Algebra PYQ — Page 37

If $|\vec{a}| = 3$ and $|\vec{b}| = 4$, then a value of $\lambda$ for which $\vec{a} + \lambda \vec{b}$ and $\vec{a} - \lambda \vec{b}$ are perpendicular is :

If a set P contains 5 elements and the set Q contains 8 elements, then the number of one-one functions from A to B is :

If $A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ 3 & -4 \\ 2 & 4 \end{bmatrix}$ then product AB is :

The variance of number of heads in three tosses of a coin is :

The corner points of the feasible region determined by the following system of linear inequalities : $2x + y \leq 10$, $x + 3y \leq 15$, $x, y \geq 0$ are (0, 0), (5, 0), (3, 4) and (0, 5). Let $z = px + qy$, where $p, q > 0$ condition on p and q so that maximum of z occurs at both (3, 4) and (0, 5) is :

For the following probability distribution : | X | 1 | 2 | 3 | 4 | |---|---|---|---|---| | P(X) | 1/10 | 1/5 | 3/10 | 2/5 | $E(X^2)$ is equal to :

Which of the following statements is incorrect regarding matrices ? For any matrices A and B of suitable orders,

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $f(x) = \frac{1}{x}, f : \mathbf{R} - \{0\} \to \mathbf{R} - \{0\}$ | (I) | neither injective nor surjective | | (B) | $f(x) = x^2, f : \mathbf{N} \to \mathbf{N}$ | (II) | surjective but not injective | | (C) | $f(x) = x^2, f : \mathbf{R} \to \mathbf{R}$ | (III) | injective but not surjective | | (D) | $f : \{1, 2, 3\} \to \{1, 2\}$ defined as $f : \{(1, 1), (2, 2), (3, 1)\}$ | (IV) | injective and surjective | Choose the **correct** answer from the options given below :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | Area of triangle $\Delta$ with adjacent sides $\vec{a}$ and $\vec{b}$ | (I) | $\vec{a} \times \vec{b}$ | | (B) | Area of parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$ | (II) | $\frac{1}{2}\lvert \vec{a} \times \vec{b} \rvert$ | | (C) | $(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})$ | (III) | $\lvert \vec{a} \times \vec{b} \rvert$ | | (D) | $\lvert \vec{a} \rvert \lvert \vec{b} \rvert \sin\theta \hat{n}$, where symbols have their usual meaning | (IV) | $2(\vec{a} \times \vec{b})$ | Choose the **correct** answer from the options given below :

Urn I contains 6 red balls and 4 black balls and Urn II contains 4 red balls and 6 black balls. One ball is drawn at random from Urn I and placed in Urn II. If one ball is drawn at random from Urn II, then the probability that it is a red ball is :

Which of the following statements are **correct** ? (A) $|A'| = |A|$, where A is the transpose of matrix A (B) If $A = [a_{ij}]_{3 \times 3}$, then $|4A| = 64|A|$ (C) $|A| = |\text{adj } A|^{n-1}$, where n is the order of the matrix (D) If A is an invertible matrix of order 2, then $\det(A^{-1})$ is equal to $\frac{1}{\det(A)}$ Choose the **correct** answer from the options given below :

If a fair coin is tossed 10 times, then the probability of obtaining at least one head is :

If $5x + y \leq 100$, $x + y \leq 60$, $x \geq 0$, $y \geq 0$. Then one of the corner points of the feasible region is :

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ & $\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}$ the angle between $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is :

The relation $R = \{(a, b) : a \leq b^2\}$ on the set of real numbers is:

A doctor is to visit a patient. It is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3}$ and $\frac{1}{12}$, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he arrives late. The probability that he comes by bus is:

Let A be the square matrix of order 3, then |kA|, where k is a scalar, is equal to:

Let $f(x) = x^3$ be a function with domain {0, 1, 2, 3} then domain of $f^{-1}$ is :

A manufacturing company makes two models M$_1$ and M$_2$ of a product. Each piece of M$_1$ requires 9 labour hours for fabricating and one labour hour for finishing. Each piece of M$_2$ require 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs.800 on each piece of M$_1$ and Rs.1200 on each piece of M$_2$ The maximum profit will be at the point

The number of all onto functions from the set {1, 2, .......n} to itself is

The unit vector in the direction of $\vec{a} + \vec{b}$ if $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ & $\vec{b} = -\hat{i} + \hat{j} + -\hat{k}$ is :

If a fair coin is tossed 10 times the probability of atleast 6 heads is:

In $\triangle ABC$ : (A) $\vec{AB} + \vec{BC} + \vec{CA} = \vec{O}$ (B) $\vec{AB} + \vec{BC} - \vec{AC} = \vec{O}$ (C) $\vec{AB} + \vec{BC} - \vec{CA} = \vec{O}$ (D) $\vec{AB} - \vec{CB} + \vec{CA} = \vec{O}$ (E) $\vec{AB} - \vec{CB} - \vec{CA} = \vec{O}$ Choose the correct answer from the options given below :

Which one of the following options is incorrect? For a square matrix A in the matrix equation AX = B.