, rth and sth terms of a geometric progression, show that P, Q, R, S are in geometric progression.

4. Let a, b, c, d be distinct real numbers and S(x) be a quadratic expression in real variable x, with the sum of the squares of a, b, c as leading coefficient, the sum of the squares of b, c, d as constant term and -2(ab+bc+cd) as the coefficient of x. If S(x) is not positive for any value of x, show that a, b, c, d are in geometric progression.

5. Let f(x)=2x+1. Show that the unequal numbers f(x), f(2x), f(4x) can be in geometric progression for no value of x.

General Term of Harmonic Progression

6. Let T(r) denotes the rth term of a progression. If T(r)-T(r+1) bears a constant ratio with T(r).T(r+1), show that T(1), T(2), T(3), ... are in harmonic progression.

7. Let T(r) denotes the rth term of a progression. If the ratio of T(2).T(3) to T(1).T(4) and that of T(2)+T(3) to T(1)+T(4), both be equal to three times the ratio of T(2)-T(3) to T(1)-T(4); show that T(1), T(2), T(3), ... are in harmonic progression.

8. Show that the square roots of any three prime numbers cannot be any three terms of an arithmetic, a geometric or a harmonic progression.

Sunday, November 5, 2017

Sheet-5

Angles

Alternate Interior Angles

1. Show that an angle of one of the two triangles formed by joining a diagonal of a trapezium equals an angle of the other.

2. Four triangles are formed when the diagonals of a parallelogram are joined. Show that the triangles based on opposite sides of the parallelogram have same angles.

3. D and E are points on the sides AB and AC respectively of a triangle ABC. A line through C parallel to AB intersects DE produced at F. Show that the triangles ADE and CEF have same set of angles.

4. D, E and F are points on the sides BC, CA and AB respectively of a triangle ABC such that DE, EF and EF are parallel to AB, BC and CA respectively. Show that the triangle DEF has same set of angles as the triangle ABC has.

Angle Sum Property of Triangles

5. Show that if two angles of a triangle are equal, then these are acute. Can the triangle be obtuse?

6. If a triangle has all the three angles different, show that nether the smallest nor the greatest of them can be 60 degrees.

7. The bisectors of the angles A and B meet at O. Show that angle AOB is obtuse.

8. Angle A of a triangle ABC is 100 degrees. What are the measures of angles B and C if the greatest of the differences A-B and A-C has the least value? What is this least value?