List of practice Questions

The point $A(3, 1, 6)$ is the mirror image of the point $B(1, 3, 4)$ in the plane $x - y + z = 5$. The plane $x - y + z = 5$ bisects the line segment joining $A(3, 1, 6)$ and $B(1, 3, 4)$.

Let $S_{1} = \sum\limits^{10}_{j = 1}j\left(j-1\right) ^{10}C_{j} , S_{2} = \sum\limits^{10}_{j = 1}j ^{10}C_{j}$ and $S_{3} =\sum\limits^{10}_{j = 1}j^{2} \,^{10}C_{j}.$ $S_3 = 55 \times $2^9$$ $S_1 = 90 \times 2^8$ and $S_2 = 10 \times 2^8$.

If two tangents drawn from a point P to the parabola $y^2 = 4x$ are at right angles, then the locus of P is

The number of $3 \times 3$ non-singular matrices, with four entries as $ 1 $ and all other entries as $0$, is

Let $\vec{a} = \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$. Then vector $\vec{b}$ satisfying $\vec{a} \times\vec{b}+\vec{c} = \vec{0}$ and $\vec{a} \cdot\vec{b} = 3$ is

The equation of the tangent to the curve $y = x +\frac{4}{x^{2}}, $ that is parallel to the $x-axis$, is

Solution of the differential equation $cos\, x\, dy = y(sin\, x - y) dx,\, 0 < x < \frac{\pi}{2}$ is

The area bounded by the curves $y =$ cos $x$ and $y=$ sin $x$ between the ordinates $x = 0$ and $x = \frac{3\pi}{2}$ is

A person is to count $4500$ currency notes. Let an denote the number of notes he counts in the nth minute. If $a_1 = a_2 = ...... = a_{10} = 150$ and $a_{10}, a_{11}, ......$ are in $A.P.$ with common difference $-2,$ then the time taken by him to count all notes is

A liquid is kept in a cylindrical vessel which is being rotated about a vertical axis through the centre of the circular base. If the radius of the vessel is $r$ and angular velocity of rotation is $\omega$, then the difference in the heights of the liquid at the centre of the vessel and the edge is

A hydrogen atom emits a photon corresponding to an electron transition from $n=5$ to $n=1$. The recoil speed of hydrogen atom is almost (mass of proton $\approx 1.6 \times 10^{-27} \,kg$ )

Spirulina belongs to kingdom

A factory produces $40\, kg$ of calcium in two hours by electrolysis. How much aluminium can be produced by same current in $2 \,hours$ if current efficiency is $50\%$?

Which factor controls the population density ?

The power factor of an $R-L$ circuit is $ \frac{ 1}{ \sqrt 2} $ . If the frequency of AC is doubled, what will be the power factor?

A galvanometer of resistance $ 15O $ gives full scale deflection for a current of $2\, mA$. Calculate the shunt resistance needed to convert it into an ammeter of range $5\,A$

Generally the path of a projectile from the earth is parabolic but it is elliptical for projectile going to a very large height. Up to ordinary height the projectile moves under a uniform gravitational force, but for great heights, projectile moves under a variable force.

One-fourth length of a spring of force constant k is cut away. The force constant of the remaining spring will be

A cyclist rides up a hill at a constant velocity. Determine the power developed by the cyclist if the length of the connecting rod of the pedal is $r = 25 \,cm$, the time of revolution of the rod is $t = 2 \,s$ and the mean force exerted by his foot on the pedal is $F = 15\, kgf$.

If $E=at+bt^2, $ what is the neutral temperature?

The output form of a full wave rectifier is

Genetically engineered bacteria are being employed for production of

A capillary tube of radius $r$ is immersed in water and water rises in it to a height $h$. The mass of water in the capillary tube is $5\,g$. Another capillary tube of radius $2\,r$ is immersed in water. The mass of water that will rise in this tube is

$ {A (C10H_18O) ->[{HCL}] P}$

Degree of unsaturation of $A = 2$, it contains no double or triple bonds.

A bullet of mass $m$ moving with velocity $v$ strikes a suspended wooden block of mass $M$. If the block rises to a $S$ height the initial velocity of the block will be