# Quant Boosters by Anurag Singh Chauhan - Set 1

• Q.1 find a natural number n for which 3^9+3^12+3^15+3^n is a  perfect cube.

Solution: First of all we will try to find out cube

Now, 3^9+ 3^12 +3^15 +3^n = 3^9( 1 + 3^3 + 3^6 +3^n-9 )

= (3^3)^3 [ 1+ 3.3^2 + 3.(3^2)^2 +(3^2)^3 + 3^(n-9) – 3(3^2)^2]

= (3^3)^3[(1+3^2)^3]

Since, 3^(n-9) – 3^5 = 0

3^(n-9) = 3^5

n-9 = 5

so, n = 14

hence given number is a pefect cube for n = 14

Q.2  if a, b, c, d, e are positive real numbers such that  a+b+c+d+e = 8 and a^2 + b^2 + c^2 + d^2 + e^2 = 16 find range of e.

Solution: Concept -  We know that these of type of questions can be solved by inequality.

If a, b, c, d are positive real numbers then [( a+ b+ c + d)/4]^2 ≤  (a^2 +b^2 + c^2 + d^2)/4    ... eqn(1) { by Techebycheff’s Inequality}

Now,  a + b + c + d + e = 8

a + b + c + d  = 8 - e

a^2 + b^2 + c^2 + d^2  = 16 – e^2

putting above value in equation (1)

(8-e)^2/ 16 ≤  ( 16 – e^2)/ 4

On solving this ,   e(5e – 16) ≤ 0

Using number line rule .

0 ≤ e  ≤ 16/5

Q.3 find the period of f(x) = Sin^4 x + cos^4 x

Solution.  Since Sin x and Cos x both has a period Pi, but f(x) is even function and Sin x and Cos x are complementary.

So f(x) has period = 1/2 {LCM of (Pi, Pi) } = Pi/2

Q. 4. How many values of x in -19 < x < 98 satisfy Cos^2 x + 2Sin^2 x = 1

Sol. This type of questions are frequently asked in mocks and MBA exams. Like CAT, MAT, GMAT.

Now we have, Cos^2 x + 2Sin^2 x = 1

We know , Cos^2 x + Sin^2 x = 1

On subtracting we get ,

Sin^2 x = 0

Sin x = 0 then x = nPi {general solution}

So, -19 < n Pi < 98

Dividing by Pi we get, -6.04 < n < 31.19

So, total number solutions equal to n = 38

Q.5 The cost of 3 Pizzas,  5Samosas,  1 order of milk ata a restaurant is 23.5 rupees . In the same restaurant the cost of 5 pizzas, 9 samosas,  and 1 order of milk is 39.50 rupees. What is the cost of 2 pizzas, 2 samosas, 2 milk at restaurant.

Solution.  Also these of questions often seen in management exams, Now we will discuss two  approaches for this questions

Let,  Pizza cost = P, Samosa cost  = S, Milk cost = M

Now,  Making equation

3P + 5S + M = 23.5  …… (A)

5P + 9S + M = 39.5  …… (B )

Multiplying equation (A) by 2 & subtracting equation (B )

P + S + M = 7.5

So required answer = 2* 7.5 = 15

Second approach -    Solve equation (A) & (B ) by elimination method and get values of P &S .   It is bit lengthy .

So this method will give you exact value of P , S & M .   try yourself .

Q. 6 Find all Positive integers x &y satisfying 1/root(y) +  1/root(x)  =  1/root(50)

Solution.

1/root(y) = - 1/root(x)  + 1/root(50) = (root(x) - root(50))/ root(50x)

Squaring both sides, 1/y = {x + 50- 2*5 * root(2)x}/ 50x

Now L.H.S is rational so also R.H.S must be rational So root(2x) must rational now x e N

So 2x = (2a)^2

x= 2a^2

Also y = 2b^2

Now equation reduces ,

1/aroot(2)  + 1/broot(2) = 1/5root(2)

1/a + 1/b = 1/5

5( a + b) = ab

5a + 5b – ab = 0

5a + 5b – ab + 25 – 25 = 0

a(5 – b) + 5( b – 5) = -25

(b-5)(a-5) = 25

a = 10 , b = 10  or (10, 10) and (30,6) and (6, 30)

Q.7- if a, b, c, d are the sides of a quadrilateral find the minimum value of  (a^2 +b^2+c^2)/d^2 .

Solution- these types of questions can be easily derived by help of inequality

We know that

(a-b)^2 + (b-c)^2 + (c-a)^2 ≥ 0

2(a^2 +b^2 +c^2) ≥ 2 (ab+bc+ca)

3(a^2+b^2+c^2) ≥ (a^2+ b^2+c^2) + 2(ab+bc+ca)

3(a^2+b^2+c^2) ≥ (a+b+c)^2

Since sum of any three sides of an quadrilateral is greater than fourth sides

3(a^2+b^2+c^2) ≥ (a+b+c)^2 > d^2

(a^2+b^2+c^2)/d^2  > 1/3 so minimum value is 1/3

Q.8– Find x^5 + y^5 + z^5 if it is given that x+y+z=1 , x^2+y^2+z^2=2, x^3+y^3+z^3= 3  .

Solution-  these types of question are frequently asked in many competitive exam

Let us we solve it by cubic equation

Now we know if x,y,z are the roots of the equation

at^3+bt^2+ct+d = 0 , (a≠0)

we are given

x+y+z = 1 = (S1)

x^2+y^2+z^2 = S2

x^3+y^3+z^3 = S3

we know , S1= -b/a

aS1+b = 0

a*1 + b = 0

b = -a

(x+y+z)^2  - 2(xy+yz+zx) = (x^2+y^2+z^2)

1^2 – 2*c/a = S2

1-2c/a = 2

c= -a/2

similarly by using identity

d = -a/6

now putting values b,c,d in cubic equation

at^3 – at^2 – at/2 – a/6 = 0

dividing by a

t^3- t^2- t/2-1/6 = 0  equation………. (A)

6t^3 – 6t^2 – 3t – 1= 0

Multiplying both sides by t

6t^4 – 6t^3 – 3t^2 – t = 0

Putting t = x, y, z in succession and adding

6 S4 – 6 S3 – 3 S2- S1 = 0

So ,S4 = 25/6

Multiplying by t^2

6t^5 – 6t^4 – 3t^3- t^2 = 0

Putting t= x,y,z in succession and adding

6 S5 – 6 S4 – 3 S3 – S2 = 0

6 S5 = 36

S5 = 6

Q.9  Find all pairs of (a, b) of positive integers such that 2a + 7b divides 7a + 2b .

Solution: These type of questions can be solved by hit and trial method but this method gives you approximation value not exact values. So we will try to find by proper method

Now, let H be the H.C.F of (a, b)

So, let    a = hx  ,b =hy

Now,  2a+7b  divides 7a+2b

Also 2x+7y divides 7x+2y

2x+7y divides 49x+14y

And  2x+7y divides  2(2x+7y)

Now 2x+7y divides 49x+14y – 4x-14y i.e 45x

Similarly  2x+7y divides 45y

Hence 2x+7y divides h.c.f of 45x, 45y

So 2x+7y divides 45

2x+7y should be 9,15,45

Possible pair of x,y (1,1) (4,1) (19,1) (12,3) (5,5) so all solutions are (h,h) (4h,h) (19h,h) (12h,3h) (5h,5h)

Q 10- A leaf is torn from back of a book. The sum of remaining pages is 15000 . what are the page numbers on the torn leaf.

Solution – this question often asked in mocks and aptitude  exams like CAT, MAT, GMAT etc..

Now, let the numbers of pages be n

Since , number of pages after a leaf is torn  is 15000, then the sum of the numbers on the pages will be greater than 15000

Now, n(n+1)/2 > 15000

n (n+1) > 30000

(n+1)^2 > n (n+1) > 30000 > 173^2

So , 172 < n < 176

n  could be 173, 174, 175

n = 173 then n(n+1)/2 = 173*174/2 = 15051

Sum of the numbers on torn pages.

= 15051 – 15000 = 51

And this would be y + ( y +1)

= 2y + 1 = 51

So page numbers are 25, 26

If n = 174 the n(n+1)/2 = 15225

Sum on torn pages

15225 – 15000 = 225

Page numbers on torn pages are 112, 113

When = 175, n(n+1)/2 = 15400

Sum on the torn pages = 400   which is even .  So not possible .

Now , we have to know that the smaller number on the torn pages should be odd. So, the numbers on the torn pages are 25 and 26 and the number of pages is 173.

Q.11  find all positive integers for which  n^2 + 96 is  perfect square

Solution- these types of question can be asked in GMAT , CAT, NMAT etc.

We solve it by two approaches

Now suppose k is a positive integer such that

n^2+ 96 = k^2

k^2 –n^2 = 96

(k-n) (k+n) = 96

Since (k-n) , (k+n) must be both odd and even

But odd case is not possible

Only possible cases are..

k-n = 2, k+n = 48

k-n = 4, k+n = 24

k-n = 6, k+n = 16

k-n = 8 , k+n = 12

on solving these equations we get

pairs of k,n are… (25,23) (14,10) (11,5) (10,2)

second approach-

positive solutions = (no of factors of 96 after dividing by 4) /2

= 2^3*3=total factors are 8

So positive solutions = 8/2=4

Q.12  Find all integral solutions of a^4 + b^4 + c^4 = 1991 + d^ 4.

Solution. In this type of question we would apply divisibility on L.H.S and R.H.S in most cases  also these questions can be solved orally

Now, we will learn new concept.

Let n be an integer, When it is odd.

Let  n^4 – 1 = ( n^2- 1)(n^2 +1)

= (n-1)(n+1)(n^2+1)

It must be divisible by 16 for any odd n.

Also (n-1) , (n+1) are consecutive even integers and one of them is divisible by 4.

If n is even- n^4 must be divisible by 16.

Thus n^4 gives remainder 0 and 1 .

Possible remainders of a^4 + b^4 + c^4 – d^4 when divided by 16 are { -1, 0, 1, 2, 3}

But 1991 gives remainder 7 when divided by 16  so no integral solution exist because L.H.S & R.H.S have different remainders

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