XII Maths DPP (36) - Prev Chaps
XII Maths DPP (36) - Prev Chaps
XII Maths DPP (36) - Prev Chaps
ANSWERKEY
DPP No. : 80 (JEE-ADVANCED)
1. (ACD) 2. (CD) 3. (ABC) 4_. (AB) 5_. (ABC) 6. (BCD)
cos sin 0
7*. (AC) 8. cos sin 0
1 0 0
Comprehension (1 to 3)
sin i cos
5_. If A() = , then which of the following is true ?
i cos sin
(A*) A()–1 = A( – ) (B*) A() + A( +) is a null matrix
(C*) A() is invertiable for all R (D) A()–1 = A(–)
sin i cos
;fn A() = gks] rks fuEu esa ls dkSulk lR; gS ?
i cos sin
(A*) A()–1 = A( – ) (B*) A() + A( +) ,d 'kwU; vkO;wg gS
(C*) lHkh R ds fy, A() O;qRØe.kh; gS (D) A()–1 = A(–)
Sol. We have, |A()| = 1
Hence, A is invertible
sin i cos – sin i cos
A( + ) = = = –A()
i cos sin i cos – sin
– sin i cos sin –i cos
adj (A()) = A()–1 = = A( – )
i cos – sin –i cos sin
6. Which of the following statements is/are true about square matrix A of order n ?
(A) (–A)–1 is equal to – A–1 when n is odd only
(B*) If An = O, then I + A + A2 + . . . + An–1 = (I – A)–1
(C*) If A is skew-symmetric matrix of odd order, then its inverse does not exist.
(D*) If A is non-singular, then (AT)–1 = (A–1)T holds always
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fuEu esa ls dkSulk dFku n dksfV dh oxZ vkO;wg A ds fy, lR; gS ?
(A) (–A)–1 = – A–1 tc n dsoy fo"ke gSA
(B*) ;fn An = O gks] rks I + A + A2 + . . . + An–1 = (I – A)–1
(C*) ;fn A fo"ke dksfV dk fo"ke lefer vkO;wg gS rc bldk O;qRØe fo|eku ugha gksxkA
(D*) ;fn A O;qRØe.kh; gS] rc (AT)–1 = (A–1)T
n–1
adj –A –1 adj(A) adj A
Sol. (–A) =
–1
= n
= = – A–1 (For any value of n)
| –A | (–1) | A | –|A|
Given An = O
Now
(I – A)(I + A + A2 + . . . . + An–1) = I – An = I
(I – A)–1 = I + A + A2 + . . . + An–1
7*. One vertex of a triangle of maximum area that can be inscribed in the curve |z – 2i| = 2 is 2 + 2i. Then
the remaining vertices is / are
oØ |z – 2i| = 2 ds vUrxZr vf/kdre {ks=kQy ds f=kHkqt dk ,d 'kh"kZ 2 + 2i gS] rc 'ks"k 'kh"kZ gSµ
(A*) – 1 + i 2 3 (B) – 1 – i 2 3
(C*) – 1 + i 2 – 3
(D) – 1 – i 2 – 3
Sol. Triangle must be equilateral. Use rotation theorem. vf/kdre {ks=kQy ds fy, f=kHkqt dk leckgq gksuk vko';d
gS ?kq.kZu izes; ls
1 cos( ) cos
8. The determinant D = cos( ) 1 cos is a square of the determinant A, then determinant
cos cos 1
A is equal to_________.
1 cos( ) cos
lkjf.kd D = cos( ) 1 cos ,d lkjf.kd A dk oxZ gks] rks lkjf.kd A _________ gSA
cos cos 1
cos sin 0
Ans. cos sin 0
1 0 0
Corporate Office : CG Tower, A-46 & 52, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.) - 324005
Website: www.resonance.ac.in | E-mail : contact@resonance.ac.in
PAGE NO.-3
Toll Free : 1800 200 2244 | 1800 258 5555 | CIN: U80302RJ2007PTC024029