# IPMAT (I) 2021 QASA

**1. IPMAT (I) 2021 QASA | Algebra - Number Theory**

The number of positive integers that divide (1890) × (130) × (170) and are not divisible by 45 is:

Answer: 320

**Explanation** :

We first need to find all factors of (1890) × (130) × (170) and then remove those factors which are divisible by 45.

(1890) × (130) × (170) = (2 × 3^{3} × 5 × 7) × (13 × 2 × 5) × (17 × 2 × 5)

⇒ 1890 × 130 × 170 = 2^{3} × 3^{3} × 5^{3} × 7 × 13 × 17

∴ Total factors = (3 + 1)(3 + 1)(3 + 1)(1 + 1)(1 + 1)(1 + 1) = 512

Now for factors that are divisible by 45, least power of 3 should be 2 and that of 5 should be 1. Powers of 2, 7, 13 and 17 can be anything.

∴ Total factors which are divisible by 45 = 4 × 2 × 3 × 2 × 2 × 2 = 192

⇒ Number of factors which are not divisible by 45 = 512 - 192 = 320.

Hence, 320.

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**2. IPMAT (I) 2021 QASA | Algebra - Progressions**

The sum up to 10 terms of the series 1 × 3 + 5 × 7 + 9 × 11 + . . is

Answer: 5310

**Explanation** :

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**3. IPMAT (I) 2021 QASA | Algebra - Progressions**

It is given that the sequence {x_{n}} satisfies x_{1} = 0, x_{n+1} = x_{n} + 1 + $2\sqrt{1+{x}_{n}}$ for n = 1,2, . . . . . Then x_{31} is

Answer: 960

**Explanation** :

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**4. IPMAT (I) 2021 QASA | Modern Math - Permutation & Combination**

There are 5 parallel lines on the plane. On the same plane, there are ‘n’ other lines that are perpendicular to the 5 parallel lines. If the number of distinct rectangles formed by these lines is 360, what is the value of n?

Answer: 9

**Explanation** :

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**5. IPMAT (I) 2021 QASA | Time & Work**

There are two taps, T_{1} and T_{2}, at the bottom of a water tank, either or both of which may be opened to empty the water tank, each at a constant rate. If T_{1} is opened keeping T_{1} closed, the water tank (initially full) becomes empty in half an hour. If both T_{1} and T_{2} are kept open, the water tank (initially full) becomes empty in 20 minutes. Then, the time (in minutes) it takes for the water tank (initially full) to become empty if T_{2} is opened while T_{1} is closed is

Answer: 60

**Explanation** :

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**6. IPMAT (I) 2021 QASA | Average, Mixture & Alligation**

A class consists of 30 students. Each of them has registered for 5 courses. Each course instructor conducts an exam out of 200 marks. The average percentage marks of all 30 students across all courses they have registered for, is 80%. Two of them apply for revaluation in a course. If none of their marks reduce, and the average of all 30 students across all courses becomes 80.02%, the maximum possible increase in marks for either of the 2 students is

Answer: 6

**Explanation** :

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**7. IPMAT (I) 2021 QASA | Algebra - Number Theory**

What is the minimum number of weights which enable us to weigh any integer number of grams of gold from 1 to 100 on a standard balance with two pans? (Weights can be placed only on the left pan)

Answer: 7

**Explanation** :

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**8. IPMAT (I) 2021 QASA | Geo - Coordinate Geometry**

If one of the lines given by the equation 2𝑥^{2} + axy + 3y^{2} = 0 coincides with one of those given by 2x^{2}+ b𝑥𝑦 - 3𝑦^{2} = 0 and the other lines represented by them are perpendicular then 𝑎^{2} + 𝑏^{2 }=

Answer: 26

**Explanation** :

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**9. IPMAT (I) 2021 QASA | Algebra - Functions & Graphs**

If a function f(a) = max (a, 0) then the smallest integer value of ‘x’ for which the equation f(x - 3) + 2f(x + 1) = 8 holds true is:

Answer: 3

**Explanation** :

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**10. IPMAT (I) 2021 QASA | Miscellaneous**

In a class, 60% and 68% of students passed their Physics and Mathematics examinations respectively. Then atleast ________ percentage of students passed both their Physics and Mathematics examinations.

Answer: 28

**Explanation** :

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