For what value of $x$, is $A = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0 \end{bmatrix}$ a skew symmetric matrix?
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Matrices - ISC Class 12 Mathematics Questions with Answers, Page 2
50 past-paper questions on Matrices 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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In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed.
(i)[1.0]
A matrix which is both symmetric and skew symmetric matrix is a / an: [Understanding]
- (a)triangular matrix
- (b)identity matrix
- (c)diagonal matrix
- (d)null matrix
(ii)[1.0]
The value of $\int a^x \cdot e^x \\, dx$ equals [Recall]
- (a)$(a^x \cdot \log_e a)e^x + c$
- (b)$\frac{a^x \cdot e^x}{\log_e(ae)} + c$
- (c)$\frac{a^x \cdot e^x}{\log_{ae} e} + c$
- (d)$\log_e(ae)(ae)^x + c$
(iii)[1.0]
The trigonometric equation $\tan^{-1} x = 3\tan^{-1} a$ has solution for [Application]
- (a)$|a| \le \frac{1}{\sqrt{3}}$
- (b)$|a| > \frac{1}{\sqrt{3}}$
- (c)$|a| < \frac{1}{\sqrt{3}}$
- (d)all real value of a.
(iv)[1.0]
Assertion: Degree of the differential equation: $a\left(\frac{dy}{dx}\right)^2 + b\frac{dx}{dy} = c$, is 3
Reason: If each term involving derivatives of a differential equation is a polynomial (or can be expressed as polynomial) then highest exponent of the highest order derivative is called the degree of the differential equation. [Analysis]
Which of the following is correct?
- (a)Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
- (b)Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
- (c)Assertion is true and Reason is false.
- (d)Assertion is false and Reason is true.
(v)[1.0]
Five numbers $x_1, x_2, x_3, x_4, x_5$ are randomly selected from the numbers $1, 2, 3, \dots, 18$ and are arranged in the increasing order such that $x_1 < x_2 < x_3 < x_4 < x_5$. What is the probability that $x_2 = 7$ and $x_4 = 11$? [Application]
- (a)$\frac{26}{51}$
- (b)$\frac{3}{104}$
- (c)$\frac{1}{68}$
- (d)$\frac{1}{34}$
(vi)[1.0]
If $\begin{vmatrix} a & b & c \\\\ m & n & p \\\\ x & y & z \end{vmatrix} = k$, then what is the value of $\begin{vmatrix} 6a & 2b & 2c \\\\ 3m & n & p \\\\ 3x & y & z \end{vmatrix}$? [Understanding]
- (a)$\frac{k}{6}$
- (b)$2k$
- (c)$3k$
- (d)$6k$
(vii)[1.0]
Consider the graph $y = x^{1/3}$
Statement 1: The above graph is continuous at $x = 0$
Statement 2: The above graph is differentiable at $x = 0$ [Application]
Which of the following is correct?
- (a)Statement 1 is true, and Statement 2 is false.
- (b)Statement 2 is true, and Statement 1 is false.
- (c)Both the statements are true.
- (d)Both the statements are false.
(viii)[1.0]
The value of $\frac{dy}{dx}$ if $y = |x-1| + |x-4|$ at $x = 3$ is
- (a)$-2$
- (b)$0$
- (c)$2$
- (d)$4$
(ix)[1.0]
Statement 1: The intersection of two equivalence relations is always an equivalence relation.
Statement 2: The Union of two equivalence relations is always an equivalence relation.
Which one of the following is correct? [Understanding]
- (a)Statement 1 implies Statement 2.
- (b)Statement 2 implies Statement 1.
- (c)Statement 1 is true only if Statement 2 is true.
- (d)Statement 1 and 2 are independent of each other.
(x)[1.0]
Assertion: Matrix ‘A’ is not invertible.
Reason: Determinant A = 0
In a third order matrix $a_{ij}$ denotes the element of the $i^{\text{th}}$ row and the $j^{\text{th}}$ column.
$A = [a_{ij}] = \begin{cases} 0, & \text{for } i = j \\\\ 1, & \text{for } i > j \\\\ -1, & \text{for } i < j \end{cases}$
Which of the following is correct? [Application]
- (a)Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
- (b)Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
- (c)Assertion is true and Reason is false.
- (d)Assertion is false and Reason is true.
(xi)[1.0]
Given two events A and B such that $P(A/B) = 0.25$ and $P(A \cap B) = 0.12$.
The value $P(A \cap B')$ is: [Understanding]
- (a)$0 \cdot 36$
- (b)$0 \cdot 48$
- (c)$0 \cdot 88$
- (d)$0 \cdot 036$
(xii)[1.0]
The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of $C^2$? [Analysis]
(xiii)[1.0]
If a relation R on the set $\{a, b, c\}$ defined by $R = \\{(b,b)\\}$, then classify the relation. [Understanding]
(xiv)[1.0]
The given function $f: \mathbb{R} \to \mathbb{R}$ is not ‘onto’ function. Give reason. [Understanding]
(xv)[1.0]
There are three machines and 2 of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. What is the probability that only two tests are needed to identify the faulty machines? [Application]
No answer yet.
