If A = a i j is a square matrix of even order such that a i j = i 2 − j 2 , then
If the system of equations a x + y + z = 0 , x + b y + z = 0 , x + y + c z = 0 ( a , b , c ≠ 1 ) has a non-zero solution. Then 1 1 − a + 1 1 − b + 1 1 − c =
The system of linear equations x + λ y − z = 0 , λ x − y − z = 0 , x + y − λ z = 0 has a non-trivial solution for
The value of a for which the equations 3 x − y + a z = 1 , 2 x + y + z = 2 , x + 2 y − a z = − 1 fail to have unique solution is
If the system of equations 2 x + 3 k y + ( 3 k + 4 ) z = 0 , x + ( k + 4 ) y + ( 4 k + 2 ) z = 0 , x + 2 ( k + 4 ) y + ( 3 k + 4 ) z = 0 has a non-trivial solution, then k =
If the system of linear equations ax + ky + 2az = 0 ; bx + ky + 3bz = 0 ; cx + ky + 4cz = 0 where k,a,b,c ∈ R has non- zero solution then
If the system of equations x + y + z = 5 , x + 2 y + 3 z = 9 , x + 3 y + α z = β has infinitely many solutions , then β − α equals
If f ( x ) = a + bx + cx 2 and α , β , γ are the roots of the equation x 3 = 1 , then a b c b c a c a b is equal to
In a triangle, A B C if Δ = 1 a b 1 c a 1 b c = 0 then sin 2 A + sin 2 B + sin 2 C is
If ω ≠ 1 is a cube root of unity and Δ = x + ω 2 ω 1 ω ω 2 1 + x 1 x + ω ω 2 = 0 , then value of x is
Let Δ 1 = a x b y c z x 2 y 2 z 2 1 1 1 and Δ 2 = a b c x y z y z z x x y then , ∆ 1 – ∆ 2 equal
If the system of linear equations x + 2 a y + a z = 0 , x + 3 b y + b z = 0 , x + 4 c y + c z = 0 ,has a non-zero solution, then a , b , c
Suppose A and B are two orthogonal matrices of the same size such that det ( A ) + det ( B ) = 0 then
If A is a square matrix of order 3 such that A 2 = 2 A , then | A | 2 is equal to
Suppose a , b are two real numbers and f ( n ) = α n + β n . Let Δ = 3 1 + f ( 1 ) 1 + f ( 2 ) 1 + f ( 1 ) 1 + f ( 2 ) 1 + f ( 3 ) 1 + f ( 2 ) 1 + f ( 3 ) 1 + f ( 4 ) If Δ = k ( α − 1 ) 2 ( β − 1 ) 2 ( α − β ) 2 , then k is equal to
Let A = a b c d , where a , b , c , d ∈ R Then
If x 1 , x 2 , x 3 , … , x 13 are in A.P. then the value of e x 1 e x 4 e x 7 e x 4 e x 7 e x 10 e x 7 e x 10 e x 13 is
If Δ = 1 + y 1 − y 1 − y 1 − y 1 + y 1 − y 1 − y 1 − y 1 + y = 0 , then value of y are
Consider the system of equations a x + b y + c z = 2 , b x + c y + a z = 2 , c x + a y + b z = 2 , where a , b , c are real numbers such that a + b + c = 0 . Then the system
If x = a , y = b , z = c is a solution of the system of linear equations x + 8 y + 7 z = 0 , 9 x + 2 y + 3 z = 0 , x + y + z = 0 such that the point ( a , b , c ) lies on the plane x + 2 y + z = 6 then 2 a + b + c =
If the system of equations 2 x − 3 y + 4 = 0 , 5 x − 2 y − 1 = 0 and 21 x − 8 y + λ = 0 is consistent. Then λ is
The system of equations − 2 x + y + z = a , x − 2 y + z = b , x + y − 2 z = c is consistent if
The number of values of K for which the system of equations ( k + 1 ) x + 8 y = 4 k , k x + ( k + 3 ) y = 3 k − 1 has no solution, is
The system of equations x + y + z = 6 , x + 2 y + 3 z = 10 , x + 2 y + λ z = k is inconsistent if λ = …. , k ≠ ….
For which of the following ordered pairs k , λ ,the system of linear equations x + 2 y + 3 z = k , 3 x + 4 y + 5 z = 4 , 4 x + 4 y + 4 z = λ is inconsistent
The sum of values of ‘k’ for which the linear equations 4 x + k y + 2 z = 0 , k x + 4 y + z = 0 , 2 x + 2 y + z = 0 possess a non-zero solution is
The system of equations x + 4 y − 2 z = 3 , 3 x + y + 5 z = 7 , 2 x + 3 y + z = 5 has
The absolute difference of values of α 10 for which the system of equations x + y + z = 1 , x + 2 y + 4 z = α , x + 4 y + 10 z = α 2 is consistent is,
If x , y , z are different from zero and Δ = a b − y c − z a − x b c − z a − x b − y c = 0 , then the value of the expression a x + b y + c z is
Consider the system of linear equations: x 1 + 2 x 2 + x 3 = 3 , 2 x 1 + 3 x 2 + x 3 = 3 , 3 x 1 + 5 x 2 + 2 x 3 = 1 . The system has
If A and B are two non-singular matrices of order 3 such that AA T = 2 I and A − 1 = A T − A . adj 2 B − 1 , then det (B) equals
Let A and B be two square matrices of order 3 such that det . ( A ) = 3 and det. ( B ) = 2 , then the value of det. adj ⋅ B − 1 A − 1 − 1 is
If A is a square matrix such that A . a d j A = 4 0 0 0 4 0 0 0 4 then 3 | adj A | =
If the system of equations x + y + z = 0 , x + 2 y + 3 z = 1 , λ x + μ y + 4 z = 1 has infinitely many solutions then ( λ , μ ) =
If A = 1 3 2 − 1 then adj A =
The inverse of A = 3 5 7 2 − 3 1 1 1 2 is
If A = 0 1 2 1 2 3 3 x 1 and A − 1 = 1 / 2 − 1 / 2 1 / 2 − 4 3 y 5 / 2 − 3 / 2 1 / 2 , then
If A is singular matrix, then adj A is
The inverse of a symmetric matrix ( if it exists ) is
If the system of equations 2 x − 3 y + 4 = 0 , 5 x − 2 y − 1 = 0 and 21 x − 8 y + λ = 0 is consistent , then λ is
If the system of equations x + y + z = 6 , x + 2 y + λ z = 0 , x + 2 y + 3 z = 10 has no solution then λ =
If the system of linear equations x − 2 y + k z = 1 , 2 x + y + z = 2 , 3 x − y − k z = 3 has a solution x , y , x , z ≠ o , then x , y lies on the straight line whose equation is
The greatest value of c ∈ R for which the system of linear equations x − c y − c z = 0 , c x − y + c z = 0 , c x + c y − z = 0 has a non – trivial solution , is
The number of value of k, for which the system of equation k + 1 x + 8 y = 4 k , k x + k + 3 y = 3 k − 1 has no solution is
If the system of equations 2 x + 3 y − z = 0 , x + k y − 2 z = 3 and 2 x − y + z = 0 has a non-trivial solution x , y , z then x y + y z + z x + k is equal to
If the system of equations x + a y = 0 , a z + y = 0 a n d a x + z = 0 has infinite solution then the value of a is
If α , β , γ are the roots of px 3 + qx 2 + r = 0 , then the value of the determinant αβ βγ γα βγ γα αβ γα αβ βγ is
