Download App

Arithmetic Progressions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsArithmetic Progressions5 Marks
Question
If $\text{a},\ \text{b},\ \text{c}$ are in A.P., then show that:
$\text{bc}-\text{a}^2,\ \text{ca}-\text{b}^2,\ \text{ab}-\text{c}^2$ are in A.P.
✓

Answer

To prove $\text{bc}-\text{a}^2,\ \text{ca}-\text{b}^2,\ \text{ab}-\text{c}^2$ are in A.P
$(\text{ca}-\text{b}^2)-(\text{bc}-\text{a}^2)=(\text{ab}-\text{c}^2)-(\text{ca}-\text{b}^2)$
$\text{LHS}=(\text{a}-\text{b}^2-\text{ca}+\text{a}^2)$
$=(\text{a}-\text{b})[\text{a}+\text{b}+\text{c}]\ ......(1)$
$\text{RHS}=\text{ab}-\text{c}^2-\text{ca}+\text{b}^2$
$=(\text{b}-\text{c})[\text{a}+\text{b}+\text{c}]\ .....(2)$
and since a, b, c are in ab
$\text{b}-\text{c}=\text{a}-\text{b}$
$\therefore\text{LHS}=\text{RHS}$
and
Thus, $\text{bc}-\text{a}^2,\ \text{ca}-\text{b}^2,\ \text{ab}-\text{c}^2$ are in A.P

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Arithmetic Progressions→Arithmetic Progressions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the equation of the parabola whose focus is $(1, -1)$ and whose vertex is $(2, 1)$. Also find its axis and latus rectum.
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{1}}\frac{1-\text{x}^2}{\sin\pi\text{ x}}$
→
Evaluate: $\lim _{x \rightarrow 2} \frac{x^3+3 x^2-9 x-2}{x^3-x-6}$
→
Solve the following equations:
$2\sin^{2}\text{x}=3\cos\text{x},0\leq\text{x}\leq2\pi$
→
Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}\frac{1^2+2^2+\ \cdots+\text{n}^2}{\text{n}^3}$
→
In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three subjects.
→
Mean and standard deviation of 100 observations were found to be 40 and 10 respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.
→
A person observes the angle of elevation of the peak of a hill from a station to be $\alpha.$ He walks c metres along a slope inclined at an angle $\beta$ and finds the angle of elevation of the peak of the hill to be $\gamma.$ Show that the height of the peak above the ground is $\frac{\text{c}\sin\alpha\sin(\gamma-\beta)}{(\sin\gamma-\alpha)}.$
→
Find equation of the line through the point (0, 2) making an angle $\frac{{2\pi }}{3}$ with the positive x-axis. Also, find the equation of line parallel to it an crossing the y-axis at a distance of 2 units below the origin.
→
If P(11, r) = P(12, r − 1) find r.
→