is equal to - (A) abc (B) 2abc (C) 0 (D) 4abc (sinx) + a 2 (sin 2 x) a n (A) 1 (B) 1 (C) 0 (D) 2 is equal to -
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1 J-Mthemtics XRCIS - 0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). The vlue of determinnt c c c c c c (A) c (B) c (C) 0 (D) 4c. If sin x cos x cos 4x cos x cos x sin x 4 cos x sin x sin x = 0 + (sinx) + (sin x) n (sin n x) then the vlue of 0 is - (A) (B) (C) 0 (D) 3. The vlue of the determinnt (A) 0 (B) ( )( c)(c ) (C) ( + )( + c)(c + ) (D) 4c 4. For ny ABC, the vlue of determinnt sin A cot A sin B cot B sin C cot C (A) 0 (B) (C) sin A sin B sin C (D) sin A + sin B + sin C 5. If D p = p 5 8 p p 5 0, then D + D + D 3 + D 4 + D 5 (A) 0 (B) 5 (C) 65 (D) none of these _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 sin(a B C) sin B cos C 6. If A + B + C =, then sin B 0 tn A cos(a B) tn A 0 (A) 0 (B) sin B tn A cos C (C) (D) none of these x 3x x 7. The numer of rel vlues of x stisfying x 4x 3x = 0 is - 7x 7x 6 x (A) 3 (B) 0 (C) (D) infinite log p 8. If,, c re pth, qth nd rth terms of GP, then log q log c r (A) 0 (B) (C) log c (D) pqr 9. If,,... n, n+,... re in GP nd i > 0 i, then log log log n n n 4 log log log n 6 n 8 n 0 log log log n n 4 n 6 (A) 0 (B) n log n (C) n(n + ) log n (D) none of these 7
2 J-Mthemtics x 3x x x 3 0. If px 4 + qx 3 + rx + sx + t = x x x 3 then t x 3 x 4 3x (A) 33 (B) 0 (C) (D) none. For positive numers x, y nd z, the numericl vlue of the determinnt log y log z log x log z y log x log y (A) 0 (B) log xyz (C) log(x + y + z) (D) logx logy logz z x z x y is -. If,, c > 0 nd x, y, z R, then the determinnt x x ( ) x x ( ) y y ( ) y y ( ) z z (c c ) z z (c c ) (A) x y c x (B) x y c z (C) x y c z (D) zero 3. For non-zero rel, nd c c c c c c = c, then the vlues of is - (A) 4 (B) 0 (C) (D) 4 4. The eqution ( x) ( x) ( x ) ( x) x x x 3x 5 x ( x) 3x x x x 3x x 3x x 3 = 0 (A) hs no rel solution (C) hs two rel nd two non-rel solutions (B) hs 4 rel solutions (D) hs infinite numer of solutions, rel or non-rel 5. Let determinnt is given y A = p q r x y z nd suppose determinnt A = 6. If B = 8 p x q y r z x y c z p q c r then - (A) det. B = 6 (B) det. B = 6 (C) det. B = (D) det. B = 6. If c nd c c = 0 then - (A) + + c = 0 (B) + c + c = 0 (C) + + c = + c + c (D) c = 0 7. If,, & c re nonzero rel numers, then c c c c c c (A) c ( + + c) (B) c( + + c) (C) zero (D) none of these 8. If f(x) = x x x x(x ) (x )x 3x(x ) x(x )(x ) (x )x (x ), then f (00) [J 98] (A) 0 (B) (C) 00 (D) 00 _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65
3 9. The vlue of the determinnt J-Mthemtics (A) 3 3 (B) (C) 0 (D) none of these 0. An equilterl tringle hs ech of its sides of length 6 cm. If (x, y ); (x, y ) & (x 3, y 3 ) re its vertices then the vlue of the determinnt, x y x y x y 3 3 (A) 9 (B) 43 (C) 486 (D) 97. If the system of equtions x + y + 3z =4, x + py + z = 3, x + 4y + z = 3 hs n infinite numer of solutions, then - (A) p =, µ = 3 (B) p =, µ= 4 (C) 3p = µ (D) none of these SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS). If x m m m 0, then x my e equl to - x (A) (B) (C) + (D) m _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 3. If x sin x e sin x x cos x sin x x cos x x, then the vlue of n cos (Dx) will e - D(x) cos x sin x e x x e cos x e e x x x (A) independent of x (B) dependent on x (C) 0 (D) non-existent 4. The vlue of the determinnt x n x is (A) independent of (B) independent of n (C) (x )(x ) (D)(x )(x n) 5. If the system of liner equtions x + y + z = 0, x + y + z = 0, x + cy + cz = 0 hs non-zero solution then (A) System hs lwys non-trivil solutions. (B) System is consistent only when = = c (C) If c then x = 0, y = t, z= t t R (D) If = = c then y = t, z = t, x = (t + t ) t,t R 6. If the system of equtions x + y 3 = 0, ( + K ) x + ( + K ) y 8 = 0 & x ( + K) y + ( + K) = 0 is consistent then the vlue of K my e - (A) (B) 3 5 (C) 5 3 CHCK YOUR GRASP ANSWR KY X R CI S - Que A ns. C A D A D A D A A C Que A ns. A D D D C A C A B D Que A ns. D A, B A,C B, C A,C,D A,C (D) 9
4 J-Mthemtics XRCIS - 0 BRAIN TASRS SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS). Which of the following determinnt(s) vnish(es)? (A) c c( c) c c(c ) ( ) (B) c c c c (C) 0 c 0 c c c 0 (D) log xyz log y log z x x x log xyz log z y log xyz log y z z y. If f'(x) = mx mx p mx p n n p n p mx n mx n p mx n p, then y = f(x) represents - (A) stright line prllel to x xis (C) prol (B) stright line prllel to y xis (D) stright line with negtive slope 3. The determinnt ( c) c (c ) c c c ( ) is divisile y - (A) + + c (B) ( + ) ( + c) (c + ) (C) + + c (D) ( )( c) (c ) 4. The determinnt c c c 0 (A),, c re in AP (C) is root of the eqution x +x+c=0 5. Let f(x) = sin x cos x 4 sin x sin x cos x 4 sin x sin x cos x 4 sin x is equl to zero, if - 0 (B),, c re in GP (D) (x ) is fctor of x + x + c, then the mximum vlue of f(x) = (A) (B) 4 (C) 6 (D) 8 6. The prmeter on which the vlue of the determinnt is- cos(p d)x cos px cos(p d)x sin(p d)x sin px sin(p d)x (A) (B) p (C) d (D) x 7. If + + c = - nd x ( )x ( c )x f(x) ( )x x ( c )x ( )x ( )x c x, then f(x) is polynomil of degree- (A) (B) 3 (C) 0 (D) does not depend upon _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65
5 J-Mthemtics 8. Given tht q pr < 0, p > 0, then the vlue of p q px qy q r qx ry px qy qx ry 0 is- (A) zero (B) positive (C) negtive (D) q + pr 9. The vlue of lying etween & nd 0 A nd stisfying the eqution 4 sin A cos A sin 4 sin A cos A sin 4 sin A cos A sin 4 = 0 re - (A) A =, (B) A = 3 (C) A, (D) 0. The set of equtions x y + 3z =, x y + z = 4, x y + z = 3 hs - (A) unique solution only for = 0 (B) unique solution for 8 (C) infinite numer of solutions of = 8 (D) no solution for = 8 3 A, 6 8 _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 BRAIN TASRS ANSWR KY X R CI S - Que A ns. A,B,C,D A A,C,D B, D C B A C A,B,C,D B, D
