List of practice Questions

If set \( A \) contains 8 elements, then the number of subsets of \( A \) which contain at least 6 elements is:
If \(x^2 + 5ax + 6 = 0\) and \(x^2 + 3ax + 2 = 0\) have a common root, then that common root is:

If \[ A = \begin{bmatrix} 1 & 0 & 2\\ 2 & 1 & 3 \\3 & 2 & 4 \end{bmatrix}, \] then evaluate \( A^2 - 5A + 6I \)=

Find the equation of a line passing through the intersection of \( 3x + y - 4 = 0 \) and \( x - y = 0 \), and making a \( 45^\circ \) angle with \( x - 3y + 5 = 0 \).
A spring of spring constant \( 200 \, {N/m} \) is initially stretched by \( 10 \) cm from the unstretched position. The work to be done to stretch the spring further by another \( 10 \) cm is:
If \( Z = x + iy \) is a complex number, then the number of distinct solutions of the equation \[ z^3 + \bar{z} = 0 \] is:
The range of the real valued function \( f(x) = \frac{15}{3 \sin x + 4 \cos x + 10} \) is:
The square root of the independent term in the expansion of \[ \left( \frac{2x^2}{5} + \frac{5}{\sqrt{x}} \right)^{10} \] is:
The equilibrium constant for the dissociation of HI at 773 K is:
200 g of 20\% w/w urea solution is mixed with 400 g of 40\% w/w urea solution. What is the weight percentage (w/w \%) of resultant solution?
In \( \triangle ABC \), if \( a = 13 \), \( b = 14 \), and \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \), then \( 2r_1 = \):
If \( l, m, n \) are the direction cosines of a line that is perpendicular to the lines having the direction ratios \( (1,2,-1) \) and \( (-2,1,1) \), then \( (l+m+n)^2 \) =
If \( y = f(x) \) is a thrice differentiable function and a bijection, then \[ \frac{d^2x}{dy^2} \left(\frac{dy}{dx}\right)^3 + \frac{d^2y}{dx^2} = ? \]
In a certain code, ‘WRONG’ is written as ‘FMNQV’. How is ‘RIGHT’ written in that code?
Complete the sentences given in List-I with the appropriate prepositions given in List-II:
Table
If \( P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e \) is a polynomial such that: \[ P(0) = 1, \quad P(1) = 2, \quad P(2) = 5, \] \[ P(3) = 10, \quad P(4) = 17, \] then find the value of \( P(5) \)=
The capacitance of an isolated sphere of radius \( r_1 \) is increased by 5 times when enclosed by an earthed concentric sphere of radius \( r_2 \). The ratio \( \frac{r_1}{r_2} \) is:
A car travelling at 80 kmph can be stopped at a distance of 60 m by applying brakes. If the same car travels at 160 kmph and the same braking force is applied, the stopping distance is:
If \( p \) and \( q \) be the longest and the shortest distance respectively of the point (-7,2) from any point (\(\alpha, \beta\)) on the curve whose equation is \[ x^2 + y^2 - 10x - 14y - 51 = 0 \] then the geometric mean (G.M.) of \( p \) is:
From a point A(0,3) on the circle \[ (x + 2)^2 + (y - 3)^2 = 4 \] a chord AB is drawn and extended to a point Q such that AQ = 2AB. Then the locus of Q is:
If the focus of the parabola \[ (y - k)^2 = 4(x - h) \] always lies between the lines \(x + y = 1\) and \(x + y = 3\) then:
Let \(L_1\) be the length of the common chord of the curves \[ x^2 + y^2 = 9 \quad {and} \quad y^2 = 8x \] and let \(L_2\) be the length of the latus rectum of \(y^2 = 8x\). Then:
The foci of the hyperbola \[ 4x^2 - 9y^2 - 1 = 0 \] are:
Given a real-valued function \( f \) such that: \[ f(x) = \begin{cases} \frac{\tan^2\{x\}}{x^2 - \lfloor x \rfloor^2}, & \text{for } x > 0 \\ 1, & \text{for } x = 0 \\ \sqrt{\{x\} \cot\{x\}}, & \text{for } x < 0 \end{cases} \] Then:
Let \( f(x) = \sin x \), \( g(x) = \cos x \), and \( h(x) = x^2 \). Then, evaluate: \[ \lim\limits_{x \to 1} \frac{f(g(h(x))) - f(g(h(1)))}{x - 1} \]