# Free Mixed Bag 1 Practice Test - CAT

What is the remainder when  1992  is divided by 92?

A. 0
B. 1
C. 48
D. None of these

#### SOLUTION

Solution : D

Euler’s number of  92=92(1123)(112)=88

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

Find the value of g, given that 8x2+2x+g=0 and 2x2+gx+1=0 have a common root?

A. 1
B. 2
C. 3
D. a and b
E. None of the above

#### SOLUTION

Solution : E

OPTION E

To find the common root, equate the two equations

2x2+gx+1=8x2+2x+g

-6x2+x(g-2)+(1-g)=0

Now 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 : g2 - 28g +28 = 0

Check with answer options at this stage

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

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 =?

A. 2:3:4
B. 1:2:4
C. 1:2:3
D. 4:6:7

#### 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

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

A. 1
B. 3
C. 5
D. 6
E. None of these

#### 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.

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.

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]

A. 9
B. 3
C. 6
D. 7

#### SOLUTION

Solution : A

Option (a)

Digit Sum = Remainder  when  a  number is  divided by 9

25!9  Rem=0 Digit Sum=9

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?

A. 30 sec
B. 45 sec
C. 60 sec
D. 90 sec

#### SOLUTION

Solution : B

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\)
.

A. 30
B. 60
C. 45
D. 15

#### 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 .

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.

A. 2
B. 1
C. 3
D. 4
E. 5

#### 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.

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

A. a<2
B. a>-5/2
C. a<-1/2
D. a>-1/2
E. a < -2

#### 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)

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?

A. 12
B. 15
C. -10
D. -2

#### 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).

Take option c

Sum to 1st 6 terms is given by the formula n2(2a+[n1]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

Find the last two digits of  (148)1084

A. 06
B. 66
C.  56
D. 16

#### 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

Consider a,b,c : real numbers. X=a+bab,Y=b+cbc,Z=c+aca . 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=4

X= -5Y= -7Z= 3

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

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

A. 15cm
B. 30cm
C. 25cm
D. 252 cm

#### 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)

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.

A. 4x3x1
B. 13
C. 2x+1x
D. 23

#### 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.