The value of β γ β γ ' + β ' γ β ' γ ' γ α γ α ' + γ ' α γ ' α ' α β α β ' + α ' β α ' β ' is

α ' β - αβ ' β ' γ - βγ ' ( γα ' - γ ' α )

αα - ββ ' ββ ' - γγ ( γγ - αα )

αβ ' + αβ ( βγ + βγ ) γα α + γα

E M B I B E

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 3

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JEE Main

IMPORTANT

Earn 100

Area of a triangle whose vertices are $(a, a^{2}), (b, b^{2})$ and $(c, c^{2})$ is $\frac{1}{2}$ , and area of another triangle whose vertices are $(p, p^{2}), (q, q^{2})$ and $(r, r^{2})$ is $4$ , then the value of $|\begin{matrix} (1 + a p)^{2} & (1 + b p)^{2} & (1 + c p)^{2} \\ (1 + a q)^{2} & (1 + b q)^{2} & (1 + c q)^{2} \\ (1 + a r)^{2} & (1 + b r)^{2} & (1 + c r)^{2} \end{matrix}|$ can be

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Important Questions on Determinant

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 4

The value of

|\begin{matrix} β γ & β γ^{'} + β^{'} γ & β^{'} γ^{'} \\ γ α & γ α^{'} + γ^{'} α & γ^{'} α^{'} \\ α β & α β^{'} + α^{'} β & α^{'} β^{'} \end{matrix}|

is

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 5

If

|\begin{array}{l} a & b & 1 \\ b & c & 1 \\ c & a & 1 \end{array}| = 2010

, and if

|\begin{array}{l} c - a & c - b & a b \\ a - b & a - c & b c \\ b - c & b - a & c a \end{array}| - |\begin{array}{l} c - a & c - b & c^{2} \\ a - b & a - c & a^{2} \\ b - c & b - a & b^{2} \end{array}| = p

, then the number of positive divisors of

p

is

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 6

If

|\begin{array}{l} a & l & m \\ l & b & n \\ m & n & c \end{array}| |\begin{array}{l} b c - n^{2} & m n - l c & l n - b m \\ m n - l c & a c - m^{2} & l m - a n \\ l n - b m & l m - a n & a b - l^{2} \end{array}| = 64

, then the value of

|\begin{matrix} 2 a + 3 l & 3 l + 5 m & 5 m + 4 a \\ 2 l + 3 b & 3 b + 5 n & 5 n + 4 l \\ 2 m + 3 n & 3 n + 5 c & 5 c + 4 m \end{matrix}|

can be equal to

MEDIUM

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 7

The value of

|\begin{matrix} x^{2} + y^{2} & a x + b y & x + y \\ a x + b y & a^{2} + b^{2} & a + b \\ x + y & a + b & 2 \end{matrix}|

depends on

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 8

If

u = a x + b y + c z, v = a y + b z + c x, w = a z + b x + c y

, then the value of

|\begin{array}{l} a & b & c \\ b & c & a \\ c & a & b \end{array}| \times |\begin{array}{l} x & y & z \\ y & z & x \\ z & x & y \end{array}|

is

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 9

If the number of positive integral solutions of

u + v + w = n

, is denoted by

P_{n}

, then the absolute value of

|\begin{matrix} P_{n} & P_{n + 1} & P_{n + 2} \\ P_{n + 1} & P_{n + 2} & P_{n + 3} \\ P_{n + 2} & P_{n + 3} & P_{n + 4} \end{matrix}|

is

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 8 - Determinant>DPP>Q 10

If $f (x), g (x)$ and $h (x)$ are polynomials of degree $4$ and $|\begin{matrix} f (x) & g (x) & h (x) \\ a & b & c \\ p & q & r \end{matrix}| = m x^{4} + n x^{3} + r x^{2} + s x + t$ be an identity in $x,$ then $|\begin{matrix} f^{'''} (0) - f^{''} (0) & g^{'''} (0) - g^{''} (0) & h^{'''} (0) - h^{''} (0) \\ a & b & c \\ p & q & r \end{matrix}|$ is