PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

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.

If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is

If, S₁ is the sum of an arithmetic progression of 'n' odd number of terms and S₂ the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2}\] = 

If in an A.P., S_n = n²p and S_m = m²p, where S_r denotes the sum of r terms of the A.P., then S_p is equal to

If the 6th, 7th and 8th terms in the expansion of (x + a)ⁿ are respectively 112, 7 and 1/4, find x, a, n.

Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is

Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is

If the 2nd, 3rd and 4th terms in the expansion of (x + a)ⁿ are 240, 720 and 1080 respectively, find x, a, n.

Find a, b and n in the expansion of (a + b)ⁿ, if the first three terms in the expansion are 729, 7290 and 30375 respectively.

Mark the correct alternative in the following question:

\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P  . , then }S_q \text { equals }\]