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If a,b,c are in arithmetic progression , then show that the following are in arithematic progression :-

1. b+c , c+a , a+b

2. a+3k , b+3k , c+3k

3. a\7k , b\7k , c\7k

4. b+c-a , c+a-b , a+b-c

Mr. Shashank’s saving bank account details are as follows:

$\begin{array}{ccccc}Date& Particulars& Withdrawal\left(inRs\right)& Deposit\left(inRs\right)& Balances\left(inRs\right)\\ January{1}^{st}2006& B/F& -& -& 1270\\ January{7}^{th}2006& ByCheque& -& 2310& 3580\\ March{9}^{th}2006& ToSelf& 2000& -& 1580\\ June{26}^{th}2006& Bycash& -& 6200& 7780\\ June{10}^{th}2006& ToCheque& 4500& -& 3280\\ July{15}^{th}2006& ByClearing& -& 2630& 5910\\ October{18}^{th}2006& ToCheque& 530& -& 5380\\ October{27}^{th}2006& ToSelf& 2690& -& 2690\\ November{3}^{rd}2006& ByCash& -& 1500& 4190\\ December{6}^{th}2006& ToCheque& 950& -& 3240\\ December{23}^{rd}2006& ByTransfer& -& 2920& 6260\end{array}$

If he receives 198 as interest on 1st January, 2007, then what is the rate of interest paid by the bank?

Find the roots of the equation $5x^2–6x–2=0$ by completion of the square method.

Find the roots of the following quadratic equations by the factorisation method.
$\frac{2}{5}{x}^2-x-\frac{3}{5}=0$

The solution of the quadratic equation: x² -4x + 13 = 0

Draw the graph of xy = 20, x, y > 0. Use the graph to find y when x = 5, and to find x when y = 10

A motor cycle is a marked for sale for Rs 17600, which includes sale tax at 10%. Calculate the original price of the motor cycle.

The following pairs of linear equations 2x + 5y = 3 and 6x + 15y = 12 represent :

In the given figure, PB & QA are ⊥ AB. If PO = 6 cm, QO = 9 cm & area of ∆ POB = 120 cm²,find area ∆ QOA.

The ratio of the corresponding sides of two similar triangles is 1:3. The ratio of their corresponding heights is –

In the given figure, the circle touches the sides AB, BC, CD and DA of a quadrilateral ABCD at the points P, Q, R, S respectively. If AB = 11 cm, BC =$x$cm, CR = 4 cm and AS = 6 cm, the value of $x$ is

___ cm.

The total surface area of a right circular cylinder of height 9 m is 440 sq. meter. Find the radius of the cylinder. $(Take\pi =\frac{22}{7})$

What should be added to each of the numbers 1, 21, and 121 so that the resulting numbers are in continued proportion?
___

If α and β are zeroes of the polynomial x² ​​​​​​ - a(x + 1) - b such that (α + 1) ( β + 1) = 0, find the value of b.

Solve the quadratic equation by factorisation method:

25m²=9

In the given figure, XY is a diameter of the circle, PQ is a tangent to the circle at Y. Given that $\angle$AXB = ${50}^0$ and $\angle$ABX = ${70}^0$, calculate $\angle$BAY

The $9^{th}$ term of an A.P is 449 and ${449}^{th}$ term is 9. The term which is equal to zero is

Find the coordinates of the centroid of the ∆ whose vertices are A (–1, 9), B (1, –1), C (6, 1)

Prove the following trigonometric identities:

$\frac{tanA}{1+secA}-\frac{tanA}{1-secA}=2cosecA$

For what value of n, 5ⁿ ends with a zero

A bag(1) contains 6 red beads and 4 green beads. Another bag(2) contains 2 red and 2 green beads more. The probability of getting a red bead from which bag is more?

Which term of the AP : 3, 8, 13, 18, . . . , is 78?

In the given figure $\angle ABC={40}^{\circ }$, then angle $\angle BEA$ is equal to?

SOLVE:

If 2a+b=11 and ab=24 , find the value of 4a²+b².

If $x=\sqrt{a}sin\alpha$ and $y=\sqrt{a}cos\alpha$. Find the value of $y^2$ + $x^2$.

___

A man invests ₹20306 in buying shares of nominal value ₹26 at $10\%$ premium. Calculate the number of shares he buys.

Using the formula cos2θ = 1 - 2 sin²θ the value of sin15^o is

A point on the line x + y = 0 can be represented as

Which of the following advantageous features is common in direct tax and indirect tax?

If the sphere has a surface area of 256π cm${}^2$, what is its volume (approximated value)?

The line $x=0$ refers to ____

In the process of construction of ${45}^{\circ }$, what is the sum of all the other angles you would be constructing during the course?

A trader was moving along a road selling eggs. An idler who didn't have much work to do, started to get the trader into a wordy duel. This grew into a fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke. The trader requested the Panchayat to ask the idler to pay for the broken eggs. The Panchayat asked the trader how many eggs were broken. He gave the following response:

If counted in pairs, one will remain;
If counted in threes, two will remain;
If counted in fours, three will remain;
If counted in fives, four will remain;
If counted in sixes, five will remain;
If counted in sevens, nothing will remain;
My basket cannot accommodate more than 150 eggs.

So, how many eggs were there?

In a polygon, the interior angles are in A.P. with common difference $5^{\circ }$ and smallest angle ${120}^{\circ }$, then the number of sides is:

The area of the quadrilateral ABCD whose vertices are respectively A(1,1), B(7,-3), C(12,2) and D(7,21) is ... Sq. units

The 12^th term of the A.P. with first term 1/2 and common difference 1/12 is …

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the rim. When lead shots, each of which is a sphere of radius 0.5 cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

The sum of three numbers in an AP is 9. What is the common difference if the $5^{th}$ term is 35.

Find the length of the tangent AP

The sum of all the zeroes of polynomial p(x) = $5x^5-20x^4+5x^3+50x^2-20x-40$ is ___.

If $m=(cos\theta -sin\theta )$ and $n=(cos\theta +sin\theta )$ then show that

$\sqrt{\frac{m}{n}}+\sqrt{\frac{n}{m}}=\frac{2}{\sqrt{1-ta{n}^2\theta }}$