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y² = 4x and x² = 4y
any tangent of y² = 4x is y = mx + 1/m
it also tangent for x² = 4y
1/m = -m² ⇒ m = -1
∴ common tangent is y = -x - 1, it also touches x² + y² = c²
∴ 1 = c² ⋅ (1+1) ⇒ c² = 1/2

n (C) = 73, n (T) = 65, n (C ∩ T) = x
n (C ∪ T) ≤ 100
⇒ n (C) + n (T) - n (C ∩ T) ≤ 100
⇒ x ≥ 38
n (C ∩ T) ≤ min (n (C), n (T) ⇒ x ≤ 65
⇒ 38 ≤ x ≤ 65

The college offers three UG and one PG courses, such as BA, BBA, MA, etc. The students can choose the major during the second year at TSLAS (for some majors, the prerequisites are offered during the second semester). Students can check the table below to know the detailed course-wise eligibility:

a, b, c are in Andhra Pradesh then
2b = a + c
28 = 3^ (2sin2θ-1) + 3^ (4-2sin2θ)
Put 3^ (2sin2θ) = x
28 = x/3 + 81/x ⇒ x² - 84x + 243 = 0
(x-3) (x-81) = 0
3^ (2sin2θ) = 3 or 3?
2sin2θ = 1 or 4
sin2θ = 1/2
terms are 1, 14, 27,
then T? = 1 + 5 (13)

Assuming the equation is (x + iy)² = I (x² + y²)
x² - y² + 2ixy = I (x² + y²)
Compare real and imaginary parts
x² - y² = 0 ⇒ x = y or x = -y
2xy = x² + y²
If x=y, then 2x² = x² + x², which is true for all x.
If x=-y, then -2y² = y² + y² = 2y², which implies 4y²=0, so y=0 and x=0.
The non-trivial solution is x = y.

The last round cutoff for the BSc course at IHM Chandigarh in the year 2025 stood at 7905. Further, for the year 2024, the cutoff was on the rank 9972, and 3650 in the year 2023. Thus, the NCHMCT cutoff shows a fluctuating trend over three years. For a more detailed view, refer to the table below:

I = ∫ (e²? + 2e? - e? - 1)e^ (e? +e? ) dx
I = ∫ (e²? + e? - 1)e^ (e? +e? ) dx + ∫ (e? - e? )e^ (e? +e? ) dx
I = ∫ (e? + 1 - e? )e^ (e? +e? ) dx + e^ (e? +e? )
(e? - e? + 1)dx = du
I = e^ (e? +e? ) + e^ (e? +e? ) = e^ (e? +e? ) (e? + 1) then g (x) = e? + 1
g (0) = 2

The cutoff scores/ ranks change every year, as they depend on varying factors. The factors affecting the NCHMCT JEE cutoff for the BSc course at IHM Chandigarh are mentioned below:

For the parabola y = x², the tangent at (2,4) is given by (y+4)/2 = 2x, which simplifies to 4x - y - 4 = 0.
The equation of a circle touching the line 4x - y - 4 = 0 at the point (2,4) is
(x-2)² + (y-4)² + λ (4x-y-4) = 0.
It passes through (0,1).
∴ (0-2)² + (1-4)² + λ (4 (0) - 1 - 4) = 0
4 + 9 + λ (-5) = 0 ⇒ 13 = 5λ ⇒ λ = 13/5
∴ the circle is x² - 4x + 4 + y² - 8y + 16 + (13/5) (4x-y-4) = 0
x² + y² + (-4 + 52/5)x + (-8 - 13/5)y + (20 - 52/5) = 0
x² + y² + (32/5)x - (53/5)y + 48/5 = 0
∴ centre is (-g, -f) = (-16/5, 53/10).

Here D = | 2 -4 λ |
| 1 -6 1 | = (λ-3) (3λ+2)
| λ -10 4 |
D = 0 ⇒ λ = 3, -2/3

The general term is T? = ¹? C? (K/x²)? (√x)¹?
= ¹? C? K? x? ²? x? /² = ¹? C? K? x? /²
For the constant term, the power of x is 0.
5 - 5r/2 = 0 ⇒ r = 2
The term is T? = ¹? C? · K² = 405
45 · K² = 405
⇒ K² = 9 ⇒ |K| = 3

The Uttar Pradesh Public Service Commission conducted the UPPSC Prelims Exam on October 12, 2025. The Commission released the UPPSC Prelims Answer Key on October 18, 2025. The final key will be released soon.

S' = 2¹? + 2? ⋅ 3 + 2? ⋅ 3² + . + 2 ⋅ 3? + 3¹?
G.P. → a = 2¹? , r = 3/2, n = 11
S' = 2¹? ⋅ [ (3/2)¹¹ - 1)/ (3/2 - 1)] = 2¹¹ (3¹¹/2¹¹) - 1)
= 3¹¹ - 2¹¹

The MBA course fees at Chandigarh University Online ranges between INR 2 Lacs - INR 2.1 Lacs in the year 2025. Candidates who are able to avail the Early Bird Scholarship get a discount of around 25% on the course fees. The course fees can be charged on both annual basis at INR 79,000 - INR 90,000 or per semester basis at INR 39,500 and INR 45,000 respectively. The aforementioned instalments are calculated from the course fee after applying the Early Bird Scholarship discount.

Negation of x ↔~ y
≡ ~ (x ↔ ~y)
≡ x ↔ ~ (~y)
≡ x ↔ y
≡ (x ∧ y) ∨ (~x ∧ ~y)

Johns Hopkins University offers diverse programs in areas like Public Policy, Health, Health Science, International Relations, and more. International applicants can take admission to undergraduate programs in Early or Regular modes, where postgraduate deadlines vary based on the program. The Johns Hopkins University application deadlines for UG and PG are given below:

• UG: Jan 2, 2026 (Early Decision 2/Regular Decision)
• MS: Feb 15, 2026 (Whiting School of Engineering)

International students can track their application on the application portal through which they have applied. UG applicants can apply through the Common App or Coalition App, where graduate applicants can apply via the program-specific portals. Students can also receive the decision mail through the same portal.

Chandigarh University Online MBA placement data is currently unavailable. The Chandigarh University Online does have a placement support cell to help students secure placement opportunities. Candidates looking to compare Chandigarh University Online MBA courses can compare the institute through other metrics such as course fees, student ratings, infrastructure and faculty quality, etc.