30 questions in the pattern of XAT. Each question is marked with the exams it suits. Answer it, then open the step-by-step solution.
What is the highest power of 12 that divides 50! exactly?
12 = 2² × 3. Powers in 50!: of 2 = 25 + 12 + 6 + 3 + 1 = 47; of 3 = 16 + 5 + 1 = 22.
Each 12 needs two 2s and one 3: 2s allow ⌊47/2⌋ = 23 twelves, 3s allow 22. The smaller limit wins: 22.
What is the remainder when 7²⁰²⁶ is divided by 100?
7⁴ = 2401 ≡ 1 (mod 100), so powers of 7 repeat every 4.
2026 = 4 × 506 + 2, so 7²⁰²⁶ ≡ 7² = 49 (mod 100).
How many ordered pairs of positive integers (x, y) satisfy 1/x + 1/y = 1/12?
Multiply out: 12(x + y) = xy, so xy − 12x − 12y + 144 = 144, that is (x − 12)(y − 12) = 144.
Both factors must be positive (if both were negative, x and y would be less than 12 and the product could not reach 144). So each divisor d of 144 gives one pair: x = 12 + d, y = 12 + 144/d.
144 = 2⁴ × 3² has (4 + 1)(2 + 1) = 15 divisors, so 15 ordered pairs.
What is the sum of all three-digit numbers that leave a remainder of 3 when divided by 7?
The numbers are 101, 108, …, 997 (101 = 7 × 14 + 3 and 997 = 7 × 142 + 3).
Count = (997 − 101)/7 + 1 = 129. Sum = 129 × (101 + 997)/2 = 129 × 549 = 70,821.
If x + 1/x = 3, what is the value of x⁵ + 1/x⁵?
Let a(n) = xⁿ + 1/xⁿ. Then a(n+1) = 3a(n) − a(n−1), with a(0) = 2 and a(1) = 3.
a(2) = 7, a(3) = 18, a(4) = 47, a(5) = 3 × 47 − 18 = 123.
For how many integer values of k does x² − kx + 36 = 0 have both roots as integers?
If the roots are integers p and q, then pq = 36 and k = p + q.
Positive pairs: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) give k = 37, 20, 15, 13, 12. The negative pairs give −37, −20, −15, −13, −12.
That is 10 different values of k.
f(x) = |x − 1| + |x − 3| + |x − 8|. What is the least value of f(x)?
A sum of distances |x − a| is smallest at the median of the points, here x = 3.
f(3) = 2 + 0 + 5 = 7.
How many integers x with |x| ≤ 10 satisfy (x − 2)(x − 5)(x − 9) < 0?
The product is negative when x < 2 (three negative factors) or 5 < x < 9 (one negative factor).
x < 2 with x ≥ −10: −10 to 1, that is 12 integers. 5 < x < 9: 6, 7, 8, that is 3. Total 15.
What is the value of 1×2 + 2×3 + 3×4 + … + 20×21?
Σ n(n + 1) for n = 1 to N equals N(N + 1)(N + 2)/3.
For N = 20: 20 × 21 × 22 / 3 = 3,080.
If log₂x + log₄x + log₁₆x = 21/4, what is x?
log₄x = (log₂x)/2 and log₁₆x = (log₂x)/4, so the left side is (7/4) log₂x.
(7/4) log₂x = 21/4 gives log₂x = 3, so x = 8.
Two trains, 240 m and 160 m long, cross each other in 10 seconds when moving in opposite directions, and the faster one overtakes the slower in 50 seconds when they move in the same direction. What is the speed of the faster train?
Total length = 400 m. Opposite directions: sum of speeds = 400/10 = 40 m/s. Same direction: difference = 400/50 = 8 m/s.
Faster speed = (40 + 8)/2 = 24 m/s = 24 × 18/5 = 86.4 km/h.
A boat goes 30 km upstream and 44 km downstream in 10 hours, and 40 km upstream and 55 km downstream in 13 hours. What is the speed of the boat in still water?
Let u = 1/(upstream speed) and d = 1/(downstream speed): 30u + 44d = 10 and 40u + 55d = 13.
4 × first: 120u + 176d = 40; 3 × second: 120u + 165d = 39. Subtracting, 11d = 1, so downstream speed = 11 km/h. Then 30u = 10 − 4 = 6, so upstream speed = 5 km/h.
Boat speed = (11 + 5)/2 = 8 km/h.
A and B together can finish a job in 12 days, B and C in 15 days, and A and C in 20 days. How many days would A alone take?
Add the three rates: 2(A + B + C) = 1/12 + 1/15 + 1/20 = (5 + 4 + 3)/60 = 1/5, so A + B + C = 1/10 a day.
A = 1/10 − (B + C) = 1/10 − 1/15 = 1/30, so A alone takes 30 days.
