# Problem Solving Select Many Practice Test 5

#### Q.1

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

• A. 1
• B. -1
• C. -5
• D. -6
• E. -10
• (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.

#### Q.2

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

• A. (3,4) lies in the first quadrant
• B. (5,-2) lies in the fourth quadrant
• C. (3,0) lies on the y-axis
• D. (0,3) lies of the x-axis
• E. (-2,-2) lies in the fourth quadrant
• (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.

#### Q.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.

• A. His loss was Rs.2000
• B. His loss % was 100/51%
• C. His selling price was Rs.500 per table
• D. The selling price cannot be determined
• E. The total loss cannot be determined
• Answer: B, D and E
• 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.

#### Q.4

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

• A. n = 7
• B. n = -2
• C. n can have only positive values
• D. n is negative
• E. n has two values
• 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.

#### Q.5

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

• A. r =9
• B. n = 9
• C. n = 8
• D. r = 3
• E. r = 7
• 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.

#### Q.6

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

• A. 7!
• B. 3!
• C. 5!
• D. 5040
• E. 12
• 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

#### Q.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.

• A. The first term of the AP is m/n
• B. The first term of the AP is n/m
• C. The first term of the AP is 1/mn
• D. The common difference of the AP is 1/mn
• E. The first term of the AP is equal to the common difference.
• Answer: C, D and E
• 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.

#### Q.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]

• A. f(-2) = f(1)
• B. f(-1) = f(1)
• C. f(-1) = f(2)
• D. f(3) = 23
• E. f(3) = f(2) + 7
• Answer: A, C and D
• 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]

#### Q.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.

• A. One number is 12
• B. The smaller number is 12
• C. The sum of the numbers is less than 22
• D. The smaller number is 10
• E. The two numbers are multiples of 4
• 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

#### Q.10

If (n+1)! = 12 (n ??? 1)!, then which of the following is true for n? Indicate all correct options. [n^2=n*n]

• A. n = 4
• B. n = 3
• C. n = -4
• D. n > 0
• E. n < 0
• (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.

#### Q.11

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

• A. 9!*10 = 90!
• B. 4!*5!*6 = 6!
• C. 7!*8*9 = 9!
• D. (5-1)!5! = [(4!)^2]*5
• E. 4! = 12*3!
• 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.

#### Q.12

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

• A. Every rectangle is a square
• B. Every square is a rectangle
• C. Every parallelogram is a rhombus
• D. Every rhombus is a parallelogram
• E. Every square is a rhombus
• Answer: B, D and E
• 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.

#### Q.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.

• A. There are two possible values of x
• B. There are two positive values of x
• C. x = -3
• D. x = 20
• E. x = -20
• 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]

#### Q.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.

• A. The faster bus travels at a speed of 40 km/hr
• B. The difference between the speeds of the buses is 20 km/hr
• C. The faster bus is twice as fast as the slower bus
• D. The slower bus travels at a speed of 40 km/hr
• E. The speeds of the buses cannot be determined
• Answer: A, B and C
• 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.

#### Q.15

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

• A. Sqrt(2) is an integer
• B. Sqrt(2) is an irrational number
• C. Sqrt(2) is a real number
• D. -5/4 is an integer
• E. 5 is an integer
• Answer: B, C and E
• 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.