Problem Solving Select Many Practice Test 5

Question 1

Which of the following values of x satisfies the equations (2-3x)/4 < 9 and x < -5? Indicate all correct options.

1. 1
2. -1
3. 5
4. -6
5. 10

Explanation:

(2-3x)/4 < 9
2-3x < 36
-3x < 36-2 = 34
x>-34/3 and x < -5

Only x = -6 and -10 satisfy the given equations.

Question 2

Which of the following is true? Indicate all correct options.

1. (3,4) lies in the first quadrant
2. (5,-2) lies in the fourth quadrant
3. (3,0) lies on the y-axis
4. (0,3) lies of the x-axis
5. (-2,-2) lies in the fourth quadrant

Explanation:

(3,4) lies in the first quadrant. Option A is true.
(5,-2) lies in the fourth quadrant. Option B is true.
(3,0) lies on the x-axis and (0,3) lies on the y-axis. Options C and D are false.
(-2,-2) lies in the third quadrant. Option E is false.

Question 3

A shopkeeper sold 200 tables at a loss. His loss was equal to the selling price of 4 tables. Which of the following statements is true?
Indicate all correct options.

1. His loss was Rs.2000
2. His loss % was 100/51%
3. His selling price was Rs.500 per table
4. The selling price cannot be determined
5. The total loss cannot be determined

Correct Answer: B, D and E

Explanation:

Let the selling price of one table be Rs.x.
We are given that the loss is equal to the selling price of 4 tables.
Hence, the total loss can be written in terms of the selling price i.e. 4x.
200 tables were sold and the cost price of 200 tables can be calculated as
C.P. = SP + loss
= 200x + 4x = 204x
Since the loss percent is calculated with respect to the cost price, it is independent of x
Loss % = loss/CP*100
= (4x/204x) *100
= 100/51%
Opions B, D and E are true.

Question 4

P(n,4) = 20 P(n,2). Which of the following is true? Indicate all correct options.
P is Probability

1. n = 7
2. n = -2
3. n can have only positive values
4. n is negative
5. n has two values

Explanation:

P(n,4) = 20 P(n,2)
n!/(n-4)! = 20*n!/(n-2)!
(n-2)!=20(n-4)!
(n-2)(n-3)(n-4)!=20(n-4)!
(n-2)(n-3)=20
n2 - 5n + 6 - 20 = 0
n2 - 5n - 14 = 0
n2 - 7n + 2n - 14 = 0
n(n-7) + 2(n-7) = 0
(n+2)(n-7) = 0
n = -2,7
Since n is positive, n = 7.
Options A and C are true.

Question 5

C(n,r) = 120 and P(n,r) = 720. Which of the following is true? Indicate all correct options.

1. r = 9
2. n = 9
3. n = 8
4. r = 3
5. r = 7

Explanation:

C(n,r) = 120
n!/[(n-r)!r!] = 120...(1)
P(n,r) = 720
n!/(n-r)! = 720...(2)
Dividing (2) by (1), we get
[n!/(n-r)!] / n!/[(n-r)!r!] = 720/120
r! = 6 = 3!
r = 3

Putting r = 3 in (2), we get
n!/(n-3)! = 720
n(n-1)(n-2)(n-3)!/(n-3)! = 720
n(n-1)(n-2) = 720
Clearly, n = 8
Options C and D are true.

Question 6

Which of the following is the LCM of 3!, 5! and 7!? Indicate all correct options.

1. 7!
2. 3!
3. 5!
4. 5040
5. 12

Explanation:

LCM of 3!, 5! and 7! = LCM of 3!, 5*4*3! and 7*6*5*4*3!
= 3!*4*5*6*7 = 7! = 5040

Question 7

The mth term of an A.P. is 1/n. The nth term is 1/m. Which of the following is true? Indicate all correct options.

1. The first term of the AP is m/n
2. The first term of the AP is n/m
3. The first term of the AP is 1/mn
4. The common difference of the AP is 1/mn
5. The first term of the AP is equal to the common difference.

Correct Answer: C, D and E

Explanation:

Let a be the first term and d be the common difference of the AP
According to the given conditions, we have
1/n = a + (m-1)d
1/m = a + (n-1)d
Subtracting one equation from the other, we get
1/n - 1/m = (m-n)d
(m-n)/mn = (m-n)d
d = 1/mn
Putting this value of d in any of the above equations, we get
1/n = a + (m-1)*1/mn
a = 1/n - (m-1)*1/mn
= (m-m+1)/mn = 1/mn
The first term of the AP is 1/mn and the common difference is 1/mn
Options C, D and E are true.

Question 8

f is a real function defined by f(x) = x3 - 3x + 5. Which of the following is true? Indicate all correct options.
[x3=x*x*x]

1. f(-2) = f(1)
2. f(-1) = f(1)
3. f(-1) = f(2)
4. f(3) = 23
5. f(3) = f(2) + 7

Correct Answer: A, C and D

Explanation:

f(-2) = (-2)3 -3*(-2) + 5 = -8 + 6 + 5 = 3
f(-1) = (-1)3 - 3*(-1) + 5 = -1 + 3 + 5 = 7
f(1) = 13 -3*1 + 5 = 1 - 3 + 5 = 3
f(2) = 23 - 3*2 + 5 = 8 - 6 + 5 = 7
f(3) = 33 - 3*3 + 5 = 27 - 9 + 5 = 23

Options A, C and D are true.
[23=2*2*2]

Question 9

The sum of two consecutive even positive numbers is less than 25 and each number is larger than 8. Which of the following is true?
Indicate all correct options.

