# Free Mixed Bag 1 Practice Test - CAT

### Question 1

What is the remainder when 1992 is divided by 92?

#### SOLUTION

Solution :D

Euler’s number of 92=92(1−123)(1−12)=88

198892|r×19492|r=1×4992|r=49 (from frequency)

### Question 2

Find the value of g, given that 8x^{2}+2x+g=0 and 2x^{2}+gx+1=0 have a common root?

#### SOLUTION

Solution :E

OPTION E

To find the common root, equate the two equations

2x

^{2}+gx+1=8x^{2}+2x+g-6x

^{2}+x(g-2)+(1-g)=0Now find the discriminant of this equation and equate that to 0(D=0 means that the roots are equal and there exists a single root, acording to the question also, there is a single root which is common, hence D=0)

we will get it as : g

^{2}- 28g +28 = 0Check with answer options at this stage

None of the values of x is being satisfied, hence option e.

### Question 3

If a, b, c, all distinct are such that a,b, and c are in AP and b-a, c-b, and a are in GP then a:b:c =?

#### SOLUTION

Solution :C

Going from answer options is the best choice.

**Observe that only options (a) and (c) are in an AP.

By Assumption technique; taking a=2, b=3 and c=4

b-a=1, c-b=1 and a=2, which does not form a GP. Hence this option can never be true

Taking a=1, b=2 and c=3.

b-a=1, c-b=1 and a=1 is in a GP. Hence, answer option (c) is the right one

### Question 4

How many different Arithmetic Progressions can you form with atleast three terms with first term 2 and last term 999?

#### SOLUTION

Solution :A

Option e

The number of AP’s = (Number of factors of 997 - 1), which is prime.

Hence only one AP’s of 998 terms can be formed.

Answer is option a

### Question 5

In the figure below, C is the center of the square. Find the area of the shaded area?

#### SOLUTION

Solution :Two perpendicular lines intersecting at the center of the square will divide the square in four equal areas. Area of the square = 100. So, area of shaded region = 1004 = 25.

### Question 6

Find the sum of the sum of the sum of digits (i.e. digit sum) of 25! [e.g. digit sum of 378 = 3+7+8 = 18 : 1+8 = 9]

#### SOLUTION

Solution :A

Option (a)

Digit Sum = Remainder when a number is divided by 9

25!9 Rem=0⇒ Digit Sum=9

### Question 7

A man can walk up a moving "up" escalator in 30 seconds. The same man can walk down this moving "up" escalator in 90 seconds. Assume that his walking speed is same upwards and downwards. How much time will he take to walk up the escalator, when it is not moving?

#### SOLUTION

Solution :B

### Question 8

In the given figure AB is a diameter of the circle, CD is a chord parallel to AB, and AC intersects BD at E. If the ratio of area of triangles AEB and DEC= 4: 1, then find out the value of (\theta\)

.

#### SOLUTION

Solution :B

Join AD. As AB is the diameter ∠ ADB = 90. And hence ΔADB is a right angle triangle.

Triangles AEB and DEC are similar hence if the ratio of area is 4: 1, ratio of sides will be 2:1.

So, AE: DE = 2: 1. In triangle ADE, hypotenuse AE = 2, DE = 1 AD2 = 4 – 1 = 3. AD = √3

So, ∠ AED = 60 ∘ .

### Question 9

There are a certain number of tins that are filled with milk from a machine. If the machine fills at the rate of 5 tins/min, 3 tins are left empty; at the rate of 8 tins/min, 5 tins are left empty and and the rate of 7 tins/min, 4 tins are left empty. In this setup it is possible to have N different number of tins. Find N, given that (100<N<1000). There is no leakage or wastage of milk and the machine can work only for integral number of minutes.

#### SOLUTION

Solution :C

Option c

Using CRT

5A+3=8B+5

5A=8B+2, B=1 and A=2 .the AP is of the form 40K +13

40K+13=7C+4

7C=40K+9

7c=5K+2 ⇒ K=1 and C=7. The AP is of the form 53+280N

Three digit numbers possible in this AP are 53+280=333 , 53+2(280)=613 and 53+3(280)=893.

### Question 10

If |a-2|<|a-3|, then for what real values of ‘a’, will the inequality hold true?

#### SOLUTION

Solution :A

option (a)

Take a=0. the inequality satisfies for this value as(2<3). Hence option (c) and option (e) are ruled out.

Take a= 3 => 1<0, Which is not true. Hence, option (b) and option (d) are ruled out. Answer is option (a)

### Question 11

What is the common difference of an AP which has its first term as 100 and the sum of its first 6 terms = 5 times the sum of its next six terms?

#### SOLUTION

Solution :C

Option a and b can be eliminated directly, as from the statement sum of its first 6 terms = 5 times the sum of its next six terms, we can deduce that the Common Difference is negative (decreasing AP).

Go from answer options :

Take option c

Sum to 1st 6 terms is given by the formula n2(2a+[n−1]d)

= 3(200-50)= 450

Sum to 2nd 6 terms = Sum to 12 terms – sum to 6 terms

= 6( 200-110) -450

= 540 – 450 = 90

5 × Sum to 2nd 6 terms = 450

Hence option c

### Question 12

Find the last two digits of (148)1084

#### SOLUTION

Solution :D

(148)1084=(37)1084×(4)1084

=(374)271×(2)2168

=(372×372)271×....56

=(..69×..69)271×.....56

=(61)271×....56=....61×....56=......16

### Question 13

Consider a,b,c : real numbers. X=a+ba−b,Y=b+cb−c,Z=c+ac−a . The value of XY+YZ+XZ is

#### SOLUTION

Solution :Assume a=1, b=2, c=3

X=-3Y=-5Z=2

XY+YZ+XZ = 15-10-6 = -1

Assume a=2,b=3 and c=4X= -5Y= -7Z= 3

XY+YZ+XZ= 35-21-15= -1

### Question 14

A cyclic quadrilateral has one of its angles as 90∘ . Also, its radius is 12.5 cm. What is the length of its diagonal?

#### SOLUTION

Solution :C

Option(c)

If one of the angles of a cyclic quadrilateral is 90∘ , the diagonal will be as long as the diameter = 25 cm (12.5 x 2)

### Question 15

There is an arithmetic progression with first term = common difference. The number of terms in this series is 2x. A new series is formed taking the odd terms of the first series upto (2x-1)th term. Find the ratio of the sum of the terms in the old series to that in the new series.

#### SOLUTION

Solution :C

Option (c)

Let the first term = 1, cd = 1

Take x = 2, 2x = 4, the progression is 1, 2, 3, 4

New progression = 1, 3

Sum to x terms of progression 1 = 1+2+3+4 = 10

Sum to new progression = 4

Ratio=104=52.

Only option c satisfies this condition.