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A ladder, inclined at 60 ° with the horizontal is leaning against a vertical wall. The foot of the ladder is 3 meters away from the foot of the wall. A boy climbs the ladder such that his distance z meters with respect to the foot of the ladder is given by z = 6t, where tis the time in seconds. Find the rate at which his vertical distance from the ground changes with respect to ¢. Find the rate at which his distance from the foot of the wall is changing with respect to t when he is 3 m away from the foot of the ladder.


A box of miniature cars contains 5 red cars, 7 blue cars, and 6 black cars. Two cars are drawn, but the first car drawn is not replaced. What is the probability of getting a red car on the first draw and a black car on the second draw?


Solve the following 2 × 5 game by graphical

method

Player B

1 –5 5 0 –1 8

Player A

2 8 –4 –1 6 –5



Let R be the ring of Gaussian integers as in Exercise 11, and let I = {a + bi |3 divided a and 3 divides b}.



a. Show that I is an ideal of R.



b. Show that I is not a maximal ideal of R.

Let X denote the random variable distribution that gives 2,3,4,5,6,7,8,9 and 10 in a deck


of cards. Find the probability of X.

From a paddy field, 12 plants randomly selected. The length of panicle (cm) (x) and no. of grains/panicle (y) is recorded as follows.

x - 22.9,23.9,24.8,21.2,22.2,22.7,23,24,20.6,21,24,23.1


y -112,131,147,90,100,106,127,145,85,94,142,111

Calculate ‘r’ by direct and change of origin and change of scale.



Provide all necessary steps and evaluate the following integrals:




(a) ∫ (𝑥^2√(2 + 𝑥)) 𝑑𝑥




(b)∫ (2^𝑡/(2^𝑡 + 3)) 𝑑𝑡




(c) ∫ 𝑑𝑡/(cos^2(𝑡 √1 + tan 𝑡))




A population consists of the numbers 3,5,9,10 and 6. Given the sample size of 3 Determine the following:



A. List all the possible samples


B. Compute the mean of each sample.


C. Construct the sampling distribution


D. Consryya histogram


E. Compute the mean of the sampling distribution of the sample means


F. Compute of the variance sampling distribution of the sample means



𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥



Suppose 𝑓 is differentiable on ℝ and has two roots. Show that 𝑓′ has at least one root.

Show that the minimum and maximum points of every curve in the family of polynomials


𝑓(𝑥) = 2𝑥^3 + 𝑐𝑥^2 + 2𝑥 lie on the curve 𝑦 = 𝑥 − 𝑥^3


.

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