JEE/BITSAT LEVEL TEST

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1 JEE/BITSAT LEVEL TEST Booklet Code A/B/C/D Test Code : 00 Matrices & Determinants Answer Key/Hints Q. i 0 A =, then A A is equal to 0 i (a.) I (b.) -ia (c.) -I (d.) ia i 0 i 0 0 Sol. We have AA I 0 i 0 i 0 Q. cos cos sin Let E( ). and cos sin sin /, then E ( ) E ( ) is a differs by an odd multiple of (a.) null matri (b.) unit matri (c.) Diagonal Matri Sol. We have E( )E( ) (d.) Orthogonal Matri cos cos sin cos cos sin cos cos cos( ) cos sin cos( ) cos sin sin cos sin sin sin cos cos( ) sin sin cos( ) As and differ by an odd multiple of /, (n ) / for some integer n. Thus, cos [(n ) /] 0 E( )E( ) =0 Q.3 5, y 4 then the value of y y is equal to: Sol. (a.) 5 (b.) 3 (c.) 5 y 5 y 4 5 y 5, y 4, y y 4 y 4 y. 3 6 (d.) 3 6 Q.4 for the matrices A and B, AB = A and BA = B, then A is equal to (a.) I (b.) A (c.) B (d.) none of these Sol. AB = A, BA = BA = AA = (AB)A = A (BA) = AB = A Q.5 The value of for which the matri / / is singular is

2 (a.) (b.) (c.) 3 (d.) None of these Sol.. As A = / (0) ( ) ( ) ( ) ( ) = Now, A =0. Q.6 4 A., then A is equal to: (a.) 3 (b.) 3 (c.) 3 (d.) 3 Sol. Let B 3 3 adj B 3 T Also, B = 6 = 7 B 7 3 Also, B = 6 = 7 B 7 We have, AB A B Q.7 cos sin 0 cos 0 sin A( ) = sin cos 0 and A( ) = 0 0, then (A( ). A ( )) sin 0 cos (a.) A( ). A( ) (b.) A(- ). A(- ) (c.) A( ). A( ) (d.) A(- ). A(- ) Sol. A( ) = cos + sin = adj (A( )) = cos sin 0 sin cos T cos sin 0 sin cos A adj A A cos sin 0 sin cos A Now, A cos sin adj A cos 0 sin 0 0 sin 0 cos T cos 0 sin 0 0 A sin 0 cos Finally, (A( ). A( )) - = A - ( ). A - ( ) = A(- ). A(- ) Q.8 A sin cos sin cos sin cos, then ( ) is equal to: ( Cancelled) (a.) 0 (b.) n, n I (c.) n, n I (d.) n, n I Sol.

3 Q.9 Sol. 5 5 Let A= A 5 then = (a.) 5 (b.) (c.) /5 (d.) 5 A 5 A 5 (5 ) 5 5 /5 Q.0 C is a 3 3 matri satisfying the relation C +C=I, then C is given by (a.) C (b.) 3C- (c.) C (d.) I+C Sol. C C I C(C I) I C C I Thus, C (C I) C C I = I - C + C + I = I + C Q. Sol. A, B and C are three square matrices of the same size such that B=CA 3 C, then CA C is equal to (a.) B (b.) B (c.) 3 B (d.) 9 B 3 3 CA C (CAC )(CAC )(CAC ) B Q. a a 3 The Values of a for which the matri A = a a 4 is symmetric 3 4a (a.) - (b.) - (c.) 3 (d.) Sol. A A a a 3 a a 3 a 4a a a 4 a a and 4a a 4 3 a 4 3 4a a 4a 4 6 3a a. Q.3 the system of equations a+y=3, +y=3, 3+4y=7 is consistent, then value of a is given by (a.) (b.) (c.) - (d.) 0 Sol. Solving +y=3, 3+4y=7 =, y= a 3 a=. Q.4 tan tan a b, then tan tan b a (a.) a = b = (b.) a cos, b sin (c.) a sin,b cos (d.) a, b sin 3

4 Q.5 Let M be a 3 3 matri satisfying of the diagonal entries of M is 0 0 M, M, and M then sum (a.) 0 (b.) -3 (c.) 6 (d.) 9 Q.6 3 A, 0 then A -3 is (a.) (b.) (c.) (d.) Sol. We have A - = A -3 8 = Q.7 p q The matri A = is orthogonal if and only if q p (a.) P q (b.) P q (c.) P q (d.) None of these Q.8 i i A andb then A 8 equals i i (a.) 8 B (b.) 3 B (c.) 6 B (d.) 64 B Sol. A = ib A =i B = (-) =-B A 4 = (-B) = 4B = 4(B)=8B A 8 = (8B) = 64B =64(B)= 8B Q.9 The values of k for which the system of equations ky 3z 0, 3 ky z 0, 3y 4z 0 has a non trivial solution is (are) (a.) 0 (b.) 3 0 Sol. From () and (), +z=0 =-z. 4 (c.) -5 (d.) 4 From (3) 3y=5z 5k 5k 7 Substituting this in () we get z z 3z 0. z Q.0 cos sin, R, then the determinant Δ = sin cos cos( ) sin( ) 0 Now, 5 z, y z 3 z 0 k. 0 lies in the interval (a.), (b.), (c.), (d.), Sol. Applying R3 R3 cos R sin R

