List of practice Questions

Let $ \alpha $ and $ \beta $ be the roots of $ a{{x}^{2}}+bx+c=0 $ . Then, $ \underset{x\to \alpha }{\mathop{\lim }}\,\frac{1-\cos (a{{x}^{2}}+bx+c)}{{{(x-\alpha )}^{2}}} $ is equal to

Standard deviation of first $n$ odd natural numbers is

The $A$.$M$. of $9$ terms is $15$. If one more term is added to this series, then the $A$.$M$. becomes $16$. The value of the added term is

The angle between the straight lines $x-1=\frac{2y+3}{3}=\frac{z+5}{2}$ and $x-3r+2; y=-2r-1; z=2,$ where $r$ is a parameter, is

The argument of the complex number $ \left( \frac{i}{2}-\frac{2}{i} \right) $ is equal to

The locus of a point which is equidistant from the points $(1,1)$ and $(3, 3)$ is

The slope of the normal to the curve $x=t^{2}+3t-8, y=2t^{2}-2t-5$ at the point $(2,-1)$ is

If the distance between the two points $(-1, a )$ and $(-1, -4a )$ is $10$ units, then the values of $a$ are

If the mean of six numbers is $41$, then the sum of these numbers is

If $ y={{\tan }^{-1}}\left( \frac{\cos x}{1+\sin x} \right), $ then $ \frac{dy}{dx} $ is equal to

Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?

The area of the triangle formed by the points $(2, 2), (5, 5), (6, 7)$ is equal to (in square units)

If $ n=5, $ then $ {{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+..... $ $ +{{{{(}^{n}}{{C}_{5}})}^{2}} $ is equal to

The number of words that can be formed by using all the letters of the word $PROBLEM$ only one is

The value of $\displaystyle \lim_{y \to \infty} \left[y \, sin \left(\frac{1}{y}\right) - \frac{1}{y} \right]$ is equal to

Factorise 

1.  \(a^ 4 - b^ 4\) 
2.  \(p^ 4 - 81\) 
3.  \(x^ 4 - (y + z)^ 4\) 
4.  \(x^ 4 - (x - z)^ 4\) 
5.  \(a ^4 - 2a ^2b^ 2 + b ^4\)

A $50\, Hz \,AC$ current of peak value $2\, A$ flows through one of the pair of coils. If the mutual inductance between the pair of coils is $150\, mH$, then the peak value of voltage induced in the second coil is

A body of mass $20\, g$ connected to spring of constant $k$ executes simple harmonic motion with a frequency of $\bigg(\frac{5}{\pi}\bigg)Hz$. The value of spring constant is

A comet orbits around Sun in an elliptical orbit. Which of the following quantities remains constant during the course of its motion?

A concave lens of focal length 20 cm produces an image half the size of the real object. The distance of the real object is

A monochromatic source of wavelength $60\, nm$ was used in Young?s double slit experiment to produce interference pattern. $I_1$ is the intensity of light at a point on the screen where the path difference is $150\, nm.$ The intensity of light at a point where the path difference is $200\, nm$ is given by

A particle of mass $m$ and charge $q$ is placed at rest in uniform electric field $E$ and then released. The kinetic energy attained by the particle after moving a distance $y$ is

From a circular ring of mass $M$ and radius $R$, an arc corresponding to a $ {{90}^{o}} $ sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is $k$ times $ M{{R}^{2}} $ . Then the value of $k$ is

A body is thrown up with a speed $u$, at an angle of projection $\theta$. If the speed of the projectile becomes $\frac{u}{\sqrt{2}}$ on reaching the maximum height, then the maximum vertical height attained by the projectile is