Practice (Basic)

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Find the number of solutions to the equation $7\sin x + 2\cos^2 x = 5$ for $0\le x < 2\pi$.


Find the point on the circle $(x − 5)^2 + (y − 4)^2 = 4$ which is closest to the circle $(x − 1)^2 + (y − 1)^2 = 1$.


Let function $f(x)$ satisfy:  $$\int^1_0 3f (x) dx +\int^2_1 2f (x) dx = 7$$

and $$\int^2_0 f (x) dx + \int^2_1 f (x) dx = 1$$

Find the value of $$\int^2_0 f (x) dx$$


Find the area of the region bounded by the curve $y=\sqrt{x}$, the line line $y=x-2$, and the $x-$ axis.


Let $a$, $b$, $c$ and $d$ be real numbers. Find the relation of these four numbers such that the two curves $y=ax^2+c$ and $y=bx^2 + d$ have exactly two points of intersections.

Find the value of $c$ such that two parabolas $y=x^2+c$ and $y^2=x$ touch at a single point.


Explain why we cannot apply the cut-the-rope technique to count the non-negative integer solutions to the equation $$x_1 + x_2 + \cdots + x_k = n$$

For example, can we allow two cuts in the same interval thus to model one of the $x_i$ is zero?


Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?


How many even $4$- digit integers are there whose digits are distinct?


Derive the permutation formula $P_n^n=n\times (n-1)\times\cdots\times 2\times 1$ using the recursion method.


Find the minimal value of $4^m + 4^n$ if $m+n=3$.


Show that the $(k+1)$ leading digits of the number $\underbrace{333\cdots 3}_{k}4^2$ are all $1$s. Here $k$ is any positive integer.