Problem Solving Select Many Practice Test 4
Q.1
If f(x) = x + 1/x, then which of the following is true? Indicate all such statements.
 A. f(2) = 2.05
 B. f(5) = f(0.5)
 C. f(1/x) = f(x)
 D. f(4) = 4.25
 E. f(3) is equal to infinity
 Answer: C and D

f(2) = 2 + 1/2 = 2 + 0.5 = 2.5 Option (A) is false. f(5) = 5 + 1/5 = 5 + 0.2 = 5.2 f(0.5) = 0.5 + 1/0.5 = 0.5 + 2 = 2.5 Option (B) is false. f(1/x) = 1/x+1/1/x = 1/x + x = f(x) Option (C) is true. f(4) = 4 + 1/4 = 4 + 0.25 = 4.25 Option (D) is true. f(3) = 3 + 1/3 = 3+0.3333 = 3.3333 Option (E) is false.
Q.2
A triangle has its vertices at A(0,1), B(5,2) and C(7,7). Which of the following statements is true? Indicate all correct options.
 A. AB>BC
 B. BC>AC
 C. Angle A > angle B
 D. Angle C > angle B
 E. AB is the longest side
 Answer: B and C

AB = sqrt[(05)^2+(12)^2] = sqrt[ 25+1] = sqrt(26) BC = sqrt[(75)^2+(72)^2] = sqrt[144+81] = sqrt(225) CA = sqrt[(70)^2 + (71)^2] = sqrt[49+64] = sqrt(113) BC>CA>AB Options A and E are false and B is true. Angle A is opposite side BC Angle B is opposite side AC Angle C is opposite side AB Angle A> angle B> angle C Option C is true and D is false. [2^2 = 2*2]
Q.3
40% of a 2:3 solution of substance A and substance B is replaced with substance B. Which of the following is true? Indicate all correct options.
 A. The concentration of substance B increases
 B. The concentration of substance A remains the same
 C. The ratio of A to B becomes 1:3
 D. The ratio of A to B becomes 5:11
 E. The ratio of A to B becomes 6:19
 Answer: A and E

Let there be 10 liters of the solution initially. There will be 2*10/(2+3) = 20/5 = 4 liters of substance A and 104 = 6 liters of substance B in the solution. 40% of the solution is removed 40% of 10 liters = 40*10/100 = 4 liters These 4 liters contain 1.6 liters of substance A and 2.4 liters of substance B Substance B is then added Quantity of substance A in the new solution = 4  1.6 = 2.4 liters Quantity of substance B in the new solution = 6  2.4 + 4 = 7.6 liters Ratio of A to B in the resulting solution = 2.4/7.6 = 6/19 Options A and E are true.
Q.4
x+1/x = 3. Which of the following is true? Indicate all correct options. [x^2=x*x]
 A. x^2 + 1/x^2 = 9
 B. x^2 + 1/x^2 = 7
 C. x^4 + 1/x^4 = 47
 D. x^4 + 1/x^4 = 49
 E. x^4 + 1/x^4 = 81
 Answer: B and C

x+1/x = 3 Squaring both sides, we getx^2 + 1/x^2 + 2*x*(1/x) = 9 x^2 + 1/x^2 = 92 = 7 (x^2+1/x^2)^2=49 x^4 + 1/x^4 = 492 = 47 Options B and C are true.
Q.5
A sum of Rs.2000 is invested at 5% rate of interest. Which of the following is true? Indicate all correct options.
 A. The compound interest for 2 years is Rs.102.50
 B. The simple interest for 2 years is Rs.200
 C. The compound interest for 4 years is more than the simple interest for 2 years
 D. The compound interest is directly proportional to the sum invested
 E. The simple interest is inversely proportional to the period of investment.
 Answer: B, C and D

