NOVEMBER/DECEMBER 2021
திருவண்ணாமலை மாவட்டம் அரசு கலைக் கல்லூரி இரண்டாம் ஆண்டு கணிதவியல் மாணவர்களுக்கு நடைபெற்றுவரும் மூன்றாம் பருவம் செமஸ்டர் தேர்வுகளுக்கான விடைகள் கேள்விகளுடன் கீழே கொடுக்கப்பட்டுள்ளது.
CSMA32 — MATHEMATICS FOR
COMPETITIVE EXAMINATIONS – I
Time : Three hours Maximum : 75 marks.
SECTION- A (10 × 2 = 20 marks)
Answer ALL questions.
1) SIMPLIFY :
ANS :
2) Find the L.C.M. of 72, 108 and 2100.
Ans :
To calculate the LCM of 72, 108, 2100 by listing out the common multiples, we can follow the given below steps:
Step 1: List a few multiples of 72 (72, 144, 216, 288, 360 . . .), 108 (108, 216, 324, 432, 540 . . .), and 2100 (2100, 4200, 6300, 8400, 10500 . . .).
Step 2: The common multiples from the multiples of 72, 108, and 2100 are 37800, 75600,
Step 3: The smallest common multiple of 72, 108, and 2100 is 37800.∴ The least common multiple of 72, 108, and 2100 = 37800.
3) SIMPLIFY : 108÷36 OF
ANS :
4) Find the average of all prime numbers between 30
and 50.
ANS :
Find the average of all prime numbers between 30 and 50.
there are five prime numbers between 30 and 50. They are 31,37,41,43 and 47.
Therefore the required average=(31+37+41+43+47)/5
* 199/5
* 39.8.
5) Three members are in the ratio 4 : 5 : 6 and their
average is 25. Find the largest number.
ANS :
Let us assume the numbers are 4x, 5x, 6x.
From the question, their average is 25.
To find x,
(4x+5x+6x)/3=25
(15x/3)=25.
5x=25.
X=5.
Now, the numbers are 4x, 5x, 6x.
4×5=20
5×5=25
6×5=30.
The numbers are 20,25,30.
The largest number is 30.
6) The ratio between the present ages of P and Q is
6 : 7. If Q is 4 years old than P . What will be the
ratio of the ages of P and Q after 4 years.
ANS :
The ratio of P and Q is 6:7. This means P/Q = 6/7.
Q is 4 years older than P, so Q = P + 4.
Therefore, P/Q = P/(P+4) = 6/7.
Multiplying both sides by 7/(P + 4) gives
7P = 6(P + 4) = 6P + 24
Subtract 6P from both sides gives
P = 24
Q = P + 4
So Q = 24 + 4 = 28
After 4 years
P/Q = (24 + 4)/(28 + 4) = 28/32
So after 4 years the ratio of their ages will be 7/8.
7) SIMPLIFY :
ANS :
 |
8) How will the ratio 5 : 4 expressed as a percent?
ANS :
Percent=[(Fraction)×100] %
We know that,5:4=5/4
5:4=5/4 So, by substituting the value of the fraction as 5/4 as given in the question in the above equation,
we get,Percent=5/4×100 %
Percent=500/4 %
Percent=125 %
9) If a:b=5:9 and b:c = 4 :7 find a:b:c and a:c ?
ANS :
Solution :
a : b = 5 : 9
b : c = 4 : 7
Observe that the figures against b are 9 and 4 .
Find LCM of 9 and 4. This is 36.
Now make figures in both ratios such that figure against b is 36.
Therefore ,
a : b = (Multiply by 4) 5×4 : 9×4 = 20 : 36
B : c = (multiply by 9) 4×9 : 7×9 = 36 : 63
When figure against b is made same (as b =36 above),
Ratio a: b : c = 20 : 36 : 63 and a:c = 20 : 63
10 ) P and Q started a business investing Rs. 85,000
and Rs. 15,000 respectively. In what ratio the
profit earned after 2 years be divided between P
and Q respectively.
ANS :
SECTION- B (5 × 5 = 25 marks)
Answer ALL questions.
11. (a) Find the sum of all even natural numbers
less than 75.
ANS :
All natural numbers before 74 can be arranged to form an AP, Where
ANS :
The square root of 1471369 by long division method is 1213.
Step-by-step explanation:
* To find the Square root of 1471369, take long division of the number,
*. Multiplying by 1, we get the remainder 47
* multiplying the quotient by 2, and to divide 47 by the nearest two digit
number, multiply by 22
* Now we get the 313 as the remainder, multiplying the quotient by 2, we get
24, multiplying the number by 241
* now the remainder is 7269, multiplying by 2423 the remainder becomes O.
* Square root of 1471369 = 1213.
13) b) Ramya’s age after 15 years will be 5 times her age 5 years back. Find the present age of Ramya.
ANS :
Let us suppose Ramya’s current age to be = x years.
5 years back his age would have been = x – 5
15 years time his age will be = x + 15
So, his present age will be
5(x- 5) =x + 15
5x-25=x+15
4x =. 40
so Ramya’s age is x = 10
14) b) If 20% of A = B and 40% of B = C find 60%
of (A + B) .
