AILET entrance exam Question Bank Solutions

15 men can do a work in 20 days. If there are 25 men, how many days will it take to complete the work?

Ram can do a work in 60 days and the same work Sohan can finish in 40 days. If both of them work together, then in how many days will the work be over?

Sanju has two watches with a 12-hour cycle. One of these watches gains one minute a day and the other loses 1 and `1/2` minutes per day. If Sanju sets both the watches at the correct time, how long will it be before they again show the correct time together?

How many three-digit positive integers exist that when divided by 6 leave a reminder of 4?

A positive integer is written on each face of a cube. Each vertex is then assigned the product of the numbers written on the three faces intersecting the vertex. The sum of the numbers assigned to all the vertices is equal to 1001. Find the sum of the numbers written on the faces of the cube.

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If x, y, z are positive real numbers, such that x + y + z = 25, then the maximum value of `1/x + 1/y + 1/z` is

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Let A = { 3, 6, 9, 12, ......., 696, 699} & B = {7, 14, 21, .........., 287, 294}
Find no. of ordered pairs of (a, b) such that a ∈ A, b ∈ B, a ≠ b & a + b is odd.

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If a, b, c > 0, then find the minimum value of (a + b + c)`(1/(a + b) + 1/(b + c) + 1/(c + a))`.

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Using the digits 1,2 find the number of natural numbers, which are smaller than 2 × 10⁸.

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Consider positive integers x and y such that the difference between x² + y and x + y² is a prime number. How many different values can the ordered pair (x, y) take?

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a, b, c and d are positive real numbers such that a + b + c + d = p, where ‘p’ is a constant. Find the maximum value of (p – a) (p – b) (p – c) (p – d)?

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Find 3x²y² if x and y are integers such that y² + 3x²y² = 30x² + 517.

If (1+ x)ⁿ = C₀ + C₁x + C₂x² + …..C_nxⁿ , then C₀C² + C₁C³ + C₂C⁴ + …..=C_n-2 C_n is equal to:

If (1 + x)ⁿ = c₀ + c₁x + C₂x² +……+ C_nxⁿ , then C₀ + `"C"_1/2 + "C"_1/3 + ……+ "C"_1/("n" + 1)` is equal to:

If ωω the cube root of unity then the value of `|(1, ωω, ωω^2),(ωω,ωω^2,1),(ωω^2,1,ω)|` will be:

The value of the determination `|("cos" alpha,"sin" alpha , 0),(-"sin" alpha , "cos" alpha , 0),(0,0,1)|` is :

If a + b = a – b , then the angle between a and b will be:

`pi pi^2 0 "sin x"/sqrt(1 + "sin 2x")`dx is equal to :

If y = sec^-1 `("x+1"/"x - 1") + "sin"^-1 ("x - 1"/"x + 1") "then" "dy"/"dx"` is equal to:

x e^x dx is equal to: