CSIR -NET Dec 2014: Part A (Solved) Mathematical Science and Life Science ~ APRAJITA'S BLOG

1. What is the 94th term of the following sequence?
1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4, ...

Solution:

1 \to 2

$1\to 2$ times

2 \to 4

$2\to 4$ times

3 \to 6

$3\to 6$ times

4 \to 8

$4\to 8$ times
So to find when the sum of sequence

2, 4, 6, 8 \dots

$2,4,6,8\cdots$ reached

94

$94$ .

\frac{n}{2} \times (2 \times 2 + (n - 1) \times 2)

$\frac{n}{2} \times (2\times 2+(n-1)\times 2)$ =

94

$94$ Solving we get,

n^{2} = 94

$n^2=94$ Which is between

9

$9$ and

10

$10$ .
But in the sequence

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, . . .

$1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,...$ number of term where

8

$8$ finished can be found.
number of term=

4 \times (2 + 16))

$4\times (2+16))$ =

72

$72$ . (found by considering sequence

2, 4...

$2,4...$ )
So when

9

$9$ finished number of term will be

72 + 18 = 90

$72+18=90$
So, answer is

10

$10$

2. Which of the following numbers is a perfect square?

1022121
2042122
3063126
4083128

Solution: The sum of digits of a perfect square is

1, 9, 4, 7

$1,9,4,7$ So

3, 4

$3,4$ can't be a perfect square. Also, the last digit of a perfect square cannot be

2

$2$ . So

1 s t

$1st$ option Is correct.

3. The equation $m^2 -33n + 1 =0$ , where m & n are integers, has

no solution
exactly one solution
exactly two solutions
Infinitely many solutions

Solution:
No Solution!
Since

m^{2} = 33 n - 1

$m^2 =33n-1$ .

A multiple of

3

$3$ has a Digital root =

3, 6, 9

$3,6,9$

Digital root is a digit obtained by adding all digits of a number till a single digit is obtained.

So , Digital root of

33 n - 1

$33n-1$ . is

2, 5,

$2,5,$ or

8

$8$ But it is equal to

m^{2}

$m^2$ . Which is a perfect Square (A perfect Square has digital root

= 1, 9, 4, 7

$=1,9,4,7$ ) So no solution.
4. The following graphs depict variation in the value of Dollar and Euro in terms of the Rupee over six months. Which of the following statements is true?

Values of Dollar and Euro rose steadily from January to June
Values of Dollar and Euro rose by equal rate between January to March
the rise in the value of Dollar from April to May is three times the fall in Euro during same time period.
Values of Dollar and Euro rose equally between May and June.

Solution: 2nd option is correct.

5. What is maximum number of whole ladoos having diameter of 6 cm that can be packed in a box whose inner dimensions
are 24 X 18 X 17 cm?

Solution:

So, total number of ladoos =

2 \times 3 \times 4

$2\times 3\times 4$ =

24

$24$

6. Which of the following figures best shows that y is inversely proportional to x?

Solution: 4th option is correct.
7. What is the next term in the sequence?
7, 11, 13, 17, 19, 23, 29…..

Solution: Sequence of prime numbers, Next prime is:

31

$31$

8. What is the area of the triangle bound by lines $y=2x, y=-2x$ and $y=6$ ?

Solution:

Height =

6

$6$ cm and Width =

6

$6$ cm So area is

\frac{1}{2} \times 6 \times 6

$\frac{1}{2}\times 6 \times 6$ =

18

$18$ .

9. Three volumes of a Hindi book, identical in shape and size, are next to each other in a shelf, all upright, so that their pines
are visible, left to right: I, II and III. A worm starts eating from the outside front cover of volume I, and eats its way horizontally to the outside back cover of volume III. What is the distance travelled by the worm, if each volume is 6 cm thick?

6 cm
12 cm
18 cm
a little more than 18 cm

Solution:

Worm crossed 3 books So answer is

18

$18$ .
10. If N, E and T are distinct positive integers such that N x E xT = 2013, then which of the following is the maximum possiblesum of N, E and T?

39
2015
675
671

Solution: Factors of

2013 = 1, 3, 11, 61

$2013=1, 3, 11,61$ So possible sum =

1 + 33 + 61 = 95

$1+33+61=95$ or

1 + 3 + 671 = 675

$1+3+671=675$

2015

$2015$ is not possible since factors must be distinct. So answer is

675

$675$ .

11. Every month the price of a particular commodity falls in this order: 1024, 640, 400, 250…
What is next value?

156.5
Approximately 39
64
40

Solution: Every term is

\frac{5}{8} \times

$\frac{5}{8}\times$ Previous term. So answer is

250 \times \frac{5}{8}

$250\times \frac{5}{8}$ =

156.5

$156.5$

12. We define a function f (N) = sum of digits of N, expressed as decimal number. Eg. $f (137) = 1+3+7=11$ . Evaluate $f (2^7 3^5 5^6)$

Solution:

2^{7} \times 3^{5} \times 5^{6}

$2^7\times 3^5 \times 5^6$ =

{(2 \times 5)}^{6} \times 2 \times 3^{5}

${(2\times 5)}^6\times 2 \times 3^5$ =

10^{6} \times 486

$10^6\times 486$ So answer is

4 + 8 + 6

$4+8+6$ =

18

$18$ .

