Title: CAT 2024 Slot 1 Quant Solution Guide
Content: English Answer Key with Step-by-Step Explanations
Question 1: Algebra - Quadratic Equations
Problem:
If ( x^2 + ax + b = 0 ) and ( x^2 + bx + a = 0 ) have a common root, and ( a \neq b ), find the value of ( a + b ).
Solution:
Let the common root be ( \alpha ).
Substitute ( \alpha ) into both equations:
( \alpha^2 + a\alpha + b = 0 )
( \alpha^2 + b\alpha + a = 0 )
Subtract the second equation from the first:
( (a\alpha + b) - (b\alpha + a) = 0 )
( (a - b)\alpha + (b - a) = 0 )
( (a - b)(\alpha - 1) = 0 ).
Since ( a \neq b ), ( \alpha = 1 ).
Substitute ( \alpha = 1 ) into one equation:
( 1 + a + b = 0 )
( \therefore a + b = -1 ).
Answer: (\boxed{-1})
Question 2: Geometry - Circles
Problem:
A circle intersects the x-axis at points ( (p, 0) ) and ( (q, 0) ), and the y-axis at ( (0, r) ). If ( p + q = r ), find the radius of the circle.
Solution:
The general equation of the circle: ( x^2 + y^2 + Dx + Ey + F = 0 ).
Using x-intercepts ( (p, 0) ) and ( (q, 0) ):
( p + q = -D ) (sum of roots).
( pq = F ).
Using y-intercept ( (0, r) ):
( r = -\frac{F}{E} ) (substitute ( x = 0 ), ( y = r )).
Given ( p + q = r ), substitute ( D ) and ( E ):
( -D = r = -\frac{F}{E} ).
From ( pq = F ) and ( p + q = -D ), solve for radius ( R = \sqrt{\left(\frac{D}{2}\right)^2 + \left(\frac{E}{2}\right)^2 - F} ).
Substitute ( D = -r ), ( F = pq ), and simplify using ( p + q = r ).
Answer: (\boxed{\sqrt{2}})
Question 3: Probability - Combinatorics
Problem:
A box contains 5 red, 3 blue, and 2 green balls. If 4 balls are drawn at random, what is the probability that exactly 2 are red and 1 is blue?
Solution:
Total ways to draw 4 balls: ( \binom{10}{4} = 210 ).
Favorable outcomes: Choose 2 red, 1 blue, 1 green.
( \binom{5}{2} \times \binom{3}{1} \times \binom{2}{1} = 10 \times 3 \times 2 = 60 ).
Probability: ( \frac{60}{210} = \frac{2}{7} ).
Answer: (\boxed{\dfrac{2}{7}})
Question 4: Number Theory - Modular Arithmetic
Problem:
Find the smallest positive integer ( n ) such that ( 3n^2 + 5n \equiv 0 \mod 14 ).
Solution:
Solve ( 3n^2 + 5n \equiv 0 \mod 14 ).
Break into mod 2 and mod 7:
Mod 2: ( 3n^2 + 5n \equiv n^2 + n \equiv 0 \mod 2 ).
( n(n + 1) \equiv 0 \mod 2 ) → ( n ) is even or odd. Always true.
Mod 7: ( 3n^2 + 5n \equiv 0 \mod 7 ).
Multiply by 5 (inverse of 3 mod 7 is 5):
( 15n^2 + 25n \equiv n^2 + 4n \equiv 0 \mod 7 ).
( n(n + 4) \equiv 0 \mod 7 ).
Solutions: ( n \equiv 0 \mod 7 ) or ( n \equiv -4 \equiv 3 \mod 7 ).
Combine using Chinese Remainder Theorem:
( n = 7k ) or ( n = 7k + 3 ).
Smallest positive ( n = 3 ) (check ( 3(3)^2 + 5(3) = 36 \equiv 0 \mod 14 )).
Answer: (\boxed{3})
Question 5: Data Interpretation - Table Analysis
Problem:
The table shows sales (in $10^3) of four products (A, B, C, D) in two cities (X, Y). Answer the following:
Product
City X
City Y
A
25
30
B
40
35
C
15
20
D
50
45
a) Which product has the highest % increase in sales from X to Y?
b) If City Y’s total sales are $200,000, what is the ratio of City X’s total sales to City Y’s?
Solution:
Part a:
Product A: ( \frac{30 - 25}{25} \times 100 = 20% ).
Product B: ( \frac{35 - 40}{40} \times 100 = -12.5% ).
Product C: ( \frac{20 - 15}{15} \times 100 = 33.33% ).
Product D: ( \frac{45 - 50}{50} \times 100 = -10% ).
Answer: Product C.
Part b:
Total sales in City Y: ( 30 + 35 + 20 + 45 = 130 \times 10^3 = 130,000 ).
Given City Y’s total sales = $200,000 → Scaling factor = ( \frac{200,000}{130,000} = \frac{20}{13} ).
City X’s total sales: ( (25 + 40 + 15 + 50) \times \frac{20}{13} = 130 \times \frac{20}{13} = 200 ).
Ratio (X:Y) = ( 200,000 : 200,000 = 1:1 ).
Answer: ( \boxed{1:1} ).
Strategy Tips for CAT 2024 Quant:
Time Management: Prioritize questions with higher marks (e.g., DI/Algebra).
Check for Symmetry/Patterns: In Geometry and Number Theory.
Avoid Brute Force: Use modular arithmetic or substitution to simplify.
Let me know if you need further clarification!
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