Mathematics Test - 1

1/e

let f(x)=logx/x

to get max of f(x)... fi(x)=0 u get some value of x

substitute it in f(x) u get max value of f(x)

The line which makes equal distance from the two fixed points will definitely pass through the midpoint of line joining the two points and will definitely perpendicular to the line formed by joining the two points.

Let's first find the probability of not getting a club card. There are 52 cards in a deck, with 13 clubs and 39 non-clubs:

P(not getting a club) = 39/52 = 3/4.

Now, we're looking for the number of times a card must be drawn so that the probability of getting at least one club is greater than 3/4.

Let n be the number of times a card is drawn. The probability of not getting a club in n draws is (3/4)^n. Since we want the probability of getting at least one club, we will consider the complementary probability, which is 1 - (3/4)^n.

We want this probability to be greater than 3/4:

1 - (3/4)^n > 3/4.

Now, we can solve for n:

(3/4)^n < 1/4.

Since (3/4)^n is a decreasing function, we can find the smallest integer n that satisfies the inequality by trial and error:

n = 1: (3/4)^1 = 3/4 (not less than 1/4)
n = 2: (3/4)^2 = 9/16 (not less than 1/4)
n = 3: (3/4)^3 = 27/64 (not less than 1/4)
n = 4: (3/4)^4 = 81/256 < 1/4.

Therefore, the smallest number of times a card must be drawn so that the probability of getting at least one club is greater than 3/4 is n = 4. The answer is D. 4.

ANSWER :- b

Solution :- f′(1-0)= lim h→0 f(1-0-h)−f(1-0)

= lim h→0 (sinπ[1-0-h]−sinπ[1-0])/h

​= lim h→0 (sinπ−sinπ)/h=0