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Question 1:Â The problem below consists of a question and two statements numbered
1 & 2.
You have to decide whether the data provided in the statements are sufficient to answer the question.
Rahim is riding upstream on a boat, from point A to B, at a constant speed. The distance from A to B is 30 km. One minute after Rahim leaves from point A, a speedboat starts from point A to go to point B. It crosses Rahimâ€™s boat after 4 minutes. If the speed of the speedboat is constant from A to B, what is Rahimâ€™s speed in still water?
1. The speed of the speedboat in still water is 30 km/hour.
2. Rahim takes three hours to reach point B from point A.

a)Â Statement 1 alone is sufficient to answer the question, but statement 2 alone is not
sufficient

b)Â Statement 2 alone is sufficient to answer the question, but statement 1 alone is not
sufficient

c)Â Each statement alone is sufficient

d)Â Both statements together are sufficient, but neither of them alone is sufficient

e)Â Statements 1 & 2 together are not sufficient

Solution:

Let ‘a’ and ‘b’ be the speeds of Rahim and the speed boat in still water, respectively

Let ‘c’ beÂ the speed of the stream.

Speed boat starts after 1 minute of Rahim’s departure and crosses him after 4 minutes.

It implies that the speed boat covered the same distance in 4 minutes that Rahim took 5 minutes.

Hence, the ratio of upstream speeds of Rahim and Speed boat =Â $\frac{a-c}{b-c}=\frac{4}{5}$

Using Statement 1, we get $b=30\ kmph$. But, it’s not sufficient to get the value of ‘a’.

Using Statement 2, we getÂ $a-c=\frac{30}{3}=10\ kmph$. But it’s also not sufficientÂ to get the value of ‘a’.

using both statements, we get

$\ \frac{\ a-c\ }{b-c}=\frac{4}{5}$

$\ \frac{\ 10\ }{30-c}=\frac{4}{5}$

$\ \ 50\ =120-4c$

$\ \ c\ =17.5$

$\ \ a=17.5+10=27.5\ kmph$

Question 2:Â The problem below consists of a question and two statements numbered
1 & 2.
You have to decide whether the data provided in the statements are sufficient to answer the question.
In a cricket match, three slip fielders are positioned on a straight line. The distance between 1st slip and 2nd slip is the same as the distance between 2nd slip and the 3rd slip. The player X, who is not on the same line of slip fielders, throws a ball to the 3rd slip and the ball takes 5 seconds to reach the player at the 3rd slip. If he had thrown the ball at the same speed to the 1st slip or to the 2nd slip, it would have taken 3 seconds or 4 seconds, respectively. What is the distance between the 2nd
slip and the player X?
1. The ball travels at a speed of 3.6 km/hour.
2. The distance between the 1st slip and the 3rd slip is 2 meters.

a)Â Statement 1 alone is sufficient to answer the question, but statement 2 alone is not sufficient.

b)Â Statement 2 alone is sufficient to answer the question, but statement 1 alone is not sufficien

c)Â Each statement alone is sufficient

d)Â Both statements together are sufficient, but neither of them alone is sufficient

e)Â Statements 1 & 2 together are not sufficient

Solution:

Let players at slips 1, 2 and 3 be a, b, and c, respectively.

Let the speed of the ball be m.

The trajectory of the ball is not specified in the question. But let’s take it as straight line.

The distance between the 1st and 2nd slips is the same as the distance between the 2nd and 3rd slips.

The length of ‘ab’ = The length of ‘bc’. It implies Xb is the median of triangle Xac.

It takes 3 sec, 4 sec and 5 sec for the ball to reach a, b and c, respectively.

Hence, Xa = 3m, Xb = 4m and Xc = 5m.

Using Apollonius’s theorem,

$2\times\ \left(\left(4m\right)^2+\left(ab\right)^2\right)=\left(3m\right)^2+\left(5m\right)^2$

$16m^2+\left(ab\right)^2=17m^2$

$ab=m$

Hence,Â $ac=2m$

Using the properties of triangle, the sum of two sides should be greater than the third side,

But, Xa + ac = Xc

3m + 2m = 5m.

Hence, this arrangement is not possible.

As thisÂ question has ambiguity, XAT officials awarded full marks to all candidates for this question.

Question 3:Â Nadeemâ€™s age is a two-digit number X, squaring which yields a three-digit number,whose last digit is Y. Consider the statements below:
Statement I: Y is a prime number
Statement II: Y is one-third of X

a)Â either of I and II, by itself, is sufficient.

b)Â only II is sufficient, but I is not.

c)Â only I is sufficient, but II is not.

d)Â it is necessary and sufficient to take I and II together.

e)Â even taking I and II together is not sufficient.