Let P, R and t be the principle, rate and time respectively. Let CI and SI be the compound interest and simple interest respectively. CI = P[(1+R/100)^t1] =2000[(1+5/100)^21] =2000*(441/4001)=205 Option A is false SI = P*R*T/100 = 2000*5*2/100= 200 Option B is true. CI = P[(1+R/100)^t1] =2000[(1+5/100)^41] =431.0125 > 200 Option C is true. From the formulas it is clear that compound interest is directly proportional to the sum invested. Simple interest is directly proportional to the period of investment. Option D is true and E is false. [100^t=100*100*...t tmes]
Q.6
Which of the following is not a linear equation in two variables? Indicate all such choices.
 A. x + y + 35 = 0
 B. 15L = M
 C. 5 ??? 12x = 244 + 31x
 D. z + x = 2y
 E. x + y = 2x + 2y
 Answer: C and D

x + y + 35 = 0 is a linear equation in two variables. 15L = M is a linear equation in two variables. 5 ??? 12x = 244 + 31x is a linear equation in one variable. z + x = 2y is a linear equation in three variables. x + y = 2x + 2y is a linear equation in two variables. (C) and (D) are not linear equations in two variables.
Q.7
A boat takes 6 hours to go downstream and takes 8 hours to go the same distance upstream. The speed of the stream is 6 km/hr. Which of the following is true? Indicate all correct options.
 A. Speed of the boat upstream is 42 km/hr
 B. Speed of the boat downstream is 48 km/hr
 C. Distance travelled one way is 288 km
 D. Distance travelled one way is 28 km
 E. Speed of the boat in still water is 42 km/hr
 Answer: B, C and E

Let the speed of the boat in still water be x km/hr. Speed of the boat upstream = x  6 Speed of the boat downstream = x + 6 Distance = speed * time Since the distance travelled is the same 6(x+6) = 8(x6) 6x + 36 = 8x  48 2x = 36+48 = 84 x = 84/2 = 42 km/hr Hence, speed of the boat in still water is 42 km/hr Speed of the boat upstream = 426 = 36 km/hr Speed of the boat downstream = 42+6 = 48 km/hr Distance travelled = 6(42+6) = 6*48 = 288 km Options B, C and E are true.
Q.8
In a group of 800 people, 550 can speak Arabic and 450 can speak Spanish and each person can speak at least one of the two languages. Which of the following is true? Indicate all correct options.
 A. 350 people do not speak Spanish
 B. 250 people do not speak Arabic but speak Spanish
 C. 250 people speak both Arabic and Spanish
 D. 200 people speak both Arabic and Spanish
 E. 350 people speak English
 Answer: A and D

Let A and S denote the sets of people speaking Arabic and Spanish respectively. n(A) = 550, n(S) = 450, n(AUS) = 800 People who do not speak Spanish = n(S') = 800  450 = 350 Option A is true and B is false. n(A intersection S) = n(A) + n(S)  n(AUS) = 550 + 450  800 = 200 Hence, 200 people speak both Arabic and Spanish Option D is true and C and E are false. [AUB = Unions of sets A and B]
Q.9
Solve the following equations 3x  5y = 1 and x + 2 = 0, y<0. Which of the following statements is true? Indicate all correct options.
 A. x = 2
 B. y = 0
 C. y has multiple values
 D. y = 1
 E. y < 1
 Answer: A and D

3x  5y = 1...(1) x + 2 = 0, y<0...(2) From (2), we know that x = 2. Putting this value in (1), we get 3(2)  5y = 1 6  5y = 1 5y = 1 + 6 5y = 5 y = 1 x = 2, y = 1 Options A and D are true.
Q.10
x = 2 is a solution of the equation (k^2)(x^2)  2x + 3=0 and k>0. Which of the following is true? Indicate all correct options. [k^2=k*k]
 A. k = 1/2
 B. k = 4
 C. k > 1/4
 D. k = 1/2
 E. k < 1
 Answer: C, D and E

x = 2 is a solution of the equation Clearly, this is a quadratic equation in x. x = 2 is a solution of the equation (k^2)(2^2)  2*2 + 3 = 0 4k^24+3=0 4k^2 = 1 k^2 = 1/4 k = 1/2, 1/2 Since k>0, we discard the negative value k = 1/2 Options C, D and E are true.
Q.11
Which of the following is a value of x if 3x + 17 < 2(1x)? Indicate all correct options.
 A. 11
 B. 1
 C. 2
 D. 5
 E. 7
 Answer: A, D and E

