Question 1
Let $n$ be the least positive integer such that 168 is a factor of $1134^n$. If $m$ is the least positive integer such that $1134^n$ is a factor of $168^m$, then $m + n$ equals
- 9
- 15
- 12
- 24
Correct answer: B · 15
All 22 Quant questions from the CAT 2023 · Slot 1 paper, with the correct answer for each.
Let $n$ be the least positive integer such that 168 is a factor of $1134^n$. If $m$ is the least positive integer such that $1134^n$ is a factor of $168^m$, then $m + n$ equals
Correct answer: B · 15
If $x$ and $y$ are positive real numbers such that $\log_x(x^2 + 12) = 4$ and $3\log_y(x) = 1$, then $x + y$ equals
Correct answer: C · 10
If $\sqrt{5x+9} + \sqrt{5x-9} = 3(2 + \sqrt{2})$, then $\sqrt{10x+9}$ is equal to
Correct answer: C · $3\sqrt{7}$
If $x$ and $y$ are real numbers such that $x^2 + (x - 2y - 1)^2 = -4y(x + y)$, then the value of $x - 2y$ is
Correct answer: B · 1
The number of integer solutions of the equation $2|x|(x^2 + 1) = 5x^2$ is (in numerical value)
Correct answer: 3
The equation $x^3 + (2r+1)x^2 + (4r-1)x + 2 = 0$ has $-2$ as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of $r$ is (in numerical value)
Correct answer: 2
Let $\alpha$ and $\beta$ be the two distinct roots of the equation $2x^2 - 6x + k = 0$, such that $(\alpha + \beta)$ and $\alpha\beta$ are the distinct roots of the equation $x^2 + px + p = 0$. Then the value of $8(k - p)$ is (in numerical value)
Correct answer: 6
A mixture P is formed by removing a certain amount of coffee from a coffee jar and replacing it with the same amount of cocoa powder. The same amount is again removed from mixture P and replaced with cocoa powder to form a new mixture Q. If the ratio of coffee and cocoa in mixture Q is 16 : 9, then the ratio of cocoa in mixture P to that in mixture Q is
Correct answer: C · 5 : 9
The minor angle between the hour hand and minute hand of a clock was observed at 8:48 am. The minimum duration, in minutes, after 8:48 am when this angle increases by 50% is
Correct answer: D · 24/11
Gita sells two objects A and B at the same price such that she makes a profit of 20% on object A and a loss of 10% on object B. If she increases the selling price such that objects A and B are still sold at an equal price and a profit of 10% is made on object B, then the profit made on object A will be nearest to
Correct answer: C · 47%
Brishti went on an 8-hour trip in a car. Before the trip the car had travelled a total of x km, where x is a whole number that is palindromic. At the end of the trip the car had travelled a total of 26862 km, again palindromic. If Brishti never drove at more than 110 km/hr, then the greatest possible average speed at which she drove during the trip, in km/hr, was
Correct answer: C · 100
In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks and each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is
Correct answer: A · 21
The salaries of three friends Sita, Gita and Mita are initially in the ratio 5 : 6 : 7. In the first year they get salary hikes of 20%, 25% and 20% respectively. In the second year Sita and Mita get salary hikes of 40% and 25% respectively, and the salary of Gita becomes equal to the mean salary of the three friends. The salary hike of Gita in the second year is
Correct answer: C · 26%
Arvind travels from town A to town B, and Surbhi from town B to town A, both starting at the same time along the same route. After meeting each other, Arvind takes 6 hours to reach town B while Surbhi takes 24 hours to reach town A. If Arvind travelled at a speed of 54 km/h, then the distance, in km, between town A and town B is (in numerical value)
Correct answer: 972
Anil invests Rs. 22000 for 6 years in a scheme with 4% interest per annum, compounded half-yearly. Sunil invests in the same scheme for 5 years, and then reinvests the entire amount received at the end of 5 years for one year at 10% simple interest. If the amounts received by both at the end of 6 years are equal, then the initial investment made by Sunil, in rupees, is (in numerical value)
Correct answer: 20808
The amounts of a job that Amal, Sunil and Kamal can individually do in a day are in harmonic progression. Kamal takes twice as much time as Amal to do the same amount of job. If Amal and Sunil work for 4 days and 9 days respectively, Kamal needs to work for 16 days to finish the remaining job. Then the number of days Sunil will take to finish the job working alone is (in numerical value)
Correct answer: 27
Let $C$ be the circle $x^2 + y^2 + 4x - 6y - 3 = 0$, and $L$ be the locus of the point of intersection of a pair of tangents to $C$ with the angle between the two tangents equal to 60 degrees. Then the point at which $L$ touches the line $x = 6$ is
Correct answer: B · (6, 3)
A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals
Correct answer: C · 8 : 5
In a right-angled triangle ABC, the altitude AB is 5 cm and the base BC is 12 cm. P and Q are two points on BC such that the areas of triangles ABP, ABQ and ABC are in arithmetic progression. If the area of triangle ABC is 1.5 times the area of triangle ABP, then the length of PQ, in cm, is (in numerical value)
Correct answer: 2
For some positive and distinct real numbers $x$, $y$ and $z$, if $\frac{1}{\sqrt{y}+\sqrt{z}}$ is the arithmetic mean of $\frac{1}{\sqrt{x}+\sqrt{y}}$ and $\frac{1}{\sqrt{z}+\sqrt{x}}$, then the relationship which will always hold true is
Correct answer: B · $y$, $x$ and $z$ are in arithmetic progression
The number of all natural numbers up to 1000 with non-repeating digits is
Correct answer: C · 738
A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the nth day exceeds one million, then the lowest possible value of n is (in numerical value)
Correct answer: 19