Solution:

The age of Nadeem is a two-digit number. When squared yields a three-digit number whose last digit is Y. Y is a prime number.

Using statement 1:

When a number is squared :

The last digit of the number can be :

1, 2, 3, 4, 5, 6, 7, 8, 9. When squared the last digit can be :

For a number ending with 1: 1

For a number ending with 2: 4

For a number ending with 3: 9

For a number ending with 4: 6

For a number ending with 5: 5

For a number ending with 6: 6

For a number ending with 7: 9

For a number ending with 8: 4

For a number ending with 9: 1

The only possible prime number is 5.

Hence the last digit of X is 5 and Y is 5.

Using statement 2 : Y = X/3.

This alone cannot beÂ sufficient to determine the possibilities for Y and X.

Combining both the statements :

Since Y = 5, then the value of X = 15.

The age is equal to 15.

Question 4:Â Given below are two statements
Statement I : $(543)_6$ is equivalent to $(317)_8$
Statement II : The last 4 bits in the binary representation of a multiple of 16 is 1000.
In light of the above statements, choose the correct answer from the options given below

a)Â Both Statement I and Statement II are true

b)Â Both Statement I and Statement II are false

c)Â Statement I is true but Statement II is false

d)Â Statement I is false but Statement II is true

Solution:

543 in base 6 expressed in decimal form =5(36)+4(6)+3 =207
317 in base 8 expressed in decimal form = 3(64)+8+7 =207
So statement 1 is correct
Now 16 in base 2 = 10000
32 in base 2 = 100000
Therefore statement 2 is false
Hence C is the correct answer

Question 5:Â Given below are two statements
Statement I: In the sequence of numbers 30, 90, 182, 306, 462,P…….. . the term P is 650.
Statement II : There are 8 digits in $9^{8}$ when it is expressed in decimal form.
In light of the above statements, choose the correct answer from the options given below

a)Â Both Statement I and Statement II are true

b)Â Both Statement I and Statement II are false

c)Â Statement I is true but Statement II is false

d)Â Statement I is false but Statement II is true

Solution:

We have :
30, 90, 182, 306, 462,P.
Now 90-30 =60Â  (1)
182-90 = 92 (2)
306-182 = 124 Â  (3)
462 -306 = 156
If you observe , the first difference of the terms form an arithmetic progression
So we can say the next term will be
462 +156+32 = 462 +188 = 650
So P is 650
So statement 1 is true .
Taking statement 2 :
we get 9^8
Now number of digits = [log9^8 +1]
we get [8log9+1]= 8
so number of digits in 9^8 is 8

Question 6:Â Which of the following statements regarding quadratic equations are true?
(A) Solution set for the equation $6x^{2}-5x=4$ is $\left\{-\frac{1}{2},\frac{4}{3}\right\}$
(B) The nature of the roots of the equation $9x^{2}+6x+1=0$ is equal, real and rational.
(C) If one root of $4x^{2}-3x+K=0$ is 3 times the other, then $K=\frac{27}{256}$

a)Â (A),(B) and (C)

b)Â (A) and (B) only

c)Â (B) and (C) only

d)Â (C) and (A) only

Solution:

We have :
$6x^2-5x-4\ =0\$
=$6x^2-8x+3x-4\ =0\$
(2x+1)(3x-4)=0
so solution set is {-1/2 ,4/3}

$9x^2+6x+1\ =0$
D = 36-36 =0
So we can say roots are real and equal

$4x^2-3x+k$=0
let roots be p and 3p
we get 4p =3/4
p =3/16
Now k/4 = 3p^2
we get k = 12p^2
we get k = 27/64
So A and B are true and C is false

Question 7:Â Given below are two statements:
Statement I : Acoin is tossed three times. The probability of getting exactly two heads is 3/8
Statement II: In tossing of 10 coins, the probability of getting exactly 5 heads is 63/256
In the light of the above statements, choose the correct answer from the options given below.

a)Â Both Statement I and Statement II are true

b)Â Both Statement I and Statement II are false

c)Â Statement is true but Statement II is false

d)Â Statement I is false but Statement II is true

Solution:

Statement I:

Probability of getting head = P(H) =Â $\frac{1}{2}$

Probability of getting tail = P(T) = $\frac{1}{2}$

3 coins are tossed and probability of getting 2 heads =Â $3_{C_2}\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)$ =Â $\frac{3}{8}$

Statement I is true.

