Linear Equations

1) If $9^{2 x – 1} – 81^{x – 1} = 1944$, then $x$ is

(1) 3
(2) 9/4
(3) 4/9
(4) 1/3

2) How many different pairs $(a, b)$ of positive integers are there such that $a \le b$ and $\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{9}$?

Answer:

3) While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is

Answer:

4) The arithmetic mean of $x$, $y$ and $z$ is 80, and that of $x$, $y$, $z$, $u$ and $v$ is 75, where $u$ = ($x$ + $y$)/2 and $v$ = ($y$ + $z$)/2. If $x \ge z$, then the minimum possible value of $x$ is

Answer:

5) In an examination, Rama's score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by 6. The revised scores of Anjali, Mohan, and Rama were in the ratio 11:10:3. Then Anjali's score exceeded Rama's score by

(1) 32
(2) 35
(3) 24
(4) 26

6) In a six-digit number, the sixth, that is, the rightmost, digit is the sum of the first three digits, the fifth digit is the sum of first two digits, the third digit is equal to the first digit, the second digit is twice the first digit and the fourth digit is the sum of fifth and sixth digits. Then, the largest possible value of the fourth digit is

Answer:

7) A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then the half of the remaining stock along with one extra to the second and continues giving away in this fashion. His total stock exhausts after he takes care of 5 children. How many toffees were there in his stock initially?

Answer:

8) Aron bought some pencils and sharpeners. Spending the same amount of money as Aron, Aditya bought twice as many pencils and 10 less sharpeners. If the cost of one sharpener is ₹ 2 more than the cost of a pencil, then the minimum possible number of pencils bought by Aron and Aditya together is

(1) 36
(2) 30
(3) 27
(4) 33

9) The number of pairs of integers (x, y) satisfying $x \ge y \ge -20$ and $2x + 5y = 99$ is

Answer:

10) If $x$ and $y$ are non-negative integers such that $x + 9 = z$, $y + 1 = z$ and $x + y \lt z + 5$, then the maximum possible value of $2x + y$ equals

Answer:

11) Let $k$ be a constant. The equations $kx + y = 3$ and $4x + ky = 4$ have a unique solution if and only if

(1) k = 2
(2) |k| ≠ 2
(3) |k| = 2
(4) k ≠ 2

12) A basket of 2 apples, 4 oranges and 6 mangoes costs the same as a basket of 1 apple, 4 oranges and 8 mangoes, or a basket of 8 oranges and 7 mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

(1) 12
(2) 13
(3) 10
(4) 11

13) A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is

Answer:

14) A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR 50 less than that of a large shirt. She paid a total of INR 5000 for the large shirts, and a total of INR 1800 for the small shirts. Then, the price of a large shirt and a small shirt together, in INR, is

(1) 175
(2) 150
(3) 225
(4) 200

CAT 2025 Lesson : Quant: Jan '25 to Feb '25 - Algebra III Solving - 09 Feb 2025