# IPMAT (I) 2019 QASA

**1. IPMAT (I) 2019 QASA | Geometry**

The sum of the interior angles of a convex n-sided polygon is less than 2019°. The maximum possible value of n is

Answer: 13

**Explanation** :

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**2. IPMAT (I) 2019 QASA | Algebra - Logarithms**

Suppose that a, b, and c are real numbers greater than 1. Then the value of $\frac{1}{1+{\mathrm{log}}_{{\mathrm{a}}^{2}\mathrm{b}}\left({\displaystyle \frac{\mathrm{c}}{\mathrm{a}}}\right)}$ + $\frac{1}{1+{\mathrm{log}}_{{\mathrm{b}}^{2}\mathrm{c}}\left({\displaystyle \frac{\mathrm{a}}{\mathrm{b}}}\right)}$ + $\frac{1}{1+{\mathrm{log}}_{{\mathrm{c}}^{2}\mathrm{a}}\left({\displaystyle \frac{\mathrm{b}}{\mathrm{c}}}\right)}$ is

Answer: 3

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**3. IPMAT (I) 2019 QASA | Algebra - Functions & Graphs**

A real-valued function f satisfies the relation f(x)f(y) = f(2xy + 3) + 3f(x + y) - 3f(y) + 6y, for all real numbers x and y, then the value of f(8) is

Answer: 19

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**4. IPMAT (I) 2019 QASA | Modern Math - Determinants & Metrices**

Let A, B, C be three 4 X 4 matrices such that det A = 5, det B = -3, and det C = 1/2. Then the det (2AB^{-1}C^{3}B^{T}) is

Answer: 10

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**5. IPMAT (I) 2019 QASA | Modern Math - Determinants & Metrices**

If A is a 3 X 3 non-zero matrix such that A^{2} = 0 then determinant of [(1 + A)^{2} - 50A] is equal to

Answer: 3

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**6. IPMAT (I) 2019 QASA | Ratio, Proportion & Variation**

Three friends divided some apples in the ratio 3 : 5 : 7 among themselves. After consuming 16 apples they found that the remaining number of apples with them was equal to largest number of apples received by one of them at the beginning. Total number of apples these friends initially had was

Answer: 30

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**7. IPMAT (I) 2019 QASA | Percentage, Profit & Loss**

A shopkeeper reduces the price of a pen by 25% as a result of which the sales quantity increased by 20%. If the revenue made by the shopkeeper decreases by x% then x is

Answer: 10

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**8. IPMAT (I) 2019 QASA | Algebra - Quadratic Equations**

For all real values of x, $\frac{3{x}^{2}-6x+12}{{x}^{2}+2x+4}$ lies between 1 and k, and does not take any value above k. Then k equals

Answer: 9

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**9. IPMAT (I) 2019 QASA | Coordinate Geometry**

The maximum distance between the point (-5, 0) and a point on the circle x^{2} + y^{2} = 4 is

Answer: 7

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**10. IPMAT (I) 2019 QASA | Algebra - Logarithms | Algebra - Progressions**

If x, y, z are positive real numbers such that x^{12} = y^{16} = z^{24},and the three quantities 3log_{y}x, 4log_{z}y, nlog_{x}z are in arithmetic progression, then the value of n is

Answer: 16

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**11. IPMAT (I) 2019 QASA | Trigonometry**

The number of pairs (x, y) satisfying the equation sinx + siny = sin(x + y) and |x| + |y| = 1 is

Answer: 6

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**12. IPMAT (I) 2019 QASA | Coordinate Geometry**

The circle x^{2} + y^{2} - 6x - 10y + k = 0 does not touch or intersect the coordinate axes. If the point (1, 4) does not lie outside the circle, and the range of k is (a, b] then a + b is

Answer: 54

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**13. IPMAT (I) 2019 QASA | Modern Math - Determinants & Metrices**

If a 3 X 3 matrix is filled with +1 's and - 1 's such that the sum of each row and column of the matrix is 1, then the absolute value of its determinant is

Answer: 4

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**14. IPMAT (I) 2019 QASA | Modern Math - Sets**

Let the set = {2,3,4,..., 25}. For each k ∈ P, define Q(k) = {x ∈ P such that x > k and k divides x}. Then the number of elements in the set P − ${U}_{k=2}^{25}$ Q(k) is

Answer: 9

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**15. IPMAT (I) 2019 QASA | Geometry**

The number of whole metallic tiles that can be produced by melting and recasting a circular metallic plate, if each of the tiles has a shape of a right-angled isosceles triangle and the circular plate has a radius equal in length to the longest side of the tile (Assume that the tiles and plate are of uniform thickness, and there is no loss of material in the melting and recasting process) is

Answer: 12

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**16. IPMAT (I) 2019 QASA | Algebra - Inequalities & Modulus**

If |x| < 100 and |y| < 100, then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is

Answer: 29

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**17. IPMAT (I) 2019 QASA | Average, Mixture & Alligation**

The average of five distinct integers is 110 and the smallest number among them is 100. The maximum possible value of the largest integer is

Answer: 144

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**18. IPMAT (I) 2019 QASA | Algebra - Progressions**

Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The 6389^{th} digit in this sequence is

Answer: 4

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**19. IPMAT (I) 2019 QASA | Algebra - Number Theory**

The number of pairs of integers whose sums are equal to their products is

Answer: 2

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**20. IPMAT (I) 2019 QASA | Algebra - Number Theory**

You have been asked to select a positive integer N which is less than 1000 , such that it is either a multiple of 4, or a multiple of 6, or an odd multiple of 9. The number of such numbers is

Answer: 388

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