If x 2 + x x + 1 x − 2 2 x 2 + 3 x − 1 3 x 3 x − 3 x 2 + 2 x + 3 2 x − 1 2 x − 1 = A x + B then A is equal to:
If x = – 9 is a root of x 3 7 2 x 2 7 6 x = 0 then the other two roots are
If a , b , c ∈ R the number of real roots of the equation x c − b − c x a b − a x = 0 is
The inverse of a symmetric matrix (if it exists) is
If A , B , C , P , Q , R ∈ R and Δ = cos ( A + P ) cos ( A + Q ) cos ( A + R ) cos ( B + P ) cos ( B + Q ) cos ( B + R ) cos ( C + P ) cos ( C + Q ) cos ( C + R )
If A = a i j 3 × 3 is a matrix satisfying the equation x 3 − 3 x + 1 = 0 then
If 1 − tan θ tan θ 1 1 tan θ − tan θ 1 − 1 = a − b b a then
A root of the equation Δ = 0 x − a x − b x + a 0 x − c x + b x + c 0 = 0 is
Let x = cos π 3 + i sin π 3 and Δ = 1 x x 2 x 3 1 x 1 x 2 1 then numerical value of ∆ is
Let A and B be two 2×2 matrices. Consider the statements (i) A B = O ⇒ A = O or B = O (ii) A B = I 2 ⇒ A = B − 1 (iii) ( A + B ) 2 = A 2 + 2 A B + B 2 Then
Let Δ r = 2 r − 1 2 3 r − 1 4 5 r − 1 α β γ 2 n − 1 3 n − 1 5 n − 1 , for r = 1 , 2 , … , n . r .Then ∑ r = 1 n Δ r
Let A 2 − A + I = O then inverse of A is
If 2 1 7 4 A − 3 2 5 − 3 = 1 0 0 1 , then matrix A equals
If the system of equations a x + y = 3 , x + 2 y = 3 , 3 x + 4 y = 7 is consistent, then value of a is given by
Let λ and α be real. The set of all values of for which the system of linear equations λ x + ( sin α ) y + ( cos α ) z = 0 , x + ( cos α ) y + ( sin α ) z = 0 , − x + ( sin α ) y − ( cos α ) z = 0 has a non-trival solution is
The system of equations 2 x + 6 y + 11 = 0 , 6 y − 18 z + 1 = 0 , 6 x + 20 y − 6 z + 3 = 0 is
Given that a α 2 + 2 b α + c ≠ 0 and that the system of equations ( a α + b ) x + a y + b z = 0 , ( b α + c ) x + b y + c z = 0 , ( a α + b ) y + ( b α + c ) z = 0 has a non trivial solution, then a , b , c are in
For the equations x + 2 y + 3 z = 1 , 2 x + y + 3 z = 2 , 5 x + 5 y + 9 z = 4
If the system of equations ( k + 1 ) 3 x + ( k + 2 ) 3 y = ( k + 3 ) 3 , ( k + 1 ) x + ( k + 2 ) y = k + 3 , x + y = 1 is consistent. Then the value of k is
If the system of equations x + y + z = 6 , x + 2 y + λ z = 0 , x + 2 y + 3 z = 10 has no solution, Then λ =
If the system of linear equations x − 4 y + 7 z = g , 3 y − 5 z = h , − 2 x + 5 y − 9 z = k is consistent, then
Let λ be a real number for which the system of linear equations x + y + z = 6 , 4 x + λ y − λ z = λ − 2 , 3 x + 2 y − 4 z = − 5 has infinitely many solutions. Then λ is a root of the quadratic equation
The greatest value of c ∈ R for which the system of linear equations x − c y − c z = 0 , c x − y + c z = 0 , c x + c y − z = 0 has a non-trivial solution is
It the system of equations x + y + 2 z = 3 , x + 2 y + 3 z = 4 , x + c y + 2 c z = 5 is inconsistent, then
If the system of linear equations x + 2 a y + a z = 0 , x + 3 b y + b z = 0 , x + 4 c y + c z = 0 , has non-zero solution. Then a , b , c
If x = α , y = β , z = γ is the solution, for the system of equations 2 x − y + 8 z = 13 , 3 x + 4 y + 5 z = 18 , 5 x − 2 y + 7 z = 20 . Then α β + β γ + γ α =
The number of values of k for which the linear equations 4 x + k y + 2 z = 0 , k x + 4 y + z = 0 , 2 x + 2 y + z = 0 possess a non-zero solution is
If the equations b + c x + c + a y + a + b z = 0 , c x + a y + b z = 0 , a x + b y + c z = 0 have non zero solutions, then a relation among a,b,c is
If system of linear equations x + y + z = 6 , x + 2 y + 3 z = 14 , 2 x + 5 y + λ z = μ ( λ , μ ∈ R ) has a unique solution, then
The value of θ for which the system sin 3 θ x − y + z = 0 , cos 2 θ x + 4 y + 3 z = 0 , 2 x + 7 y + 7 z = 0 has a non-trivial solution is (where π 2 < θ < π , ( π = 3.14 ) )
The system of linear equations x − 4 y + 7 z = g , 3 y − 5 z = h , − 2 x + 5 y − 9 z = k is consistent, then
The following system of linear equations 7 x − 2 y + 6 z = 0 , 3 x + 2 y + 4 z = 0 , x − 6 y − 2 z = 0 has
The value of λ for which the system of equations x − 2 y − 2 z = λ x , x + 2 y + z = λ y , − x − y − λ z = 0 has a non-trivial solution is
If the values of λ for which the system of linear equations 2 x 1 − 2 x 2 + x 3 = λ x 1 , 2 x 1 − 3 x 2 + 2 x 3 = λ x 2 , − x 1 + 2 x 2 = λ x 3 has non – trivial solution are λ 1 , λ 2 , λ 3 then λ 1 2 + λ 2 2 + λ 3 2 / 1000 =
If the system of equations x + y + z = 5 , x + 2 y + 3 z = 9 , x + 3 y + α z = β has infinitely many solutions, then β 3 − α 3 =
If the system of linear equations, x + y + z = 6 x + 2 y + 3 z = 10 3 x + 2 y + λ z = μ has more than two solutions, then μ – λ 2 is equal to
The system of linear equations x + y + z = 2 , 2 x + 3 y + 2 z = 5 , 2 x + 3 y + α 2 − 1 z = α + 1
If A = 3 2 4 3 and B = − 1 7 3 5 , then find the sum of the absolute value of the entries of the matrices X and Y satisfying A X = B and Y A = B is
If A is 3 order square matrix such that | A | = 2 then ad j ( a d j ( a d j A ) ) is
Let α , β and γ be the roots of the equation x 3 − x 2 + 3 x + 1 = 0 . Then the value of γ β α + 2 β + 2 γ α γ β + 2 α + 2 γ β α γ + 2 α + 2 β is equal to
If A is a non-zero matrix such that A 2 = A , satisfving ( I − λ A ) − 1 = I − 3 A , where I is unit matrix of same order as that of A , then the value of λ , where | I − λ A | ≠ 0 is
The matrix 1 − 1 2 3 2 1 5 2 3 is
If A = 5 a − b 3 2 and A adj A = A A T , then 5 a + b is equal to
If A is the square matrix order 3 such that | A | = 2 , then | a d j ( adj ( adj A ) ) | is
If A = 1 3 3 1 3 4 1 4 3 , Then A has
If the system of equations x + 2 y + 3 z = λ x , 3 x + y + 2 z = λ y , 2 x + 3 y + z = λ z has non-trivial solution , then λ =……..
