List of practice Questions

If the line $ ax + 2y = 1 $, where $ a \in \mathbb{R} $, does not meet the hyperbola $ x^2 - 9y^2 = 9 $, then a possible value of $ a $ is:

Let $ PQR $ be a triangle such that \[ \vec{PQ}=-2\hat i-\hat j+2\hat k,\quad \vec{PR}=a\hat i+b\hat j-4\hat k,\ a,b\in\mathbb{Z}. \] Let $ S $ be the point on $ QR $ which is equidistant from the lines $ PQ $ and $ PR $. If \[ |\vec{PR}|=9 \quad \text{and} \quad \vec{PS}=\hat i-7\hat j+2\hat k, \] then the value of $ 3a-4b $ is:

Let $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to

In a microscope, the objective has a focal length $f_o=2$ cm and the eye-piece has a focal length $f_e=4$ cm. The tube length is 32 cm. The magnification produced by this microscope for normal adjustment is_____.

Given below are two statements:
Statement I: An object moves from position $ \vec{r}_1 $ to position $ \vec{r}_2 $ under a conservative force field $ \vec{F} $. The work done by the force is \[ W = -\int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r}. \]
Statement II: Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below:

Four persons measure the length of a rod as 20.00 cm, 19.75 cm, 17.01 cm and 18.25 cm. The relative error in the measurement of average length of the rod is :

The value of \[ \sum_{k=1}^{\infty} (-1)^{k+1}\left(\frac{k(k+1)}{k!}\right) \] is:

An atom $ ^8_3X $ is bombarded by a shower of fundamental particles and in 10 s this atom absorbed 10 electrons, 10 protons and 9 neutrons. The percentage growth in the surface area of the nucleus is recorded by:

If $y=y(x)$ satisfies the differential equation \[ 16(\sqrt{x}+9\sqrt{x})(4+\sqrt{9+\sqrt{x}})\cos y\,dy=(1+2\sin y)\,dx,\quad x>0 \] and \[ y(256)=\frac{\pi}{2},\quad y(49)=\alpha, \] then $2\sin\alpha$ is equal to

Given below are two statements:
Statement I: $CH_3)_3C^+$ is more stable than $CH_3CH_2^+$ as nine hyperconjugation interactions are possible in $(CH_3)_3C^+$.
Statement II: $CH_3^+$ is less stable than $(CH_3)_3C^+$ as only three hyperconjugation interactions are possible in $CH_3^+$.
In the light of the above statements, choose the correct answer...

Let $\alpha, \beta$ be the roots of the quadratic equation \[ 12x^2 - 20x + 3\lambda = 0,\ \lambda \in \mathbb{Z}. \] If \[ \frac{1}{2} \le |\beta-\alpha| \le \frac{3}{2}, \] then the sum of all possible values of $\lambda$ is

For a closed circuit Daniell cell, which of the following plots is the accurate one at a given temperature ?

Statement-I: An element ‘X’ of P-block forms a hydride H–X, which has the longest bond length, then element ‘X’ will have the shortest covalent radius.
Statement-II: An element ‘E’ of Group 15 forms hydride EH$_3$, that has least B.P. The maximum covalency of E is 4.

Choose the correct statements in respect of hydrides of group 15.
(A) Reducing power increases down the group
(B) Basic nature increases down the group
(C) Stability decreases down the group
(D) Boiling point decreases regularly down the group.

Three charges $+2q$, $+3q$ and $-4q$ are situated at $(0,-3a)$, $(2a,0)$ and $(-2a,0)$ respectively in the $x$-$y$ plane. The resultant dipole moment about origin is ___.

The correct sequence of reagents for the above conversion of X to Y is :

The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of 1 $\Omega$ is connected across these points is ______ J.

A bag contains 10 balls out of which $ k $ are red and $ (10-k) $ are black, where $ 0 \le k \le 10 $. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:

The strain-stress plot for materials A, B, C and D is shown in the figure. Which material has the largest Young's modulus?

A small bob A of mass m is attached to a massless rigid rod of length 1 m pivoted at point P and kept at an angle of 60° with vertical. At 1 m below P, bob B is kept on a smooth surface. If bob B just manages to complete the circular path of radius R after being hit elastically by A, then radius R is_______ m :

A thin prism with angle 5° of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion without deviation. The angle of second prism is :

The following diagram shows a Zener diode as a voltage regulator. The Zener diode is rated at $V_z = 5$ V and the desired current in load is 5 mA. The unregulated voltage source can supply up to 25 V. Considering the Zener diode can withstand four times of the load current, the value of resistor $R_s$ (shown in circuit) should be_______ $\Omega$.

In a perfectly inelastic collision, two spheres made of the same material with masses 15 kg and 25 kg, moving in opposite directions with speeds of 10 m/s and 30 m/s, respectively, strike each other and stick together. The rise in temperature (in °C), if all the heat produced during the collision is retained by these spheres, is (specific heat 31 cal/kg.°C and 1 cal = 4.2 J) :

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is:

A wire of uniform resistance $\lambda$ $\Omega$/m is bent into a circle of radius r and another piece of wire with length 2r is connected between points A and B (ACB) as shown in figure. The equivalent resistance between points A and B is_______ $\Omega$.