# 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
- 5
- -6
- 10

**Correct Answer: D and E**

**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.**

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

**Correct Answer: A and B **

**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.**

- His loss was Rs.2000
- His loss % was 100/51%
- His selling price was Rs.500 per table
- The selling price cannot be determined
- 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**

- n = 7
- n = -2
- n can have only positive values
- n is negative
- n has two values

**Correct Answer: A and C**

**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

n^{2} - 5n + 6 - 20 = 0

n^{2} - 5n - 14 = 0

n^{2} - 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.**

- r = 9
- n = 9
- n = 8
- r = 3
- r = 7

**Correct Answer: C and D**

**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.**

- 7!
- 3!
- 5!
- 5040
- 12

**Correct Answer: A and D**

**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.**

- The first term of the AP is m/n
- The first term of the AP is n/m
- The first term of the AP is 1/mn
- The common difference of the AP is 1/mn
- 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) = x ^{3} - 3x + 5. Which of the following is true? Indicate all correct options.[x^{3}=x*x*x]**

- f(-2) = f(1)
- f(-1) = f(1)
- f(-1) = f(2)
- f(3) = 23
- 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) = 1^{3} -3*1 + 5 = 1 - 3 + 5 = 3

f(2) = 2^{3} - 3*2 + 5 = 8 - 6 + 5 = 7

f(3) = 3^{3} - 3*3 + 5 = 27 - 9 + 5 = 23

Options A, C and D are true.

[2^{3}=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.**

- One number is 12
- The smaller number is 12
- The sum of the numbers is less than 22
- The smaller number is 10
- The two numbers are multiples of 4

**Correct Answer: A and D**

**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.**

- n = 4
- n = 3
- n = -4
- n > 0
- n < 0

**Correct Answer: B and D**

**Explanation:**

(n+1)! = 12 (n – 1)!

(n+1).n.(n-1)!=12(n-1)!

(n+1).n = 12

n^{2} + n -12=0

n^{2} + 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!]**

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

**Correct Answer: C and D**

**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.**

- Every rectangle is a square
- Every square is a rectangle
- Every parallelogram is a rhombus
- Every rhombus is a parallelogram
- 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.**

- There are two possible values of x
- There are two positive values of x
- x = -3
- x = 20
- x = -20

**Correct Answer: A and E**

**Explanation:**

x+17 = 60*1/x

x^{2} + 17x = 60

x^{2} + 17x - 60=0

x^{2} +20x - 3x - 60=0

x(x+20) - 3(x+20) = 0

x = 3, -20

Options A and E are true.

[x^{2}=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.**

- The faster bus travels at a speed of 40 km/hr
- The difference between the speeds of the buses is 20 km/hr
- The faster bus is twice as fast as the slower bus
- The slower bus travels at a speed of 40 km/hr
- 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.**

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