1. One number is 12
2. The smaller number is 12
3. The sum of the numbers is less than 22
4. The smaller number is 10
5. The two numbers are multiples of 4

Explanation:

Let one number be x. The other number will be x+2
x+x+2 < 25
2x+2 < 25
2x < 25-2=23
x < 23/2 = 11.5
8 < x < 11.5
The numbers are 10 and 10+2 = 12
The smaller number is 10 and the larger number is 12
The sum of the numbers is 10+12 = 22
The two numbers are not multiples of 4
Hence, options A and D are true.

Question 10

If (n+1)! = 12 (n – 1)!, then which of the following is true for n? Indicate all correct options.

1. n = 4
2. n = 3
3. n = -4
4. n > 0
5. n < 0

Explanation:

(n+1)! = 12 (n – 1)!
(n+1).n.(n-1)!=12(n-1)!
(n+1).n = 12
n2 + n -12=0
n2 + 4n - 3n -12=0
n(n+4) -3(n+4) = 0
(n-3)(n+4) =0
n = 3, -4
Since n cannot be negtive, n = 3
Options B and D are true.

Question 11

Which of the following statements is true? Indicate all such statements.
[(4!)2 = 4!*4!]

1. 9!*10 = 90!
2. 4!*5!*6 = 6!
3. 7!*8*9 = 9!
4. (5-1)!5! = [(4!)2]*5
5. 4! = 12*3!

Explanation:

9!*10 = 10!
Option (A) is not true.

4!*5!*6 = 4!*(4!*5)*6
= 4!*6!
Option (B) is not true.

7!*8*9 = 9!
Option (C) is true.

(5-1)!5! = 4!*5! = 4!*4!*5 = [(4!)2]*5
Option (D) is true.

4! = 3!*4
12*3! = 3*4*3! = 3*4!
Option (E) is not true.

Question 12

Which of the following statements is true? Indicate all correct options.

1. Every rectangle is a square
2. Every square is a rectangle
3. Every parallelogram is a rhombus
4. Every rhombus is a parallelogram
5. Every square is a rhombus

Correct Answer: B, D and E

Explanation:

In a rectangle opposite sides are equal and each angle is a right angle. In a square all sides are equal.
Option A is false and B is true.
A parallelogram is a quadrilateral in which both the pairs of opposite sides are parallel and equal.
A rhombus is a parallelogram in which all sides are equal.
Option C is false and D and E are true.

Question 13

When a number x is increased by 17, it equals 60 times its reciprocal. Which of the following statements is true? Indicate all correct options.

1. There are two possible values of x
2. There are two positive values of x
3. x = -3
4. x = 20
5. x = -20

Explanation:

x+17 = 60*1/x
x2 + 17x = 60
x2 + 17x - 60=0
x2 +20x - 3x - 60=0
x(x+20) - 3(x+20) = 0
x = 3, -20
Options A and E are true.
[x2=x*x]

Question 14

If two buses starting from points A and B go in the same direction, then they meet in 6 hours. If they go in opposite directions, then they meet in 2 hours. The distance between points A and B is 120 km. Which of the following is true? Indicate all correct options.

1. The faster bus travels at a speed of 40 km/hr
2. The difference between the speeds of the buses is 20 km/hr
3. The faster bus is twice as fast as the slower bus
4. The slower bus travels at a speed of 40 km/hr
5. The speeds of the buses cannot be determined

Correct Answer: A, B and C

Explanation:

Let the buses start in the same direction, from A towards B and beyond. They meet in 6 hours at a point at a distance of say x km from point B.
Distance travelled by bus at A = (120 + x) km
Distance travelled by bus at B = x km in 6 hours.
Let the buses start in the opposite directions, towards each other. They meet in 2 hours at a point at a distance of say y km from point B.
Distance travelled by bus at A = (120 - y) km
Distance travelled by bus at B = y km in 2 hours.
Speed =distance/time
We equate the speeds of the buses in the two situations
(120+x)/6 = (120-y)/2 and x/6 = y/2
120+x = 360-3y and x = 3y
120 + 3y = 360 - 3y
6y = 360 - 120
y = 240/6 = 40
Speed of bus at A = (120+3*40)/6
= 240/6 = 40 km/hr
Speed of bus at B = y/2 = 40/2
= 20 km/hr
Options A, B and C are true.

Question 15

Which of the following is true? Indicate all correct options.

1. Sqrt(2) is an integer
2. Sqrt(2) is an irrational number
3. Sqrt(2) is a real number
4. -5/4 is an integer
5. 5 is an integer

Correct Answer: B, C and E

Explanation:

Sqrt(2) = 1.41421 is a non-terminating repeating decimal. Hence, it is not an integer and it is an irrational real number.
Options B and C are true.
-5/4 is not an integer and 5 is an integer.
Option E is true. All other options are false.