The system of linear equations x + y + z = 2 , 2 x + y − z = 3 , 3 x + 2 y + k z = 4 has a unique solution if
If the system of linear equations x + 2 a y + a z = 0 , x + 3 b y + b z = 0 , x + 4 c y + c z = 0 has a non-zero solution, then a,b,c
If the system of linear equations sin 3 θ x − y + z = 0 , cos 2 θ x + 4 y + 3 z = 0 , 2 x + 7 y + 7 z = 0 has a non-trivial solution then the values of θ are
The system of equations sin 3 θ x − y + z = 0 , cos 2 θ x + 4 y + 3 z = 0 , 2 x + 7 y + 7 z = 0 has non-trivial solutions if
If A,B,C are the angles of a triangle , the system of equations sin A x + y + z = cos A , x + sin B y + z = cos B , x + y + sin C z = 1 − cos C has
If the system of equations x = c y + b z , y = a z + c x , z = b x + a y has a nonzero solution then a 2 + b 2 + c 2 + 2 a b c =
By eliminating a , b , c from the homogenous equations x = a b − c , y = b c − a , z = c a − b where a , b , c not all zero
An ordered pair α , β for which the system of linear equations 1 + α x + β y + z = 2 α x + 1 + β y + z = 3 a x + β y + 2 z = 2 Has a unique solution , is
The system of linear equations x + y + z = 2 , 2 x + 3 y + 2 z = 5 2 x + 3 y + a 2 − 1 z = a + 1
If A = 5 4 1 2 and if ( A + 2 I ) − 1 = k 1 A + k 2 I then k 1 , k 2 =
If θ ∈ R , then maximum value of Δ = 1 1 1 1 1 + sin θ 1 1 1 1 + cos θ is
If [ ] denotes the greatest integer less than or equal to the real number under consideration, and − 1 ≤ x < 0 , 0 ≤ y < 1 , 1 ≤ z < 2 , then the value of the determinant [ x ] + 1 [ y ] [ z ] [ x ] [ y ] + 1 [ z ] [ x ] [ y ] [ z ] + 1 is
Suppose A = a i j 3 × 3 , where a i j ∈ R If det ( adj A ) = 25 then | d e t ( A ) | equals:
Suppose a , b , c are distinct real numbers and Δ = a a 2 b + c b b 2 c + a c c 2 a + b = 0 .Then a + b + c equals
The value of x for which the matrix A = 2 / x − 1 2 1 x 2 x 2 1 1 / x 2 is singular is
Let A be 3×3 matrix such that A is orthogonal and idempotent, then
Let A = 2 – 1 3 4 , B = 5 2 7 4 , C = 2 5 3 8 .Let D be a matrix such that CD=AB, then D equals
Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form A = 1 a b ω 1 c ω 2 ω 1 where a , b , c are either ω or ω 2 . Then number of distinct matrices in the set S is
If 1 − 3 4 − 5 x + 2 2 4 1 x − 6 = 0 then x equals
Suppose A, B are two 3×3 matrices such that A –1 exists. Then ( A − B ) A − 1 ( A + B ) is equal to
If a matrix A is both symmetric and skew-symmetric, then
If a 2 b 2 c 2 ( a + λ ) 2 ( b + λ ) 2 ( c + λ ) 2 ( a − λ ) 2 ( b − λ ) 2 ( c − λ ) 2 = k λ a 2 b 2 c 2 a b c 1 1 1 λ ≠ 0 then k is equal to:
Let x = cos π 3 + i sin π 3 and Δ = 1 x x 2 x 2 1 x 1 x 2 1 then numerical value of ∆ is
If Δ 1 = x b b a x b a a x and Δ 2 = x b a x then
Let P ( x ) = 7 6 x − 10 2 x − 10 5 x − 10 3 4 sum of zeros of P ( x ) is
If Δ ( x ) = 1 x x + 1 2 x x ( x − 1 ) x ( x + 1 ) 3 x ( x − 1 ) x ( x − 1 ) ( x − 2 ) x x 2 − 1 then ∆ ( 100 ) equals
Let A = a i j 3 × 3 , where a i j ∈ C the set of complex numbers. If det ( A ) = 2 − 3 i , then det A − 1 equals:
If a , b , c are distinct, and 1 1 1 a b c a 3 b 3 c 3 = ( b − c ) ( c − a ) ( a − b ) ( a + b + c ) then Δ = 1 1 1 ( x − a ) 2 ( x − b ) 2 ( x − c ) 2 ( x − b ) ( x − c ) ( x − c ) ( x − a ) ( x − a ) ( x − b ) vanishes if
Suppose a , b and c are distinct real numbers. Let Δ = a a + c a − b b − c b a + b c + b c − a c = 0 .Then the straight line a ( x − 5 ) + b ( y − 2 ) + c = 0 passes through the fixed point
Suppose a , b , c > 0 and a , b , c c are the p t h , q t h , r t h terms of a G.P. Let Δ = 1 p log a 1 q log b 1 r log c then numerical value of ∆ is
The equation Δ = x − a x − b x − c x − b x − c x − a x − c x − a x − b = 0 is satisfied when
If x , y , z are different from zero and Δ = a b – y c – z a – x b c – z a – x b – y c = 0 then the value of the expression a x + b y + c z is
If a , b , c are the sides of a ∆ A B C opposite angles A , B , C respectively, and Δ = a 2 b sin A c sin A b sin A 1 cos ( B − C ) c sin A cos ( B − C ) 1 , then ∆ equals
If α , β , γ are the roots of x 3 + p x 2 + q = 0 where q ≠ 0 and Δ = 1 / α 1 / β 1 / γ 1 / β 1 / γ 1 / α 1 / γ 1 / α 1 / β then ∆ equal
If Δ 1 = b + c a – b a c + a b – c b a + b c – a c and Δ 2 = a b c b c a c a b then ∆ 1 – ∆ 2 equal
If a, b, c are three complex numbers such that a 2 + b 2 + c 2 = 0 and Δ = b 2 + c 2 a b a c a b c 2 + a 2 b c a c b c a 2 + b 2 = k a 2 b 2 c 2 , then the value of k is
If p + q + r = a + b + c = 0 , then the determinant Δ = p a g b r c q c r a p b r b p c q a equal
Let Δ = 1 − 4 20 1 − 2 5 1 2 x 5 x 2 Solution set of ∆ = 0 is
If Δ = − a 2 b 0 0 − a 2 b 2 b 0 − a = 0 then
The number of 2 × 2 matrices A = a b c d or which a b c d − 1 = 1 / a 1 / b 1 / c 1 / d , ( a , b , c , d ∈ R ) is
Let ω = 1 2 ( − 1 + 3 i ) and Δ = 1 1 1 1 − 1 − ω 2 ω 2 1 ω 2 ω 4 then Δ equals
If a + b + c = 0 , then a root of the equation Δ = a – x c b c b x b a c – x = 0 is
Let Δ = 1 sin θ 1 − sin θ 1 sin θ − 1 − sin θ 1 , 0 ≤ θ ≤ 2 π . The
If Δ = 6 i − 3 i w 4 3 i − w 20 3 i w = x + i y then
Let A be a 3×3 matrix such that det ( A ) = − 2 . Then det − 2 A − 1 is equal to
Let ω ≠ 1 be a cube root of unity and Δ = 1 − ω − ω 2 2 2 2 ω ω − ω 2 − 1 2 ω 2 ω 2 2 ω 2 ω 2 − 1 − ω then ∆ equals
The determinant Δ = b 2 − a b b − c b c − a c a b − a 2 a − b b 2 − a b b c − a c c − a a b − a 2 equals
Suppose a , b , ∈ R and a , b ≠ 1 . If the system of equation a x + y + z = 0 , x + b y + z = 0 , x + y + 2 z = 0 has a non-trivial solution, then
Let A and B be two non-zero 3 × 3 matrices such that AB = O. Then
Let A = a b c d , be a 2×2 matrix where a , b , c , d ∈ { 0 , 1 } . The number of such matrices which have inverse is
The number of matrices A = a b c d (where a , b , c , d ∈ R ) such that A − 1 = − A is:
The inverse of a skew-symmetric matrix (if it exists) is
If A and B are two matrices such that A + B = A B , then
If A = a + i b c + i d − c + i d a − i b , where a 2 + b 2 + c 2 + d 2 =1 then A –1 is equal to
Let a, b and c be three real numbers satisfying a b c 1 9 7 8 2 7 7 3 7 = 0 0 0 (1) If the point P ( a , b , c ) with reference to (1), lies on the plane 2 x + y + z = 1 , then the value of 7 a + b + c is
Consider the system of linear equations : x 1 + 2 x 2 + x 3 = 3 2 x 1 + 3 x 2 + x 3 = 3 3 x 1 + 5 x 2 + 2 x 3 = 1 The system has
If A = 3 2 0 1 , then A – 3 is
Let A t = 1 3 2 2 5 t 4 7 − t − 6 then the value(s) of t for which inverse of A t does not exist.