5 Q. cos sin sin cos 0 0 sin cos sin cos sin( / 4) or. (sin cos )(cos sin ) As sin( / 4), sin( / 4) Given - y + z =, y z 4, y z 4, then the values of such that the given system equations has no solution, is (a.) 3 (b.) (c.) 0 (d.) -3 Sol. Form the first two equation +y=6-z Putting the value in the last equation we get 6+(-+ )z=4=( -)z=-. For the system of equations to have no solution = Q. a c b a +b + c=0, then a root of the equation Δ = c b a 0 b a c is (a.) (b.) - (c.) Sol. Applying C C C C3 a b c c b c b a b c b a = b a a b c a c a c a b c (d.) 0 [ abc0] clearly equals 0 when = 0 Q.3 The values of for which the system of equations +y-3=0, ( ) ( ) 8 0, ( )y ( ) 0 has a non-trivial solutions, (a.) -5/3, (b.) /3,-3 (c.) -/3,-3 (d.) 0 Sol. The given system of equations has a non ( ) trivial solution if 0 3 Applying C C C and C3 C3 3C, we ge (5 ) ( )(3 5) /3 or. Q.4 the system of equations -ky-z=0, k y-z=0,+y-z=0 has a non-zero solution, then the possible values of k are (a.) -, (b.), (c.) 0, (d.) -, Sol. As the given system has a non-zero solution, k k k 0 k k 0 0 using C C C, C C C3 ] 0=(-)[(+k)(-)-(+k)(-k-)] 0=(+k)(-+k+) k=-, 5

6 Q.5 The number of real values of a for which the system of equations +ay-z=0,-y+az=0,a+y+z=0 has a non-trivial solution, is (a.) 3 (b.) (c.) 0 (d.) infinite Q.6 a b 0 Δ = 0 a b 0, then b 0 a (a.) /b is a cube root of unity (b.) a is one of the cube roots of unity (c.) b is one of the cube roots of 8 (d.) a/b is a cube roots of 8 Sol. Show 3 3 a 8b Thus, a 0 8 b 3 a/b is a cube root of 8. Q.7 ab a b a, b, c are non-zero real numbers, the Δ = 6 bc b c equals ca c a (a.) 0 (b.) bc+ca+ab (c.) a b c (d.) abc- Sol. Write R R R and R3 R3 R that = 0 Q.8 Y Y Y Δ = Y Y Y Y Y Y abc =0,then values of y are c a b a b c b c a and use C C C, to show (a.) 0,3 (b.), (c.) -,3 (d.) 0, Sol. Using C C C C C3 y y ( y) y y y y y y ( y) 0 y 0 ( y)(4y ) 0 y y Applying R3 R3 R, R R R 0 y 0 or y. Q.9 A is symmetric matri and B is a skew symmetric matri, then for any n N, which of the following is not correct? (a.) A n is symmetric (b.) A n is symmetric, if n is odd. (c.) B n is skew symmetric, if n is odd (d.) B n is symmetric, if n is even.

7 Sol. A is symmetric, then every integral power of A is symmetric. B is skew symmetric then every odd integral powers of B are skew symmetric and every even integral powers of B are symmetric. Q.30 i 3 i 3 A i i, i i 3 i 3 i i and f() = +, then f(a) is equal to: (a.) 0 0 (b.) 3 i (c.) 5 i (d.) i Sol. 3 i and i 3 Also, 3 and Then, i i 0 A i A i 0 i i f 0 0 f A A I i i Q.3,, are the roots of then equals 3 p q 0, where q 0, and / / / / / / / / / (a.) -p/q (b.) /q (c.) Sol. We have 0. We can write as = 0 [all zero property pq (d.) 0 [using [ C C C C 3 ] 7

8 Q.3 b c a b a c a b c b a b c a c a b c and b c a, then equal c a b (a.) 0 (b.) 3abc (c.) 6abc (d.) (a b c ) Sol. Using C C C3 and C C C3 in Q.33 a b c b a a b c c b a b c a c c b a a b c a b c a c b b c a b c a b a c c a b c a b Using C C C C3,, are real numbers, then the determinant (a.) 0 (b.) - 0. sin cos cos sin cos cos sin cos cos equals (c.) sin sin sin (d.) None of these Sol. Use C C (C3 C ) to show that C consists of all zero. Q.34 R, the determinant cos 0 cos sin cos 0 sin( / 4 equals (a.) 0 (b.) - (c.) (d.) None of these Sol. Using R R R cos 0 cos 0 0 sin cos Using R3 R3 R 0 sin cos 0 (sin cos ) 0 0 Q.35 R, maimum value of sin cos is (a.) / (b.) 3 / (c.) (d.) 3 / 4 Sol. Use R R R, R3 R3 R to obtain sincos sin( ) Q.36 = - 7 and , then 8