6 J-Mthemtics XRCIS - 03 MISCLLANOUS TYP QUSTIONS TRU / FALS. If,, c re sides of sclene tringle, then the vlue of c c is positive.. If x + hxy + y + gx + fy + c ( x + m y + n ) ( x + m y + n ), then h g h f g f c 3. If x = cy + z, y = z + cx, z = x + y, where x, y, z re not ll zero, then + + c + c + = If i i i i i i x y z nd x y y z z x 0 then i i i i i i i i i 3 x x x y y y 3 z z z 5. Consider the system of equtions i x + i y + c i z = d i where i =,, 3. If = d c d c d c = d c d c d c then the system of equtions hs infinite solutions. MATCH TH COLUMN = d d d Following question contins sttements given in two columns, which hve to e mtched. The sttements in Column-I re lelled s A, B, C nd D while the sttements in Column-II re lelled s p, q, r nd s. Any given sttement in Column-I cn hve correct mtching with ON sttement in Column-II.. Column-I Column-II (A) If the determinnt p x u f q m y v g c r n z w h (p) 3 splits into exctly K determinnts of order 3, ech element of which contins only one term, then the vlues of K is (B) The vlues of for which the system of equtions (q) 8 x + y + z = 6, x + y + 3z = 0 & x + y + z = is inconsistent (C) If x, y, z re in A.P. then the (r) 5 vlue of the determinnt 3 x 3 4 y 4 5 z is (D) Let p e the sum of ll possile (s) 0 determinnts of order hving 0,, & 3 s their four elements (without repetition of digits). The vlue of 'p' is = 0 = 0. _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65
7 J-Mthemtics _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 ASSRTION & RASON These questions contin, Sttement I (ssertion) nd Sttement II (reson). (A) Sttement-I is true, Sttement-II is true ; Sttement-II is correct explntion for Sttement-I. (B) Sttement-I is true, Sttement-II is true ; Sttement-II is NOT correct explntion for sttement-i. (C) Sttement-I is true, Sttement-II is flse. (D) Sttement-I is flse, Sttement-II is true.. Sttement - I : Consider D = 3 3 c c c 3 Let B, B, B 3 e the co-fctors of,, nd 3 respectively then B + B + 3 B 3 = 0 B e c u s e Sttement - II : If ny two rows (or columns) in determinnt re identicl then vlue of determinnt is zero. (A) A (B) B (C) C (D) D. Sttement - I : Consider the system of equtions, x + 3y + 4z = 5 x + y + z = x + y + 3z = 4 This system of equtions hs infinite solutions. B e c u s e Sttement - II : If the system of equtions is e : x + y + c z d = 0 e : x + y + c z d = 0 e 3 : e + e = 0, where R & Then such system of equtions hs infinite solutions. (A) A (B) B (C) C (D) D 3. Sttement - I : If,, c R nd c nd x,y,z re non zero. Then the system of equtions x + y + cz = 0 x + cy + z = 0 cx + y + z = 0 hs infinite solutions. B e c u s e Sttement - II : If the homogeneous system of equtions hs non trivil solution, then it hs infinitely mny solutions. (A) A (B) B (C) C (D) D COMPRHNSION BASD QUSTIONS Comprehension # Let x, y, z R + & D = 3 4 x x x 3 4 y y y 3 4 z z z On the sis of ove informtion, nswer the following questions :. If x y z & x, y, z re in GP nd D = 0, then y (A) (B) (C) 4 (D) none of these. If x, y, z re the roots of t 3 t + t 343 = 0, R, then D is equl to- (A) (B) 0 (C) dependent on x, y, z (D) dt indequte 3
