Properties of Determinants

Interactive Feed

JEE Main 2014

If $α, β \neq = 0$ , and $f (n) = α^{n} + β^{n}$ 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 $K$ is equal to:

1

(B)

- 1

(C)

α β

(D)

\frac{1}{α β}

JEE Main 2012

LEVELJEE Main

Let $P$ and $Q$ be $3 \times 3$ matrices $P \neq = Q$ . If $P^{3} = Q^{3}$ and $P^{2} Q = Q^{2} P$ then determinant of $(P^{2} + Q^{2})$ is equal to:

(A)

2

(B)

1

(C)

0

(D)

- 1

JEE Main 2009

LEVELJEE Main

Let $a, b, c$ be such that $b (a + c) \neq = 0$ . If $a - b c a + 1 b + 1 c - 1 a - 1 b - 1 c + 1 + a + 1 a - 1 (- 1)^{n + 2} a b + 1 b - 1 (- 1)^{n + 1} b c - 1 c + 1 (- 1)^{n} c = 0$ , then the value of $n$ is:

If $D = 111 1 1 + x 1 11 1 + y$ for $x \neq = 0, y \neq = 0$ , then $D$ is

(A)

divisible by

x

but not

y

(B)

divisible by

y

but not

x

(C)

divisible by neither

x

nor

y

(D)

divisible by both

x

and

y

JEE Main 2007

LEVELBoard

Let $A = 500 5 α α 0 α 5 α 5$ . If $∣ A^{2} ∣ = 25$ , then $∣ α ∣$ equals

(A)

1/5

(B)

5

(C)

5^{2}

(D)

1

JEE Main 2005

LEVELJEE Main

If $a^{2} + b^{2} + c^{2} = - 2$ and $f (x) = 1 + a^{2} x (1 + a^{2}) x (1 + a^{2}) x (1 + b^{2}) x 1 + b^{2} x (1 + b^{2}) x (1 + c^{2}) x (1 + c^{2}) x 1 + c^{2} x$ , then $f (x)$ is a polynomial of degree

(A)

1

(B)

0

(C)

3

(D)

2

JEE Main 2005

LEVELJEE Main

If $a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots$ are in G.P., then the determinant $Δ = lo g a_{n} lo g a_{n + 3} lo g a_{n + 6} lo g a_{n + 1} lo g a_{n + 4} lo g a_{n + 7} lo g a_{n + 2} lo g a_{n + 5} lo g a_{n + 8}$ is equal to

(A)

1

(B)

0

(C)

4

(D)

2

JEE Main 2004

LEVELJEE Main

If $a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots$ are in G.P., then the value of the determinant $lo g a_{n} lo g a_{n + 3} lo g a_{n + 6} lo g a_{n + 1} lo g a_{n + 4} lo g a_{n + 7} lo g a_{n + 2} lo g a_{n + 5} lo g a_{n + 8}$ is

(A)

- 2

(B)

1

(C)

2

(D)

0

JEE Main 2003

LEVELJEE Main

If $a b c a^{2} b^{2} c^{2} 1 + a^{3} 1 + b^{3} 1 + c^{3} = 0$ and vectors $(1, a, a^{2}), (1, b, b^{2})$ and $(1, c, c^{2})$ are non-coplanar, then the product $ab c$ equals

$l, m, n$ are the $p$ th, $q$ th and $r$ th term of a G. P. all positive, then $lo g l lo g m lo g n p q r 111$ equals

If $a > 0$ and discriminant of $a x^{2} + 2 b x + c$ is $- v e$ , then $a b a x + b b c b x + c a x + b b x + c 0$ is equal to

(A)

+ v e

(B)

(a c - b^{2}) (a x^{2} + 2 b x + c)

(C)

- v e

(D)

0

JEE Advanced 2015

LEVELJEE Main

Which of the following values of $α$ satisfy the equation $(1 + α)^{2} (2 + α)^{2} (3 + α)^{2} (1 + 2 α)^{2} (2 + 2 α)^{2} (3 + 2 α)^{2} (1 + 3 α)^{2} (2 + 3 α)^{2} (3 + 3 α)^{2} = - 648 α$ ?

* Multiple Correct Options

(A)

- 4

(B)

9

(C)

- 9

(D)

4

JEE Advanced 2014

LEVELJEE Main

Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N = N M$ . Further, if $M \neq = N^{2}$ and $M^{2} = N^{4}$ , then

* Multiple Correct Options

(A)

determinant of

(M^{2} + M N^{2})

0

(B)

there is

3 \times 3

non-zero matrix

U

such that

(M^{2} + M N^{2}) U

is the zero matrix

(C)

determinant of

(M^{2} + M N^{2}) \geq 1

(D)

for a

3 \times 3

matrix

U

, if

(M^{2} + M N^{2}) U

equals the zero matrix then

U

is the zero matrix

JEE Advanced 2012

LEVELBoard

Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$ , where $b_{ij} = 2^{i + j} a_{ij}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