9 Sol. Note that (a.) 7 (b.) 343 (c.) 49 (d.) 49 C C C 3 C C C where Cij cofactor of (i, j)th element of 3 C C C Q.37 Suppose A is square matri such that 3 A =I, then 3 3 (A I) (A I) 6A equals (a.) I (b.) I (c.) A (d.) 3 A Q.38 m C m3 C m6 C 3 5 m C m3 C m6 C, then is equal (a.) 3 (b.) 5 (c.) 7 (d.) None of these Sol. We have m m 3 m 6 m(m ) (m 3)(m ) (m 6)(m 5) Applying C3 C3 C, C C C 0 0 m 3 3 m(m ) 3(m 3(m 4) 3 3 (m 4 m ) 3 3 Q.39 a a d a d The determinant a a d a d a 3d a d a d 0. then : (a.) d = 0 (b.) a + d = 0 (c.) d = 0 or Sol Use C 3 C 3 C, C C C. a + d = 0 (d.) none of these Q.40 there are two values of a for which values 5 a is 86, then the sum of these 0 4 a (a.) 4 (b.) 5 (c.) -4 (d.) -5 9

10 Q.4 the system of equations + y 3z =, (p + ) z = 3, (p + ) y + z = is inconsistent, the value of p is: (a.) 3 (b.) Sol. 0 0 p p p 0 p =(p + ) (p + ) (3 p 4) 9 (p + ) (c.) 0 (d.) p p = (p + ) (p + + 9) = (p + ) (p 5) Clear for p = -, = 0, 0, Q.4 Let A A B C B y y and y z C z z then z y z y (a.) yz (b.) 0 (c.) = 0 (d.) 0 Q.43 The determinant cos( y) sin( y) cos y sin cos siny cos sin cos y is (a.) Independent of y only (b.) Independent of only (c.) Independent of and y (d.) dependent on and y Q.44 The value of 3 is equal to: 3 (a.) 0 (b.) 6 (c.) 8 (d.) none of these Sol. Applying R R R, R R R Q.45 The value of C C C m C6 C7 Cm 3 C8 C9 Cm4 is equal to zero, when m is: (a.) 6 (b.) 4 (c.) 5 (d.) none of these 0

11 Sol. C C + C gives C C C m C6 C7 Cm 3 3 C8 C9 Cm4 For m = 5, C C 3 Q.46 n n n a a 0 R, where n N, then value of a is: n5 a6 n5 (a.) n (b.) n (c.) n + (d.) none of these Sol. Taking 5 common from the last row n n n 5 a a 0, R n a n a n a n (as it will make first and third row identical). Q.47 Let p q r s t be an identity in l, where p, q, r, s, t are independent of, then the value of t is: (a.) 4 (b.) 0 (c.) (d.) Sol. Put =0 it will be skew symmetric matri. Q.48 the epression is equal to a 5 + b 4 + c 3 + d + e + f, then the value of e is equal to: (a.) (b.) 3 (c.) 0 (d.) 4 Sol.. Differentiating both sides w.r.to 5a 4 + 4b 3 + 3c + d + e = + + = 3. Q.49 the equations a(y + z) =, b(z + ) = y and c( + y) = z, where a, b, c admit of non-trivial solutions, then ( + a) - +(+b) - +(+c) - is: (a.) (b.) (c.) Sol. a( + y + z) = ( + a), b( + y + z) (d.) 3 = ( + b)y, c( + y + z) = ( + c)z a In case of non-trivial solution, + y + z 0, etc. a y z Adding, a b a b c. a b c

12 Q.50 a 3 + b + c + d= 3 3 4, then a = (a.) (b.) - (c.) 0 (d.) Sol.. Apply C C C, C C C 3. Value of the determinant is independent of. a b c 0. Q.5 Choose any 9 distinct integers. These 9 integers can be arranged to form 9! determinants each of order 3. The sum of these 9! determinants is: (a.) 0 (b.) > 0 (c.) < 0 (d.) 9! Sol. Let the nine distinct digits be a, a,.. a 9. Let one of the 9! determinants, then there eists a determinants among 9! determinants, obtained by interchanging R and R in such that 0. Thus, 9! be there such that sum of each pair is zero. Required sum = 0. pairs of determinants will Q.5 The system of equations + 3y = 8, 7 5y + 3 = 0, 4 6y + = 0 is solvable, if is: (a.) 6 (b.) 8 (c.) -8 (d.) -6 Sol. Equations are solvable when they are consistent. i.e. =0 Q.53 The determinant a k ab ac k ab b k bc ac bc c k is divisible by: (a.) k (b.) k (c.) ( + k) (d.) k- Sol. 0 ' 0 0. k is divisible by k. Q.54 a y y ay y 0, 0 a y ay where a + ay + 0, represents: (a.) a straight line (b.) a circle (c.) a parabola (d.) none of these. Sol. Required curve is y =. Q.55 The number of distinct values of a determinant whose entries are from the set {-, 0, } is:

13 (a.) 3 (b.) 4 (c.) 5 (d.) 6 Sol. Possible values are -, -, 0,, i.e. 0, 0, 0 0 0, 3

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