The number of gift pack bought by Dishu is one more than the price, in rupees, of each gift pack. The amount of Rs 380 which Dishu had, fell short of the required amount. What is the amount by which he might have fallen short.
Answer: B Let the price of gift pack be 'a' Then number of packs bought = a+1 Hence total cost is a(a+1). It is given that 380 < a(a+1) If a =19, the total cost = 19*20 =380 Dishu would not have fallen short. If a =20, a(a+1) = 420. And he would have fallen short by Rs 40. This is the minimum amount by which he may have fallen short.
Q. No. 2:
Fortuner, the latest SUV by Toyota Motors, consumes diesel at the rate of 1/400*{[100/x] + x} litres per km, when driven at the speed of x km per hour. If the cost of diesel is Rs 35 per litre and the driver is paid at the rate of Rs 125 per hour then find the approximate optimal speed (in km per hour) of Fortuner that will minimize the total cost of the round trip of 800kms?
Answer: A Given that the diesel consumption is at the rate =1/400*{[100/x] + x} litres per km Cost of diesel = Rs 35 per litre Payment to the driver = Rs 125 per hour. Also given that the SUV is driven at the speed of x km per hour. Total cost (c) = 1/400*{[100/x] + x}*800*35 + (800*125)/x => 70,000/x + 70x + 10,000/x => x= 49 km per hour. Now differentiating both sides in the above equation with respect to x. dc/dx = -1,70,000/x2 +70 =0
Q. No. 3:
Three business entities X Ltd, Y Ltd. and Z Ltd, with 4, 3 and 5 employees respectively, merged intoXYZ Ltd in order to jointly raise the capital for setting up a new modern production plant in Jaipur.After two years, on the question of management decisions on the new venture at Jaipur, the employeesstarted adopting differing viewpoints and began to quarrel among themselves. Given the factthat there is not quarrel among the employees of the erstwhile X Ltd, Y Ltd and Z Ltd, what could bethe maximum number of quarrels that can take place within XYZ Ltd?
Answer: C Before the merger X Ltd. had no. of employees = 4 Y Ltd. had no. of employees = 3 Z Ltd. had no. of employees = 5 After the merger, there are quarrels amongst the employees of the erstwhile X ltd, Y ltd, & Z ltd. An employee does not quarrel with any of the employees from his parent company. Company :- X, Y, Z Employees:- 4, 3 ,5 Any employees in X can have a maximum possible no. of quarrels with the employees of Y ltd. & Z ltd. as given by:- 1 × 3 + 1 × 5 As there are 4 employees in the merged company, who came from X; So; the total number of quarrels involving employees of X company is 4 × [1 × 3 + 1 × 5] = 4 × 8 = 32 quarrels. ...(1) Further employees of Y & Z ltd. will be involved with quarrels within each other. The total number of such quarrels. = 3 × [1 × 5] = 5 × [1 × 3] = 15 quarrels ...(2) From (1) and (2) The total number of quarrels between the employees of three erstwhile companies is = 32 + 15 = 47 quarrels. Hence (C) option is correct.
Q. No. 4:
F(x) is a fourth order polynomial with integer coefficients and with no common factor. The roots of F(x) are –2, –1, 1, 2. If p is a prime number greater than 97, then the largest integer that divides F(p) for all values of p is:
Answer: D Given that F(x) = (x + 2)(x +1)(x −1)(x − 2) Putting x = P, we have F(P) = (P + 2)(P +1)(P −1)(P − 2) Now P is in the form 6K ± 1 where K is positive integer F(6K +1) = (6K + 3)(6K + 2)(6K)(6K −1) = (36)(2K + 1)(3K + 1)(K)(6K −1) …(1) F(6K −1) = (6K +1)(6K + 2)(6K)(6K − 3) = 36(6K + 1)(3K + 1)(K)(3K −1) …(2) Please note that the value of K ≥ 17 and expression F (6K + 1) and F (6K – 1) always bear the factor 10. Hence 360 is the correct choice. Therefore option (D) is the correct choice
Q. No. 5:
A shop sells two kinds of rolls- egg roll and mutton roll. Onion, tomato, carrot, chilli sauce and tomato sauce are the additional ingredients. You can have any combination of additional ingredients, or have standard rolls without any additional ingredients subject to the following constraints: (a) You can have tomato sauce if you have an egg roll, but not if you have a mutton roll. (b) If you have onion or tomato or both you can have chilli sauce, but not otherwise. How many different rolls can be ordered according to these rules?
Answer: D Let 5 additional ingredients Onion, Tomato, Carrot, Chilli Sauce and Tomato Sauce are denoted by O, T, C, CS, TS respectively. Case I: Egg Roll: Without any restriction the number of ways ‘Egg Roll’ can be Ordered = 2 × 2 × 2 × 2 ×2 = 32 ways (As each of the five additional ingredients can be selected or rejected i.e. 2 ways) The cases are: (standard ‘Egg Roll’), (O), (T), (C), (CS), (TS), (O, T), (O, C) ……………… (O, T, C, CS, TS) Out of these cases the following four cases are not possible by the condition (b) as given in the question (CS) (CS, TS) (CS, C) (CS, C, TS) =>Total number of ways Egg Roll can be ordered = 32 – 4 = 28. Case II: Mutton Roll: Total number of cases for Mutton roll must be half of the total numbers of cases for Egg Roll as mutton roll will never have the ingredient TS. =>Total cases for Mutton Roll = 28/2 = 14 Required number of cases = 28 + 14 = 42.
Q. No. 6:
In a cricket match, Team A scored 232 runs without losing a wicket. The score consisted of byes, wides and runs scored by two opening batsmen: Ram and Shyam. The runs scored by the two batsmen are 26 times wides. There are 8 more byes than wides. If the ratio of the runs scored by Ram and Shyam is 6:7, then the runs scored by Ram is
Answer: B Let the number of runs scored by byes, wides and runs be x, y and z respectively. x + y + z = 232 …(i) The runs scored by the two batsmen are 26 times the wides z = 26y …(ii) There are 8 more byes than wides x = y + 8 …(iii) Substituting equations (iii) and (ii) in equation (i), we get, y = 8, z = 208 The runs scored by Ram and Shyam were in the ratio 6 : 7 Let the runs scored by Ram be 6r and by Shyam be 7r. 13r = 208 => r = 16 Runs scored by Ram is 96.