8 J-Mthemtics 3. If x y z & x, y, z re in A.P. nd D = 0, then xy z + x z is equl to- (A) (B) (C) 3 (D) none of these Comprehension # Consider the system of liner equtions x + y + z = m x + y + z = n nd x + y + z = p On the sis of ove informtion, nswer the following questions :. If, then the system hs - (A) no solution (B) infinte solutions (C) unique solution (D) trivil solution if m n p. If = & m + n + p 0 then system of liner equtions hs - (A) no solution (B) infinite solutions (C) unique solution (D) finitely mny solution 3. If = & m p then the system of liner equtions hs - (A) no solution (B) infinite solutions (C) unique solution (D) unique solution if p = n MISCLLANOUS TYP QUSTION ANSWR KY X R CI S - 3 Tr ue / Flse. F. T 3. F 4. T 5. F Mtch the Column. (A) (q); (B) (p); (C) (s); (D) (s) Assertion & Reson. A. A 3. A Comprehension Bsed Questions Comprehension # :. A. B 3. C Comprehension # :. C. A 3. A 4 _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65
9 J-Mthemtics XRCIS - 04 [A]. Without expnding the determinnt prove tht : 0 c () 0 = 0 () c 0. Prove tht : x y c z () x y z = x y z () y z z x x y CONCPTUAL SUBJCTIV XRCIS 0 p q p r q p 0 q r = 0 r p r q 0 c c c c = 0 3. Prove tht : = ( ) Using properties of determinnts or otherwise evlute If D = c c nd D = c c c c c c then prove tht D= D. 6. Prove tht = ( + ² + ²) 3. _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 c c 7. Prove tht c c ( c)( c ) c x x 3 3 x 4 8. Solve for x, x 3 3 x 4 4 x 5 = 0. 3 x 5 5 x 8 0 x 7 x c 9. If + + c = 0, solve for x : c x = 0. c x c c ' 'c 'c ' 0. Prove tht c c ' c ' c ' ' ( ' ')(c ' 'c)(c ' c '). ' ' ' '. Let the three digit numers A8, 3B9, nd 6C, where A, B, nd C re integers etween 0 nd 9, e A 3 6 divisile y fixed integer k. Show tht the determinnt 8 9 C is divisile y k. B 5
10 J-Mthemtics n! (n )! (n )!. For fixed positive integer n, if D = (n )! (n )! (n 3)! y n. (n )! (n 3)! (n 4)! r r r 3. If D r = x y z then prove tht n n n 3 5 n r D r = 0. D then show tht 3 (n!) 4 is divisile 4. Find the vlue of the determinnt (x y) z z z (y z) x x x y (y z) x y z y(x y) x z xz xz 5. Prove tht = 64( ) ( )( ) ( ) ( ) ( ) 6. Show tht l m l m l m 3 3 l m l m l m 3 3 l m l m l m Solve the following sets of equtions using Crmer s rule nd remrk out their consistency. () x y z 6 0 x y z 0 x y z 3 0 () x y z 3 x y z 6 x y 0 (c) 6 = 0. x 3 y z 3 x y z 6 5 x y 3 z 3 (d) 7x 7 y 5 z 3 3 x y 5z 7 x 3y 5 z 5 8. Investigte for wht vlues of, the simultneous equtions x + y + z = 6 ; x + y + 3 z = 0 & x + y + z = hve : () A unique solution. () An infinite numer of solutions. (c) No solution. 9. Find the vlues of c for which the equtions x +3y = 0 (c + ) x + (c + 4)y = c + 6 (c + ) x + (c + 4) y = (c + 6) re consistent. Also solve ove equtions for these vlues of c. 0. Let, nd, e the roots of x + x + c = 0 nd px + qx + r = 0 respectively. If the system of c equtions y + z = 0 nd y + z = 0 hs non-trivil solution, then prove tht q pr. CONCPTUAL SUBJCTIV XRCIS ANSWR KY X R C IS - 4 ( A ) x = or x = 9. x = 0 or x = ± () x =, y =, z = 3; consistent () x =, y =, z = ; consistent (c) x = 3 3, y = 7 6, z = () 3 () = 3, = 0 (c) = 3, 0 ; consistent (d) inconsistent 9. c = 6,, for c = 6, x = 0 = y & for c=, x = 5, y = 0 3 _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65
