CUET PG 2023 — Mathematics PYQ

Analysis of Statement A

Condition: $a\cos A=b\cos B$

Sine Rule ke mutabiq, hum jante hain ki $a=2R\sin A$ aur $b=2R\sin B$. Isse equation mein rakhte hain:

Dono taraf $2$ se multiply karne par:

Iska matlab hai:

1. $2A=2B⟹A=B$ (Isosceles Triangle)

2. $2A=180^{\circ }-2B⟹2A+2B=180^{\circ }⟹A+B=90^{\circ }$. Iska matlab $C=90^{\circ }$ (Right Angled Triangle)

Verdict: Statement A sahi hai.

Analysis of Statement B

Condition: $\cos A\cos B+\sin A\sin B\sin C=1$

Hum jante hain ki $\sin C\le 1$. Isliye:

Sawal ke mutabiq value $1$ honi chahiye. Yeh tabhi mumkin hai jab:

1. $\cos (A-B)=1⟹A-B=0⟹A=B$

2. $\sin C=1⟹C=90^{\circ }$

Agar $C=90^{\circ }$ hai aur $A=B$ hai, toh $A=B=45^{\circ }$. Yeh ek Isosceles Right Angled Triangle hai.

Verdict: Statement B sahi hai.

Analysis of Statement C

Condition: $r_1,r_2,r_3$ are in H.P.

Ex-radii ka formula hota hai $r_1=\frac{\Delta }{s-a}$, $r_2=\frac{\Delta }{s-b}$, aur $r_3=\frac{\Delta }{s-c}$.

Agar yeh H.P. mein hain, toh inka reciprocal A.P. mein hoga:

Iska matlab hai: