tag:blogger.com,1999:blog-91759328746951684182018-03-20T00:34:39.455+05:30Mathematics MentorPresents Sheets of Problems on Various Concepts of MathematicsSanjeev Kumarnoreply@blogger.comBlogger6125tag:blogger.com,1999:blog-9175932874695168418.post-55556080945998149152017-11-09T13:24:00.000+05:302017-11-09T13:24:23.379+05:30Sheet-6<div dir="ltr" style="text-align: left;" trbidi="on">Progressions<br /><br />General Term of Arithmetic Progression<br /><br />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.<br /><br />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.<br /><br />General Term of Geometric Progression<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />General Term of Harmonic Progression<br /><br />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.<br /><br />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.<br /><br />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.</div>Sanjeev Kumarhttps://plus.google.com/102586937696118125212noreply@blogger.comtag:blogger.com,1999:blog-9175932874695168418.post-47627732214237716382017-11-05T11:54:00.000+05:302017-11-27T18:46:45.028+05:30Sheet-5<div dir="ltr" style="text-align: left;" trbidi="on">Angles<br /><br />Alternate Interior Angles<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />Angle Sum Property of Triangles<br /><br />5. Show that if two angles of a triangle are equal, then these are acute. Can the triangle be obtuse?<br /><br />6. If a triangle has all the three angles different, show that nether the smallest nor the greatest of them can be 60 degrees.<br /><br />7. The bisectors of the angles A and B meet at O. Show that angle AOB is obtuse.<br /><br />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?<br /><br /><br /></div>Sanjeev Kumarhttps://plus.google.com/102586937696118125212noreply@blogger.comtag:blogger.com,1999:blog-9175932874695168418.post-77869948585252058492017-11-03T10:28:00.000+05:302017-11-07T10:30:27.236+05:30Sheet-4<div dir="ltr" style="text-align: left;" trbidi="on">Progressions<br /><br />Geometric Mean<br /><br />1. Let g be the geometric mean between two numbers m and n; e and f be the arithmetic mean of a,g and g,b respectively. Evaluate<br />(i) (m/e)+(n/f)<br />(ii) g(e+f)/ef<br /><br />2. If one geometric mean G and two arithmetic means K and L be inserted between two given numbers, show that G is the geometric mean between 2K-L and 2L-K.<br />n Geometric Means between Two Numbers <br /><br />3. Two geometric means are inserted between given numbers m and n. Show that the sum of the cubes of these means equals<br />(m+n)mn<br /><br />Harmonic Mean<br /><br />4. If A and H be respectively the arithmetic and harmonic means between two distinct numbers a and b, then prove that<br />H(A-a)(A-b)=A(H-a)(H-b)<br /><br />n Harmonic Means between Two Numbers<br /><br />5. Let n harmonic means be inserted between two distinct numbers a and b. Let F be the first and L be the last harmonic mean. Show that<br />[(F+a)/(F-a)]+[(L+b)/(L-b)]=2n<br /><br />6. If 9 arithmetic means and 9 harmonic means be inserted between 2 and 3, show that A+(6/H) is a constant,where A is any of the 9 arithmetic means and H be the corresponding harmonic mean. What is the value of this constant?<br /><br />7. An odd number of arithmetic, geometric and harmonic means are inserted between two numbers. Show that the middle ones are in geometric progression.<br /><br /><br /></div>Sanjeev Kumarhttps://plus.google.com/102586937696118125212noreply@blogger.comtag:blogger.com,1999:blog-9175932874695168418.post-14343553434319044932017-11-02T13:16:00.000+05:302017-11-07T08:25:20.622+05:30Sheet-3<div dir="ltr" style="text-align: left;" trbidi="on">Angles<br /><br />Angles of Polygons<br /><br />1. The angles of a five-sided figure are x, 2x, x+30, x-10, x+40 degrees, find x.<br /><br />2. The angles of a pentagon are in the ratio 1:2:3:4:5; find them.<br /><br />3. Three of the angles of a quadrilateral are equal; the fourth angle is 120 degrees; find the others.<br /><br />4. In the quadrilateral ABCD, angle ABC equals 140 degrees, angle ADC equals 20 degrees; the lines bisecting the angles BAD, BCD meet at O; calculate angle AOC.<br /><br />5. If the angles of a quadrilateral taken in order are in the ratio 1:3:5:7, prove that two of its sides are parallel.<br /><br />6. Each angle of a polygon is 140 degrees, how many sides has it?<br /><br />7. Find the sum of the interior angles of a 12-sided convex polygon.<br /><br />8. Find the interior angle of a regular 20-sided figure.<br /><br />9. Prove that the sum of the interior angles of an 8-sided convex polygon is twice the sum of those of a pentagon.<br /><br />10. Each angle of a regular polygon of x sides is 3/4 of each angle of a regular polygon of y sides; express y in terms of x, and find any values of x, y which will fit.