If $A = \begin{bmatrix} 0 & a \\ 0 & 0 \end{bmatrix}$, then $A^{16}$ is:
- (a)Unit matrix
- (b)Null matrix
- (c)Diagonal matrix
- (d)Skew matrix
No answer yet.
A furniture factory uses three types of wood namely, teakwood, rosewood and satinwood for manufacturing three types of furniture, that are, table, chair and cot.
The wood requirements (in tonnes) for each type of furniture are given below:
It is found that 29 tonnes of teakwood, 13 tonnes of rosewood and 16 tonnes of satinwood are available to make all three types of furniture.
Using the above information, answer the following questions:
| Table | Chair | Cot | |
|---|---|---|---|
| Teakwood | 2 | 3 | 4 |
| Rosewood | 1 | 1 | 2 |
| Satinwood | 3 | 2 | 1 |
(i)
Express the data given in the table above in the form of a set of simultaneous equations.
(ii)
Solve the set of simultaneous equations formed in subpart (i) by matrix method.
(iii)
Hence, find the number of table(s), chair(s) and cot(s) produced.
No answer yet.
If $A$ and $B$ are symmetric matrices of the same order, then $AB - BA$ is:
- (a)Skew-symmetric matrix
- (b)Symmetric matrix
- (c)Diagonal matrix
- (d)Identity matrix
No answer yet.
Assertion: Let the matrices $A = \begin{bmatrix} -3 & 2 \\ -5 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -2 \\ 5 & -3 \end{bmatrix}$ be such that $A^{100}B = BA^{100}$.
Reason: $AB = BA$ implies $A^n B = BA^n$ for all positive integers $n$.
- (a)Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
- (b)Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
- (c)Assertion is true and Reason is false.
- (d)Assertion is false and Reason is true.
No answer yet.
Use matrix method to solve the following system of equations:
$\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4$
$\frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1$
$\frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2$
No answer yet.
If $A = \begin{pmatrix} 1 & b \\\\ 0 & 1 \end{pmatrix}$, then $A^n$ ($n \in \mathbb{N}$) is equal to
- (a)$\begin{pmatrix} 1 & nb \\\\ 0 & 1 \end{pmatrix}$
- (b)$\begin{pmatrix} 1 & b^n \\\\ 0 & 1 \end{pmatrix}$
- (c)$\begin{pmatrix} 1 & nb \\\\ 0 & 1 \end{pmatrix}$
- (d)$\begin{pmatrix} 1 & nb \\\\ 0 & 0 \end{pmatrix}$
No answer yet.
For what value of $k$ the matrix $\begin{bmatrix} 0 & k \\ -6 & 0 \end{bmatrix}$ is a skew symmetric matrix?
No answer yet.
Find $A^{-1}$, if $A = \begin{pmatrix} 1 & 2 & 0 \\\\ -2 & -1 & -2 \\\\ 0 & -1 & 1 \end{pmatrix}$. Using $A^{-1}$, solve the system of linear equations: $x - 2y = 10$, $2x - y - z = 8$, $-2y + z = 7$
No answer yet.
If $\begin{pmatrix} 3 & 5 \\\\ 7 & 9 \end{pmatrix} = X + Y$, where $X$ is skew-symmetric matrix and $Y$ is symmetric matrix. Find $|X|$.
No answer yet.
What is the multiplicative inverse of matrix $A$?
- (a)$A$
- (b)$A^2$
- (c)$|A|$
- (d)$\frac{\operatorname{adj} A}{|A|}$
No answer yet.
From the matrix equation $AB = AC$, it can be concluded that $B = C$ provided:
- (a)$A$ is singular matrix
- (b)$A$ is non-singular matrix
- (c)$A$ is a symmetric matrix
- (d)$A$ is a skew symmetric matrix
No answer yet.
If $A$ and $B$ are two non singular matrices, and $AB$ exists, then $(AB)^{-1}$ is:
- (a)$A^{-1}B^{-1}$
- (b)$B^{-1}A^{-1}$
- (c)$AB$
- (d)None of the above
No answer yet.
What is the transpose of a column matrix?
- (a)Zero matrix
- (b)Diagonal matrix
- (c)Column matrix
- (d)Row matrix
No answer yet.
Solve the matrix equation: $A \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 15 \\ 1 & -2 \end{pmatrix}$.
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If $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, then $A^2$ is equal to:
- (a)$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
- (b)$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
- (c)$\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$
- (d)$\begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}$
No answer yet.
Using matrices, solve the following system of equations:
$2x - 3y + 5z = 11$, $3x + 2y - 4z = -5$ and $x + y - 2z = -3$
No answer yet.
Solve the following system of linear equations using matrices:
$x - 2y = 10, \quad 2x - y - z = 8, \quad -2y + z = 7$
No answer yet.
If $\begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \begin{pmatrix} 1 & -3 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -4 & 6 \\ -9 & x \end{pmatrix}$, find $x$.
No answer yet.