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a) Find "\\ lim_{x\\to 0}\\frac{\\left(tanxsec^2x-x\\right)}{x^3}"
b) Examine whether the equation, x3- 11x +9 =0 has a real root in the interval [0,1]
c) Check whether the following series are convergent or not (4)
i) "\\sum _{n=1}^{\\infty }\\:\\frac{\\left(3n-1\\right)}{7^n}"
(ii) "\\sum _{n=1}^{\\infty }\\frac{\\left(\\:\\sqrt{n^2+3}-\\sqrt{n^2-3}\\right)}{\\sqrt{n}}\\:"
Are the following statements true or false? Give reasons for your answers.
a) 2 is a limit point of the interval [1,4]
b) Every bounded sequence is convergent.
c) The function, f :R → R defined by f(x)= |x −1| + |3 − x |is differentiable at x =4
d) The function f(x) = [x] − x is not integrable in [0,3] where [x] denotes the greatest
integer function.
e) The function f defined by f(x)= (ex+ e-x )/2 when x not equal to 0 and 1/2 when x =0 is continuous in the closed interval, [-1,1 ]
In a certain Algebra 2 class of 29 students, 18 of them play basketball and 5 of them play baseball. There are 9 students who play neither sport. What is the probability that a student chosen randomly from the class plays basketball or baseball?
A water tank with vertical sides has a
horizontal base in the shape of a rectangle with semi-circular ends. The total inside length of the tank is 7 m, its width 4m and its height 2 m. Calculate:
(a) the surface area of the vertical walls of the
tank in m2?
(b) the area of the base in m2
(c) the number of liters of water in the tank when the depth of water is 1.56m
f(x) = kx * e ^ (- x / 2); k is a constant, x > 0 a) Find the value of k for f(x) to be a valid probability function
One of the psychological tests conducted by the guidance counselors in a public school is the Survey of Study Habits and Attitudes (SSHA) that measures student’s attitudes toward studying. The mean score of Senior High students is with standard deviation of . Mrs. Suacillo suspects that older students have better attitudes toward school. He randomly selects Grade students who are at least years old and gives them SSHA. The test result has a mean score of points. Is there a reason to believe that the claim of the guidance counselors is correct? Assume that the population mean score is normally distributed. Carry out a significance test at level. \
Evaluate the following integrals using residue theorem: (a) ∫∞−∞ x² /(x² +9)(x² +4)²dx
The time taken to resolve complaints reported to a call centre is exponentially distributed with an average complaint resolution time of 4.3 days. Approximately 25% of complaints are resolved in less than:
The daily net profit for spaza shop follows a normal distribution with a mean of R220 and a standard deviation of R175.
a) The probability that the daily net profit exceeds R250 is (answer to 2 decimal places)
b) The cut-off value for lowest 40 % of daily net profits is (in Rands and cents)
The monthly amount spent on stationery in a large company has a normal distribution with mean R2500 and standard deviation R150. How much should the company budget for stationery if the chance of exceeding the budget has to be only 2.5%?
