JEE Main 2024 PYQ: A is the set of natural-number solutions to 2x+3y=23, and B is the set of their x-values. Find the number of one-one functions from A to B.
JEE Main 2024 PYQ: find the number of triangles whose vertices are vertices of a regular octagon, but none of whose sides coincide with a side of the octagon.
JEE Main 2024 PYQ: A(α,β), B(1,0), C(γ,δ), D(1,2) are vertices of a parallelogram. Find 2(α+β+γ+δ) using the diagonal-bisection property of a parallelogram.
JEE Main 2024 PYQ: given z1+z2=5 and z1³+z2³=20+15i, find |z1⁴+z2⁴| by working through z1z2, then z1²+z2², using the standard algebraic identities.
A square is inscribed in the circle x²+y²−10x−6y+30=0 with one side parallel to y=x+3. Find Σ(xᵢ²+yᵢ²) over the four vertices, using the fact that the square’s diagonals lie along the horizontal and vertical through the centre.
If the length of the minor axis of an ellipse equals half the distance between its foci, find the eccentricity using c=2b and a²=b²+c².
Circle C1: x²+y²−2(x+y)+1=0 and C2 have centre (−1,0), radius 2. Their common chord meets the y-axis at P. Find the square of the distance from P to the centre of C1, using the radical axis.
Find the sum of all possible values of θ ∈ [-π, 2π] for which (1+i cosθ)/(1-2i cosθ) is purely imaginary. Multiple-choice: 3π, 2π, 5π, or 4π.
Find the sum of the squares of the modulus of complex numbers z=a+ib (a,b integers) satisfying |z−1|≤1 and |z−5|≤|z−5i|, by finding the disk and half-plane the condition defines.
Given α, β are the roots of x²−x+2=0 with Im(α)>Im(β), find α⁶+α⁴+β⁴−5α² by repeatedly reducing powers using α²=α−2.