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If a + b + c = 0, then prove the following

(a) (b + c) (b − c) + a(a + 2b) = 0

(b) a(a² − bc) + b(b² − ca) + c(c² − ab) = 0

(c) a(b² + c²) + b(c² + a²) + c(a² + b²) = −3abc

(d)

Question 25

Discount = Discount % of ___

Express
$(cos{83}^{\circ }-sec{76}^{\circ })$
in terms of trigonometric ratios of angles between $0^{\circ }$ and ${45}^{\circ }.$

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that

To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships warned. [Use π = 3.14]

A boy attends school for 'r' days in a month. The number of days he is absent in a month is two more than one sixth of the days he attends the school. The mathematical expression for the number of days he remains absent in a month is

In the assumed mean method, if A is the assumed mean, then deviation $d_i$ is :

An AP has the first term as 2 and the common difference as 4. Which term in the AP is the first multiple of 23 to appear in the AP?

The angle of elevation of the building from the top of the bus which is 3m above the ground is 30º and the angle of depression of the foot of the building from the same point is 60º. Find the height of the building.

In $△ABC$, right angled at C, AC = 4 cm and $\angle BAC={60}^o$, find the length of side BC.

A cylinder, a cone and a hemisphere have equal base and equal height. What is the ratio in their volumes?

The sum of roots of the equation (x+1)(x+2) + (x+3)(x+4) + (x+5)(x+6) +…. (x+9)(x+10) = 0 is

Question 6

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

The zeros of the polynomial $x^2-\sqrt{2}x-12$ are

In the given figure, the circles touch the lines at X , Y , Z , B , C and D.

If PQ = 8cm , QA = 9cm and PA = 7cm, then find the length PX , AZ and QY.

What is the maximum value of k for which 2013 can be written as a sum of k consecutive positive integers?

Question 2(i)

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

$t^2-3,2t^4+3t^3-2t^2-9t-12$

A bag contains 3 white, 4 red and 5 black balls. One ball is drawn at random. What is the probability that the ball drawn is neither black nor white?
(a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) $\frac{1}{3}$ (d) $\frac{3}{4}$

Find $\frac{x}{y}$ ; when $x^2+6y^2=5xy$.

Which of the following is a zero of $(49x^2-1)+(1+7x{)}^2$ ?

If the coordinates of the point of intersection of less than ogive and more than ogive is (12.5, 20) then the value of median is