For x = 11, 3(11)+17<2(1+11) 33 +17 < 2*12=24 16<24, which is true. x = 11 For x = 1, 3(1)+17<2(1+1) 3 +17 < 2*2=4 14<4, which is not true. x is not equal to 1 For x = 2, 3(2)+17<2(1+2) 6 +17 < 2*3=6 11<6, which is not true. x is not equal to 2 For x = 5, 3(5)+17<2(1+5) 15 +17 < 2*6=12 2<12, which is true. x = 5 For x = 7, 3(7)+17<2(1+7) 21 +17 < 2*8=16 4<16, which is true. x = 7
Q.12
In a triangle PQR, LM is parallel to QR with L lying on PQ and M lying on PR. PL/LQ = 3/5 and PR = 5.6. Which of the following statements is true? Indicate all correct options.
 A. PM/MR=PL/LQ
 B. PR/MR=8/5
 C. MR = 5.6
 D. MR = PRPM
 E. PL/PM = PQ/MR
 Answer: A, B and D

Since LM is parallel to QR, by applying the basic proportionality theorem, we get, PM/MR = PL/LQ Option A is true. PM/MR + 1 = PL/LQ + 1 = 3/5 + 1 PR/MR = PQ/LQ = 8/5 Option B is true MR = PR*5/8 = 5.6*5/8 = 3.5 Option C is false. Option D is true and E is false.
Q.13
In a twodigit number, the sum of the digits is 12. The smaller digit subtracted from the larger digit gives us 4. Which of the following is the number? Indicate all correct options.
 A. 48
 B. 75
 C. 84
 D. 57
 E. 93
 Answer: A and C

Let x and y be the two digits of the twodigit number. Let us assume that x>y. According to the conditions, x + y = 12...(1) x ??? y = 4 ...(2) Adding the two we get, 2x = 12 + 4 = 16 x = 8 and hence y = 12 ??? 8 = 4 Now the twodigit number can be either 48 or 84. Options A and C are true.
Q.14
An empty rectangular tank has length and breadth of the base as 44 cm and 7 cm respectively. A cylindrical bucket full of water has radius of the base as 14 cm and height as 17 cm. The water from the cylindrical bucket is poured in the rectangular tank. Which of the following is true? Indicate all correct options.
 A. The level of water in the tank goes up by 4 cm
 B. The height of water in the tank is 34 cm
 C. The height of water in the bucket is 17 cm
 D. The volume of the water is 10472 cc
 E. The area of the base of the tank is 400 sq.cm
 Answer: B, C and D

Area of the base of the tank = Length*breadth = 44*7 = 308 sq.cm. Volume of water in the bucket = pi*r^2*h = 22/7*14*14*17 = 10472 cc Height of water in the tank = Volume of water/area of base of the tank = 10472/308 = 34 cm Options B, C and D are correct. [pi=22/7, r^2=r*r]
Q.15
A single man can do the work in 25 days. 5 men and 3 women can complete the work in 2 days. Which of the following is true? Indicate all correct options.
 A. 5 men and 3 women complete half the work in one day
 B. 1 woman takes 10 days to complete the work
 C. 1 woman completes half the work in 2 days
 D. 1 woman completes half the work in 10 days
 E. 1 woman completes half the work in 5 days
 Answer: A, B and E

A single man can do the work in 25 days. Work done by one man in one day = 1/25 Work done by 5 men and 3 women in one day = 1/2 Work done by 5 men in one day = 5*1/25 = 1/5 Work done by 3 women in one day = work done by 5 men and 3 women in one day ??? work done by 5 men in one day = 1/21/5 = (52)/10 = 3/10 Work done by 1 woman in one day = 3/10*1/3 = 1/10 Time taken by 1 woman to complete the work = 10 days Options A, B and E are true.