November 14, 2017
Answer the questions independently of each other.
Start the test!
Congratulations - you have completed Mathematics Test-21.
You scored %%SCORE%% out of %%TOTAL%%.
Your performance has been rated as %%RATING%%
Your answers are highlighted below.
The 288th term of the series a, b, b, c, c, c, d, d, d, d, e, e, e, e, e, f, f, f, f, f, f…. is
Question 1 Explanation:
The number of terms of the series forms the sum of first n natural numbers i.e. n(n + 1)/2. Thus the first 23 letters will account for the first (23 x 24)/2 = 276 terms of the series. The 288th term will be the 24th letter viz. x.
Three horses are grazing within a semi-circular field. In the diagram given below, AB is the diameter of the semi-circular field with centre at O. Horses are tied up at P, R and S such that PO and RO are the radii of semi-circles with centers at P and R respectively, and S is the centre of the circle touching the two semi-circles with diameters AO and OB. The horses tied at P and R can graze within the respective semi-circles and the horse tied at S can graze within the circle centred at S. The percentage of the area of the semi-circle with diameter AB that cannot be grazed by the horses is nearest to ...
Let p and q be the roots of the quadratic equation x2 – (? - 2) x - ? - 1 = 0. What is the minimum possible value of p2 + q2?
Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true? note: read a2 as a square 2
The maximum possible value of a2 + b2 + c2 + d2 is 4m2 - 2m + 1
The minimum possible value of a2 + b2 + c2 + d2 is 4m2 + 2m + 1
The maximum possible value of a2 + b2 + c2 + d2 is 4m2 + 2m + 1
The minimum possible value of a2 + b2 + c2 + d2 is 4m2 – 2m + 1
Question 4 Explanation:
(a + b + c + d)2 = (4m + 1)2 Thus, a2 + b2 + c2 + d2 + 2(ab + ac + ad + bc + bd + cd) = 16m2 + 8m + 1 a2 + b2 + c2 + d2 will have the minimum value if (ab + ac + ad + bc + bd + cd) is the maximum. This is possible if a = b = c = d = (m + 0.25) ……….since a + b + c + d = 4m + 1 In that case 2((ab + ac + ad + bc + bd + cd) = 12(m + 0.25)2 = 12m2 + 6m + 0.75 Thus, the minimum value of a2 + b2 + c2 + d2 = (16m2 + 8m + 1) – 2(ab + ac + ad + bc + bd + cd) = (16m2 + 8m + 1) – (12m2 + 6m + 0.75) = 4m2 + 2m + 0.25 Since it is an integer, the actual minimum value = 4m2 + 2m + 1
The number of non-negative real roots of 2x – x – 1 = 0 equals
Once you are finished, click the button below. Any items you have not completed will be marked incorrect. Get Results
There are 5 questions to complete.
You have completed
Your score is
You have not finished your quiz. If you leave this page, your progress will be lost.
Final Score on Quiz
Attempted Questions Correct
Attempted Questions Wrong
Questions Not Attempted
Total Questions on Quiz
Answer Choice(s) Selected
Need more practice!