The number of 3 x 3 non-singular matrices, with four entries as 1 and all other entries as 0 is
If A is a singular matrix, then adj A is
If a b c ≠ 0 and the system of equations x + 7 a y + 2 a z = 0 , x + 6 b y + 2 b z = 0 , x + 5 c y + 2 c z = 0 has a non – trivial solution, Then a , b , c are in
If a system of three linear equations in three unknowns which is in the matrix equation form of A X = D , is inconsistent, then r a n k o f A r a n k o f [ A D ] is
The equations x − y + 2 z = 4 , 3 x + y + 4 z = 6 , x + y + z = 1 have
The equations 2 x + y − 4 z = 0 , x − 2 y + 3 z = 0 , x − y + z = 0 have
The solution of the system of equations whose augmented matrix 1 1 1 2 1 2 3 1 3 1 – 5 4 is
The equations x + y + z = 6 , x + 2 y + 3 z = 10 , x + 2 y + λ z = μ have unique solution if
The system of equations α x + y + z = α − 1 , x + α y + z = α − 1 x + y + α z = α − 1 has no solution if α is
The number of values of K for which the system of equations ( K + 2 ) x + 10 y = K , K x + ( k + 3 ) y = K − 1 has no solution is
T h e s y s t e m o f e q u a t i o n s 3 x – y + 4 z = 3 , x + 2 y – 3 z = – 2 a n d 6 x + 5 y + λ z = – 3 h a s a t l e a s t o n e s o l u t i o n w h e n
If x + y + z = 1 , a x + b y + c z = k , a 2 x + b 2 y + c 2 z = k 2 has unique solution Then x =
Solution of the system of equations 2 x + 3 y + 10 z = 4 , 4 x − 6 y + 5 z = 1 , 6 x + 9 y − 20 z = 2 , ( x , y , z ) =
If ω is cube root of unity and x + y + z = a , x + ω y + ω 2 z = b , x + ω 2 y + ω z = c then which of the following is correct
x , y , z not all zeros and the equations x + y + z = 0 , ( 1 + a ) x + ( 2 + a ) y − 8 z = 0 , x − ( 1 + a ) y + ( 2 + a ) z = 0 have non-trivial solution. Then a =
Let A X = B be a system of non – homogeneous equations and det A = 0 Then the system has
If the trivial solution is the only solution of the system of equations x − k y + z = 0 , k x + 3 y − k z = 0 , 3 x + y − z = 0 . Then the set of all values of k is
If the system of linear equations x + k y + 3 z = 0 , 3 x + k y − 2 z = 0 , 2 x + 4 y − 3 z = 0 has a non – zero solution x , y , z Then x z y 2 is equal to
If the system of equations x = c y + b z , y = a z + c x , z = b x + a y has a nonzero solution then a 2 + b 2 + c 2 + 2 a b c =
If the system of linear equations ( sin 3 θ ) x − y + z = 0 , ( cos 2 θ ) x + 4 y + 3 z = 0 , 2 x + 7 y + 7 z = 0 has a non-trivial solution, then the values of θ are
An ordered pair ( α , β ) for which the system of linear equations ( 1 + α ) x + β y + z = 2 , α x + ( 1 + β ) y + z = 3 , α x + β y + 2 z = 2 has a unique solution is
The system of linear equations x + y + z = 2 , 2 x + y − z = 3 , 3 x + 2 y + k z = 4 has a unique solution if k is
If the system of linear equations x − 2 y + k z = 1 2 x + y + z = 2 , 3 x − y − k z = 3 has a solution ( x , y , z ) , z ≠ 0 , Then ( x , y ) lies on the straight line whose equation is
If the system of linear equations 2 x + 2 y + 3 z = a , 3 x − y + 5 z = b , x − 3 y + 2 z = c where a , b , c non – zero real numbers, has more than one solution, Then
By elimination of a , b , c from the homogeneous equations x = a b − c , y = b c − a , z = c a − b where a , b , c not all zero.
If the system of linear equations x + y + z = 5 , x + 2 y + 2 z = 6 , x + 3 y + λ z = u has infinitely many solutions, then the value of λ + u is
If the system of linear equations x + y + z = a , x − y + b z = 2 , 2 x + 3 y − z = 1 has infinitely many solutions, then b − 5 a =
If a , b , c are all different and the equations a x + a 2 y + ( a 3 + 1 ) z = 0 , b x + b 2 y + ( b 3 + 1 ) z = 0 , c x + c 2 y + ( c 3 + 1 ) z = 0 have a non-zero solution, then
If the system of equations 2 x + 3 y − z = 0 , x + k y − 2 z = 0 and 2 x − y + z = 0 has a non trivial solution. Then x y + y z + z x + k =
If A and B are the two real values of k for which the system of equations x + 2 y + z = 1 , x + 3 y + 4 z = k , x + 5 y + 10 z = k 2 is consistent, then Α + B =
The equations 2 x + y − 4 z = 0 , x − 2 y + 3 z = 0 , x − y + z = 0 have
by eliminating a,b,c from the homogeneous equations x = a b − c , y = b c − a , z = c a − b , where a,b,c not all zero
The system of equations ( sin 3 θ ) x − y + z = 0 , ( cos 2 θ ) x + 4 y + 3 z = 0 , 2 x + 7 y + 7 z = 0 has a non-trivial solution if
The values of k for which the system of equations x + k y − 3 z = 0 , 3 x + k y − 2 z = 0 , 2 x + 3 y − 4 z = 0 has a non-trivial solution is
The set of all values of λ for which the system of linear equations 2 x 1 − 2 x 2 + x 3 = λ x 1 , 2 x 1 − 3 x 2 + 2 x 3 = λ x 2 , − x 1 + 2 x 3 = λ x 3 has a non-trivial solution
The values of a for which the system of equations x + y + z = 1 , x + 2 y + 4 z = α , x + 4 y + 10 z = α 2 is consistent is given by
For the equations x + 2 y + 3 z = 1 , 2 x + y + 3 z = 2 , 5 x + 5 y + 9 z = 4
I f t h e s y s t e m o f e q u a t i o n s x + 2 y + 3 z = λ x , 3 x + y + 2 z = λ y , 2 x + 3 y + z = λ z h a s n o n – t r i v i a l s o l u t i o n t h e n λ =
If the system of equations x + y + z = 6 , x + 2 y + 3 z = 0 , x + 2 y + λ z = 0 has a unique solution then λ =
Let x , y , z be a non-zero solution of the equation x + λ y + 2 z = 0 , 2 x + λ z = 0 a n d 2 λ x − 2 y + 3 z = 0 where λ ∈ R then the value of x + y − z y =
If the system of equations x + y + z = 6 , x + 2 y + λ z = 0 , x + 2 y + 3 z = 10 has no solution, then λ =
x,y,z not all zeros and the equations x = c y + b z , y = a z + c x , z = b x + a y are consistent then selection among a,b,c is
The value of ‘a’ for which the equations 3 x − y + a z = 1 , 2 x + y + z = 2 , x + 2 y − a z = − 1 fail to have unique solution
a ≠ b ≠ c ≠ 1 , a x + y + z = 0 , x + b y + z = 0 , x + y + c z = 0 have non-trivial solutions then a + b + c − a b c =
Suppose a,b,c ∈ R and a b c , α ≠ 0 If the system of equations a + α x + α y + α z = 0 – – – – – ( 1 ) α x + b + α y + α z = 0 – – – – – ( 2 ) α x + α y + α + c z = 0 – – – – – – ( 3 ) has a non-trivial solution, then α 1 a + 1 b + 1 c i s =
The values of λ for which the system of equations λ + 5 x + λ − 4 y + z = 0 , λ − 2 x + λ + 3 y + z = 0 , λ x + λ y + z = 0 has a non-trivial solution is
If the system of equations a x + y = 3 , x + 2 y = 3 , 3 x + 4 y = 7 is consistent , then value of a is given by
The system of equations − 2 x + y + k z = 1 , 1 − 2 y + 3 z = 2 , x + y − 2 z = 3 is consistent if
The value of K so that the system of equations x + k y + 3 z = 0 , 3 x + k y − 2 z = 0 , 2 x + 3 y − 4 z = 0 have non zero solution then
