There are 30 students in a mathematics class. 20 of the students have studied probability, 14 have studied set theory, and two people have studied neither. Find the number of students who: a. have studied both topics b. have studied exactly one of these subjects c. have studied set theory, but not probability.

In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the probability that (a) the student took mathematics or history; (b) the student did not take either of these subjects; (c) the student took history but not mathematics.

The table shows the distance travelled by train between various places in India and the cost of the journey.

$$\begin{matrix} \text{Distance (km)} & \text{204} & \text{1407} & \text{1461} & \text{793} & \text{1542} & \text{343} & \text{663} & \text{780}\\ \text{Cost (rupees)} & \text{390} & \text{2200} & \text{2270} & \text{1390} & \text{2280} & \text{490} & \text{1200} & \text{1272}\\ \end{matrix}$$

a. Find the correlation coefficient, r and comment on your result. b. Find the equation of the regression line. c. Use your equation to estimate the cost of a 1000 km train journey.

Consider the function $f(x)=3 x-x^{2}-1$. a. Write down $f^{\prime}(x)$. b. Show that at x = 1, $f(x)=f^{\prime}(x)$. c. Find the coordinates of a second point on the curve y = f(x) for which $f(x)=f^{\prime}(x)$.

Let

$$w = ( 4 x - 3 y + 2 z ) ^ { 5 }$$

. Find

$$\frac { \partial ^ { 2 } w } { \partial x \partial z }$$

What factors affect how strongly an atom is attracted to its electrons?

Given 3y + x = 30, find the maximum/minimum value of 2xy and determine its nature.

Explain why the pH of a solution containing both acetic acid and sodium acetate is higher than that of a solution containing the same concentration of acetic acid alone.

Choose the word or words that best complete the sentence. People who are descendants of a region's first inhabitants are ________ people. A. ruling B. religious C. indigenous D. conquering

Write each rational number as a decimal. State whether the decimal is terminating or repeating. If the decimal repeats, state the block of digits that repeats. -1/6

/hat is the difference between delivery cycle time and throughput ti.me? What four elements make up throughput time? What elements of throughput time are value-added and what elements are non-value-added?

What is the difference between tangential and radial acceleration for a point on a rotating body?

$$9x^2-1=0$$

Where is the tropical wet and dry climate found?