<p> Numeric Entry Practice Test 3</p><p>Question 1 Find r when C(n,r) = 35 and P(n,r) = 840 Correct Answer: 4 Explanation: From the given conditions, we get C(n,r) = n!/[r!(n-r)!] = 35...(1) P(n,r) = n!/(n-r)! = 840...(2)</p><p>Dividing (2) by (1), we get n!/(n-r)!/n!/[r!(n-r)!] = 840/35 r! = 24 r! = 1*2*3*4 = 24 r = 4</p><p>Question 2 If (n+1)! = 30*(n-1)!, then find n. Correct Answer: 5 Explanation: (n+1)! = 30*(n-1)! (n+1)*n*(n-1)! = 30(n-1)! (n+1)*n = 30 n2 + n - 30 = 0 n2 +6n-5n - 30 =0 n(n+6) - 5(n+6) = 0 n = -6, 5 Since n cannot be negative, n = 5</p><p>Question 3 Find x if (0,0), (3, sqrt(3)) and (x, 2*sqrt(3)) are the vertices of an equilateral triangle in the first quadrant. Correct Answer: 0 Explanation: Let the vertices (0,0), (3, sqrt(3)) and (x, 2*sqrt(3)) be A, B and C respectively. Since the triangle is equilateral, we have AB = BC = CA AB2 = BC2 = CA2 (3-0)2+(sqrt(3)-0)2 = (3-x)2 + (2*sqrt(3) - sqrt(3))2 = (x-0)2 + (2*sqrt(3)-0)2 9 + 3 = 9 - 2x + x2 +3 = x2 + 12 Hence, x2+12 = 12 x = 0</p><p>[AB2=AB*AB]</p><p>Question 4 C can complete the work in 18 days if he works alone. B takes 3 days lesser than C and A takes 5 days lesser than B to complete the work. On which day will they complete the work if C works for the initial two days only? Correct Answer: 6 Explanation: C takes 18 days to complete the work. B takes 3 days lesser. Hence, B takes 18-3 = 15 days. A takes 5 days lesser than B. Hence, A takes 15-5 = 10 days. Work done by A in one day = 1/10 Work done by B in one day = 1/15 Work done by C in one day = 1/18 Work done by the three of them in two days = 2(1/18+1/10+1/15) = 2(5+9+6)/90 = 2*20/90 = 4/9 Remaining work = 1-4/9 = 5/9 Work done by A and B in one day = 1/10+1/15 = (3+2)/30 =1/6 Time taken by A and B to complete the remaining work = 5/9*6/1 = 10/3 = 3.33 The work in completed on 2+3.33 = 6th day</p><p>Question 5 If A and B are two mutually exclusive events and P(A) = 2/3 and P(B) = 4/9, then find the probability of the occurrance of A and B together. Correct Answer: 0 Explanation: Since A and B are mutually exclusive events, they do not occur together at all. Hence, the required probability is zero.</p><p>Question 6 Find the diagonal of a cuboid whose volume is 144 cc and the dimensions of its base are 12cm and 4cm. Correct Answer: 13cm Explanation: The length (l) and breadth (b) of the base are 12 cm and 4 cm respectively. Height (h) of the cuboid = volume/(l*b) = 144/(12*4) = 3 cm Diagonal of the cuboid = sqrt(l2+b2+h2) = sqrt(122+42+32) = sqrt(144+16+9) = sqrt(169) = 13 cm The diagonal of the cuboid is 13 cm.</p><p>[l2=l*l]</p><p>Question 7 Find the value of k for which both the equations 12x2+4kx+3=0 and (k+1)x2-2(k-1)x+1=0 have equal real roots.</p><p>[x2=x*x] Correct Answer: 3 Explanation: Since the two equations have equal real roots, their discriminants are 0. For 12x2+4kx+3=0, D = 0 D2 = 0 b2-4ac=0 (4k)2 - 4*12*3 = 0 16k2 - 144=0 k2 = 144/16 = 9 k = -3, 3 For (k+1)x2-2(k-1)x+1=0 b2-4ac=0 [2*(k-1)]2-4*(k+1)*1=0 4(k2-2k+1)-4(k+1)=0 4[k2-2k+1-k-1]=0 k2 - 3k = 0 k(k-3)=0 k=0,3 Hence, we have k = 3 [k2=k*k]</p><p>Question 8 What percent is the first number of the second if the first number is 120% of the third and the second number is 150% of the third? Correct Answer: 80% Explanation: Let the first, second and the third numbers be x, y and z respectively. x = 120% of z = 120/100*z y = 150% of z = 150/100*z x/y*100 = (120z/100)/(150z/100)*100 = 80% x is 80% of y.