# IPMAT (I) 2021 QA MCQ

**1. IPMAT (I) 2021 QA MCQ | Algebra - Functions & Graphs**

Suppose that a real-valued function f(x) of real numbers satisfies f(x + xy) = f(x) + f(xy) for all real x, y, and that f(2020) = 1. Compute f(2021).

- A.
2021/2020

- B.
2020/2019

- C.
1

- D.
2020/2021

Answer: Option A

**Explanation** :

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**2. IPMAT (I) 2021 QA MCQ | Algebra - Logarithms**

Suppose that log_{2}[log_{3}(log_{4}a)] = log_{3}[log_{4}(log_{2}b)] = log_{4}[log_{2}(log_{3}c)] = 0 then the value of a + b + c is

- A.
105

- B.
71

- C.
89

- D.
37

Answer: Option C

**Explanation** :

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

Let S_{n} be sum of the first n terms of an A.P. {a_{n}}. If S_{5} = S9 , what is the ratio of a_{3} : a_{5}

- A.
9 : 5

- B.
5 : 9

- C.
3 : 5

- D.
5 : 3

Answer: Option A

**Explanation** :

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**4. IPMAT (I) 2021 QA MCQ | Modern Math - Determinants & Metrices**

If A, B and A + B are non singular matrices and AB = BA then 2A - B - A(A + B)^{-1}A + B(A + B)^{-1}B equals

- A.
A

- B.
B

- C.
A + B

- D.
I

Answer: Option D

**Explanation** :

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**5. IPMAT (I) 2021 QA MCQ | Geo - Triangles**

If the angles A, B, C of a triangle are in arithmetic progression such that sin(2A + B) = 1/2 then sin(B + 2C) is equal to

- A.
-1/2

- B.
1/2

- C.
-1/√2

- D.
3/√2

Answer: Option A

**Explanation** :

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**6. IPMAT (I) 2021 QA MCQ | Algebra - Number Theory**

The unit digit in (743)^{85} – (525)^{37} + (987)^{96} is ________

- A.
9

- B.
3

- C.
1

- D.
5

Answer: Option A

**Explanation** :

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**7. IPMAT (I) 2021 QA MCQ | Geo - Trigonometry**

The set of all real value of p for which the equation 3sin^{2}x + 12cosx – 3 = p has one solution is

- A.
[-12, 12]

- B.
[-12, 9]

- C.
[-15, 9]

- D.
[-15, 12]

Answer: Option D

**Explanation** :

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**8. IPMAT (I) 2021 QA MCQ | Geo - Quadrilaterals & Polygons**

ABCD is a quadrilateral whose diagonals AC and BD intersect at O. If triangles AOB and COD have areas 4 and 9 respectively, then the minimum area that ABCD can have is

- A.
26

- B.
25

- C.
21

- D.
16

Answer: Option B

**Explanation** :

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**9. IPMAT (I) 2021 QA MCQ | Algebra - Number Theory**

The highest possible value of the ratio of a four-digit number and the sum of its four digits is

- A.
1000

- B.
277.75

- C.
900.1

- D.
999

Answer: Option A

**Explanation** :

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**10. IPMAT (I) 2021 QA MCQ | Algebra - Functions & Graphs**

Consider the polynomials f(x) = ax^{2} + bx + c, where a > 0, b, c are real, g(x) = -2x. If f(x) cuts the x-axis at (-2, 0) and g(x) passes through (a, b), then the minimum value of f(x) + 9a + 1 is

- A.
0

- B.
1

- C.
2

- D.
3

Answer: Option B

**Explanation** :

g(x) = -2x passes through (a, b)

⇒ b = -2a ...(1)

Now, f(x) = ax^{2} + bx + c

⇒ f(x) = ax^{2} - 2ax + c [from (1)]

f(x) passes through (2, 0)

⇒ a(-2)^{2} - 2a(-2) + c = 0

⇒ 8a + c = 0

⇒ c = -8a

∴ f(x) = ax^{2} - 2ax - 8a

Now, we have to find the least value of f(x) + 9a + 1

= ax^{2} - 2ax - 8a + 9a + 1

= ax^{2} - 2ax + a + 1

= a(x^{2} - 2x + 1) + 1

= a(x - 1)^{2} + 1

Least value of this expression will be 1, when x = 1.