(A)

(a)

2^{10}

(B)

(b)

2^{11}

(C)

(c)

2^{12}

(D)

(d)

2^{13}

JEE Advanced 2011

LEVELJEE Main

Let $ω \neq = 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $1 ω ω^{2} a 1 ω b c 1$ where each of $a, b$ and $c$ is either $ω$ or $ω^{2}$ . Then the number of distinct matrices in the set $S$ is

If $A = [α 2 2 α]$ and $∣ A^{3} ∣ = 125$ then the value of $α$ is

(A)

(a)

\pm 1

(B)

(b)

\pm 2

(C)

(c)

\pm 3

(D)

(d)

\pm 5

JEE Advanced 2001

LEVELJEE Main

The number of distinct real roots of $sin x cos x cos x cos x sin x cos x cos x cos x sin x = 0$ in the interval $- π /4 \leq x \leq π /4$ is

(A)

0

(B)

2

(C)

1

(D)

3

JEE Advanced 1999

LEVELJEE Main

If $f (x) = 1 2 x 3 x (x - 1) x x (x - 1) x (x - 1) (x - 2) x + 1 (x + 1) x (x + 1) x (x - 1)$ then $f (100)$ is equal to

- 100

JEE Advanced 1998

LEVELBoard

If $6 i 420 - 3 i 3 i 3 1 - 1 i = x + i y$ , then

If $6 i 420 - 3 i 3 i 3 1 - 1 i = x + i y$ , then

(A)

x = 3, y = 1

(B)

x = 1, y = 3

(C)

x = 0, y = 3

(D)

x = 0, y = 0

JEE Advanced 1997

LEVELJEE Main

The parameter, on which the value of the determinant $1 cos (p - d) x sin (p - d) x a cos p x sin p x a^{2} cos (p + d) x sin (p + d) x$ does not depend upon is

(A)

(a)

a

(B)

(b)

p

(C)

(c)

d

(D)

(d)

x

JEE Advanced 1995

LEVELJEE Main

If $ω (\neq = 1)$ is a cube root of unity, then $1 1 - i - i 1 + i + ω^{2} - 1 - i + ω - 1 ω^{2} ω^{2} - 1 - 1 =$

i

(D)

(d)

ω

JEE Advanced 1988

LEVELJEE Main

The values of $θ$ lying between $θ = 0$ and $θ = π /2$ and satisfying the equation $1 + sin^{2} θ sin^{2} θ sin^{2} θ cos^{2} θ 1 + cos^{2} θ cos^{2} θ 4 sin 4 θ 4 sin 4 θ 1 + 4 sin 4 θ = 0$ are

* Multiple Correct Options

(A)

7 π /24

(B)

5 π /24

(C)

11 π /24

(D)

π /24

JEE Advanced 1986

LEVELJEE Main

The determinant $a b a α + b b c b α + c a α + b b α + c 0$ is equal to zero, if

* Multiple Correct Options

(A)

a, b, c

are in A. P.

(B)

a, b, c

are in G. P.

(C)

a, b, c

are in H. P.

(D)

α

is a root of the equation

a x^{2} + b x + c = 0

(E)

(x - α)

is a factor of

a x^{2} + 2 b x + c

JEE Advanced 1981

LEVELJEE Main

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

(A)

(a)

C

is empty

(B)

(b)

B

has as many elements as

C

(C)

(c)

A = B \cup C

(D)

(d)

B

has twice as many elements as

C

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α,β=0, and f(n)=αn+βn and ​31+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 is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P and Q be 3×3 matrices P=Q. If P3=Q3 and P2Q=Q2P then determinant of (P2+Q2) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If D=​111​11+x1​111+y​​ for x=0,y=0, then D is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​500​5αα0​α5α5​​. If ∣A2∣=25, then ∣α∣ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a2+b2+c2=−2 and f(x)=​1+a2x(1+a2)x(1+a2)x​(1+b2)x1+b2x(1+b2)x​(1+c2)x(1+c2)x1+c2x​​, then f(x) is a polynomial of degree