ANS :
Let A =100,
Then, according to question 20% of A =100×20/100 = 20 i.e., B = 20,
Now again according to question 40% of B = 20×40/100 = 8i.e., C=8
Hence, (A+B )=100 + 20
= 120
Now, 60% of (A + B) = 120 x 60/100
=72
Now 72/8100 = 900 i.e., 900% of C
60% of (A + B) = 900% of C
you can prof it by taking number such as A= 60 or any other number
Let’s see C = 8, therefore 900% of 8 = 8x 900/100 = 72 it’s proved.
15) (b) A, B and C start a business each investing Rs. 20,000. After 5 months A withdraw Rs. 5,000, B withdrew Rs. 4,000 and C invests Rs. 6,000 more. At the end of the year, a total profit of Rs. 69,900 was recorded. Find the share of each.
A, B and C is total investment for first 5 months:
A- 5 x 20000= 1,00,000
B- 5 x 20000 = 1,00,000
C- 5 x 20000 = 1,00,000
After 5 months investments (per month) are like this
A- 20000 – 6000 = 14,000
B – 20000 – 4000 = 16,000
C – 20000 + 6000 = 26,000
A, B and C%s total investment for remaining 7 months:
A- 7x 14,000 = 98,000
B- 7 x 16,000 = 1,12,000
C- 7 x 26,000 = 1,82,000
A, B and C is total investment for 12 months:
A – 1,00,000 + 98,000 = 1,98,000
B – 1,00,000 + 1,12,000 = 2,12,000
C – 1,00,000 + 1,82,000 = 2,82,000
Profit ratio is equal to investment ratio. So,
198000 : 212000 : 282000 = 99: 106: 141 = 346
- A’s share in profit – 69,900 x 99 / 346 = Rs. 20,000.3
- B’s share in profit 69,900 x 106 / 346 = Rs. 21,414.5
- C’s share in profit 69,900 – 20000.3 – 21,414.5 = Rs. 28485.2
Solution:
The highest number that divides 513, 1134, and 1215 exactly is
their highest common factor.
e Factors of 513 =1, 3, 9,19, 27, 57, 171, 513
e Factors of 1134 =1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 81,
126, 162, 189, 378, 567, 1134
e Factors of 1215 = 1, 3, 5, 9,15, 27, 45, 81, 135, 243, 405, 1215
The HCF of 513, 1134, and 1215 is 27.
-. The highest number that divides 513, 1134, and 1215 is 27.
To calculate the LCM of 16, 24, 36, and 54 by the division method, we will
divide the numbers(16, 24, 36, 54) by their prime factors (preferably
common). The product of these divisors gives the LCM of 16, 24, 36, and
54.
the numbers, 16, 24, 36, and 54. Write this prime number(2) on the
left of the given numbers(16, 24, 36, and 54), separated as per the
ladder arrangement.
Step 2: If any of the given numbers (16, 24, 36, 54) is a multiple of 2,
divide it by 2 and write the quotient below it. Bring down any number
that is not divisible by the prime number.
e Step 3: Continue the steps until only 1s are left in the last row.
The LCM of 16, 24, 36, and 54 is the product of all prime numbers on the
left, i.e. LCM(16, 24, 36, 54) by division method=2x2x2x2x3x3x3
= 432.
18 ) a) The sum of a number and its reciprocal is 13/6 Find the number.
ANS :
Solution :-
Let the number be x.
According to the Question
=> x + (1/x) = 13/6
= (x2 + 1)/x = 13/6
= 6 x 2 -13x +6=0
=> 6 x 2 – 9x – 4x +6=0
=> (3x – 2) (2x – 3) =0
=> X = 2/3 or x = 3/2
Hence, the required number is 2/3 or 3/2.
b) The ratio of the ages of a husband and his wife is 4 : 3.
After 4 years, this ratio will be 9 : 7. If at the time of marriage, the ratio was
5 : 3, then how many years ago were they married.
ANS :
Present age of his wife = 3x years
After 4 years
4x + 4/3x+4=9/7
= 28x + 28 = 27x + 36
= 28x – 27x = 36-28
therefore, x = 8
Present age of man = 4x
=4×8=32 years
Present age of his wife = 3x
=3×8=24 years
Suppose, they were married y years ago
According to question,
32-y/24-y = 5/3
= 96 – 3y = 120-5y
= 5y – 3y = 120-96
=2y=24
y=12
19) (a) The C.P. of 21 articles is equal to S.P. of 18 articles. Find the gain or loss percent.
CP of 21 articles=x
CP of 1 article=x/21
SP of 18 articles=x
SP of 1 article=x/18
since x/21 <x/18, there is a gain
gain=SP-CP
=x/18-x/21
=(7x-6x)/126
=x/126
gain%=gain/CPx100
=(x/126=x/21)x100
=(x/126*21/x)x100
=100/6
=50/3
gain% =16 2/3% ,,
Let the cost price of an article = Rs 100
|) cost price of 3/4 the article = (3/4 100)
c.p1=Rs 75
gain = g = 20%
ii ) cost price of remaining article = 100 – 25
c.p2 = Rs 25
iii ) Total selling price = cp1(100+g)/100+cp2
s.p = 75[( 100+20)/100] + 25
s.p= 90 + 25
= Rs115
amount gained by
whole transaction = s.p – c.p
total gain = 115 – 100
=Rs 15
Therefore ,
Total gain % = [ ( total gain )/c.p ]x 100
= (15/100) x 100
Total gain = 15% ,,