Sign in
Please select an account to continue using cracku.in
↓ →
Join Our JEE Preparation Group
Prep with like-minded aspirants; Get access to free daily tests and study material.
If $$b(a^2 - bc)(1 - ac) = a(b^2 - ca)(1 - bc)$$ where $$a \neq b$$ and $$abc \neq 0$$, prove that $$a + b + c = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$$.
a, b, c are three distinct positive integers. Show that among the numbers $$a^5b - ab^5$$, $$b^5c - bc^5$$, $$c^5a - ca^5$$ there must be one which is divisible by 8.
Loading solution...
There are four points P, Q, R, S on a plane such that no three of them are collinear. Can the triangles PQR, PQS, PRS and QRS be such that at least one has an interior angle less than or equal to $$45^\circ$$ ? If so, how? If not, why?
Loading solution...
.webp)
A straight line $$\ell$$ is drawn through the vertex C of an equilateral triangle ABC, wholly lying outside the triangle. AL, BM are drawn perpendiculars to the straight line $$\ell$$. If N is the midpoint of AB, prove that $$\triangle LMN$$ is an equilateral triangle.
Loading solution...
ABCD is a parallelogram. Through C, a straight line is drawn outside the parallelogram. AP, BQ and DR are drawn perpendicular to this line. Show that $$AP = BQ + DR$$. If the line through C cuts one side internally, then will the same result hold? If so prove it. If not, what is the corresponding result? Justify your answer.
Loading solution...
m, n are non-negative real numbers whose sum is 1. Prove that the maximum and minimum values of $$\frac{m^3 + n^3}{m^2 + n^2}$$ are respectively 1 and $$\frac{1}{2}$$.
Loading solution...
.webp)
(a) Solve for x the equation $$\frac{x+5}{2018} + \frac{x+4}{2019} + \frac{x+3}{2020} + \frac{x+2}{2021} + \frac{x+1}{2022} + \frac{x}{2023} = -6$$.
(b) $$\frac{a^2+b^2}{725} = \frac{b^2+c^2}{149} = \frac{c^2+a^2}{674}$$ and $$a - c = 18$$, find the value of $$(a + b + c)$$.
Loading solution...
If $$a + b + c + d = 0$$, prove that $$a^3 + b^3 + c^3 + d^3 = 3(abc + bcd + cda + dab)$$.
Loading solution...
Predict your JEE Main percentile, rank & performance in seconds
Educational materials for JEE preparation
91- 6303239042