11 J-Mthemtics XRCIS - 04 [B] BRAIN STORMING SUBJCTIV XRCIS. Let,, c, d e rel numers in G.P. If u, v, w stisf y the system of equtions u + v + 3w = 6, 4u + 5v + 6w =, 6u + 9v = 4, then show tht the roots of the equtions u v w x + [ ( c) + (c ) + (d ) ] x + u + v + w = 0 nd 0x +0 ( d) x 9 = 0 re reciprocls of ech other. [J 99]. Prove tht c c c c c c c c c ( ) ( c) ( c) ( ) (c ) (c ) = 3. ( c) (c ) ( ) ( + + c) ( + c + c) 3. If + + c = then show tht the vlue of the determinnt ( c ) cos ( cos ) c( cos ) ( cos ) (c ) cos c( cos ) c( cos ) c( cos ) c ( ) cos simplifies to cos 4. Find the vlue of the determinnt cos(x y) cos(y z) cos(z x) cos(x y) cos(y z) cos(z x) sin(x y) sin(y z) sin(z x). S S S 0 5. If S r = r + r + r then show tht S S S 3 = ( ) ( ) ( ). 6. If S S 3 S 4 x ² + y ² + cz = x + y + cz = x 3 + y 3 + cz 3 = d nd x x 3 + y y 3 + cz z 3 = x 3 x + y 3 y + cz 3 z = x x + y y + cz z = f, _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 then prove tht x y z x y z x y z d f = (d f) c / (,, c 0) 7. If u = x + xy + cy, u = x + xy + cy, then prove tht- y xy x 8. Solve the system of equtions : x y x cy u u x y x cy y x y x y. where c. 3 z y x 0 3 z y x 0 3 z c y c x c 0 9. If x,y,z re not ll zero nd if x + y + cz = 0; x + cy + z = 0; cx + y + z = 0 Prove tht x : y : z = : : or : : or : :. 0. Prove tht the system of equtions in x nd y ; x + hy + g = 0, hx + y + ƒ = 0, x + hxy + y + gx + ƒy + c = t is consistent if h g t h ƒ h g ƒ c h BRAIN STORMING SUBJCTIV XRCIS ANSWR KY X R C I S - 4 ( B ) 4. sin(x y) sin(y z) sin(x z) 8. x = ( + + c), y = + c + c, z = c 7
12 J-Mthemtics XRCIS - 05 [A] J-[MAIN] : PRVIOUS YAR QUSTIONS log p. If,, c re pth, qth nd rth terms of GP, nd ll re positive then log q is equl to- [AI-00] log c r () 0 () (3) log c (4) pqr. If,, re cue roots of unity nd n 3p, p Z, then n n n n n n is equl to- [AI-003] () 0 () (3) (4) 3. If 3 3 c c c 3 = 0 nd vectors (,, ), (,, ) nd (, c, c ) re non-coplnr, then the product c equls- [AI ] () () 0 (3) (4) 4. If,,... n, n+,... re in GP nd i > 0 i, then log log log n n n 4 log log log n 6 n 8 n 0 log log log n n 4 n 6 is equl to- [AI-04,05] () 0 () n log n (3) n(n + ) log n (4) none of these 5. If + + c = nd x ( )x ( c )x f(x) ( )x x ( c )x ( )x ( )x c x, then f(x) is polynomil of degree- [AI 005] () () 3 (3) 0 (4) 6. The system of equtions x + y + z = x + y + z = x + y + z = hs no solution, If is [AI 005] () () not (3) either - or (4) 7. If D = x for x 0, y 0 then D is- [AI - 007] y () Divisile y oth x nd y () Divisile y x ut not y (3) Divisile y y ut not x (4) Divsile y neither x nor y Let A = 0 5, if A = 5 then equls- [AI - 007] () 5 () 5 (3) (4) /5 9. Let,, c e ny rel numers. Suppose tht there re rel numers x, y, z not ll zero such tht x = cy + z, y = z + cx nd z = x + y, then + + c + c is equl to [AI - 008] () () (3) 0 (4) 8 _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65