</p><p>Question 9 The difference between two numbers is 2. Each number is less than 14 and their sum is greater than 22. Find the greater number of the two. Correct Answer: 13 Explanation: Let the two numbers be x and y such that x < y. y – x = 2, x < 14, y < 14, x + y > 22 y = x + 2, x < 14, x + 2 < 14, x + x + 2 > 22 x < 14, x < 12, 2x + 2 > 22 x < 12, x +1 > 11 x < 12, x > 10 Hence, x can be 11 and the corresponding value of y is 13. The two numbers are 11 and 13.</p><p>Question 10 The sum of the squares of the two positive numbers is 68 and the square of their difference is 36. Find the sum of the numbers. Correct Answer: 10 Explanation: Let the numbers be x and y, x>y x2+y2=68 and (x-y)2=36 (x-y)2 = x2+y2-2xy 36 = 68-2xy 2xy=68-36=32 xy=32/2=16</p><p>(x+y)2=x2+y2+2xy = 68+2*16 =68+32=100 x+y = sqrt(100) = 10</p><p>[x2=x*x]</p><p>Question 11 The difference between the time when the lightening was seen and the time when the thunder was heard is 10 seconds. Sound covers a distance 330 meters in one second. Find the distance of the thundercloud from the point of observation in meters. Correct Answer: 3300 meters Explanation: Distance = speed*time = 330*10 = 3300 meters</p><p>Question 12 Six points lie on a circle. How many cyclic quadrilaterals can be formed by joining the points? Correct Answer: 15 Explanation: Since, the points lie on a circle, all the possible quadrilaterals shall be cyclic. Number of quadrilateals = C(6,4) = 6!/(2!4!) = 6*5/2 = 15 Hence, 15 cyclic quadrilaterals can be drawn.</p><p>Question 13 When a number 'a' is increased by 17, it equals 60 times its reciprocal. How many values of 'a' are possible? Correct Answer: 2 Explanation: According to the conditions, we have a+17=60*1/a a2+17a-60=0 a2 +20a-3a-60=0 a(a+20)-3(a+20)=0 (a-3)(a+20)=0 a=3, -20 When a = 3, a+17 = 3+17 = 20 = 60*1/3 When a = -20, a+17 = -20+17 = -3 = 60*(-1/20) Hence, the two values of 'a' are valid. There are two possible values of 'a'. [a2=a*a]</p><p>Question 14 The tax on a commodity decreases by 10% and the consumption increases by 20%. What is the percentage increase in the revenue? Correct Answer: 8% Explanation: Revenue = Tax*Consumption Let the original tax be x and consumption be y. According to the given conditions, the new tax and consumption will be (x-10x/100) = 90x/100 and (y+20y/100) = 120y/100 The original revenue would be xy and the new revenue would be (90x/100)*(120y/100) Perentage change in revenue = (new revenue-old revenue)/old revenue * 100 = (90x/100*120x/100-xy)/xy*100 = (1.08xy-xy)/xy*100 = 0.08*100 = 8% The increase in revenue will be 8%.</p><p>Question 15 A shopkeeper bought flowers at the rate of 8 for Rs.34. He sold them at 12 flowers for Rs.57 and gained Rs. 900. How many flowers were there? Correct Answer: 1800 Explanation: Let x be the number of flowers bought and then sold by the shopkeeper. Let C.P. and S.P. be the cost price and selling price. Total C.P. = 34/8*x = 17x/4 Total S.P. = 57/12*x = 57x/12 Gain = Total SP - Total CP 900 = 57x/12 - 17x/4 (57x-51x)/12 = x/2 = 900 x = 900*2=1800 Number of flowers = 1800.</p>