Hence, option (b).

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**11. IPMAT (I) 2021 QA MCQ | Percentage, Profit & Loss**

In a city, 50% of the population can speak in exactly one language among Hindi, English and Tamil, while 40% of the population can speak in at least two of these three languages. Moreover, the number of people who cannot speak in any of these three languages is twice the number of people who can speak in all these three languages. If 52% of the population can speak in Hindi and 25% of the population can speak exactly in one language among English and Tamil, then the percentage of the population who can speak in Hindi and in exactly one more language among English and Tamil is

- A.
22%

- B.
25%

- C.
30%

- D.
38%

Answer: Option A

**Explanation** :

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**12. IPMAT (I) 2021 QA MCQ | Time, Speed & Distance**

A train left point A at 12 noon. Two hours later, another train started from point A in the same direction. It overtook the first train at 8 PM. It is known that the sum of the speeds of the two trains is 140 km/hr. Then, at what time would the second train overtake the first train, if instead the second train had started from point A in the same direction 5 hours after the first train? Assume that both the trains travel at constant speeds.

- A.
3 am the next day

- B.
4 am the next day

- C.
8 am the next day

- D.
11 pm the same day

Answer: Option C

**Explanation** :

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**13. IPMAT (I) 2021 QA MCQ | Algebra - Number Theory**

The number of 5-digit numbers consisting of distinct digits that can be formed such that only odd digits occur at odd places is

- A.
5250

- B.
6240

- C.
2520

- D.
3360

Answer: Option C

**Explanation** :

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**14. IPMAT (I) 2021 QA MCQ | Modern Math - Permutation & Combination**

There are 10 points in the plane, of which 5 points are collinear and no three among the remaining are collinear. Then the number of distinct straight lines that can be formed out of these 10 points is

- A.
10

- B.
25

- C.
35

- D.
36

Answer: Option D

**Explanation** :

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**15. IPMAT (I) 2021 QA MCQ | Geo - Coordinate Geometry**

The x-intercept of the line that passes through the intersection of the lines x + 2y = 4 and 2x + 3y = 6, and is perpendicular to the line 3x – y = 2 is

- A.
2

- B.
0.5

- C.
4

- D.
6

Answer: Option D

**Explanation** :

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**Answer the following questions based on the information given below:**

In a football tournament six teams A, B, C, D, E, and F participated. Every pair of teams had exactly one match among them. For any team, a win fetches 2 points, a draw fetches 1 point, and a loss fetches no points. Both teams E and F ended with less than 5 points. At the end of the tournament points table is as follows (some of the entries are not shown):

It is known that: (1) team B defeated team C, and (2) team C defeated team D

**16. IPMAT (I) 2021 QA MCQ | Games & Tournaments**

Total number of matches ending in draw is

- A.
12

- B.
4

- C.
5

- D.
6

Answer: Option D

**Explanation** :

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**17. IPMAT (I) 2021 QA MCQ | Games & Tournaments**

Which team has the highest number of draws

- A.
A

- B.
C

- C.
D

- D.
E

Answer: Option D

**Explanation** :

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**18. IPMAT (I) 2021 QA MCQ | Games & Tournaments**

Total points Team F scored was

- A.
0

- B.
1

- C.
2

- D.
3

Answer: Option C

**Explanation** :

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**19. IPMAT (I) 2021 QA MCQ | Games & Tournaments**

Which team was not defeated by team A

- A.
B

- B.
C

- C.
D

- D.
F

Answer: Option C

**Explanation** :

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**20. IPMAT (I) 2021 QA MCQ | Games & Tournaments**

Team E was defeated by

- A.
Teams A and B only

- B.
Only Team A

- C.
Only Team B

- D.
Teams A, B and D only

Answer: Option C

**Explanation** :

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