List of practice Questions

Find the shortest distance between the lines \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}) \]

If \( R \) is the relation "less than" from \( A = \{1,2,3,4,5\} \) to \( B = \{1,4,5\} \), find the set of ordered pairs corresponding to \( R \). Also, define this relation from \( B \) to \( A \).

Find the value of: \[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) \]

Find the vector equation of the line which passes through the point \( (5,2,-4) \) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 8\hat{k} \).

Differentiate \( y = (\cos x)^{\sin x \).}

A die is thrown once. The number on the die is a multiple of 3 is denoted by \( E \), and the number on the die is even is denoted by \( F \). Are \( E \) and \( F \) independent events?

\( R \) is a relation on a set of natural numbers \( N \) defined by \[ R = \{(a, b): a, b \in N \text{ and } a = b^2 \} \] Is \( (a, b) \in R \), \( (b, c) \in R \Rightarrow (a, c) \in R \) true? Justify it by one example.

The probability of \( A \) winning the race is \( \frac{1}{3} \) and that of \( B \) is \( \frac{1}{4} \). In this race, find the probability that neither \( A \) nor \( B \) can win the race.

Show that the function

\[ f(x) = \begin{cases} x^2 + 2, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases} \]

is not continuous at \( x = 0 \).

(e) The degree of the differential equation \( d^2y/dx^2 = \sqrt{x} + \left(\frac{dy}{dx}\right)^2 \) is:

What is the number of straight lines and the number of triangles in the given figure.

Answer the following questions based on the information given below: The proportion of male students and the proportion of vegetarian students in a school are given below. The school has a total of 800 students, 80% of whom are in the secondary section and the rest equally divided between class 11 and 12.

Which is odd?
Jeevan Raksha Padak, ParamVir Chakra, Padma Bhusan, Bharat Ratna

If DARE is coded as 1083 and FATE is coded as 2093, then FARE is coded as?

A, B, C and D are to be seated in a row. But C and D cannot be together. Also, B cannot be at the third place. Which of the following must be false?

How many male members are there in the family?

With utmost respect to the Supreme Court, it is absolutely incorrect to state domestic violence is gender-neutral.

The leadership dialogue recognized that we are living in a period of immense turbulence, disruption and even subversion.

The efforts of the central government in ensuring food security should be replicated in States that are ought to fully implement the Act and reform their respective distribution systems.

The committee has been given a wide-ranging mandate that included examining the feasibility of a national law to ban cow slaughter by amending the Constitution.

We are always looking for ways to operate effectively to serve our members. This requires as to review our corporate structure to ensure that we are organized in the right way to best meet the needs of our members.

The parent company have given a mandate to either turn around the business or face a massive scale down, said another person aware of Walmart’s plans.

“The government has taken a position to wait and watch and there is no need to panic,” the minister said

Thousands of people were forced to evacuate their homes around Taal, one of the world’s smallest active volcano which spewed ash for the second day from its crater in the middle of the lake.