CSIR Part A: Solved: December 2015 (Mathematics)

Question 1:

A circle drawn in the $x-y$ coordinate plane passes through the origin and has chords of lengths $8$ units and $7$ units on the $x$ and $y$ axes, respectively. The coordinates of its centre are

1.    (8, 7)                                                                                                 2.     (-8, 7)
3.   (-4, 3.5)                                                                                            4.      (4, 3.5)

Solution:

The perpendicular drawn from the centre to chord, bisects the chord. So Co-ordinate of centre will be (4,3.5)

Question 2:

The probability that a ticketless traveler is caught during a trip is $0.1$ If the traveler makes 4 trips , the probability that he/she will be caught during at least one of the trips is:
1. $1-(0.9)^4$                                       2. ${(1-0.9)}^4$
3.$ 1-(1-0.9)^4$                                   4.$ (0.9)^4$

Solution:

It's a Binomial Distribution. The probability of $k$ success in $n$ trials is $\binom {n}{k} \times p^k \times q^{n-k}$. Where $p$ is probability of Success and $q$ is probability of failure. Here we are required to find out the probability that he will caught at least one of the trips ($P_{caught}(at least once)$. )

and this is equal to $1- P_{not caught}(k=4)$ =$1-0.9^4$

Question 3: The statement: “The father of my son is the only child of your parents”
1. can never be true
2. is true in only one type of relation
3. can be true for more than one type of relations
4. can be true only in a polygamous family

Solution:  "The father of my son" Is equivalent to "myself" So the statement is "Myself is the only child of your parents".

That person is talking to himself. So, It is true only one type of relation. )(A kind of Reflexive relation.)

Question 4: The base diameter of a glass is 20% smaller than the diameter at the rim. The glass is filled to half the height. The ratio of empty to filled volume of the glass is

1. $\frac{\sqrt{10}-\sqrt{9}}{\sqrt{9}-\sqrt{8}}$
2. $\frac{10-9}{9-8}$
3. $\frac{10^2-9^2}{9-8}$
4. $\frac{10^3-9^3}{9^3-8^3}$

Solution:

Let radius of  Rim=10 then radius of base=$10-20/100 \times 10$=$8$ From Similar triangles, We have $\frac{2h1+h}{h}=10/8$ Therefore, $h=8h1$ Again $\frac{9h1}{8h1}=x/8$ (similar Triangles).

Therefore, $x=9$ Therefore ratio of empty to filled volume of the glass is $\frac{1/3\pi \times [10^2\times 10 h1-9^2\times 9h1]}{1/3\pi \times [9^2\times 9 h1-8^2\times 8h1]}$=$\frac{10^3-9^3}{9^3-8^3}$.

Question 5: The number of diagonals of a convex
deodecagon (12-gon) is
1. 66                                                2. 54
3. 55                                                4. 60

Solution: The number of diagonal is $ n(n-3)/2$.

So 2 is correct answer.

This problem can be solved using combination

For diagonal 2 vertex is needed but they should not be adjacent. so $\binom {n}{2}-12$

Question 6: One is required to tile a plane with congruent regular polygons. With which of the following polygons is this possible?

1. 6-gon                                        2. 8-gon
3. 10-gon                                     4. 12-gon

Solution: 

If you tile with regular convex polygon, then at each vertex sum of interior angles of $r$ such regular polygon must add up to 360. Interior Angle of a hexagon is 120 degree. And 3 such hexagon add up to 360. So Hexagon is the right Answer.

Question 7:  Suppose three meetings of a group of professors were arranged in Mumbai, Delhi and Chennai. Each professor of the group attended exactly two meetings. 21 professors attended Mumbai meeting, 27 attended Delhi meeting and 30 attended Chennai meeting. How many of them attended both the Chennai and Delhi meetings?

1. 18
2. 24
3. 26
4. Cannot be found from the above information

Solution:           

  

Since all have attended exactly two meeting therefore, number of people attended only one meeting or all three meetings is equals to zero.

Let \(x\) Professor attended Mumbai and Chennai Meeting, then \(30-x\) will attend Chennai and Delhi meeting. (Since, total 30 have attended Chennai meeting.) Similarly Since total \(21\) people have attended Mumbai Meeting, So, \(21-x\) will attend Delhi and Mumbai meeting. since \(27\) people have attended Delhi meeting So,

\((21-x)+(30-x)=27\)

\(\Rightarrow 51-2x=27 \Rightarrow x=12\)

So, people who have attended Chennai and Delhi meeting =\(30-12=18\)

Question 8: A wheel barrow with unit spacing between its wheels is pushed along a semi-circular path of mean radius 10. The difference between distances covered by the inner and outer wheels is
1. $0$                  2. $10$         3. $\pi$                 4. $2\pi $

Solution: 

From the Picture clearly, Difference of distance covered is $\pi [10+1/2]-\pi[10-1/2]$=$\pi$.