<br /><br />11. The sum of the interior angles of an n-sided convex polygon is double the sum of the exterior angles. Find n.<br /><br /></div>Sanjeev Kumarhttps://plus.google.com/102586937696118125212noreply@blogger.comtag:blogger.com,1999:blog-9175932874695168418.post-39983178926191356822017-11-01T16:21:00.001+05:302017-11-07T10:35:09.901+05:30Sheet-2<div dir="ltr" style="text-align: left;" trbidi="on">Equation of Ellipse<br /><br />Standard Equation of Ellipse<br /><br />1. The equation to a curve is S(x,y)=0, where S(x,y) is a second degree expression in x and y, in which, the coefficients of the second degree terms in x and y are 4 and 9 respectively; those of x and y are -8 and -36 respectively; and the constant term is 4. Show that the equation represents an ellipse: Find its latus rectum, eccentricity and foci.<br /><br />2. Find the equation to the ellipse whose centre is (-2,3), whose semi-axes are 3 and 2; and the major axis is along the y-axis.<br /><br />3. Find the equation to the ellipse whose one vertex is (3,1), the nearer focus is (1,1) and the eccentricity is 2/3.<br /><br />Parametric Equations of Ellipse<br /><br />4. Find the inclination to the major axis of the diameter of the ellipse, the square of whose length is the harmonic mean between the squares of the major and minor axes.<br /><br />5. An ellipse has major and minor axes along the axes of x and y respectively; their lengths being 2a and 2b respectively. Find the length of a focal chord of it that makes an angle t with the major axis.<br /><br />Equation of Tangent<br /><br />6. Show that the distance of any tangent to the ellipse from one of its foci is inversely proportional to the distance of the tangent from the other one.<br /><br />7. Let AB be the segment of any tangent to an ellipse intercepted between the tangents to the ellipse at the extremities of its major axis. Show that the circle with AB as diameter passes through the foci of the ellipse.<br /><br />8. An ellipse has major and minor axes along the axes of x and y respectively; their lengths being 2a and 2b respectively. Another ellipse is tangent to every that chord of the given ellipse, the eccentric angles of whose extremities differ by 2(pi)/K, for some constant K. Find the equation of the latter ellipse.<br /><br />Director Circle<br /><br />9. An ellipse has major axis 4 and minor axis 2. Another ellipse has major axis √24 and minor axis √12. The major and minor axes of either ellipse are along the axes of x and y respectively. Tangents are drawn to the second ellipse at the extremities of a chord of it that is tangent to the first ellipse. Show that the tangents to the second ellipse are mutually perpendicular.<br /><br />Equation of Normal<br /><br />10. An ellipse has major and minor axes along the axes of x and y respectively; their lengths being 2a and 2b respectively. Four normals are drawn to this ellipse from some point (u,v). Find the mean of the feet of the normals.<br /><br />11. Normals are drawn to an ellipse at the extremities of a focal chord of it. Show that the line through their point of intersection parallel to the major axis of the ellipse bisects the chord.<br /><br /><br /><br /><br /></div>Sanjeev Kumarhttps://plus.google.com/102586937696118125212noreply@blogger.comtag:blogger.com,1999:blog-9175932874695168418.post-74340900830308744402017-11-01T08:53:00.000+05:302017-11-07T10:39:08.081+05:30Sheet-1<div dir="ltr" style="text-align: left;" trbidi="on">Area<br /><br />Areas of Parallelograms on Same Base and between Same Parallels<br /><br />1. ABCD is a parallelogram; a line parallel to BD cuts BC, DC at P, Q; prove that the area of the triangle ABP equals that of the triangle ADQ.<br /><br />2. ABCD is a parallelogram; P is any point on AD; prove that the sum of the areas of triangles PAB and PCD equals the area of the triangle PBC.<br /><br />Areas of Parts of Parallelogram made by Diagonal<br /><br />3. ABCD is a quadrilateral; lines are drawn through A, C parallel to BD, and through B, D parallel to AC; prove that the area of the parallelogram so obtained equals twice the area of ABCD.<br /><br />Area of Triangle: Basic Formula<br /><br />4. ABCD is a parallelogram; P is any point on BC; DQ is the perpendicular from D to AP; prove that the area(ABCD) = DQ.AP<br /><br />5. ABC is a straight line; O is a point outside it; prove that the ratio of the area of the triangle OAB to that of the triangle OBC equals the ratio of AB to BC.<br /><br />Areas of Parts of Triangle made by Median<br /><br />6. ABCD is a parallelogram; P is any point on BD; prove that the area of triangle PAB is equal to the area of the triangle PBC.<br /><br />7. P is any point on the median AD of a triangle ABC; prove that the area of the triangle APB equals that of the triangle APC.<br /><br /></div>Sanjeev Kumarhttps://plus.google.com/102586937696118125212noreply@blogger.com