List of practice Questions

If $\alpha \, and \, \beta$ are the roots of the equation ax+bx+c=0, then the value of $\alpha^{3} \, + \, \beta^{3}$ is

The volume of the tetrahedron with vertices P (-1, 2, 0), Q ( 2, 1, -3), R (1, 0, 1) and S (3, -2, 3) is

$If \, \, \vec{a}=\hat{i}+2\hat{j}+3\hat{k}, \, \vec{b}= \, -\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c} \, = \, 3\hat{i}+\hat{j}$ then t such that $\vec{a}+t\vec{b}$ is at right angle to $\vec{c}$ will be equal to

An equation of the plane passing through the line of intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0 and passing through (1, 1, 1) is

The length of the shortest distance between the lines $\vec{r}=3\hat{i}+5\hat{j}+7\hat{k}+\lambda(\hat{i}-2\hat{j}+\hat{k})$ and $\vec{r} = -\hat{i}-\hat{j}-\hat{k}+\mu (7\hat{i}-6\hat{j}+\hat{k})$ is

The particular integral of $\frac{d^2y}{dx^2}+2y=x^2$is

The value of the integral $\int\limits_{0}^{\frac{\pi}{2}} log \left(tan\,x\right)dx=$

In rolling two fair dice, what is the probability of obtaining a sum greater than 3 but not exceeding 6 ?

What is the least value of k such that the function x$^2$ + kx + 1 is strictly increasing on (1,2)

$\int\limits_{0}^{\pi /4} \log (1 + \tan \, x ) dx$ is equal to :

A can is taken out from a refrigerator at $0^{\circ} C$. The atmospheric temperature is $25^{\circ}C$. If $t_1$ is the time taken to heat from $0^{\circ} C$ to $5^{\circ} C$ and $t_2$ is the time taken from $10^{\circ} C$ to $15^{\circ}C$, then :

A body of mass $5\, kg$ is suspended by a spring balance on an inclined plane as shown in figure. The spring balance measure :

At what point of a projectile motion, acceleration and velocity are perpendicular to each other ?

The solution of $\sin^{-1}\,x-\sin^{-1}\,2x=\pm\frac{\pi}{3}$ is

An object is placed at a distance $20\, cm$ from the pole of a convex mirror of focal length $20\, cm$. The image is produced at :

A cell of constant emf first connected to a resistance $R_1$ and then connected to a resistance $R_2$. If power delivered in both cases is same then the internal resistance of the cell is :

Energy gas between valence band and conduction band of a semiconductor is :

Initially two gas samples $1$ and $2$ are at the same condition. The volume of the two are halved, one isothermally and the other adiabatically. What is the relation between the final pressure $P_1$ and $ P_2$ ?

In a $\triangle A B C$ if the sides are $a=3, b=5$ and $c=4$, then $\sin \frac{B}{2}+\cos \frac{B}{2}$ is equal to :

Angular momentum is conserved :

$(0, - 1)$ and $(0, 3)$ are two opposite vertices of a square. The other two vertices are :

A proton moving vertically downward enters a magnetic field pointing towards north. In which direction proton will deflect?

How many asymmetric carbon atoms are present in : (i) $1,2$-dimethylcyclohexane, (ii) $3$-methylcyclopentene and (iii) $3$-methylcyclohexene ?

Which of the following species has a bond order other than $3$ ?