PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

If the sum of n terms of an A.P., is 3 n² + 5 n then which of its terms is 164?

If the sum of n terms of an A.P. is 2 n² + 5 n, then its nth term is

In the expansion of (1 + x)ⁿ the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.

If a₁, a₂, a₃, .... a_n are in A.P. with common difference d, then the sum of the series sin d [cosec a₁cosec a₂ + cosec a₁ cosec a₃ + .... + cosec a_{n − 1} cosec a_n] is

Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle. 

In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is

If in the expansion of (1 + x)ⁿ, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.

If a₁, a₂, a₃, .... a_n are in A.P. with common difference d, then the sum of the series sin d [sec a₁ sec a₂ + sec a₂ sec a₃ + .... + sec a_{n − 1} sec a_n], is

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is

Let S_n denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = S_n − k S_n − 1 + S_{n − 2} , then k =

Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus. 

If 3rd, 4th 5th and 6th terms in the expansion of (x + a)ⁿ be respectively a, b, c and d, prove that `(b^2 - ac)/(c^2 - bd) = (5a)/(3c)`.

Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.

The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] ,  then k =

Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.

If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].

If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =

If the coefficients of three consecutive terms in the expansion of (1 + x)ⁿ be 76, 95 and 76, find n.