If $\operatorname{adj}(A) = \begin{bmatrix} 2 & 3 & 5 \\ x & 5 & 1 \\ 3 & 3 & 4 \end{bmatrix}$ and $|A| = 4$, then the value of $x$ is:
- (a)$16$
- (b)$12$
- (c)$32$
- (d)$10$
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Determinants - ISC Class 12 Mathematics Questions with Answers, Page 2
43 past-paper questions on Determinants from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 21-40 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.
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The existence of unique solution of the system of equations $x + y = \lambda$ and $5x + ky = 2$ depends on:
- (a)$\lambda$ only
- (b)$\frac{\lambda}{k} = 1$
- (c)both $k$ and $\lambda$
- (d)$k$ only
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Find the value of the determinant given below, without expanding it at any stage.
$\begin{vmatrix} \beta\gamma & 1 & \alpha(\beta+\gamma) \\ \gamma\alpha & 1 & \beta(\gamma+\alpha) \\ \alpha\beta & 1 & \gamma(\alpha+\beta) \end{vmatrix}$
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If $\begin{vmatrix} a & b & c \\\\ x+a & y+2a & z+3a \\\\ 1 & 2 & 3 \end{vmatrix} = \begin{vmatrix} a & b & c \\\\ x & y & z \\\\ 1 & 2 & 3 \end{vmatrix} + |B|$, then the value of $|B|$ is equal to
- (a)$a + 2b + 3c$
- (b)$a - b + 2c$
- (c)$0$
- (d)$3a$
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If $A$ is a square matrix of order 3, then $|2A|$ is equal to:
- (a)$2|A|$
- (b)$4|A|$
- (c)$8|A|$
- (d)$6|A|$
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If $A$ is a $3 \times 3$ matrix such that $A(\text{adj } A) = \begin{pmatrix} 3 & 0 & 0 \\\\ 0 & 3 & 0 \\\\ 0 & 0 & 3 \end{pmatrix}$, then the value of $|A^T|$ will be
- (a)$27$
- (b)$9$
- (c)$3$
- (d)$-3$
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Evaluate the following determinant without expanding.
$\begin{vmatrix} 5 & 5 & 5 \\ a & b & c \\ b+c & c+a & a+b \end{vmatrix}$
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What will be the value of $x$ for the determinant equation $\begin{vmatrix} 3-x & 6 & 3 \\ -6 & 3-x & 3 \\ 3 & 3 & 3-x \end{vmatrix} = 0$?
- (a)6
- (b)3
- (c)0
- (d)-6
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The value of $\begin{vmatrix} 1 & \log_a b \\ \log_b a & 1 \end{vmatrix}$ is:
- (a)$1 - \log_a b$
- (b)$1 - \frac{\log b}{\log a}$
- (c)0
- (d)$\log_a b - 1$
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Ramu purchased 5 pens, 3 bags and 1 instrument box and paid ₹ 16. From the same shop Venkat purchased 2 pens , 1 bag and 3 instrument boxes and paid ₹ 19 while Gopi purchased 1 pen, 2 bags and 4 instrument boxes and paid ₹ 25.
Using the concept of Matrices and Determinants to answer the following questions by choosing the correct option:
(i)[2.0]
If $x, y\ \&\ z$ respectively denotes the cost of pen, bag and instrument box then which of the following is true?
- (a)$5x + 3y + z = 16$
- (b)$2x + y + 3z = 19$
- (c)$x + 2y + 4z = 25$
- (d)All of the above
(ii)[2.0]
If $A = \begin{pmatrix} 5 & 3 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & 4 \end{pmatrix}$, $|A|$ is:
- (a)-22
- (b)22
- (c)0
- (d)20
(iii)[2.0]
If $A = \begin{pmatrix} 5 & 3 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & 4 \end{pmatrix}$ and $\operatorname{adj} A = \begin{pmatrix} -2 & x & 8 \\ -5 & 19 & -13 \\ 3 & -7 & y \end{pmatrix}$ then missing value of $x$ and $y$ are:
- (a)$x = -10\ \&\ y = -1$
- (b)$x = 10\ \&\ y = -1$
- (c)$x = -10\ \&\ y = 1$
- (d)$x = 10\ \&\ y = 1$
(iv)[2.0]
The cost of one pen is
- (a)₹ 2
- (b)₹ 5
- (c)₹ 1
- (d)₹ 3
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If $\Delta = \begin{vmatrix} a & b & c \\ x & y & z \\ p & q & r \end{vmatrix}$, then $\begin{vmatrix} ka & kb & kc \\ kx & ky & kz \\ kp & kq & kr \end{vmatrix}$ is:
- (a)$\Delta$
- (b)$k\Delta$
- (c)$3k\Delta$
- (d)$k^3\Delta$
No answer yet.
Which of the following statements is correct:
- (a)Determinant is number associated to a matrix.
- (b)Determinant is a square matrix.
- (c)Determinant is a number associated to a square matrix.
- (d)None of the above.
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For what value of $k$ inverse does not exist for the matrix $\begin{bmatrix} 1 & 2 \\ k & 6 \end{bmatrix}$:
- (a)0
- (b)3
- (c)6
- (d)2
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If A is square matrix of order 3, with $|A| = 4$, then write the value of $|-2A|$.
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Without expanding at any stage, find the value of the determinant:
$\Delta = \begin{vmatrix} 20 & a & b + c \\ 20 & b & a + c \\ 20 & c & a + b \end{vmatrix}$
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Using properties of determinants, show that:
$\begin{vmatrix} x & p & q \\ p & x & q \\ q & q & x \end{vmatrix} = (x - p)(x^2 + px - 2q^2)$
Given: $\begin{vmatrix} x & p & q \\ p & x & q \\ q & q & x \end{vmatrix}$
To show: $(x - p)(x^2 + px - 2q^2)$
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Using determinants, find the values of $k$, if the area of triangle with vertices $(-2, 0)$, $(0, 4)$ and $(0, k)$ is $4$ square units.
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Using properties of determinants prove that:
$\begin{vmatrix} x & x(x^2 + 1) & x + 1 \\\\ y & y(y^2 + 1) & y + 1 \\\\ z & z(z^2 + 1) & z + 1 \end{vmatrix} = (x - y)(y - z)(z - x)(x + y + z)$
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Without expanding at any stage, find the value of the determinant:
$\Delta = \begin{vmatrix} 2 & x & y+z \\ 2 & y & z+x \\ 2 & z & x+y \end{vmatrix}$
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Use properties of determinants to solve for $x$:
$\begin{vmatrix} x+a & b & c \\ c & x+b & a \\ a & b & x+c \end{vmatrix} = 0$ and $x \ne 0$.
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