A vessel holds 60 litres of milk. 6 litres are taken out and replaced with water, and this is done three times in all. How much milk is left?
Each round keeps (60 − 6)/60 = 0.9 of the milk.
Milk left = 60 × 0.9³ = 60 × 0.729 = 43.74 litres.
A trader marks goods 40% above cost and allows two successive discounts of 10% each. What is his profit percentage?
Selling price = 1.40 × 0.90 × 0.90 = 1.134 times the cost.
Profit = 13.4%.
The difference between compound interest and simple interest on a sum for 3 years at 10% a year is ₹310. What is the sum?
For 3 years at rate r: CI − SI = P[(1 + r)³ − 1 − 3r] = P(3r² + r³).
With r = 0.1: P(0.03 + 0.001) = 0.031P = 310, so P = ₹10,000.
How many four-digit numbers have their digits in strictly increasing order from left to right?
0 cannot be used (it would have to come first). Any 4 different digits from 1 to 9 can be arranged in increasing order in exactly one way.
So the count is C(9, 4) = 126.
In how many ways can 6 people sit around a round table if two particular people must not sit next to each other?
All circular arrangements: (6 − 1)! = 120.
With the two together: treat them as one unit, 4! = 24, times 2 for their order = 48.
Not together: 120 − 48 = 72.
Two fair dice are thrown. What is the probability that the sum is a prime number?
Prime sums: 2 (1 way), 3 (2), 5 (4), 7 (6), 11 (2): 15 of 36 outcomes.
Probability = 15/36 = 5/12.
Three different numbers are chosen at random from 1 to 10. What is the probability that their sum is even?
Total ways: C(10, 3) = 120. There are 5 odd and 5 even numbers.
Even sum: three even (C(5, 3) = 10) or two odd and one even (C(5, 2) × 5 = 50), that is 60.
Probability = 60/120 = 1/2.
A triangle has sides 13 cm, 14 cm and 15 cm. What is the length of the altitude to the 14 cm side?
Semi-perimeter s = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 cm².
Altitude = 2 × 84 / 14 = 12 cm.
Chords AB and CD of a circle intersect at P inside the circle. AP = 6 cm, PB = 8 cm and CP = 4 cm. What is the length of CD?
Intersecting chords: AP × PB = CP × PD, so 48 = 4 × PD and PD = 12 cm.
CD = CP + PD = 16 cm.
A cone and a hemisphere stand on equal circular bases and have equal volumes. What is the ratio of the cone's height to its base radius?
(1/3)πr²h = (2/3)πr³ gives h = 2r.
So h : r = 2 : 1.
What is the area of the triangle formed by the lines y = x, y = −x and y = 6?
The vertices are (0, 0), (6, 6) and (−6, 6).
Base along y = 6 is 12 and the height is 6, so area = (1/2) × 12 × 6 = 36.
What is the smallest positive number that leaves remainders 1, 2, 3 and 4 when divided by 2, 3, 4 and 5 respectively?
In each case the remainder is one less than the divisor, so N + 1 is divisible by 2, 3, 4 and 5.
LCM(2, 3, 4, 5) = 60, so the smallest N = 59.
How many factors of 2⁴ × 3³ × 5² are perfect squares?
A square factor uses even powers: 2⁰, 2², 2⁴ (3 choices); 3⁰, 3² (2 choices); 5⁰, 5² (2 choices).
3 × 2 × 2 = 12.
The average of 11 numbers is 50. The average of the first six is 49 and of the last six is 52. What is the sixth number?
The sixth number is counted in both groups: 6 × 49 + 6 × 52 − 11 × 50 = 294 + 312 − 550 = 56.
If a + b + c = 0 and abc = 4, what is a³ + b³ + c³?
a³ + b³ + c³ − 3abc = (a + b + c)(…), which is 0 when a + b + c = 0.
So a³ + b³ + c³ = 3abc = 12.
The third term of a geometric progression is 12 and the sixth term is 96. What is the sum of its first eight terms?
r³ = 96/12 = 8, so r = 2 and the first term = 12/4 = 3.
Sum of 8 terms = 3(2⁸ − 1)/(2 − 1) = 3 × 255 = 765.
The ratio of A's salary to B's is 3 : 5. A's salary rises by 20% and B's falls by 10%. What is the new ratio?
New salaries: 3 × 1.2 = 3.6 and 5 × 0.9 = 4.5.
3.6 : 4.5 = 4 : 5.
These are original practice questions written by Target Campus in the pattern and difficulty of MAH MBA CET, NMAT, SNAP and other MBA entrance tests. They are not previous-year papers.
Preparing for MAH CET, SNAP or NMAT? Courses, live classes and mocks by Dr. Kiran Derle at Target Campus.
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