Statement II:

10 coins are tossed and probability of getting exactly 5 heads =Â $10_{C_5}\left(\frac{1}{2}\right)^5\left(\frac{1}{2}\right)^5$ =Â $\frac{63}{256}$

Statement II is true

Question 8:Â Given below are two statements:
Statement I: $\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}=2$
Statement II: If $a+b+c=0$, then $(a^{3}+b^{3}+c^{3})\div abc=3$
In light of the above statements, choose the correct answer from the options given below

a)Â Both Statement I and Statement II are true

b)Â Both Statement I and Statement II are false

c)Â Statement I is true but Statement II is false

d)Â Statement I is false but Statement II is true

Solution:

From statement 1 we get
$\frac{\left(\sqrt{\ 7}+\sqrt{\ 5}\right)^2+\left(\sqrt{\ 7}-\sqrt{\ 5}\right)^2}{7-5}$
= 12
So statement 1 is false
Now statement 2 is an identity
so 2 is true .

Question 9:Â Given below are two statements
Statement I : The set of numbers(5,6, 7, p, 6, 7, 8, q) has an arithmetic mean of 6 and mode (most frequently occurring number) of 7. Then $p\times q=16$.
Statement II: Let p and q be two positive integers such that $p+q+p\times q=94$. Then $p+q=20$.
In light of the above statements, choose the correct answer from the options given below

a)Â Both Statement I and Statement II are true

b)Â Both Statement I and Statement II are false

c)Â Statement I is true but Statement II is false

d)Â Statement I is false but Statement II is true

Solution:

Statement 1:

It is given that the mode of the set is 7. Thus, either p or q(or both) will be 7.

Now, p*q will be a multiple of 7. But it is given that p*q = 16. Thus, this statement is false.

Statement 2:

It is given that p +Â q + p*q = 94

If pÂ + q = 20, then p*q = 74.

Thus, either (p,q) will be (2,37), (37,2), (1,74) or (74,1).

Since none of the values will give a sum of 20, this statement is also false.

Thus, the correct option is B.

Question 10:Â Study the questions and the statements given below. Decide whether any information provided in the statement(s) is redundant and/or can be dispensed with, to answer it.
If 7 is added to numerator and denominator each of fraction a/b, will the new fraction be less than the original one?
Statement I: a = 73, b = 103
Statement II: The average of a and b is less than b
Statement III: a âˆ’ 5 is greater than b âˆ’ 5

a)Â II and either I or III

b)Â Only II or III

c)Â Any two of them

d)Â Any one of them

Solution:

IfÂ $\ \frac{\ a}{\ \ b}$ is less than 1, thenÂ $\ \frac{\ a+7}{\ \ b+7}$ >Â $\ \frac{\ a}{\ \ b}$ given that a,b>0

Hence, if I is given. II or III are redundant.

Question 11:Â This question consists of a question and two statements numbered I and II. Decide whether the data given in the statements are sufficient to answer the question.
What is the 57th number in a series of numbers?
I. Each number in the series is three more than
the preceding number.
II. The tenth number in the series is 29.

a)Â The data in Statement I alone is sufficient to answer the question while the data in Statement II alone is not sufficient to answer the question.

b)Â The data in Statement II alone is sufficient to answer the question, while the data in Statement I alone is not sufficient to answer the question

c)Â If the data either in Statement I or Statement II alone are sufficient to answer the question

d)Â If the data in both Statements I and II together is necessary to answer the question

Solution:

UsingÂ $a_n$ = a+(n-1)d

=>Â $a_57$=a+56d

From 1, the value of d can be obtained.

From 2, an equation in a and d is obtained.

Hence, both 1 and 2 are necessary to calculate the value ofÂ $a_57$.

Question 12:Â If a and b are negative, and c is positive, which of the following statement/s is/are true?
I) $a – b < a – c$
II) if $a < b$, then $\frac{a}{c} < \frac{b}{c}$
III) $\frac{a}{b} > \frac{a}{c}$

a)Â I only

b)Â II only

c)Â III only

d)Â II and III only

Solution:

I) $a-c$ is negative as $a$ is negative and $c$ is positive. Thus, the value of $a-c$ will be smaller than $a$(since more negative is added to a negative number). $a-b$ can be negative or positive depending upon the value of $a$ and $b$, but it can be concluded that this number will be greater than $a$(since a positive number is added to a negative number). Thus, $(a-c)<(a-b)$. Thus, the statement I is false.

