Ncert Solutions Maths class 12th

Let f : N ® R be a function such that f(x + y) = 2f(x) f(y) for natural number x and y. If f(1) = 2, then the value of a for which ${\displaystyle \sum _{k=1}^{10}f\left(\alpha +k\right)}=\frac{512}{3}\left(2^{20}-1\right)$  holds is:

Let f : R -> R be defined as f(x) = 2x -1 and g : R - {1} -> R be defined as g(x) = $\frac{x-\frac{1}{2}}{x-1}.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> Then the composition function f(g(x)) is:

Let a, b and c be the length of sides of a triangle ABC such that $\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}$ if r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of $\frac{R}{r}$  is equal to:

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an off number of times is:

Let g(x) = x + 2| - 3|. If a denotes the number of relative minima, b denotes the number of relative maxima and c denotes the product of the zeros. Then the value of (a + 2b - c) is __________

In a geometric progression the ratio of the sum of the first 5 terms to the sum of their reciprocals is 49 and sum of the first and the third term is 35. The fifth term of the G.P. is p, the value of 4p is __________

f(1) = 2, g(1) = 2, f'(x), g'(x) exist, then the value of lim(x→1) [f(1)g(x)-f(1)-g(1)f(x)+g(1)] / [f(1)g(x)-f(x)g(1)] is __________

The lines (x-2)/1 = (y-3)/1 = (z-4)/-k and (x-1)/k = (y-4)/2 = (z-5)/1 are coplanar, if
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A student appeared in an examination consisting of 8 true – false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least ‘n’ correct answers is less than 1/2, is :
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Let f: R → R be defined as f(x+y)+f(x-y) = 2f(x)f(y), f(1/2) = -1. Then, the value of Σ₂₀_(k=_1) 1/(sin(k)sin(k+f(k))) is equal to :