PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

Find the term independent of x in the expansion of the expression: 

(x) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^6\]

Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.

If the coefficients of \[\left( 2r + 4 \right)\text{ th and } \left( r - 2 \right)\] th terms in the expansion of  \[\left( 1 + x \right)^{18}\]  are equal, find r.

Find the distance between the following pairs of points: 

P(1, –1, 0) and Q(2, 1, 2)

Find the distance between the following pairs of point: 

A(3, 2, –1) and B(–1, –1, –1).

If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.

Find the distance between the points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).

If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)⁴³ are equal, find r.

If m th term of an A.P. is n and nth term is m, then write its pth term.

Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)^{n + 1} is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)ⁿ.

Using distance formula prove that the following points are collinear:

A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)

If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.

Using distance formula prove that the following points are collinear: 

P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)

Prove that the term independent of x in the expansion of \[\left( x + \frac{1}{x} \right)^{2n}\]  is \[\frac{1 \cdot 3 \cdot 5 . . . \left( 2n - 1 \right)}{n!} . 2^n .\]

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

Using distance formula prove that the following points are collinear: 

A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)

If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be