II) Given $a<b$,Â $\frac{a}{c}<\frac{b}{c}$ as $c$ is positive and $a < b$. Therefore, statement II is true

III) $a$ is negative and $b$ is negative, this implies $\frac{a}{b}$ is positive. $a$ is negative and $c$ is positive, this impliesÂ $\frac{a}{c}$ is negative. Therefore,Â $\frac{a}{b}>\frac{a}{c}$. Statement III is true

Question 13:Â In March 2007, Computers Ltd. made a bundled offer of its Laptops together with Deskjet printers to boost Sales, though both the Laptop and the printer were also available individually. What is the price of the Printer, if purchased separately? Decide whether the information given in the two statements is sufficient to solve the problem.
Statement (1): The bundled offer price was Rs 42,600
Statement (2): The Laptop, without the bundle offer was priced at Rs 39,400

a)Â Any one of the two statements (1) or (2) is sufficient to answer the question

b)Â Each of the statement (1) or (2) taken alone, is sufficient to answer the question

c)Â Both statements, taken together, are sufficient to answer the question, but neither of them alone is sufficient.

d)Â Both the statements together are insufficient to answer the question.

Solution:

Understanding the premise of the question, we can infer that the price of the laptop when taken individually is, let’s say x and that of the printer is y.

But, when taken together under the bundle offer, the price is lesser than (x+y), because of the discount given, let’s say a%.

Statement 1 gives us the discounted price of the printer and laptop taken together, i.e.Â $\ \left(x+y\right)\left(1-\ \frac{\ a}{100}\right)$

We have three variables and cannot solve this equation.

Statement 2 gives us the value of x. But still, two variables, a and y are unknown.

Even if we use both the statements together, we will have 2 variables and only 1 equation.

Therefore, both statements are insufficient to find the value of y.

Question 14:Â The profits of Biscuits India Ltd soared by 32% in the year 2006 – 07 as compared to year 2005 – 06. By what % did Biscuits Indiaâ€™s Sales increase in 2006 – 07 compared to the previous year? ( ssume: Profit = Sales – Expenses) Decide whether the information given in the two statements is sufficient to solve the problem.
Statement (1): Expenses in 2006 – 07 were Rs 1,400 crores, as compared to Rs 1,220 crores in 2005 – 06
Statement (2): Sales in 2006 – 07 were Rs 4,300 crores

a)Â Any one of the two statements (1) or (2) taken alone, is sufficient to answer the question.

b)Â Each of the statement (1) or (2) taken alone, is sufficient to answer the question.

c)Â Both statements, taken together, are sufficient to answer the question, but neither of them alone is sufficient

d)Â Both the statements together are insufficient to answer the question.

Solution:

Considering statement 1 alone, we can make a table as follow:

We get S-1220=P and S’-1400=1.32P

Since there are 3 variables, S, S’ and P, we cannot get the values of the variable using the two equations alone.

So, Statement 1 alone is not sufficient.

Using statement 2, 1.32P= 4300

=> P= 4300/1.32. But again, two variables, S and S’ are unknown.

So,Â So, Statement 1 alone is not sufficient.

When we use both the given statements, we will have

S-1220=4300/1.32 and

S’-1400=4300.

Using these two equations, we can find S and S’ respectively and hence answer the question asked.

Therefore, both statements are necessary to answer the question.

Question 15:Â Bags I, II and III together have ten balls. If each bag contains at least one ball, how many balls does each bag have? Decide whether the data given in the statements are sufficient to answer the question.
Statement (1): Bag I contains five balls more than bag III.
Statement (2): Bag II contains half as many balls as bag I

a)Â Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

b)Â Statement (2) Alone is sufficient, but statement (1) alone is not sufficient to answer the question asked.

c)Â BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked

Solution:

Considering statement 1 alone,

Let the number of balls in bag III be x.

The number of balls in bag I becomes x+5.

And the rest, which is 10- (x+x+5)= 5-2x balls are with C.

We cannot find the value of x, so Statement 1 alone is not sufficient.

Considering statement 2 alone,

Let the number of balls in bag I be 2a

Balls in bag II will be a and the rest, i.e 10-3a will be in bag 3.

Again, we cannot find the value of a in this case, and hence, Statement 2 alone is not sufficient.

Considering both the statements and using the number of balls with each one of them as found using statement 1, we get

$5+a=2\left(5-2a\right)$

=>5+a= 10-4a

=> 5a=5, or a=1.

So, balls in Bag I= 5+a=6

balls in Bag II= 5-2a=3

balls in Bag III= a= 5.

Therefore, both statement I and II are required.

Question 16:Â Ram is taller than Shyam and Jay is shorter than Vikram. Who is the shortest among them?
(I) Ram is the tallest.
(II) Shyam is taller than Vikram.

a)Â Data in Statement I alone is sufficient to answer the question but the data in Statement II alone is not sufficient to answer the question.

b)Â Data in Statement II alone is sufficient to answer the question but the data in Statement I alone is not sufficient to answer the question.

c)Â Data in both statements I and II together are necessary to answer the question.

d)Â Data in both statements I and II together is not sufficient to answer the question.

