UP Board Maths Phase 3 - The 8-Mark Jackpot

QUESTION 7: MATRICES & LINEAR EQUATIONS 08 MARKS

Setter's Radar: Bhai, yahan Board 100% predictible hai। Q7 mein humesha "Matrix Method" ya "Inverse Proof" ka muqabla hota hai [cite: 1013, 2772, 5321]।

Option A: Matrix Equation & Inverse Proof

Prove equation: \( A^3 - 6A^2 + 5A + 11I = 0 \) and then find \( A^{-1} \) using it [cite: 7123-7126, 8336]।

अथवा (OR)

Option B: Solve System of Equations (Matrix Method)

Solve for \( x, y, z \):

3x - 2y + 3z = 8 \] \[ 2x + y - z = 1 \] \[ 4x - 3y + 2z = 4

[cite: 1013, 2772, 3187, 7132, 7552, 8348]

Topper Strategy for Q7: "Option B" (Matrix Method) humesha choose karo। Iske steps fixed hain aur calculation lambi hone par bhi logic clear hota hai। Bass \( \text{Det}(A) \neq 0 \) dikhana mat bhoolna!

QUESTION 8: AOD & DEFINITE INTEGRALS 08 MARKS

Expert Analysis: Q8 Board ka high-level logic zone hai। Isme aksar Applications of Derivatives aur Integral Properties ka combination hota hai।

Option A: Theoretical Optimization (AOD)

Prove: Maximum Volume of a cylinder/cone inscribed in a sphere of radius \( R \) is \( \frac{8}{27} \) of volume of sphere [cite: 1449, 3612, 5353, 5806, 7561, 8371]।

अथवा (OR)

Option B: Definite Integral Proofs (Property 4)

Prove:

\int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} dx = \frac{\pi^2}{4}

Ya fir \( \int_{0}^{\pi/2} \log \sin x \, dx = -\frac{\pi}{2} \log 2 \) [cite: 1024, 1470, 2307, 2732, 3709, 4321, 5808, 6287, 7172, 8358]।

Setter's Tip for Q8: AOD proof karte waqt diagram zaroor banayein। Integral proofs mein property ka reference (जैसे: \( \int_0^a f(x) dx = \int_0^a f(a-x) dx \)) side mein bracket mein zaroor likhein।

MISSION 2026: FINAL WORD

Question 7 और 8 के ये 16 नंबर तुम्हारी मेरिट डिसाइड करेंगे। "Matrix Method" और "Definite Integral Property 4" को कम से कम 5-5 बार हल कर लो। बोर्ड एग्जामिनर इन सवालों को ढूंढ-ढूंढ कर पेपर में डालता है!

UP BOARD MATHS 2026: PHASE 3 (Q7 & Q8)

QUESTION 7: MATRICES & LINEAR EQUATIONS 08 MARKS

QUESTION 8: AOD & DEFINITE INTEGRALS 08 MARKS

MISSION 2026: FINAL WORD