PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

Find the A.M. between:

12 and −8

Find the A.M. between:

(x − y) and (x + y).

Insert 4 A.M.s between 4 and 19.

Insert 7 A.M.s between 2 and 17.

Insert six A.M.s between 15 and −13.

If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.

The vertices of the triangle are A(5, 4, 6), B(1, –1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the length AD.

A point C with z-coordinate 8 lies on the line segment joining the points A(2, –3, 4) and B(8, 0, 10). Find its coordinates.

Show that the three points A(2, 3, 4), B(–1, 2 – 3) and C(–4, 1, –10) are collinear and find the ratio in which C divides AB. 

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane.

Find the ratio in which the line segment joining the points (2, –1, 3) and (–1, 2, 1) is divided by the plane x + y + z = 5. 

If the points A(3, 2, –4), B(9, 8, –10) and C(5, 4, –6) are collinear, find the ratio in which Cdivides AB.

Find the ratio of the coefficients of x^p and x^q in the expansion of \[\left( 1 + x \right)^{p + q}\] .

If a and b are the coefficients of xⁿ in the expansion of  \[\left( 1 + x \right)^{2n} \text{ and }  \left( 1 + x \right)^{2n - 1}\]  respectively, find  \[\frac{a}{b}\]

If  \[\left( 1 - x + x^2 \right)^n = a_0 + a_1 x + a_2 x^2 + . . . + a_{2n} x^{2n}\] , find the value of  \[a_0 + a_2 + a_4 + . . . + a_{2n}\] .

If in the expansion of (1 + x)²⁰, the coefficients of rth and (r + 4)th terms are equal, then ris equal to

The coefficient of  \[x^{- 17}\]  in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] is