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Two circles intersect as shown in the diagram below. The radii of the circles are $14cm$ each and $\angle AOP={45}^{\circ }$​ What is the area of the shaded region?

In the following question, two equations are given. Solve both the equations and mark the corresponding answer, if:

a) x is greater than y

b) x is less than y

c) x is greater than or equal than y

d) x is less than or equal to y

e) x = y or a definite relationship between x & y cannot be established

2x${}^2$ + 3x + 1 = 0

12y${}^2$ + 7y + 1 = 0

Which of the following sequences are in arithmetic progession ?

(i) 2,6,10,14, ........

(ii) 15,12,9,6,.........

(iii) 5,9,12,18,.......

(iv) $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},.....$

Question 18
If we rotate a right-angled triangle of height 5 cm and base 3 cm about its base, we get
a) Cone of height 3 cm and base 3 cm
b) Cone of height 5 cm and base 5 cm
c) Cone of height 5 cm and base 3 cm
d) Cone of height 3 cm and base 5 cm

A box contains 100 bolts and 50 nuts. It is given that 50% bolts and 50% nuts are rusted. Two objects are selected from the box at random. Find the probability that either both are rusted.

Expand the following, using suitable identities:
$1){(3a-7b-c)}^2$.
$2){(x+2y+4z)}^2$ [4 MARKS]

The cost of an article is Rs 6000 to a distributor. If he sells it to a trader for Rs 7500. The trader sells it to a customer for Rs 8000. If the VAT rate is 12.5%; find the VAT paid by the:

i) distributor ii) trader

Given a circle with O as the centre. Find the value of x.

Find two solutions for the equations:
(i) 2x - y = -5
(ii) y - 3x = 7 [4 MARKS]

Question 119 (vii)

Write two integers which are greater than -4 but their difference is smaller than -4.

In the given figure, if $\angle ACB=\angle CDA,AC=8cm$ and $AD=3cm$, then find $BD$.

If $\frac{x^2+y^2}{x^2-y^2}=\frac{17}{8}$, then $\frac{x}{y}$?

All theorems in the REAL NUMBERS

Solve for x and y: 3x + 2y = 11 and 2x + 3y = 4. Also find p if p = 8x + 5y

Consider A = {2x + 1: 5 $\le$ x $\le$ 10 and x $\in$ I} and B = { 2x - 3 : 5 $\le$ x $\le$ 10 and x $\in$ I}. Find A $\Delta$ B (A symmetric difference B)

If $S=4-9x+16x^2-25x^3+36x^4-49x^5+......\infty$. Find the value of S for x=$\frac{-1}{2}$.

In the given figure, a circle is inscribed in a trapezium of height 14 cm and lengths of parallel sides are equal to 25 cm and 40 cm. What is the area of the shaded region?

If n is a natural number then 8^n - 3^n is divisible by which number?

Which of the following functions are identical to f(x)?

$f\left(x\right)=\{\begin{array}{cc}x,& 1\le x<2\\ x^2,& 2\le x<3\end{array}$

(1 $\le$ x < 3 )

Which of the following are sure events?

In the given series, 1.5, 0.5, -0.5, -1.5, .... , -8.5 If $n^{th}$ term is -8.5, find n.

Circles with centres P and Q intersect at points A and B as shown in the figure.

CBD is a line segment and EBM is tangent to the circle, with centre Q, at point B. If the circles are congruent ;

show that CE = BD.

Which all statements are true :

(i) Two circles with different radii are similar.

(ii) Any two rectangles are similar.

(iii) If two triangles are similar then their corresponding angles are equal and their corresponding sides are equal.

(iv) The length of the line segment joining the midpoints of any two sides of a triangle is equal to half the length of the third side.

(v) In a $\Delta ABC,AB=6cm,\angle A={45}^{o}andAC=8cm$ and in a $\Delta DEF$ DF = 9 cm, $\angle D={45}^o$ and DE = 12 cm, then $\Delta ABC\sim \Delta DEF$.

(vi) The polygon formed by joining the midpoints of the sides of a quadrilateral is a rhombus.

(vii) The ratio of the areas of two similar triangles is equal to the ratio of their corresponding angle-bisector segments.

(viii) The ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding medians.

(ix) If O is any point inside a rectangle ABCD then $O{A}^2+O{C}^2=O{B}^2+O{D}^2$.

(x) The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals.

The tangent to a circle is perpendicular to the radius at the ___.

The sum of the sequence 1, $\left(\frac{1}{2}\right)$,$\left(\frac{1}{4}\right)$, ...... till infinity is

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

The value of q if x = 2 is a solution of 8x² + qx - 4 = 0 is _____

A straight line AB is 8 cm long.Draw and describe the locus of a point which is:

(i) always 4 cm from the line AB.

(ii) equidistant from A and B.

Mark the two points X and Y,which are 4 cm from AB and equidistant from A and B.Describe the figure AXBY.

The radii of the circles ends of a bucket of height 15 cm are 14 cm and r cm (r < 14 cm).
If the volume of bucket is 5390 $c{m}^3$, then find the value of r.
$[Use\pi =\frac{22}{7}]$

Find the centroid of the triangle with vertices A$(x_1,y_1)$, B$(x_2,y_2)$ and C$(x_3,y_3)$.

Calculate the height of a water-fall through which the fall of water leads to increase in its temperature by 1⁰C.

The first term of an A.P. is 12, the last term is -8, the common difference is -2. Find the sum of the A.P.

AB is the longest chord of a circle with centre O. P is any point on the circumference of the circle, not coinciding with A or B. Find the angle $\angle$APB in degrees.

Find the odd and even extensions of f(x) = $x^4-x^3+x^2(x<0)$ .

[ Assume the domain and range is x < 0 for the given function ]

If the roots of the equation $a{x}^2+2bx+c=0$ and $b{x}^2-2\sqrt{ac}x+b=0$ are simultaneously real then prove that $b^2=ac$.

In a village, a well with 10 m inside diameter, is dug 14 m deep. Earth taken out of it is spread all around to a width of 5 m to form an embankment. Find the height of the embankment. What value of the villagers is reflected here?

A park is in the form of a rectangle 120 m by 90 m. At the centre of the park, there is a circular lawn as shown in the figure. The area of the park excluding the lawn is 2950 m^2. Find the radius of the circular lawn. [Given, $\pi$ = 3.14]

There are two circles each with radius 5 cm. The length of tangent AB is 26 cm. Find the length of tangent CD.

S1 : A degree 10 equation can have 5 real roots .

S2 : Complex roots always occurs in pairs.

A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 2.1 cm and the height of the cone is 4 cm. The solid is placed in a cylindrical tub full of water in such a way that the whole solid is submerged in water. If the radius of the cylinder is 5 cm and its height is 9.8 cm, find the volume of the water left in the tub.

Point A (x, 5) is reflected in the origin and its image is (2, y). Write down the values of x and y.

The daily pocket expenses of 206 students in a school are given below.

$\begin{array}{ccccccccc}\text{Pocket expense}& 0-5& 5-10& 10-15& 15-20& 20-25& 25-30& 30-35& 35-40\\ \text{(in rupees)}& & & & & & & & \\ \text{Number of students}& 10& 16& 30& 42& 50& 30& 16& 12\\ \text{(Frequency)}& & & & & & & & \end{array}$
Construct a frequency polygon representing the above data. [4 MARKS]

What is the reciprocal of $cot\theta$?

Find the value of k for which the quadratic equation $(3k+1)x^2+2(k+1)x+1=0$ has equal roots. Also, find the roots.