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Which of the following metrics in R’ is not induced by a norm in R’. For x= (ಉ.ಖ) ೩೧೮.
Y= Oh, dp) ए?

८) ५८, ಖೆ /

0 if x=y

(© ०८, 9) = पपत മിമി 0) ०,७, ०) = - —

Let M be an orthonormal set in an inner product space X. If d is the metric induced by the
norm, then for distinct points x, ye M, d(x, y) is:

(8) dy(x, y)=|x, - | + | - |

(A) d(x, y)=0 (8) d(x, y)=1
© dx, y=2 ൯൬ 4७, 3) 5 ४2
The Hahn-Banach extension of h(x, y) =2x on Y ={(x, ye Rix = y} to R’ with 2-norm is :
(A) f(x, y)=38x-y (8) f@,y)=x+y
(C) f(x, y)=4x-2y ൯൬ /0ು 2) ಸುತತ)
Solution വ്‌ (കമ y—x)dy=(1+ y*)dx is:
(A) ॐ = वा 1) -1+८ ‏ءدراصر رق صلی‎ eens
(0 ysecx=xtan! y+cx* (D) None of the above

2
Solution of (1 +x)? 3 +(1+ ಖರ + y=2sinflog(1+x)] is:
2 2

(ಯ) y=e,cosx +c, sinx + log(1+x)cosx
(3) y=c,cos[log(1+ x)]+c, sin[log(1 +x)] —log(1 + x) cos[log (1+ x)]
(0) y=e,e* +c,e** log(1 +x) sin(1+ x)
(1) y=c,cos[log(1 + x)] +c, sin[log(1+x)]+ cos[log (1 +x)]
If p(x) is a continuous function, the general solution of :
2“ - ०() 0 + [7 (६) -1] 9 = 0 18:
(ಯ ‏“معدن‎ + ५०९०८१८ (3) y=c,e* + ‏تدك (:6 م ] “مره‎

ம்‌ ‏و‎ a
(0) ‏معدن‎ + கரச” “dx ൩) + =५९ tee" fel പിക]

dx
Consider the equation 2/5 ரர்‌, y(0)=0 on &-/(,3)6 ॥ 2111 <1, [9] <2}, Which of the
following is true?

(A) differential equation has unique solution

(2) /(७, 3) 5 ತ satisfies Lipschitz condition with respect to y in X

(C) differential equation has no solutions
(D) none of the above

The equation ௮ = ‏ڈرو‎ , »(0) =0 has:
2

(A) Unique solution

(B) Infinite number of solutions

(C) No solution

(D) Not unique but finite number of solutions

083/2023 12 A