X maro G Strategy | Concept of Quadratic Equations | CAT 2025 Slot 1
The number of non-negative integer values of k for which the quadratic equation x² – 5x + k = 0 has only integer roots, is
Answer
Quadratic: x² – 5x + k = 0.
For integer roots, discriminant must be a square:
D = 25 – 4k = n².
Try n = 1, 3, 5:
n = 1 → k = 6
n = 3 → k = 4
n = 5 → k = 0
All are non-negative.
Total values = 3.
Answer: 3
X maro | Visual | Inequality | Sets | EASY | CAT 2025 Slot 1
Let 3 ≤ x ≤ 6 and [x²] = ([x])², where [x] is the greatest integer not exceeding x.
If S represents all feasible values of x, then a possible subset of S is
1. [3, √10] ∪ [4, √17] ∪ {6}
2. [3, √10) ∪ [5, √26)
3. (4, √18] ∪ [5, √27) ∪ {6}
4. (3, √10] ∪ [5, √26) ∪ {6}
Answer
Using X maro
For n = 3 → x must be before 4 → [3, √10)
For n = 4 → x must be before 5 → [4, √17)
For n = 5 → x must be before 6 → [5, √26)
For n = 6: only x = 6 (since x ≤ 6)
Only Option 4 matches: (3, √10] ∪ [5, √26) ∪ {6}
Note: ideally Set should include 4 as well.
X maro Functions + Maxima Minima | EASY | CAT 2025 Slot 1
A value of c for which the minimum value of f(x) = x² – 4cx + 8c is greater than the maximum value of
g(x) = –x² + 3cx – 2c, is
A. 1/2
B. –1/7
C. 2
D. –2
Answer
Let B = number of boys, G = number of girls (both integers), and B > 10.
After departures: remaining girls = 60% of G = 3G/5, remaining boys = 40% of B = 2B/5.
Given 3G/5 = 2B/5 + 8. Multiply by 5: 3G = 2B + 40. (Equation ★)
For 3G to be integer, G must be integer. For the remaining counts 3G/5 and 2B/5 to be integers (no fractional people), G and B should both be multiples of 5.
Let B = 5m (m integer) with B > 10 ⇒ m ≥ 3. Try smallest multiples of 5 for B:
• B = 15 ⇒ 2B + 40 = 70 ⇒ G = 70/3 (not integer)
• B = 20 ⇒ 2B + 40 = 80 ⇒ G = 80/3 (not integer)
• B = 25 ⇒ 2B + 40 = 90 ⇒ G = 90/3 = 30 (integer, works)
So the smallest B (multiple of 5) that satisfies (★) is B = 25, with G = 30.
Total initial students = B + G = 25 + 30 = 55.
Therefore the minimum possible number of students initially in the class (with realistic integer people and integer remaining counts) is 55.