The number of values of k for which the system equations k + 1 x + 8 y = 4 k , k x + k + 3 y = 3 k − 1 has no solution is
Consider the system of linear equations x + 2 y + z = 3 , 2 x + 3 y + z = 3 , 3 x + 5 y + 2 z = 1 , then the system has
If the system of equations a x + y = 3 , x + 2 y = 3 , 3 x + 4 y = 7 is consistent, then the value of ‘ a ’ is given by
If trivial solution is the only solution of the system of linear equations x − k y + z = 0 , k x + 3 y − k z = 0 , 3 x + y − z = 0 then set of all values of ‘ k ’ is
Consider the system of equations, x + a y = 0 , y + a z = 0 , z + a x = 0 .Then the set of all values of a for which the system has a unique solution is
For 1 ≤ i , j ≤ 3 ,Let a i j = ∫ − π / 2 π / 2 cos ( i x ) cos ( j x ) d x and let A = [ a i j ] 3 × 3 then
The system of equations x + 4 y − 2 z = 3 , 3 x + y + 5 z = 7 and 2 x + 3 y + z = 5 has
If the system of linear equations x 1 + 2 x 2 + 3 x 3 = 6 , x 1 + 3 x 2 + 5 x 3 = 9 , 2 x 1 + 5 x 2 + a x 3 = b is consistent and has infinite number of solutions, then
If c < 1 and the system of equations x + y − 1 = 0 , 2 x − y − c = 0 and − b x + 3 b y − c = 0 is consistent, then the possible real values of b are
If the system of equations x = k y + z , y = k x − z and z = x + y has a non-zero solution, then the possible values of ‘ k ’ are
Let the homogeneous system of linear equations p x + y + z = 0 , x + q y + z = 0 , x + y + r z = 0 where p , q , r ≠ 1 have a non-trivial solution then 1 1 − p + 1 1 − q + 1 1 − r =
The system of linear equations x + λ y − z = 0 , λ x − y − z = 0 , x + y − λ z = 0 has a non-trivial solution for
If the system of equations x + k y + 3 z = 0 , k x + 2 y + 2 z = 0 , 2 x + 3 y + 4 z = 0 admits of non-trivial solution, then sum of values of ‘ k ‘
If the system of equations x + 2 y − 3 z = 1 , ( p + 2 ) z = 3 , ( 2 p + 1 ) y + z = 2 is inconsistent, then the value of 2020 500 p =
If the following system of equations possess a non-trivial solution over the set of rationals x + y − 2 z = 0 , 2 x − 3 y + z = 0 and x − 5 y + 4 z = k then k + 1 1000 =
If the system of linear equations 4 x + k y + 2 z = 0 , k x + 4 y + z = 0 , 2 x + 2 y + z = 0 possess a non-zero solution, so that k has two values k 1 , k 2 then k 1 3 + k 2 3 3 3 =
If the system of linear equations x + k y + 3 z = 0 , 3 x + k y − 2 z = 0 , 2 x + 4 y − 3 z = 0 has a non – zero solution x , y , z then x z y 2 =
If the system of equations 2 x + 3 y − z = 0 , x + k y − 2 z = 0 and 2 x − y + z = 0 has a non – trivial solution x , y , z then x y + y z + z x + k =
The number of values of θ ∈ 0 , π for which the system of equations x + 3 y + 7 z = 0 , − x + 4 y + 7 z = 0 , sin 3 θ x + cos 2 θ y + 2 z = 0 has a non-trivial solution is
The system of linear equations 4 x + λ y + 6 z = 10 , 2 λ x + 3 y + 5 z = 8 , λ x + 2 y + 2 z = 5 for λ = 2 , has
If λ be a real number of which the system of linear equations x + y + z = 6 , 4 x + λ y − λ z = λ − 2 , 3 x + 2 y − 4 z = − 5 has infinitly many solutions. Then λ is a root of the equation
The value of k, for which the system of equations k + 1 x + 8 y = 4 k , k x + k + 3 y = 3 k − 1 ,has no solution is
If the system of linear equations x − 2 y + k z = 1 , 2 x + y + z = 2 , 3 x − y − k z = 3 has a solution x , y , z , z ≠ 0 then x , y lies on the straight line
An ordered pair (a,b) for which the system of linear equations 1 + a x + b y + z = 2 , a x + 1 + b y + z = 3 , a x + b y + 2 z = 2 has a unique solution is
If for some α and β in R , the system of equations x + 4 y − 2 z = 1 , x + 7 y − 5 z = β , x + 5 y + α z = 5 has many solutions , then 2 α + β =
If the system of linear equations 2 x + 2 y + 3 z = a , 3 x − y + 5 z = b , x − 3 y + 2 z = c where a , b , c ≠ 0 , a , b , c ∈ R has more than one solution , then
The system of linear equations a x + y = 3 , x + 2 y = 3 , 3 x + 4 y = 7 are consistent , then the value of ‘a’ is
The greatest value of ‘c’, c ∈ R for which the system of linear equations x − c y − c z = 0 , c x − y + c z = 0 , c x + c y − z = 0 has a non-trivial solution is
If the system of linear equations x + y + z = 6 , x + 2 y + 3 z = 10 , 3 x + 2 y + λ z = μ has more than two solutions, then λ 3 + μ 3 =
The system of linear equations x + λ y − z = 0 , λ x − y − z = 0 ; x + y − λ z = 0 has a non-trivial solution , then sum of cubes of λ is
Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of A consisting of all determinants with value –1. Then:
Find the value of θ satisfying 1 1 sin 3 θ – 4 3 cos 2 θ 7 – 7 – 2 = 0
The maximum value of Δ = 1 1 1 1 1 + sin θ 1 1 + cos θ 1 1 is ( where , θ is real number)
If A is a matrix of order 3×3,then the number of minors in determinant of A are ……….
The value of a − b b + c a b − a c + a b c − a a + b c is
0 xyz x-z y-x 0 y-z z-x z – y 0 is equal to
If f ( x ) = ( 1 + x ) 17 ( 1 + x ) 19 ( 1 + x ) 23 ( 1 + x ) 23 ( 1 + x ) 29 ( 1 + x ) 34 ( 1 + x ) 41 ( 1 + x ) 43 ( 1 + x ) 47 = A + B x + C x 2 + … … . then A is equal to ……
If the determinant x + a p + u l + f y + b q + v m + g z + c r + w n + h splits into exactly k determinants of order 3, each element of which contains only one term, then the value of k is
If Δ = a p x b q y c r z = 16 then Δ 1 = p + x a + x a + p q + y b + y b + q r + z c + z c + r = ?
If, x , y a n d z , and are all different from zero and 1 + x 1 1 1 1 + y 1 1 1 1 + z = 0 , then the value of x − 1 + y − 1 + z − 1 is
The determinant b 2 − a b b – c b c – a c a b − a 2 a – b b 2 − a b b c − a c c – a a b – a 2 equal to
The sum of the real roots of the equation x − 6 − 1 2 − 3 x x − 3 − 3 2 x x + 2 = 0 is equal to
A value of θ ∈ 0 , π 3 for which 1 + cos 2 θ sin 2 θ 4 cos 6 θ cos 2 θ 1 + sin 2 θ 4 cos 6 θ cos 2 θ sin 2 θ 1 + 4 cos 6 θ = 0 is
if Δ 1 = x sin θ cos θ − sin θ − x 1 cos θ 1 x and Δ 2 = x sin 2 θ cos 2 θ − sin 2 θ − x 1 cos 2 θ 1 x x ≠ 0 , then for all θ ∈ 0 , π 2
if α and β are the rots of x 2 + x + 1 = 0 then for y ≠ 0 in R, y + 1 α β α y + Β 1 β 1 y + α =
If x − 4 2 x 2 x 2 x x − 4 2 x 2 x 2 x x − 4 = A + B x x − A 2 then ordered pair A , B
Let A = 2 b 1 b b 2 + 1 b 1 b 2 where b > 0. Then the minimum value of det A b is
let A and B two invertible matrices of order 3 X 3 . If det A B A T = 8 a n d det A B – 1 = 8 , then det B A − 1 B T is
if A = 2 − 3 − 4 1 Then adj 3 A 2 + 12 A is
if A = 1 0 0 0 1 1 0 − 2 4 a n d 6 A − 1 = A 2 + c A + d I , then c , d =
if P = 1 λ 3 1 3 3 2 4 4 is the adj of a 3X3 matrix A and A = 4 ,then λ =
which of the following is not the square of a 3 X 3 Matrix with real entries?
If λ , β ≠ 0 , and f n = λ n + β n and 3 1 + f 1 1 + f 2 1 + f 1 1 + f 2 1 + f 3 1 + f 2 1 + f 3 1 + f 4 = K 1 − α 2 1 − β 2 α − β 2 Then K=
if A = e t e − t cos t e − t sin t e t − e − t cos t − e − t sin t − e − t sin t + e − t cos t e t 2 e − t sin t − 2 e − t cos t Then A is
Let P = a i j be a 3 X 3 matrix and let Q = b i j where b i j = 2 i + j a i j for 1 ≤ i , j ≤ 3. If the determinant of P is 2, then the determinant of Q is
If M is a matrix of order 3 X 3 such that M M T = I and det (M) = 1 then det M − I =
if A = 5 a − b 3 2 and Aadj A = A A T Then 5 a + b =
if A = cos θ − sin θ sin θ cos θ then the matrix A − 50 when θ = π 12 is equal to
for positive numbers x , y , z the value of the determinant 1 log x y log x z log y x 1 log y z log z x log z y 1 =
let the numbers 2, b, c are in A.P and A = 1 1 1 2 b c 4 b 2 c 2 . If det A ∈ 2 , 16 then the maximum value of c is
if A = 1 sin θ 1 − sin θ 1 sin θ − 1 − sin θ 1 then for all θ ∈ 3 π 4 , 5 π 4 minimum value of det A is
cofactor of element ‘b’ in the matrix a b c 2 4 7 − 1 0 3 is
If the matrix 1 − 1 x 1 x 1 x − 1 1 has no inverse then the number of real values of x is
If the system of equations ( k + 1 ) x + ( k + 2 ) 3 y = ( k + 3 ) 3 , ( k + 1 ) x + ( k + 2 ) y = k + 3 , x + y = 1 is consistent, then k=
If the system of linear equations x + 2 a y + a z = 0 , x + 3 b y + b z = 0 , x + 4 c y + c z = 0 has a non-zero solution then a,b,c
Suppose a , b ∈ R and a , b ≠ 1 . If the system of equations a x + y + z = 0 , x + b y + z = 0 , x + y + 2 z = 0 has a non-trivial solution then
The system of equations α x + y + z = α − 1 , x + α y + z = α − 1 , x + y + α z = α − 1 has no solution, if α =
If the system of equations λ x 1 + x 2 + x 3 = 1 , x 1 + λ x 2 + x 3 = 1 , x 1 + x 2 + λ x 3 = 1 is inconsistent, then λ =
The number of real values of ‘a’ for which the system & equations x + a y − z = 0 , 2 x − y + a z = 0 , a x + y + 2 z = 0 has a non-trivial solution is
Number of real values of λ for which the system of equations λ + 3 x + λ + 2 y + z = 0 , 3 x + λ + 3 y + z = 0 , 2 x + 3 y + z = 0 possess a non-trivial solution is
The system of linear equations x + λ y − z = 0 , λ x − y − z = 0 , x + y − λ z = 0 has a non-trivial solution for
If the system of equations x + 2 y − 3 z = 1 , p + 2 z = 3 , 2 p + 1 y + z = 2 is consistent, then the value of p=
If a + b + c ≠ 0 the system of equations b + c y + z − a x = b − c , c + a z + x − b y = c − a , a + b x + y − c z = a − b has
The system of equations − 2 x + y + z = a , x − 2 y + z = b , x + y − 2 z = c is inconsistent, if
The system of equations 2 x + 6 y + 11 = 0 , 6 y − 18 z + 1 = 0 , 6 x + 20 y − 6 z + 3 = 0 ,
Given that a α 2 + 2 b α + c ≠ 0 and that the system of equations a α + b x + a y + b z = 0 , b α + c x + b y + c z = 0 , a α + b y + b α + c z = 0 has non- trivial solution then a,b,c lies in
If A,B,C are the angles of a triangle, the system of equations, sin A x + y + z = cos A , x + sin B y + z = cos B , x + y + sin C z = 1 − cos C has
The system of equations λ x + y + z = 0 , − x + λ y + z = 0 , − x − y + λ z = 0 will have a non- trivial solution for real values of λ are
Given 2 x − y + 2 z = 2 , x − 2 y + z = − 4 , x + y + λ z = 4 , then the value of λ such that the given system of equations has no solution is
If the system of equations x − k y − z = 0. k x − y − z = 0 , x + y − z = 0 has a non-zero solution then the possible value of ‘k’ are
If p ≠ a , q ≠ b , r ≠ c and the system of equations p x + a y + a z = 0 , b x + q y + b z = 0 , c x + c y + r z = 0 has a non-trivial solution, then the value of p p − a + q q − b + r r − c =
The system of linear equations x − y + z = 1 , x + y − z = 3 , x − 4 y + 4 z = α has
If the system of equations x + y + z = 0 , a x + b y + z = 0 , b x + y + z = 0 has a non-trivial solution then
The system of linear equations x + y + z = 2 , 2 x + y − z = 3 , 3 x + 2 y + k z = 4 has a unique solution, If
Consider the system of equations a x + b y + c z = 2 , b x + c y + a z = 2 , c x + a y + b z = 2 where a,b,c are real numbers such that a + b + c = 0 then the system
The number of real values of ‘t’ such that the system of homogeneous equations t x + t + 1 y + t − 1 z = 0 , t + 1 x + t y + t + 2 z = 0 , t − 1 x + t + 2 y + t z = 0 has non-trivial solution is
If the system of equations 3 x − 2 y + z = 0 , λ x − 14 y + 15 z = 0 , x + 2 y + 3 z = 0 has a non-trivial solution then λ =
The number of values of K for which the system of equations k + 1 x + 8 y = 4 k , k x + k + 3 y = 3 k − 1 has no solution is
If 2 x − 3 y + 4 z = 0 , 5 x − 2 y − z = 0 , 21 x − 8 y + a z = 0 has infinitely many solutions, then a=
The system of linear equations x + y + z = 2 , 2 x + y − z = 3 , 3 x + 2 y + k z = 4 has a unique solution if
The values of ‘ α ’ for which the system of equations x + y + z = 1 , x + 2 y + 4 z = α , x + 4 y + 10 z = α 2 is consistent are given by
If a , b , c are non-zero, then the number of solutions of 2 x 2 a 2 − y 2 b 2 − z 2 c 2 = 0 ; − x 2 a 2 + 2 y 2 b 2 − z 2 c 2 = 0 ; − x 2 a 2 − y 2 b 2 + 2 z 2 c 2 = 0 is
The number of value of ‘ k ‘for which the linear equations 4 x + k y + 2 z = 0 , k x + 4 y + z = 0 , 2 x + 2 y + z = 0 possess a non-zero solution is
The system of homogeneous equations t x + ( t + 1 ) y + ( t − 1 ) z = 0 , t + 1 x + t y + ( t + 2 ) z = 0 , t − 1 x + t + 2 y + t z = 0 has a non-trivial solution for
The number of values of ‘ k ’ for which the system of equations k + 1 x + 8 y = 4 k , k x + ( k + 3 ) y = 3 k − 1 has no solution, is
If the system of linear equations 2 x + 2 a y + a z = 0 2 x + 3 b y + b z = 0 2 x + 4 c y + c z = 0 Where a , b , c ∈ R are non-zero and distinct; has non-zero solution, then,
Let A = a i j a n d B = b i j be two 3 X 3 real matrices such that b i j = 3 i + j − 2 a j i , where i , j = 1 , 2 , 3. If the determinant of B is 81 then the determinant of A is
For which of the following ordered pairs ( μ , δ ) , the system of linear equations x + 2 y + 3 z = 1 , 3 x + 4 y + 5 z = μ a n d 4 x + 4 y + 4 z = δ i s i n c o n s i s t e n t ?
If the matrices A = 1 1 2 1 3 4 1 – 1 3 , B = a d j A and C = 3 A , then | a d j B | | C | is equal to:
If for some α and β in R, the intersection of the following three planes x+4y-2z=1 x+7y-5z= β x + 5 y + α z = 5 is a line in R 3 , then α + β is equal to:
Let λ be a real number for which the system of linear equations x+y+z=6, 4 x + λ y – λ z = λ – 2 and 3 x + 2 y – 4 z = – 5 has infinitely many solutions. Then λ is a root of the equation
Let P = – 30 20 56 90 140 112 120 60 14 and A = 2 7 ω 2 – 1 – ω 1 0 – ω – ω + 1 where ω = – 1 + i 3 2 , and I 3 be the identity matrix of order 3 . If the determinant of the matrix P – 1 A P – I 3 2 is α ω 2 , then the value of α is equal to
Let M be a square matrix of order 3 such that M M T = I and M 2 = I . Also M – 1 + adj ( M ) = O if P is another matrix such that P + 2 M = O , then the value of det P P T P – 1
If p , q , r are negative and distinct, then the determinant Δ = p q r q r p r p q is
Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations A 2 B 2 − B 2 A 2 X = 0 , where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has
If a 1 , a 2 , a 3 , 5 , 4 , a 6 , a 7 , a 8 , a 9 a r e i n H . P . a n d t h e v a l u e o f t h e d e t e r m i n a n t a 1 a 2 a 3 5 4 a 6 a 7 a 8 a 9 i s D , t h e n t h e v a l u e o f 21 D i s
The set of equations λ x − y + ( cos θ ) z = 0 , 3 x + y + 2 z = 0 , ( cos θ ) x + y + 2 z = 0 ; 0 ≤ θ < 2 π has non-trivial solution(s)
If A = 0 1 2 1 2 3 3 a 1 and A − 1 = 1 2 − 1 2 1 2 − 4 3 c 5 2 − 3 2 1 2 , then the values of a and c are respectively is equal to
If the system of linear equations x + 2 a y + a z = 0 , x + 3 b y + b z = 0 and x + 4 c y + c z = 0 has a non zero solutions; then a , b , c are in … …
If a b 1 b c 1 c a 1 = 2020 and if c − a c − b a b a − b a − c b c b − c b − a c a − c − a c − b c 2 a − b a − c a 2 b − c b − a b 2 = P , then the number of positive divisors of P is
If A = 3 − 3 4 2 − 3 4 0 − 1 1 , then the value of trace of the matrix adj adj A is
Let P , Q and R be invertible matrices of order three such that A = P Q − 1 , B = Q R − 1 and C = R P − 1 Then the value of det ( A B C + B C A + C A B ) is
If the system of equation x + 2 y − 3 z = 1 , ( P + 2 ) z = 3 , ( 2 P + 1 ) y + z = 2 is inconsistent, then the value of P is
If A = 1 + cos 2 θ sin 2 θ 4 cos 6 θ cos 2 θ 1 + sin 2 θ 4 cos 6 θ cos 2 θ sin 2 θ 1 + 4 cos 6 θ is a singular matrix and 3 θ ∈ ( 0 , π ) then θ =
Let Δ = det a b c b c a c a b . Then which of the following statements is false for a , b , c ∈ R ?
