Directions for the following three questions: Answer the questions on the basis of the information given below
In the adjoining figure I and II are circles with centres P and Q respectively, The two circles touch each other and have common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the ratio 4: 3. It is also known that the length of PO is 28 cm.
What is the ratio of the length of PQ to that of QO?
What is the radius of the circle II?
The length of SO is
Directions for the following two questions:
Answer the questions on the basis of the information given below.
$$f_1(x) = x$$ if $$0 \leq x \leq 1$$ $$f_1(x) = 1$$ if x >= 1 $$f_1(x) = 0$$ otherwise
$$f_2(x) = f_1(-x)$$ for all x
$$f_3(x) = -f_2(x)$$ for all x
$$f_4(x) = f_3(-x)$$ for all x
How many of the following products are necessarily zero for every x:
$$f_1(x)f_2(x), f_2(x)f_3(x), f_2(x)f_4(x)$$
Which of the following is necessarily true?
Directions for the following two questions: Answer the questions on the basis of the information given below.
In an examination, there are 100 questions divided into three groups A, B and C such that each group contains at least one question. Each question in group A carries 1 mark, each question in group B carries 2 marks and each question in group C carries 3 marks. It is known that the questions in group A together carry at least 60% of the total marks.
If group B contains 23 questions, then how many questions are there in group C?
If group C contains 8 questions and group B carries at least 20% of the total marks, which of the following best describes the number of questions in group B?
For the following questions answer them individually
Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide.
A rectangular sheet of paper, when halved by folding it at the mid point of its longer side, results in a rectangle, whose longer and shorter sides are in the same proportion as the longer and shorter sides of the original rectangle. If the shorter side of the original rectangle is 2, what is the area of the smaller rectangle?
If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?