Class 12th

A uniform thin rope of length 12 m and mass 6 kg hangs vertically from a rigid support and a block of mass 2 kg is attached to its free end. A transverse short wave-train of wavelength 6 cm is produced at the lower end of the rope. What is the wavelength of the wave train (in cm) when it reaches the top of the rope?

The equation of the normal to the curve y = (1+x)²? + cos²(sin?¹x) at x=0 is

If A = {X = (x,y,z)?: PX=0 and x²+y²+z²=1}, where P = [1, 2, 1; -2, 3, -4; 1, 9, -1], then the set A

If a curve y=f(x), passing through the point (1,2), is the solution of the differential equation 2x²dy = (2xy+y²)dx, then f(1/2) is equal to

Let f: (-1, ∞) → R be defined by f(0) = 1 and f(x) = (1/x)log?(1+x), x≠0. Then the function f:

The oxidation states of transition metal atoms in K₂Cr₂O₇, KMnO₄ and K₂FeO₄, respectively, are x,y and z. The sum of x,y and z is [numerical value].

For the disproportionation reaction 2Cu⁺(aq) ⇌ Cu(s) + Cu²⁺ at 298 K, ln K (where K is the equilibrium constant) is * 10⁻¹.
Given: (E⁰(Cu²⁺/Cu⁺) = 0.16V, E⁰(Cu⁺/Cu) = 0.52V, RT/F = 0.025) [numerical value].

Consider the region R = {(x,y) ∈ R²: x² ≤ y ≤ 2x}. If a line y=α divides the area of region R into two equal parts, then which of the following is true?

The molecular geometry of SF₆ is octahedral. What is the geometry of SF₄ (including lone pair(s) of electrons, if any)?

The size of a raw mango shrinks to a much smaller size when kept in a concentrated salt solution. Which one of the following process can explain this?