# Free Mixed Bag 15 Practice Test - CAT

### Question 1

Find the digit sum (till you get a single digit number) of the smallest element of Set P. The smallest element in set P can be represented as (8x)(25300), which consists of 604-609 digits, where x is a natural number.

#### SOLUTION

Solution :C

option (c)

2600×5600 will have 601 digits (1 followed by 600 zeroes). To get the minimum, 604 digits we will have to increase the power of 2 such that we get a four digit number followed by 600 zeroes. The power should also be a multiple of 3 as 8=23. The smallest such power is 212=84. Therefore, smallest element of set P=8204×5600=4096000000(604 digits). The sum of the digits of P min =19=1.

### Question 2

Find the first term of the AP given that the n^{th} term is a and the sum to first n terms is b

#### SOLUTION

Solution :C

Option C is the correct answer.

### Question 3

The perimeter of a triangle, the sum of the products of the lengths of the sides taken two at a time and the product of the sides are denoted by p, q and r respectively. What is the maximum value of (r/pq)?

#### SOLUTION

Solution :B

Option B is the correct answer.

### Question 4

In a hostel, the total expense is divided in two parts. One part is fixed and the other part is shared among the occupants. If there are 20 occupants, they have to pay Rs. 650/month, however if there were 5 more, the cost would go down by Rs. 50. How many occupants would be there if the share comes to Rs. 500?

#### SOLUTION

Solution :D

### Question 5

All possible two factor products are formed from the numbers 101,102,103,…..200. How many numbers out of the total obtained are multiples of 7?

#### SOLUTION

Solution :A

Option A is the correct answer.

### Question 6

Let a, c ∈ {2, 4, 6, 8, 10, 12} and b ∈ {22, 24, 26}, where a, b and c are distinct. Find the number of equations of the form ax^{2} + bx + c = 0, that can be formed such that the equation has real roots.

#### SOLUTION

Solution :D

### Question 7

Find 9(S_{n}- S_{n-1}) if S_{n} (1≤n≤9) is the sum to n terms of the following series 1+22+333+4444+...999999999

^{n}–n

^{2}+n

^{n}-1)

^{n}-2)

^{2}+10n

#### SOLUTION

Solution :B

Option B is the correct answer.

### Question 8

Find the value of 11+13−42+13−(12)+33−43+12−(12)

#### SOLUTION

Solution :B

Option (b)

Solving the expression using the conventional approach

After a good amount of calculation,

Given expression is equal to 47+117=157

Reverse Gear approach

Approximating, expression 1 = something >12

Similarly, Expression 2= something >32Question = slightly greater than 12+32=2

Only option (b) satisfies this.

### Question 9

There are 2 quadratic equations, a^{2} –a +p=0 and a^{2} – a + 3p=0 (p≠0). For what value of p will one root of the second equation = double the root of the first equation?

#### SOLUTION

Solution :C

Option C is the correct answer.

### Question 10

There are two equations a and b, which have “n” roots in common. What is the value of n?
A: x^{3}+2x^{2}+7x-2=0 B: 8x^{2}-4x+4=0 :x>0

#### SOLUTION

Solution :A

Option (a)

Equate a&b to get the common values

x

^{3}+2x^{2}+7x-2=8x^{2}-4x+4solving we get

(x-1)(x-2)(x-3)=0

x=1,x=2,x=3

these values do not satisfy both equations. Hence, there are 0 roots in common.