IPMAT (I) 2022 QASA questions and solutions

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1. IPMAT (I) 2022 QASA | Algebra - Logarithms

$\log_{x^{2}} (y)$ + $\log_{y^{2}} (x)$ = 1 and y = x² - 30, then the value of x² + y² is:

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Answer: 72

Video Explanation

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Explanation :

Given, $\log_{x^{2}} (y)$ + $\log_{y^{2}} (x)$ = 1

⇒ $\frac{\log_{x} (y)}{2}$ + $\frac{\log_{y} (x)}{2}$ = 1

⇒ log_xy + log_yx = 2

Now, log_xy and log_yx are reciprocal of each other and their sum is 2. Sum of a number and its reciprocal is always greater than or equal to 2. Equality is true when the number and reciprocal are both equal to 1.

∴ log_xy = 1
⇒ x = y

Now, y = x² - 30 [y = x]
⇒ x² - x - 30 = 0
⇒ (x - 6)(x + 5) = 0
⇒ x = 6 [x = -5 is rejected as log is not defined for negative numbers.]
∴ y = 6

∴ x² + y² = 6² + 6² = 72

Hence, 72.

2. IPMAT (I) 2022 QASA | Coordinate Geometry

The area enclosed by 2|x| + 3|y| ≤ 6 is ____________ sq. units.

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3. IPMAT (I) 2022 QASA | Time, Speed & Distance

When Geeta increases her speed from 12km/hr to 20km/hr, she takes one hour less than the usual time to cover the distance between her home and office. The distance between her home and office is ___________ km.

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4. IPMAT (I) 2022 QASA | Modern Math - Permutation & Combination

The number of triangles that can be formed by choosing points from 7 points on a line and 5 points on another parallel line is _________.

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5. IPMAT (I) 2022 QASA | Percentage, Profit & Loss

Aruna purchases a certain number of apples for INR 20 each and a certain number of mangoes for INR 25 each. If she sells all the apples at 10% profit and all the mangoes at 20% loss, overall she makes neither profit nor loss. Instead, if she sells all the apples at 20% loss and all the mangoes at 10% profit, overall she makes a loss of INR 150 . Then the number of apples purchased by Aruna is _________.

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6. IPMAT (I) 2022 QASA | Algebra - Functions & Graphs

Given that f(x) = |x| + 2|x−1| + |x−2| + |x−4| + |x−6| + 2|x−10|, x ∈ (−∞, ∞) the minimum value of f(x)
is _________.

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7. IPMAT (I) 2022 QASA | Modern Math - Determinants & Metrices

If A = $[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$ , then the absolute value of the determinant of (A⁹ + A⁶ + A³ + A) is __________.

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Answer: 32

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Explanation :

Given, A = $[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$

Now, A × A = $[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$ × $[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$ = $[\begin{matrix} 1 \times 1 + 0 \times 0 + 0 \times 0 & 1 \times 0 + 0 \times 0 + 0 \times 1 & 1 \times 0 + 0 \times 1 + 0 \times 0 \\ 0 \times 1 + 0 \times 0 + 1 \times 0 & 0 \times 0 + 0 \times 0 + 1 \times 1 & 0 \times 0 + 0 \times 1 + 1 \times 0 \\ 0 \times 1 + 1 \times 0 + 0 \times 0 & 0 \times 0 + 1 \times 0 + 0 \times 1 & 0 \times 0 + 1 \times 1 + 0 \times 0 \end{matrix}]$ = $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ = I

⇒ A³ = A² × A = I × A = A

⇒ A⁶ = A³ × A³ = A × A = I

⇒ A⁹ = A³ × A³ × A³ = A × A × A = A² × A = I × A = A

Now, (A⁹ + A⁶ + A³ + A)

= A + I + A + A + A

= 3A + I

= $[\begin{matrix} 3 & 0 & 0 \\ 0 & 0 & 3 \\ 0 & 3 & 0 \end{matrix}]$ + $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

= $[\begin{matrix} 4 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 3 & 1 \end{matrix}]$

∴ The absolute value of the determinant of (A⁹ + A⁶ + A³ + A) = absolute value of $[\begin{matrix} 4 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 3 & 1 \end{matrix}]$

= |4 × (1 × 1 - 3 × 3)| = |4 × (1 - 9)| = 32.

Hence, 32.

8. IPMAT (I) 2022 QASA | Modern Math - Permutation & Combination

The sum of the coefficients of all the terms in the expansion of (5x − 9)⁴ is __________.

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9. IPMAT (I) 2022 QASA | Algebra - Progressions

A new sequence is obtained from the sequence of positive integers (1,2,3,…) by deleting all the perfect squares. Then the 2022^nd term of the new sequence is ________.

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10. IPMAT (I) 2022 QASA | Trigonometry

Sin α + Sin β = $\frac{\sqrt{2}}{\sqrt{3}}$ and Cos α + Cos β = $\frac{1}{\sqrt{3}}$ , then the value of ${(20 \cos (\frac{α - β}{2}))}^{2}$ is _______.

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Answer: 100

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Explanation :

Sin α + Sin β = $\frac{\sqrt{2}}{\sqrt{3}}$ ...(1) and
Cos α + Cos β = $\frac{1}{\sqrt{3}}$ ...(2),

Squaring both the equations and adding them we get.

⇒ Sin² α + Sin² β + 2 × Sin α × Sin β + Cos² α + Cos² β + 2 × Cos α × Cos β = $\frac{2}{3}$ + $\frac{1}{3}$

⇒ Sin² α + Cos² α + Sin² β + Cos² β + 2 × Sin α × Sin β + 2 × Cos α × Cos β = 1

⇒ 1 + 1 + 2 × cos (α - β) = 1

⇒ cos (α - β) = - 1/2

⇒ 2 × ${(\cos (\frac{α - β}{2}))}^{2}$ - 1 = - 1/2

⇒ ${(\cos (\frac{α - β}{2}))}^{2}$ = 1/4

∴ 20² × ${(\cos (\frac{α - β}{2}))}^{2}$ = 400

⇒ ${(20 \cos (\frac{α - β}{2}))}^{2}$ = 100.

Hence, 100.

11. IPMAT (I) 2022 QASA | Algebra - Progressions

The 3^rd ,14^th and 69^th terms of an arithmetic progression form three distinct and consecutive terms of a geometric progression. If the next term of the geometric progression is the n^th term of the arithmetic progression, then n equals ________.

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12. IPMAT (I) 2022 QASA | Modern Math - Sets

Let P(X) denote power set of a set X. If A is the null set, then the number of elements in P(P(P(P(A)))) is _________.

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13. IPMAT (I) 2022 QASA | Algebra - Surds & Indices | Algebra - Progressions

The numbers −16, 2^x+3− 2^2x−1− 16, 2^2x−1+ 16 are in an arithmetic progression. Then x equals ________.

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14. IPMAT (I) 2022 QASA | Modern Math - Permutation & Combination

Mrs and Mr Sharma, and Mrs and Mr Ahuja along with four other persons are to be seated at a round table for dinner. If Mrs and Mr Sharma are to be seated next to each other, and Mrs and Mr Ahuja are not to be seated next to each other, then the total number of seating arrangements is _________.

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15. IPMAT (I) 2022 QASA | Average, Mixture & Alligation

Let 50 distinct positive integers be chosen such that the highest among them is 100, and the average of the largest 25 integers among them exceeds the average of the remaining integers by 50 . Then the maximum possible value of the sum of all the 50 integers is _________.

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IPMAT (I) 2022 QASA | Previous Year IPMAT - Indore Paper

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