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Draw the graphs of the following linear equations on the same graph paper:

$2x+3y=12,x-y=1$

Find the coordinates of the vertices of the triangle formed by the two straight lines and the y-axis. Also, find the area of the triangle.

In a parallelogram ABCD, if $\angle B={135}^{\circ }$ , determine the meaures of its other angles.

Two equal circles of radius r intersect such that each passes through the centre of the other. The length of the common chord of the circles is

Which of the following statements are true (T) and which are false (F) ?

(i) Sum of the three sides fo a triangle is less than the sum of its three altitudes.

(ii) Sum of any two sides of a triangle is greater than twice the median drawn to the third side.

(iii) Sum of any two sides of a triangle is greater than the third side.

(iv) Difference of any two sides of a triangle is equal to the third side.

(v) If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.

(vi) Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.

$Ifx-\frac{1}{x}=\frac{15}{4},\text{then}x+\frac{1}{x}$=

The perpendicular bisector of a line segment AB passes through the origin. If the coordinates of A are (-2, 0) then the distance of point B from the origin is ___ units.

Factorize:

$\frac{x^3}{216}-8y^3$

In the given figure, A and B are the centres of two circles having radii 5 cm and 3 cm respectively and intersecting at points P and Q respectively. If AB = 4 cm then the length of common chord PQ is

(a) 3 cm

(b) 6 cm

(c) 7.5 cm

(d) 9 cm

In the figure, prove that:

$\left(i\right)CD+DA+AB+BC>2AC$

$\left(ii\right)CD+DA+AB>BC$

Factorise:
$2x^2+11x-21$

In the given figure, $l\parallel m$ and $M$ is the mid-point of the line segment $AB.$ Which of the following options is/are correct?

If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides. Prove that the quadrilateral, so formed is cyclic.

$\text{The product}(x^2-1)(x^4+x^2+1)$ is equal to

If one angle of a triangle is greater than the sum of the other two, show that the triangle is obtuse angled.

Angles A, B, C of a triangle ABC are equal to each other. Prove that $\Delta ABC$ is equilateral.

A triangle has sides 35 cm, 54 cm and 61 cm long. Find its area. Also, find the smallest of its altitudes.

If the points A (6, 1), B (8, 2), C (9, 4) and D (p, 3) are vertices of a parallelogram, taken in order, find the value of p

The sum of digits of a two-digit number is 10. The difference between the number and the number obtained by reversing the digits is 18. Find the appropriate equations.

In the adjoining figure, AB is parallel to CD. Find the value of x.

Question 4

The range of the data 30, 61, 55, 56, 20, 26, 46, 28, 56 is

(a) 26

(b) 30

(c) 41

(d) 61

The locus of the foot of the perpendicular drawn from origin to a straight line which passes through a fixed point $P(h,k)$ is denoted by S. The tangent at $P(h,k)$ on the curve S is given by

In the figure given below, find z.

after 5 years the age of father will be two times the age of Son.write equation in two variables to represent this statement

The cork ball played in the cricket has a diameter of 7 cm .The density of cork used is $0.56gm/{cm}^3$. Calculate the weight of the ball. [3 MARKS]

If the point P lies on the line x = 5 and has its ordinate equal to –3, then its coordinates are

The factorisation of $a^4-125a$ is equal to .

In a Laboratory, the count of bacteria in a certain experiment was increasing at the rate of $2.5\%$ per hour. Find the bacteria at the end of $2$ hours if the count was initially $5,06,000$.

Find the median of following data:

83, 37, 70, 29, 45, 63, 41, 70, 34, 54

An isosceles triangle has perimeter 40 cm and each of the equal sides is 15 cm. The area of the triangle is

The product of two fractions is $20$. If one of the fractions is $\frac{1}{4}$, then what is the other fraction?

In parallelogram ABCD, if $\angle B-\angle C={40}^{\circ }$, then $\angle A$ is

The points (1, 0), (-4, 0) and (5, 0) lie on:

Find the area of the triangle formed by the lines $y=m_{1}x+c_1,y=m_{2}x+c_2$ and x = 0

Instructions for Ques 21 - Ques 23: Refer the figure below and answer the questions that follow. In the figure, line l is parallel to m

Which of the pairs of angles are equal to each other?

A bicycle wheel has a diameter of $7cm.$ What distance does the wheel cover on the ground if it completes 1 full rotation?

$(\text{Take}\pi =\frac{22}{7})$

In triangle ABC, AB = AC; BE $\perp$ AC and CF $\perp$ AB. Prove that :

(i) BE =CF

(ii) AF = AE

In the figure given below,
ABCD is a rhombus. Find the value of x.

Find the compound interest to the nearest rupee on Rs. 10,800 for $2-\frac{1}{2}$ years at 10% per annum.

A boat covers a distance of 32 km upstream and 36 km downstream in 7 hours. It also covers a distance of 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.

Solve the equations for $x$ and $y$.

$2x-3y=7$

$5x+y=9$

The cost of painting the inner curved surface area of a cylindrical container of depth 20 m is ₹ 8800. If the painter charges ₹ 40 per metre square to paint the cylindrical container, then find its radius.
[ $\text{Take}\pi =\frac{22}{7}$ ]

Solving $\frac{5}{2+x}$ + $\frac{1}{y-4}$ = 2; $\frac{6}{2+x}$ - $\frac{3}{y-4}$ = 1,
we get
x = ___
y = ___

Calculate the area of an isosceles triangle if the equal sides are 16cm each and the unequal side is 10cm.(without using heron's formula) [2 MARKS]

S1: $AM=\frac{AP}{2},BN=\frac{BP}{2}$

S2: $O{O}^{\prime }=AM+BN$

Choose the correct option.