The set of homogeneous equations t x + ( t + 1 ) y + ( t − 1 ) z = 0 ( t + 1 ) x + t y + ( t + 2 ) z = 0 ( t − 1 ) x + ( t + 2 ) y + t z = 0 has non – trivial solutions for :
If the matrix 1 − 1 2 3 2 1 − 1 α 3 is singular then α =
A = 1 − 1 3 2 4 − 2 − 3 1 2 then adj A =
If the matrix α − 2 4 1 − 3 2 4 − 2 1 is non singular then α ∈
Which of the following matrix is non singular
If A is a 4X4 matrix and det A=-2 then det ( Adj A)=
If A = − 1 − 2 − 2 2 1 − 2 2 − 2 1 , Adj A = x A T , then x =
If n t h -order square matrix A is a orthogonal , then, | a d j a d j A | is
If k ∈ R , then det a d j k I n is equal to
If A = 2 − 3 − 4 1 , then adj 3 A 2 + 12 A is equal to
If A = a i j 4 × 4 , such that a i j = 2 , wheni = j 0 , wheni ≠ j , then det ( adj ( adj A ) ) 7 is ( where {.} represents fractional part function}
The inverse of the matrix A = Sec θ − Tan θ − Tan θ Sec θ is
If A = − 6 5 − 7 6 and A B = I then B =
If x y 3 2 0 = 1 8 2 0 , Then x y 2 0 − 1 =
If A = 2 2 − 3 2 , B = 0 − 1 1 0 , then B − 1 A − 1 − 1
A is an involutary matrix given by A = 0 1 − 1 4 − 3 4 3 − 3 4 , then the inverse of A / 2 will
If 2 1 3 2 A − 3 2 5 − 3 = 1 0 0 1 , then the matrix A =
If A = 2 2 1 − 1 0 2 4 1 0 then 11 A − 1 =
If F ( x ) = cos x − sin x 0 sin x cos x 0 0 0 1 and G ( x ) cos x 0 sin x 0 1 0 − sin x 0 cos x then [ F ( x ) G ( x ) ] − 1 =
If P is an orthogonal matrix and Q = P A P T and x = P T Q 1000 P , then x − 1 is, where A is involutary matrix
If A = 1 − 2 3 0 − 1 4 − 2 2 1 , then A T − 1 =
If A = 0 1 − 1 2 1 3 3 2 1 then A ( adj A ) A − 1 A =
If A = cos α − sin α 0 sin α cos α 0 0 0 1 , then ( Adj A ) − 1 =
If S = 0 1 1 1 0 1 1 1 , A = 1 2 b + c c − a b − a c − b c + a a − b b − c a − c a + b , then S A S − 1 =
If A and B are two square matrices such that B = − A − 1 B A t h e n A + B 2 =
If the product of the matrix B = 2 6 4 1 0 1 − 1 1 − 4 with a matrix A has inverse C = − 1 0 1 1 1 3 2 0 2 , then A − 1 =
A square nonsingular matrix satisfies A 2 − A + 2 I = 0 , t h e n A − 1 =
A is square matrix satisfying the equation A 2 − 4 A − 5 I = O . Then A − 1 =
If A = 2 2 1 1 3 1 1 2 2 then A − 1 + ( A − 5 I ) ( A − I ) 2 =
The inverse of a skew symmetric matrix ( if it exists) is
The inverse of a skew symmetric matrix of odd order is
If A is an orthogonal matrix , then |A| is
In which of the following type of matrix inverse does not exist always
If A is an orthogonal matrix, then A − 1 equals
If A = a b c x y z p q r , B = q − b y − p a − x r − c z and if A is invertible. Then which of the following is true?
For two unimodular complex numbers z 1 a n d z 2 , z 1 ¯ − z 2 z 2 ¯ z 1 − 1 z 1 z 2 − z 2 ¯ z 1 ¯ − 1 is equal to
If A 3 = 0 , then I + A + A 2 equals
( − A ) − 1 is always equal to where A is n th -order square matrix)
If 1 / 25 0 x 1 / 25 = 5 0 − a 5 − 2 , then the value of x is
The inverse of a diagonal matrix is
If P is non-singular matrix, then value of a d j P − 1 in terms of P is
Let a and b be two real numbers such that a > 1 , b > 1. I f A = a 0 0 b then lim n ∞ A − n is
If B = 5 2 α 1 0 2 1 α 3 − 1 is the inverse of a 3 × 3 matrix A, then the sum of all values of α for which det(A)+1=0, is
If A = e t e t cos t e − t sin t e t − e t cos t − e − t sin t − e − t sin t + e − t cos t e t 2 e − t sin t − 2 e − t cos t then A is
Let A and B be two invertible matrices of order 3 × 3 . If det A B A T = 8 and det A B − 1 = 8 , then det B A − 1 B T is equal to
If A = cos θ − sin θ sin θ cos θ , then the matrix A − 50 when θ = π 12 , is equal to
If A is a 3 × 3 non-singular matrix such that A A T = A T A and B = A − 1 A T , then B B T =
The solution of the system of equations whose augmented matrix 1 1 1 2 1 2 3 1 3 1 − 5 4
The equations 2 x + y − 4 z = 0 , x − 2 y + 3 z = 0 , x − y + z = 0 have
For the equations x + 2 y + 3 z = 1 , 2 x + y + 3 z = 2 , 5 x + 5 y + 9 z = 4
The number of solutions of the system of equations 3 x − 1 y + z = 5 , 6 x − 4 y + 2 z = 10 and 9 x − 6 y + 3 z = 15 is
If the system of equations 3 x − 2 y + z = 0 , λ x − 14 y + 15 z = 0 , x + 2 y + 3 z = 0 has non-trivial solution , then λ =
The values of λ for which the system of equations x + y − 3 = 0 , 1 + λ x + 2 + λ y − 8 , x − 1 + λ y + 2 + λ = 0 is consistent are
The system of equations 3 x − y + 4 z = 3 , x + 2 y − 3 z = − 2 , 6 x + 5 y + λ z = − 3 has atleast one solution when
The equations x + y + z = 6 , x + 2 y + 3 z = 10 , x + 2 y + λ z = μ have unique solution if
The system of equations x + y + z = 6 , x + 2 y + 3 z = 10 , x + 2 y + λ z = μ is inconsistent if
If the system of equations a x + y + z = 0 , x + b y + z = 0 , x + y + c z = 0 , a , b , c ≠ 1 has a non trivial solution ( non-zero solution) , then 1 1 − a + 1 1 − b + 1 1 − c =
If a + b + c ≠ 0 , then system of equations b + c y + z − a x = b − c , c + a z + x − b y = c − a , a + b x + y − c z = a − b has
If a,b,c are all different and the equations a x + a 2 y + a 3 + 1 z = 0 , b x + b 2 y + b 3 + 1 z = 0 , c x + c 2 y + c 3 + 1 z = 0 have a nonzero solution, then
The system of equations − 2 x + y + z = a ; x − 2 y + z = b ; x + y − 2 z = c is consistent if
If the system of linear equations x − 4 y + 7 z = g , 3 y − 5 z = h , − 2 x + 5 y − 9 z = k is consistent, then
If x denotes the greatest integer ≤ x , then the system of linear equations sin θ x + − cos θ y = 0 , cot θ x + y = 0
Let λ be a real number for which the system of linear equations x + y + z = 6 , 4 x + λ y − λ z = λ − 2 and 3 x + 2 y − 4 z = − 5 has infinitely many solutions . Then is a root of the quadratic equation
If the system of linear equations x + k y + 3 z = 0 , 3 x + k y − 2 z = 0 2 x + 4 y − 3 z = 0 has a non-zero solution x , y , z , then x z y 2 is equal to
The set of all values of λ for which the system of linear equations 2 x 1 − 2 x 2 + x 3 = λ x 1 , 2 x 1 − 3 x 2 + 2 x 3 = λ x 2 and − x 1 + 2 x 2 = λ x 3 has a non-trivial solution
A = 3 − 4 4 1 − 2 4 1 − 1 3 t h e n A 3 − 4 A 2 + A + 8 I =
If A = 2 − 2 2 3 and if A 2 − 5 A + k I = 0 then k =
If A = 2 2 0 2 1 1 − 7 2 − 3 and if A 3 − 13 A + k I = 0 , then k = .
If p + q + r = 0 = a + b + c , then the value of the determinant pa qb rc qc ra pb rb pc qa is
If a = cos θ + isin θ , b = cos 2 θ − isin 2 θ , c = cos 3 θ + isin 3 θ and if a b c b c a c a b = 0 , then
If A = 1 2 – 1 – 1 1 2 2 – 1 1 , then det. [adj (adj A)] is
If A is a singular matrix, then adj A is
If B is a non-singular matrix and A is a square matrix, then det (B –1 AB) is equal to
Let A be an invertible matrix, which of the following is not true?