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ​abc​a2b2c2​1+a31+b31+c3​​=0 and vectors (1,a,a2),(1,b,b2) and (1,c,c2) are non-coplanar, then the product abc equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } l,m,n are the pth, qth and rth term of a G. P. all positive, then ​logllogmlogn​pqr​111​​ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0 and discriminant of ax2+2bx+c is −ve, then ​abax+b​bcbx+c​ax+bbx+c0​​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following values of α satisfy the equation ​(1+α)2(2+α)2(3+α)2​(1+2α)2(2+2α)2(3+2α)2​(1+3α)2(2+3α)2(3+3α)2​​=−648α?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let M and N be two 3×3 matrices such that MN=NM. Further, if M=N2 and M2=N4, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P=[aij​] be a 3×3 matrix and let Q=[bij​], where bij​=2i+jaij​ for 1≤i,j≤3. If the determinant of P is 2, then the determinant of the matrix Q is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A=[α2​2α​] and ∣A3∣=125 then the value of α is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of distinct real roots of ​sinxcosxcosx​cosxsinxcosx​cosxcosxsinx​​=0 in the interval −π/4≤x≤π/4 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=​12x3x(x−1)​xx(x−1)x(x−1)(x−2)​x+1(x+1)x(x+1)x(x−1)​​ then f(100) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ​6i420​−3i3i3​1−1i​​=x+iy, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ​6i420​−3i3i3​1−1i​​=x+iy, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The parameter, on which the value of the determinant ​1cos(p−d)xsin(p−d)x​acospxsinpx​a2cos(p+d)xsin(p+d)x​​ does not depend upon is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ω(=1) is a cube root of unity, then ​11−i−i​1+i+ω2−1−i+ω−1​ω2ω2−1−1​​=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The values of θ lying between θ=0 and θ=π/2 and satisfying the equation ​1+sin2θsin2θsin2θ​cos2θ1+cos2θcos2θ​4sin4θ4sin4θ1+4sin4θ​​=0 are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The determinant ​abaα+b​bcbα+c​aα+bbα+c0​​ is equal to zero, if

If $α, β \neq = 0$ , and $f (n) = α^{n} + β^{n}$ 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 $K$ is equal to:

Let $P$ and $Q$ be $3 \times 3$ matrices $P \neq = Q$ . If $P^{3} = Q^{3}$ and $P^{2} Q = Q^{2} P$ then determinant of $(P^{2} + Q^{2})$ is equal to:

If $D = 111 1 1 + x 1 11 1 + y$ for $x \neq = 0, y \neq = 0$ , then $D$ is

Let $A = 500 5 α α 0 α 5 α 5$ . If $∣ A^{2} ∣ = 25$ , then $∣ α ∣$ equals

If $a^{2} + b^{2} + c^{2} = - 2$ and $f (x) = 1 + a^{2} x (1 + a^{2}) x (1 + a^{2}) x (1 + b^{2}) x 1 + b^{2} x (1 + b^{2}) x (1 + c^{2}) x (1 + c^{2}) x 1 + c^{2} x$ , then $f (x)$ is a polynomial of degree

If $a b c a^{2} b^{2} c^{2} 1 + a^{3} 1 + b^{3} 1 + c^{3} = 0$ and vectors $(1, a, a^{2}), (1, b, b^{2})$ and $(1, c, c^{2})$ are non-coplanar, then the product $ab c$ equals

$l, m, n$ are the $p$ th, $q$ th and $r$ th term of a G. P. all positive, then $lo g l lo g m lo g n p q r 111$ equals

If $a > 0$ and discriminant of $a x^{2} + 2 b x + c$ is $- v e$ , then $a b a x + b b c b x + c a x + b b x + c 0$ is equal to

Which of the following values of $α$ satisfy the equation $(1 + α)^{2} (2 + α)^{2} (3 + α)^{2} (1 + 2 α)^{2} (2 + 2 α)^{2} (3 + 2 α)^{2} (1 + 3 α)^{2} (2 + 3 α)^{2} (3 + 3 α)^{2} = - 648 α$ ?

Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N = N M$ . Further, if $M \neq = N^{2}$ and $M^{2} = N^{4}$ , then

Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$ , where $b_{ij} = 2^{i + j} a_{ij}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

If $A = [α 2 2 α]$ and $∣ A^{3} ∣ = 125$ then the value of $α$ is

The number of distinct real roots of $sin x cos x cos x cos x sin x cos x cos x cos x sin x = 0$ in the interval $- π /4 \leq x \leq π /4$ is

If $f (x) = 1 2 x 3 x (x - 1) x x (x - 1) x (x - 1) (x - 2) x + 1 (x + 1) x (x + 1) x (x - 1)$ then $f (100)$ is equal to

If $6 i 420 - 3 i 3 i 3 1 - 1 i = x + i y$ , then

If $6 i 420 - 3 i 3 i 3 1 - 1 i = x + i y$ , then

The parameter, on which the value of the determinant $1 cos (p - d) x sin (p - d) x a cos p x sin p x a^{2} cos (p + d) x sin (p + d) x$ does not depend upon is

If $ω (\neq = 1)$ is a cube root of unity, then $1 1 - i - i 1 + i + ω^{2} - 1 - i + ω - 1 ω^{2} ω^{2} - 1 - 1 =$

The values of $θ$ lying between $θ = 0$ and $θ = π /2$ and satisfying the equation $1 + sin^{2} θ sin^{2} θ sin^{2} θ cos^{2} θ 1 + cos^{2} θ cos^{2} θ 4 sin 4 θ 4 sin 4 θ 1 + 4 sin 4 θ = 0$ are

The determinant $a b a α + b b c b α + c a α + b b α + c 0$ is equal to zero, if