How many $3 \times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is $5$ ?
- 126
- 198
- 162
- 135
The Correct Option is B
Solution and Explanation
Then sum of diagonal entries $=\sum k _{ ij }^{2}$
$\Rightarrow \sum k _{ ij }^{2}=5$
where $k _{ ij }$ are from $\{0,1,2\}$
$\Rightarrow$ Total number of matrices $={ }^{9} C _{5}+{ }^{9} C _{1} \cdot{ }^{8} C _{1}=198$
Top JEE Advanced Mathematics Questions
- Let $ \mathbb{R} $ denote the set of all real numbers. Let $ f: \mathbb{R} \to \mathbb{R} $ be defined by $$ f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \ne 0 \\ \dfrac{7}{3}, & \text{if } x = 0 \end{cases} $$ Then which of the following statements is (are) TRUE?
- JEE Advanced - 2025
- Mathematics
- Fundamental Theorem of Calculus
Let $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ be two distinct points on the ellipse $$ \frac{x^2}{9} + \frac{y^2}{4} = 1 $$ such that $ y_1 > 0 $, and $ y_2 > 0 $. Let $ C $ denote the circle $ x^2 + y^2 = 9 $, and $ M $ be the point $ (3, 0) $. Suppose the line $ x = x_1 $ intersects $ C $ at $ R $, and the line $ x = x_2 $ intersects $ C $ at $ S $, such that the $ y $-coordinates of $ R $ and $ S $ are positive. Let $ \angle ROM = \frac{\pi}{6} $ and $ \angle SOM = \frac{\pi}{3} $, where $ O $ denotes the origin $ (0, 0) $. Let $ |XY| $ denote the length of the line segment $ XY $. Then which of the following statements is (are) TRUE?
- JEE Advanced - 2025
- Mathematics
- Conic sections
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
- Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (0,4)$ be functions defined by $$ f(x) = \log_e (x^2 + 2x + 4), \quad \text{and} \quad g(x) = \frac{4}{1 + e^{-2x}}. $$ Define the composite function $f \circ g^{-1}$ by $(f \circ g^{-1})(x) = f(g^{-1}(x))$, where $g^{-1}$ is the inverse of the function $g$. Then the value of the derivative of the composite function $f \circ g^{-1}$ at $x=2$ is _____.
- JEE Advanced - 2025
- Mathematics
- Fundamental Theorem of Calculus
- Let $ x_0 $ be the real number such that $ e^{x_0} + x_0 = 0 $. For a given real number $ \alpha $, define $$ g(x) = \frac{3xe^x + 3x - \alpha e^x - \alpha x}{3(e^x + 1)} $$ for all real numbers $ x $. Then which one of the following statements is TRUE?
- JEE Advanced - 2025
- Mathematics
- Fundamental Theorem of Calculus
Top JEE Advanced Matrices Questions
- Let $ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $ and $ P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} $. Let $ Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix} $ for some non-zero real numbers $ x, y, z $, for which there is a $ 2 \times 2 $ matrix $ R $ with all entries being non-zero real numbers, such that $$ QR = RP $$ Then which of the following statements is (are) TRUE?
- Let α and β be the distinct roots of the equation x2 + x- 1 = 0. Consider the set T = {1, α, β}. For a 3 × 3 matrix M = ( aij )3×3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3.
Match each entry in List-I to the corret entry in List-II.The correct option isList - I List - II (P) The number of matrices M = ( aij )3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is (1) 1 (Q) The number of symmetric matrices M = ( aij )3×3 with all entries in T such that Cj = 0 for all j, is (2) 12 (R) Let M = ( aij )3×3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number of elements in the set
\(\left\{\begin{pmatrix} x \\ y \\ z \end{pmatrix}:x,y,z\in \R, M\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix}\right\}\) is(3) infinite (S) Let M = ( aij )3×3 be a matrix with all entries in T such that Ri = 0 for all i. Then the absolute value of the determinant of M is (4) 6 (5) 0 - Let \(S=\left\{\begin{pmatrix} 0 & 1 & c \\ 1 & a & d\\ 1 & b & e \end{pmatrix}:a,b,c,d,e\in\left\{0,1\right\}\ \text{and} |A|\in \left\{-1,1\right\}\right\}\), where |A| denotes the determinant of A. Then the number of elements in S is _______.
Let \(\beta\) be a real number Consider the
\(matrix\ A=\begin{pmatrix}\beta & 0 & 1 \\2 & 1 & -2 \\3 & 1 & -2\end{pmatrix}If A^7-(\beta-1) A^6-\beta A^5\) is a singular matrix, then the value of \(9 \beta\) is _____- Let M=(aij), i,j∈{1,2,3}, be the 3×3 matrix such that aij=1 if j+1 is divisible by i,otherwise aij=0. Then which of the following statements is(are) true?
Top JEE Advanced Questions
- Adsorption of phenol from its aqueous solution on to fly ash obeys Freundlich isotherm. At a given temperature, from 10 mg g$^{-1}$ and 16 mg g$^{-1}$ aqueous phenol solutions, the concentrations of adsorbed phenol are measured to be 4 mg g$^{-1}$ and 10 mg g$^{-1}$, respectively. At this temperature, the concentration (in mg g$^{-1}$) of adsorbed phenol from 20 mg g$^{-1}$ aqueous solution of phenol will be ____. Use: $\log_{10} 2 = 0.3$
- JEE Advanced - 2025
- Adsorption
Monocyclic compounds $ P, Q, R $ and $ S $ are the major products formed in the reaction sequences given below.
The product having the highest number of unsaturated carbon atom(s) is:
- JEE Advanced - 2025
- Organic Chemistry
For the reaction sequence given below, the correct statement(s) is(are):
- A linear octasaccharide (molar mass = 1024 g mol$^{-1}$) on complete hydrolysis produces three monosaccharides: ribose, 2-deoxyribose and glucose. The amount of 2-deoxyribose formed is 58.26 % (w/w) of the total amount of the monosaccharides produced in the hydrolyzed products. The number of ribose unit(s) present in one molecule of octasaccharide is _____.
Use: Molar mass (in g mol$^{-1}$): ribose = 150, 2-deoxyribose = 134, glucose = 180; Atomic mass (in amu): H = 1, O = 16- JEE Advanced - 2025
- Biomolecules
Consider a reaction $ A + R \rightarrow Product $. The rate of this reaction is measured to be $ k[A][R] $. At the start of the reaction, the concentration of $ R $, $[R]_0$, is 10-times the concentration of $ A $, $[A]_0$. The reaction can be considered to be a pseudo first order reaction with assumption that $ k[R] = k' $ is constant. Due to this assumption, the relative error (in %) in the rate when this reaction is 40% complete, is ____. [$k$ and $k'$ represent corresponding rate constants]
- JEE Advanced - 2025
- Chemical Kinetics
Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.