If A = 1 0 0 0 1 1 0 – 2 4 . I = 1 0 0 0 1 0 0 0 1 A – 1 = 1 6 [ A 2 + c A + d I ] where c, d ∈ R, the pair of values (c, d ) are
The value of a for which the system of equations ax + y + z = 0, x + ay + z = 0, x + y + z = 0, possess non-zero solutions are given by,
If f ( x ) = 1 x ( x + 1 ) 2 x x ( x − 1 ) ( x + 1 ) x 3 x ( x − 1 ) x ( x − 1 ) ( x − 2 ) x ( x − 1 ) ( x + 1 ) then f ( 50 ) + f ( 51 ) + … … + f ( 99 ) is equal to
If ω is a complex cube root of unity, then a root of the equation x + 1 ω ω 2 ω x + ω 2 1 ω 2 1 x + ω = 0 is
Let Δ ( x , y ) = 1 x y 1 x + y y 1 x x + y .Then Δ ( − 3 , 2 ) equal to
Let Δ = 0 b − a c − a a − b 0 c − b a − c b − c 0 , then ∆ equals
Let P ( x ) = x − 3 + 4 i 3 − 4 i x − 7 i 5 + 6 i − x 7 − 2 i − 7 − 2 i .The number of values of x for which P ( x ) = 0 is
Let Δ ( θ ) = 1 sin θ 1 − sin θ 1 sin θ − 1 − sin θ 1 , 0 ≤ θ ≤ 2 π Solution of ∆ ( θ ) = 3 is
Suppose P ( x ) = x − 51 − 71 51 x − 73 71 73 x .Product of zeros of P ( x ) is
If Δ ( x ) = 1 1 1 e x + e − x 2 π x + π − x 2 2 e x − e − x 2 π x − π − x 2 − 2 then ∆ ( x ) equal to
If α , β , γ are three real numbers such that α + β + γ = 0 , then Δ = 1 cos γ cos β cos γ 1 cos α cos β cos α 1 equals
If Δ ( x ) = 1 cos x 1 – cos x 1 + sin x cos x 1 + sin x – cos x sin x sin x 1 then ∫ 0 π / 2 Δ ( x ) d x equal
Suppose a , b , c and x are real numbers. Let Δ = 1 + a 1 + a x 1 + a x 2 1 + b 1 + b x 1 + b x 2 1 + r 1 + r x 1 + r x 2 .Then ∆ is independent of
The determinant Δ = a b a α + b b c b α + c a α + b b α + c 0 equals zero, if
Suppose a , b , c > 1 and f ( x ) = a − x a x x b − 3 x b 3 x 3 x 3 c − 5 x c 5 x 5 x 5 , x ∈ R then f is
If x is a positive integer, and ∆ ( x ) = x ! ( x + 1 ) ! ( x + 2 ) ! ( x + 1 ) ! ( x + 2 ) ! ( x + 3 ) ! ( x + 2 ) ! ( x + 3 ) ! ( x + 4 ) ! , then ∆ ( x ) is equal to
Suppose a , b , c are sides of a scalene triangle. Let Δ = a b c b c a c a b .Then
Suppose A , B , C are angles of a triangle, and let Δ = e 2 i A e − i C e − i B e − i C e 2 i B e − i A e − i B e − i A e 2 i C then value of ∆ is
Let Δ = 1 a a 2 − b c 1 b b 2 − c a 1 c c 2 − a b , then ∆ is equal to
The system of equations λ x + y + z = 0 – x + λ y + z = 0 – x – y + λ z = 0 will have a non-trivial solution if real values of λ are
The determinant Δ = 1 1 + i i 1 + i i 1 i 1 1 + i equals
The values of k for which the system of equations x + k y – 3 z = 0 , 3 x + k y – 2 z = 0 , 2 x + 3 y – 4 z = 0 has a non-trivial solution is (are)
Let A = a b c d , a , b , c , d ∈ R If A 5 = A 3 + I , then A is
The system of homogeneous equations , ( a – 1 ) x + ( a + 2 ) y + a z = 0 , ( a + 1 ) x + a y + ( a + 2 ) z = 0 , a x + ( a + 1 ) y + ( a – 1 ) z = 0 has a non-trivial solution if a equals
Let x , y , z be positive and x , y , z ≠ 1 Let Δ = 1 log x y log x z log y x 1 log y z log z x log z y 1 then numerical value of ∆ is
If ω ≠ 1 is a complex cube root of unity, and x + i y = 1 i − ω − i 1 ω 2 ω − ω 2 1 then
If a 2 + b 2 + c 2 = – 2 and f ( x ) = 1 + a 2 x 1 + b 2 x 1 + c 2 x 1 + a 2 x 1 + b 2 x 1 + c 2 x 1 + a 2 x 1 + b 2 x 1 + c 2 x then f ( x ) is a polynomial of degree
The determinant Δ = 13 + 3 2 5 5 15 + 26 5 10 3 + 65 15 5 equals
If the system of equations , x + y + z = 0 , a x + b y + z = 0 , b x + y + z = 0 ,has a non-trivial solution, then
If a , b , c are in A.P., and Δ = x + 2 x + 7 a x + 5 x + 11 b x + 8 x + 15 c then ∆ equals
The matrix A satisfying A 1 5 0 1 = 3 − 1 6 0 is
If the system of equations x − k y − z = 0 , k x − y − z = 0 , x + y − z = 0 has a non-zero solution, then the possible values of k are
First row of a matrix A is 1 3 2 . If adj A = − 2 4 α − 1 2 1 3 α − 5 − 2 then a possible value of det(A) is
If A and B are two 3 x 3 matrices and | A | ≠ 0 , , then which of the following are not true?
If D = diag d 1 , d 2 , … , d n where d i ≠ 0 , for i = 1 , 2 , … , n , then D − 1 is equal to
The inverse of a skew-symmetric matrix of odd order is
If A = 1 0 2 5 1 x 1 1 1 is a singular matrix, then x is equal to
If a , b , c are positive integers such that a > b > c and 1 1 1 a b c a 2 b 2 c 2 = − 2 then 3 a + 7 b − 10 c equals
If square matrix A is such that 3 A 3 + 2 A 2 + 5 A + I = O , then A − 1 is equal to
If a , b , c ≠ 0 and a + b + c = 0 , then the matrix 1 + 1 a 1 1 1 1 + 1 b 1 1 1 1 + 1 c is
If ω is a complex cube root of unity, then the matrix A = 1 ω 2 ω ω 2 ω 1 ω 1 ω 2 is a
Let α , β , γ be three real numbers and A = 1 cos ( β − α ) cos ( γ − α ) cos ( α − β ) 1 cos ( γ − β ) cos ( α − γ ) cos ( β − γ ) 1 then
If A = 0 1 2 1 2 3 3 x 1 and A − 1 = 1 / 2 − 1 / 2 1 / 2 − 4 3 y 5 / 2 − 3 / 2 1 / 2 , then
Let A ( θ ) = sin θ i cos θ i cos θ sin θ , then
If A = 1 2 1 3 3 − 1 , then A − 1 − A 2
Let A be a square matrix of order 3 such that | A d j A | = 100 , then | A | equals
The matrix A = 0 0 − 7 0 − 7 0 − 7 0 0 is a
Let A t = 1 3 2 2 5 t 4 7 − t − 6 then the value(s) of t for which inverse of A t does not exist.
Let P and Q be 3×3 matrices with P ≠ Q If P 3 = Q 3 and P 2 Q = Q 2 P then determinant of P 2 + Q 2 is equal to
The system of equations 3 − 2 1 5 − 8 9 2 1 a x y z = b 3 − 1 has no solution if a and b are
The system of linear equations x + y + z = 2 , 2 x + y – z = 3 , 3 x + 2 y + k z = 4 has a unique solution if
If A = 4 x + 2 2 x − 3 x + 1 is an invertible matrix, then x cannot take value
The values of α for which the system of equations x + y + z = 1 , x + 2 y + 4 z = a , x + 4 y + 10 z = α 2 is consistent, are given by
Let A be a 2×2 matrix. Statement-1: adj(adj A) = A Statement-2: |adj A| = |A|
If A = 1 0 0 0 1 1 0 â 2 4 , 6 A â 1 = A 2 + c A + d I then (c,d) is
If A, B are two n x n non-singular matrices, then
Let M be a 3 x 3 non-singular matrix with det ( M ) = α . If M − 1 adj ( adj M ) = k I , then the value of k is
Let A be an invertible matrix. Which of the following is not true?
