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

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

If the sum of n terms of an A.P. be 3 n² − n and its common difference is 6, then its first term is

Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).

Sum of all two digit numbers which when divided by 4 yield unity as remainder is

If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)²ⁿ are in A.P., show that  \[2 n^2 - 9n + 7 = 0\]

Determine the points in yz-plane and are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).

If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)ⁿ are in A.P., then find the value of n.

In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is

If in the expansion of (1 + x)ⁿ, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where  \[p \neq q\]

If S_n denotes the sum of first n terms of an A.P. < a_n > such that

The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be