List of practice Questions

Arrange the following sentences in the correct order: I. I want to change the room.
P. Last month I got a job.
Q. I had been living there for six months.
R. The office is far from the room.
S. I want to cut expenses of travelling.
6. Hopefully I will do this next week.
AUSPICIOUS
RECOMPENSE
IMPEDE
She did not believed me.
I am fine, what about you?
Through the vertex \(O\) of parabola \(y^2=4x\), chords OP and OQ are drawn at right angles to one another. The locus of the midpoint of PQ is
Let \[ f(x)= \begin{cases} \dfrac{1-\sin^3 x}{3\cos^2 x}, & x < \dfrac{\pi}{2} \\ [6pt] p, & x = \dfrac{\pi}{2} \\ [6pt] \dfrac{q(1-\sin x)}{(\pi-2x)^2}, & x > \dfrac{\pi}{2} \end{cases} \] If \(f(x)\) is continuous at \(x=\dfrac{\pi}{2}\), then \((p,q)=\)
AUGMENT
CONSOLATION
AUXILIARY
Find the angle between the line \[ \frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6} \] and the plane \(10x+2y-11z=3\).
The equation of the right bisector plane of the segment joining \((2,3,4)\) and \((6,7,8)\) is
A bag contains \(n+1\) coins. It is known that one of these coins shows heads on both sides, whereas the other coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is \(\frac{7}{12}\), then the value of \(n\) is
A coin is tossed 7 times. Each time a man calls head. Find the probability that he wins the toss on more occasions.
Consider \(\dfrac{x}{2}+\dfrac{y}{4}\ge1\) and \(\dfrac{x}{3}+\dfrac{y}{2}\le1,\; x,y\ge0\). Then number of possible solutions are
If \(A=\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\), then \(A^{100}\) is
The area bounded by the x-axis, the curve \(y=f(x)\) and the lines \(x=1,\;x=b\) is equal to \(\sqrt{b^2+1}-\sqrt{2}\) for all \(b > 1\). Then \(f(x)\) is
Solution of differential equation \[ x^2-1+\left(\frac{x}{y}\right)^{-1}\frac{dy}{dx} +\frac{x^2}{2!}\left(\frac{dy}{dx}\right)^2 +\frac{x^3}{3!}\left(\frac{dy}{dx}\right)^3+\cdots=0 \] is
If the middle points of sides BC, CA and AB of triangle ABC are respectively D, E, F. If the position vectors of A, B, C are \(\hat{i}+\hat{j},\;\hat{j}+\hat{k},\;\hat{k}+\hat{i}\) respectively, then the position vector of the centre of triangle DEF is
The angle between any two diagonals of a cube is
The value of \(\lambda\), for which the lines \(3x-4y=13\), \(8x-11y=33\) and \(2x-3y+\lambda=0\) are concurrent is
The interval in which the function \(2x^3+15\) increases less rapidly than the function \(9x^2-12x\), is
The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and cost $48 per hour at 16 miles per hour. The most economical speed if the fixed charges (i.e., salaries etc.) amount to $30 per hour is:
Evaluate: \[ \int \frac{1}{1+3\sin^2 x+8\cos^2 x}\,dx \]