13. A mouse has to go from point A to B without retracing any part of the path, and never moving backwards. What is the total number of distinct paths that the mouse may take to go from A to B?

Solution: From Multiplication rule of counting (Permutation and combination)

Answer will be

2 \times 4 \times 3 \times 2

$2\times 4\times 3 \times 2$ =

48

$48$ .

2nd option is correct.

14. The sum of first n natural numbers with one of them missed is 42. What is the number that was missed?

Solution: Let the missing number is

x

$x$ . Then

\frac{n (n + 1)}{2} - x = 42 \Rightarrow n (n + 1) = 2 x + 84

$\frac{n(n+1)}{2}-x=42\Rightarrow n(n+1)=2x+84$ .

Now if

x = 1

$x=1$ then R.H.S.=

86

$86$ . Which can't be factorized into consecutive natural numbers. Factors are

43, 2

$43,2$

x = 2

$x=2$ then R.H.S.=

88

$88$ . Which can't be factorized into consecutive natural numbers. Factors are

8, 11

$8,11$ .

x = 3

$x=3$ then R.H.S.=

90

$90$ . Which can be factorized into consecutive natural numbers. That is

90 = 9 \times 10

$90=9\times 10$ .

So, third option is correct.

15. The area of the inner circle and shaded rings are equal. The radii $r_1$ and $r_2$ are related by?

$r_1 = r_2$
$r_1 = \sqrt{2}$
$r_1 = r_2\sqrt{3}$
$r_1 = 2\times r_2$

Solution:

π {r_{2}}^{2} = π ({r_{1}}^{2} - {r_{2}}^{2})

$\pi {r_2}^2=\pi ({r_1}^2-{r_2}^2)$

\Rightarrow 2 π {r_{2}}^{2} = π {r_{1}}^{2}

$\Rightarrow 2\pi{r_2}^2=\pi{r_1}^2$

$\Rightarrow \sqrt{2}r_2=r_1$ .

4th option is correct.

16. A 2.2 m wide rectangular steel plate is corrugated as shown in the diagram. Each corrugation is a semi-circle in cross-section having a diameter of 7 cm. What will be the width of steel sheet after it is corrugated?

1.4 m
1.6 m
0.7 m
1.1 m

Solution: Let there are

$n$ semicircular pattern in corrugated sheet then we have:

$2.2\times 100$ cm=

$n\times 2\pi \frac{7}{2}$

$\Rightarrow 220=n\pi 7$

$\Rightarrow n=10$

Therefore Width of sheet =

$7\times 10$ cm=

$0.7$ m. Third option is correct.

17. Ajay, Bunty, Chinu and Deb were agent, baker, compounder and designer, but not necessarily in that order. Deb told the baker that Chinu is on his way. Ajay is sitting across the designer and next to the compounder. The designer didn't say anything. What is each person's occupation?

Ajay- compounder; Bunty-designer; Chinu- baker; Debagent
Ajay- compounder; Bunty-baker; Chinu- agent; Deb- designer
Ajay- baker; Bunty-agent; Chinu-Designer; Deb- compounder
Ajay- baker; Bunty-Designer, Chinu- Agent; Deb-compounder

Solution: Deb told the baker that Chinu is on his way

$\rightarrow$ that means chinu is not baker.Deb is not baker.

The designer didn't say anything. So Deb can't be designer.

Ajay is sitting across the designer and next to the compounder. That means Ajay is not designer or compounder.

Chinu is travelling so, Chinu is not designer or compounder.

So, 4th option is correct.

18. A cubical piece of wood was filed to make it into the largest possible sphere. What fraction of the original volume was removed?

More than ¾
½
Slightly less than ½
Slightly more than ½

Solution:

Cross Section will look like this. now remaining volume is

$\frac{4}{3}\pi a^3-a^3$ . So, first option is correct.

19.Two platforms are separated horizontally by distance A and vertically by distance B. They are to be connected by a staircase having identical steps. If the minimum permissible step length is a, and the maximum permissible step height is b, the number of steps the staircase can have is

≥ B/b
≤ A/a
≥ B/b and ≤ A/a
≤ B/b and ≥ A/a

Solution:

Let Number of steps=

$n$ . Then we have

$n\times a \leq A$ and

$n\times b \geq B$ .

$\Rightarrow n \leq \frac{A}{a}$ and

$\Rightarrow n\geq \frac{B}{b}$ . Third option is correct.

20. A certain day, which is x days before 17th August, is such that 50 days prior to that day, it was 4x days since March 30th of the same year. What is x?

Solution:

According to question:

$30$ th march+

$4x$ days

$+50+x=17$ Aug

$\Rightarrow 140-50=5x \Rightarrow x=18$ . First option is correct.