Solution:

Given that Ram > Shyam, Vikram > Jay.
Hence from this we can conclude that neither Ram nor Vikram is the shortest. And we have to find the shortest. And we have to find the shortest among them:
Consider statement alone:
We know that that Ram is not the shortest, either Shyam or Jay is the shortest.
Hence (I) alone is not sufficient.
Consider statement I alone Shyam > Vikram.
From the given information and the information in (II), we get Ram > Shyam > Vikram > Jay.
Hence, (II) alone is sufficient.

Question 17:Â What is the average height of the class?
(I) Average height of the class decreases by 1 cm if we exclude the tallest person of the class whose height is 56 cm.
(II) Average height of the class increases by 1 cm if we exclude the shortest person of the class whose height is 42 cm.

a)Â Data in Statement I alone is sufficient to answer the question but the data in Statement II alone is not sufficient to answer the question.

b)Â Data in Statement II alone is sufficient to answer the question but the data in Statement I alone is not sufficient to answer the question.

c)Â Data in statements I and II together are necessary to answer the question.

d)Â Data in statements I and II together is not sufficient to answer the question.

Solution:

Let x be the average height of the class and n be the number of students in the class.
Consider statements I alone
xn – 56 = (x -1)(n -1)
â‡’ x + n = 57Â  Â  Â Â Â  (1)
Hence, the value of x cannot be found. So, I alone is not sufficient.
Consider statement I alone:
xn – 42 = (x + 1)(n – 1)
â‡’ x – n = 41Â  Â  Â  Â  Â Â  (2)
Hence, the value of x cannot be found. So, II alone is not sufficient.
Both the statements together are sufficient as the value of x can be found by solving (1) and (2)

Question 18:Â Salary of A and B is in the ratio 3 : 4 and expenditure is in the ratio 4 : 5. What is the ratio of their savings?
(I) B’s saving is 25% of his salary.
(II) B’s salary is Rs. 2500.

a)Â Data in Statement I alone is sufficient to answer the question but the data in Statement II alone is not sufficient to answer the question.

b)Â Data in Statement II alone is sufficient to answer the question but the data in Statement I alone is not sufficient to answer the question.

c)Â Data in both statements I and II together are necessary to answer the question.

d)Â Data in both statements I and II together are not sufficient to answer the question.

Solution:

Let salary of A be 3x and salary of B be 4x
Let expenditure of A be 4y and B be 5y
Now Statement 1 says B save 25% of his salary
So savings of B = x
Now we can say : x= 4x-5y ( As Savings = Income – Expenditure )
we get : 3x =5y
y = 3x/5 Now Therefore savings of A : 3x-4y = 3x- 4(3x/5)
And then we can calculate ratio of savings .
So statement 1 alone is sufficient .
Now Statement 2 says
B’s salary is 2500
Now as we do not know anything about expenditure of B so we cannot calculate savings of B and henceforth savings of A
so Statement 2 alone is not sufficient

Question 19:Â Triangle ABC and Triangle PQR are congruent.
(I) Area of triangle ABC and triangle PQR are same
(II) Triangle ABC and triangle PQR are right angle triangles.

a)Â Data in Statement I alone is sufficient to answer the question but the data in Statement II alone is not sufficient to answer the question.

b)Â Data in Statement II alone is sufficient to answer the question but the data in Statement I alone is not sufficient to answer the question.

c)Â Data in statements I and II together are necessary to answer the question.

d)Â Data in statements I and II together are not sufficient to answer the question.

Solution:

Statement I:

If area of triangle ABC and PQR are equal, we cannot infer that they are congruent. Hence, statement I alone is not sufficient to answer the question.

Statement II:

If both triangles ABC and PQR are right angled triangles, we cannot say whether they are congruent or not. Hence, statement II alone is not sufficient to answer the question.

Considering both the statements, we do not have any condition regarding sides or angles of triangles. Therefore, statement I and statement II together are not sufficient to answer the question.

Question 20:Â Ram got Rs. 1,500 as dividend from a company. What is the rate of interest given by the company?
(I) The dividend paid last year was 10%.
(II) Ram has 350 shares of Rs. 10 denomination.

a)Â Statement I alone is sufficient to answer the question.

b)Â Statement II alone is sufficient to answer the question.

c)Â Both statements I and II together are necessary to answer the question.

d)Â Both statements I and II together are not sufficient to answer the question.

$\therefore\$Â Statement II alone is sufficient to answer the question.