So, 3rd option is correct.

Question :9 Find the missing number:

Solution:

in the second row ,sum of absolution value of entries is double of entry in the first row. Applying the same logic we can find the missing number. Which is -19

Question 10:  Decode

1. GENT STUDENTS CAUSE LITTLE HEART BURNS
2. STUDENTS ARE INTELLIGENT BUT PROBLEM IS NOT SOLVABLE
3. THIS PROBLEM IS UNSOLVABLE BY ANY STUDENT
4. THIS PROBLEM IS SOLVABLE BY INTELLIGENT STUDENTS

Solution:

Read along this line. 4th option is correct.

Question 11:  Three circles of equal diameters are placed such that their centers make an equilateral triangle as in the figure

Within each circle, 50 points are randomly scattered. The frequency distribution of distances between all possible pairs of points will look as

Solution:
 

So, 3rd option is correct.

 Question 12: How many digits are there in $3^{16}$ when it is expressed in the decimal form?

1. Three                                    2. Six                                        3. Seven                                                4. Eight

Solution: $3^{16}$=$9^8$=${[10-1]}^8$=$10^8+1-20$ =$10000000-19$

So correct answer is seven.

Question 13: Most Indian tropical fruit trees produce fruits in April-May. The best possible explanation for this is
1. optimum water availability for fruit production.
2. the heat allows quicker ripening of fruit.
3. animals have no other source of food in summer.
4. the impending monsoon provides optimum conditions for propagation.

Solution:  4th option is correct.  As water is most available in monsoon session July to September, So 1st option is wrong.  Temperature is maximum in the month of June -July and production of fruits does not depends on ripening of fruits. So 2nd option is correct as well. 3rd is false (It's obvious).

Question 14: There is an inner circle and an outer circle around a square. What is the ratio of the area of the outer circle to that of the inner circle?

1. $\sqrt {2}$             2. $2$                   3. $2 \sqrt {2}$               4. $\sqrt {3/2}$

Solution:

From the above picture it's clear that radius of smaller circle is $1/2$ and bigger circle is $1/ \sqrt{2} $. So ratio is 2  So 2 is correct option.

Question 15: Write d =1 degree, r = 1 radian and g = 1 grad. Then which of the following is true?
(100 grad = a right angle)
1. cos d < cos r < cos g
2. cos r < cos g < cos d
3. cos r < cos d < cos g
4. cos g < cos d < cos r

Solution:

$1$ g=$\pi /200$ $1$d=$\pi /180 $ $\pi /200 < \pi/180 <1$ and Cos$(x)$ is decreasing function in $[0,\pi /2]$. so$cos r < cos d < cos g$ . 3rd option is correct.

Question 16: A vendor sells articles having a cost price of Rs.$100$ each. He sells these articles at a premium price during first eight months, and at a sale price, which is half of the premium price, during next four months. He makes a net profit of$ 20%$ at the end of the year. Assuming that equal numbers of articles are sold each month, what is the premium price of the article?
1.   $122 $                                                     2.  $ 144$
3.   $150$                                                     4.  $ 160$

Solution:  Let total articles =$12x$. So C.P.=$12x\times 100$. Let Premium price=$P$. then S.P.= $8x\times P+4x\times {P/2}$. =$10x\times P$. So $10x\times P-12x\times 100$=$\frac{20}{100} \times 12x \times 100$

Solving we get $P=144$. So, 2 is correct choice.

Question 17: The minimum number of straight lines required to connect the nine points above without lifting the pen or retracing is

1.   3                                                 2.     4
3.  5                                                  4.     6

Solution:

For $n \times n$ such grid number of lines required is $2\times[n-1]$.

Question 18: “The clue is hidden in this statement”, read the note handed to Sherlock by Moriarty, who hid the stolen treasure in one of the ten pillars. Which pillar is it?
1. X                                                 2. II
3. III                                              4. IX

Answer:   The clue is hidden in this statement. So 4th option is correct.

Question  19:  Three boxes are coloured red, blue and green and so are three balls. In how many ways can one put the balls one in each box such that no ball goes into the box of its own colour?
1. 1                                      2. 2
3. 3                                    4. 4

Solution: Ways we can arrange $3$ objects=3!

red-> red , blue->green , green->blue and similar three situation is not allowed and all three ball of same color in same colored box is not allowed, So total $4$ cases is not allowed.

So, answer is $3!-3-1=2$. So, 2nd option is correct.

Question 20: Let A, B be the ends of the longest diagonal of the unit cube. The length of the shortest path from A to B along the surface is
1. $\sqrt {3}$                2. $1+\sqrt {2} $
3. $\sqrt {5}$               4. 3

Solution: 

After  unfolding the surface of cube, it will look like fig drawn above and the shortest distance is $\sqrt{1^2+2^2}$= $\sqrt 5$. So, 3  is the correct Choice.