13 J-Mthemtics 0. Let,, c e such tht ( + c) 0. If c c c + c c n ( ) n ( ) ( ) n c = 0, then the vlue of n is :- [AI - 009] () Any odd integer () Any integer (3) Zero (4) Any even integer. Consider the system of liner equtions : x + x + x 3 = 3 x + 3x + x 3 = 3 3x + 5x + x 3 = The system hs [AI - 00] () Infinite numer of solutions () xctly 3 solutions (3) A unique solution (4) No solution. The numer of vlues of k for which the liner equtions 4x + ky + z = 0 kx + 4y + z = 0 x + y + z = 0 possess non-zero solution is :- [AI - 0] () () zero (3) 3 (4) 3. If the trivil solution is the only solution of the system of equtions x ky + z = 0 kx + 3y kz = 0 3x + y z = 0 Then the set of ll vlues of k is: [AI - 0] () {, 3} () R {, 3} (3) R {} (4) R { 3} _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 PRVIOUS YARS QUSTIONS ANSWR KY XRCIS-5 [A] Que A ns Que. 3 A ns
14 J-Mthemtics XRCIS - 05 [B] J-[ADVANCD] : PRVIOUS YAR QUSTIONS. Solve for x the eqution sin(n )x sin nx sin(n )x 0 cos(n )x cos nx cos(n )x [R 00, (Mins), 3 out 00]. Test the consistency nd solve them when consistent, the following system of equtions for ll vlues of x + y + z = x + 3y z = 3x +( + )y 3z = + [R 00,(Mins), 5 out 00] 3. Let,, c, e rel numers with + + c =, Show tht the eqution x y c x y cx x y x y c cy 0 cx cy x y c represents stright line. 4. The numer of vlues of k for which the system of equtions (k +) x + 8y = 4k kx + (k +3)y = 3k 30 [J 00,(Mins), 6 out 00] hs infinitely mny solutions is [J 00,(Screening), 3] (A) 0 (B) (C) (D) infinite 5. The vlue of for which the system of equtions x y z =, x y + z = 4, x + y + z = 4 hs no solution is [J 004 (Screening)] (A) 3 (B) 3 (C) (D) 6. () Consider three point P = ( sin( ), cos), Q = (cos( ), sin) nd R = (cos(), sin()), where 0 < /4 (A) P lies on the line segment RQ (C) R lies on the line segment QP (B) Q lies on the line segment PR (D) P, Q, R re non colliner () Consider the system of equtions x y + 3z = ; x + y z = k; x 3y + 4z =. Sttement-I : The system of equtions hs no solution for k 3. n d Sttement-II : The determinnt 3 k 4 0, for k 3. [J 008, 3+3] (A) Sttement-I is true, Sttement-II is true ; Sttement-II is correct explntion for Sttement-I. (B) Sttement-I is true, Sttement-II is true ; Sttement-II is NOT correct explntion for sttement-i. (C) Sttement-I is true, Sttement-II is flse. (D) Sttement-I is flse, Sttement-II is true. _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65
15 J-Mthemtics 7. The numer of ll possile vlues of, where 0 <, for which the system of equtions (y + z)cos3 = (xyz)sin3 cos 3 sin 3 x sin 3 y z (xyz)sin3 = (y + z)cos3 + ysin3 hve solution (x 0, y 0, z 0 ) with y 0 z 0 0, is [J 00, 3] _NOD6 ()\Dt\04\Kot\J-Advnced\SMP\Mths\Unit#0\NG\Prt-\0.Determinnts\0.XRCISS.p65 PRVIOUS YARS QUSTIONS. x = n, n I ANSWR KY. If = 5, system is consistent with infinite solution given y z = K, y (3K 4) nd x (5K ) where K R If 5, system is consistent with unique solution given y z ( ); x ( ) B 5. D 6. () D; () A 7. 3 XRCIS-5 [B] nd y = 0. 3
(B) a + (D) (A) A.P. (B) G.P. (C) H.P. (D) none of these. (A) A.P. (B) G.P. (C) H.P. (D) none of these
J-Mathematics XRCIS - 01 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR) 1. The roots of the quadratic equation (a + b c) (a b c) + (a b + c) = 0 are - (A) a + b + c & a b + c (B) 1/ &
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