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: CD
- Explaination: 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: BC
- Explaination: AB = sqrt[(0-5)^2+(1-2)^2] = sqrt[ 25+1] = sqrt(26) BC = sqrt[(-7-5)^2+(-7-2)^2] = sqrt[144+81] = sqrt(225) CA = sqrt[(-7-0)^2 + (-7-1)^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: AE
- Explaination: Let there be 10 liters of the solution initially. There will be 2*10/(2+3) = 20/5 = 4 liters of substance A and 10-4 = 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: BC
- Explaination: x+1/x = 3 Squaring both sides, we getx^2 + 1/x^2 + 2*x*(1/x) = 9 x^2 + 1/x^2 = 9-2 = 7 (x^2+1/x^2)^2=49 x^4 + 1/x^4 = 49-2 = 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: BCD
- Explaination: 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)^t-1] =2000[(1+5/100)^2-1] =2000*(441/400-1)=205 Option A is false SI = P*R*T/100 = 2000*5*2/100= 200 Option B is true. CI = P[(1+R/100)^t-1] =2000[(1+5/100)^4-1] =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: CD
- Explaination: 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: BCE
- Explaination: 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(x-6) 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 = 42-6 = 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: AD
- Explaination: 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
- Answer: AD
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
- e. k < 1
- Answer: CDE