List of practice Questions

Two tangent lines l₁ and l₂ are drawn from the point (2, 0) to the parabola 2y² = – x. If the lines l₁ and l₂ are also tangent to the circle (x – 5)² + y² = r, then 17r is equal to _________.

Let a curve y = y(x) pass through the point (3, 3) and the area of the region under this curve, above the x-axis and between the abscissae 3 and
$x(>3)\ be\ (\frac{y}{x})^3$
. If this curve also passes through the point (α,6√10) in the first quadrant, then α is equal to _______.

Given below are two statements:
Statement I: On heating with KHSO₄, glycerol is dehydrated and acrolein is formed.
Statement II: Acrolein has fruity odour and can be used to test glycerol’s presence.
Choose the correct option

Chlorophyll extracted from the crushed green leaves was dissolved in water to make 2 L solution of Mg of concentration 48 ppm. The number of atoms of Mg in this solution is $x × 10^{20}$ atoms. The value of $x$ is ________. (Nearest integer)
(Given : Atomic mass of Mg is 24 g mol^–1; N_A = 6.02 × 10²³ mol^–1)

A diatomic gas$ (γ = 1.4)$ does $400$ $J$ of work when it is expanded isobarically. The heat given to the gas in the process is ________ $J$.

Let A and B be two $3 × 3$ matrices such that AB = I and |A| =$\frac{1}{8}$. Then |adj (B adj(2A))| is equal to

A bullet of mass $200 g$ having initial kinetic energy $90 J$ is shot inside a long swimming pool as shown in the figure. If it’s kinetic energy reduces to $40 J$ within $1 s$, the minimum length of the pool, the bullet has to travel so that it completely comes to rest is

Let the hyperbola $H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ pass through ($2\sqrt2,-2\sqrt2$ ). A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

Let 3, 6, 9, 12, … upto 78 terms and 5, 9, 13, 17, … upto 59 terms be two series. Then, the sum of terms common to both the series is equal to _____.

For n ∈ N let $S_n$={z∈C:|z−3+2i|=$\frac{n}{4}$} and $T_n$={z ∈ C:|z−2+3i|=$\frac{1}{n}$}.Then the number of elements in the set

A particle experiences a variable force F=(4x$\hat i$+3y²$\hat j$) in a horizontal x–y plane. Assume distance in meters and force is Newton.If the particle moves from point (1, 2) to point (2, 3) in the x–y plane; then Kinetic Energy changes by

$2NOCl(g) ⇌ 2NO(g) + Cl_2(g)$
In an experiment, $2.0$ moles of $NOCl$ was placed in a one-litre flask and the concentration of $NO$ after equilibrium established, was found to be $0.4$ mol/ L. The equilibrium constant at $30°C$ is ____ $×10^{–4}$.

Let the solution curve y = y(x) of the differential equation (4 + x²)dy – 2x(x² + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.

Let P be the plane passing through the intersection of the planes

r→.(i+3k−k)=5 and r→ .(2i−j+k)=3,

and the point (2, 1, –2). Let the position vectors of the points X and Y be

i−2j+4k and 5i−j+2k

respectively. Then the points

‘Jurisprudence is as big as law and bigger’. This statement is given by

If the absolute maximum value of the function
f(x)=(x²–2x+7)e(4x³−12x²−180x+31) in the interval [–3, 0] is f(α), then

All black money funds terrorist activities. All terrorist activities are financed solely by corrupt politicians. If the Lokpal Bill is passed, all assets and financial sources of all corrupt politicians will be frozen. If demonetisation is declared, all black money will disappear. If the above statements are true, which of the following follow(s) ?
I. If demonetisation is declared, funding of terrorist activities will completely stop.
II. If the Lokpal Bill is passed, funding of terrorist activities will completely stop.
III. If demonetisation is declared and the Lokpal Bill is passed, funding of terrorist activities will completely stop.
Choose the most appropriate answer

If a₁ (> 0), a₂, a₃, a₄, a₅ are in a G.P., a₂ + a₄ = 2a₃ + 1 and 3a₂ + a₃ = 2a₄, then a₂ + a₄ + 2a₅ is equal to _______.

Capitalism is a philosophy. All ideas are fictional objects. Only fictional objects are philosophy. Examine the following statements :
I. Capitalism is an idea.
II. Capitalism is a fictional object.
III. Capitalism is not an idea.
Choose the most appropriate answer.

An electric bulb is rated as 200 W. What will be the peak magnetic field at 4 m distance produced by the radiations coming from this bulb? Consider this bulb as a point source with 3.5% efficiency.

$y=tan^{-1}(sec \;x^3 - tan\; x^3), \frac{π}{2} < x^3< \frac{3π}{2},$ then

From the top of a tower, a ball is thrown vertically upward which reaches the ground in 6 s. A second ball thrown vertically downward from the same position with the same speed reaches the ground in 1.5 s. A third ball released, from the rest from the same location, will reach the ground in ____ s.

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $-\hat{ i }+2 \hat{ j }-2 \hat{ k }$ is $30$, then $\alpha$ is equal to

A student in the laboratory measures thickness of a wire using screw gauge. The readings are 1.22 mm, 1.23 mm, 1.19 mm and 1.20 mm. The percentage error is $\frac{x}{121}$%. The value of x is _________.

Given two statements below:
Statement I : In $Cl_2$ molecule the covalent radius is double of the atomic radius of chlorine.
Statement II : Radius of anionic species is always greater than their parent atomic radius.
Choose the most appropriate answer from options given below: