Algebra PYQ — Page 5

Let E and F are events associated with an experiment. If $P(E) = 0.4$, $P(F) = 0.8$ and $P(F|E) = 0.6$, then $P(E|F)$ is

If $c_{ij}$ denotes the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & -3 & 2 \\ 4 & 2 & 3 \end{bmatrix}$ then the value of $c_{21} \cdot c_{33}$ is

The function $f: \mathbb{R} \rightarrow \mathbb{R}, f(x) = |x|$ ($\mathbb{R}$ is the set of real numbers) is

A relation $f: N \rightarrow N$ be defined by $f(x) = x^2$, $x \in N$ (Set of Natural numbers). Then $f(x)$ is

Match List-I with List-II Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$. | List-I | List-II | | --- | --- | | (A) $\vec{a} \cdot \vec{b}$ | (I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2} \vec{b}$ | | (B) $\vec{a} \times \vec{b}$ | (II) $\vec{a} \cdot \vec{b} = 0$ | | (C) Projection vector of $\vec{a}$ on $\vec{b}$ ($\ne{0}$) | (III) $\vert \vec{a}\vert \vert \vec{b}\vert \sin \theta \, \hat{n}$ where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ | | (D) $\vec{a}$ and $\vec{b}$ are orthogonal vectors | (IV) $\vert \vec{a}\vert \vert \vec{b}\vert \cos \theta$ |

Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$. Then match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\sin \theta$ | (I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert \vert \vec{b}\vert }$ | | (B) $\cos \theta$ | (II) $\vert \vec{a} \times \vec{b}\vert $ | | (C) Area of the parallelogram with adjacent sides represented by $\vec{a}$ and $\vec{b}$ | (III) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert }$ | | (D) Projection of $\vec{a}$ on $\vec{b}$ | (IV) $\dfrac{\vert \vec{a} \times \vec{b}\vert }{\vert \vec{a}\vert \vert \vec{b}\vert }$ | Choose the correct answer from the options given below:

If A and B are square matrices of the same order 3, such that det (A) = 3 and AB = 3I, where I is an identity matrix of order 3. Then the value of det (B) is:

If A is a square matrix and I is an identity matrix of same order such that $A^2 = A$, then $(I + A)^3 - 8I$ is equal to

Let $A = \{a, b, c\}$. Then number of relations containing $(a,b)$ and $(b, c)$ which are reflexive and transitive but not symmetric is

Nitin has taken the subjects mathematics, physics and chemistry. The probability of him getting grade A in these subjects are respectively 0.2, 0.3 and 0.9. Getting grades in different subjects are regarded as independent events. The probability of getting A grade by him, either in mathematics or physics, is

For what value of $k$, the following system have a unique solution? (where $\mathbb{R}$ is set of real numbers) $x + y + z = 1$ $2x + 3y + 4z = 3$ $x - y + kz = 5$

Three students A, B and C can respectively solve 50%, 25% and 20% of the problems in a book. A particular problem is selected at random from the book. The probability that at least one of them will solve the problem is

Let A = [aᵢⱼ]ₙₓₙ and B = [bᵢⱼ]ₙₓₙ. Then which of the following is/are true? (A) AB = BA (B) (AB)⁻¹ = B⁻¹ A⁻¹ (C) $(AB)^T = B^T A^T$ (D) AB = 0 ⇒ A = 0 or B = 0 Choose the correct answer from the options given below:

Which of the following statements is incorrect?

In an LPP, the feasible region represented by the set off constraints $2x + 3y \leq 18$, $x + y \leq 10$, $x \geq 0$, $y \geq 0$ is 

Let A be any square matrix of order n, then which of the following are true? (A) $| \text{adj } A| = |A|^{n-1}$ (B) $|A^{-1}| = \frac{1}{|A|}$ (C) $| \text{adj } A| = |A|^n$ (D) $(A^T)^{-1} = (A^{-1})^T$ Choose the correct answer from the options given below:

If A is a square matrix and I is the identity matrix of same order such that $A^2 = I$, then $(A - I)^3 + (A + I)^3 - 3A$ is equal to

If a matrix $A = \begin{bmatrix} 5 & -8 \\ -3 & 5 \end{bmatrix}$ then which of the following is / are TRUE? (A) $|A| = 1$ (B) $A$ is a singular matrix. (C) $-2A = \begin{bmatrix} 10 & -16 \\ -6 & 10 \end{bmatrix}$ (D) $AI = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$ $I$ is an identity matrix of order 2. Choose the correct answer from the options given below:

For LPP: Maximize $z = 2x + 3y$ subject to the constraints $x + y \geq 2$, $x + 2y \geq 3$, $x \geq 0$, $y \geq 0$, which of the following graph represents the feasible region of the above LPP as shaded portion?

If A and B are skew-symmetric matrices, then which of the following is not true?

Match List-I with List-II Let $A$ be any invertible square matrix. Then | List-I | List-II | | --- | --- | | (A) $A - A^T$ | (I) $\vert A\vert A^{-1}$ | | (B) $A A^T$ | (II) Skew-symmetric | | (C) $\det (A^{-1})$ | (III) Symmetric | | (D) $\text{adj} A$ | (IV) $[\det(A)]^{-1}$ | Choose the correct answer from the options given below:

If $A$ and $B$ are invertible matrices of order $3$ then match List-I with List-II | List-I | List-II | | --- | --- | | (A) $\text{adj}(A)$ | (I) $B^{-1} A^{-1}$ | | (B) $(AB)^{-1}$ | (II) $\vert A\vert ^{-1}$ | | (C) $\vert A^{-1}\vert $ | (III) $\vert A\vert ^2$ | | (D) $\vert \text{adj} A\vert $ | (IV) $\vert A\vert A^{-1}$ | Choose the correct answer from the options given below:

Which of the following statements is/are true? (A) The vector sum of the three sides of a triangle in order is $\vec{0}$ (B) The magnitude $(r)$, direction ratios $(a, b, c)$ and direction cosines $(l, m, n)$ of any vector $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$ are related as $l = \frac{a}{r}, m = \frac{b}{r}, n = \frac{c}{r}$ (C) If θ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then their cross product is given as $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin \theta$ (D) The cross product of two vectors is commutative Choose the correct answer from the options given below:

If $y = -4$ is a root of $\begin{vmatrix} y & 2 & 3 \\ 1 & y & 1 \\ 3 & 2 & y \end{vmatrix} = 0$, then the product of the other two roots is