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When the length and width of a rectangle are increased by 5 cm each, its area increases by 100 cm². However, if the length of the rectangle is increased by 1 cm and its breadth is decreased by 1 cm, then the area of the rectangle decreases by 6 cm².

What is the original area of the rectangle?

In a quadrilateral PQRS, PQ = RS and QR = SP and one angle between two adjacent sides is 90°.

The quadrilateral is definitely a

If $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$ in the system of equations $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$

S1 : The following is condition for inconsistent equations

S2 : There exists infinitely many solutions

S3 : The equations satisfying the condition are parallel

Which of the following statements are true ?

Kevin hired a cab and followed the path as shown in the figure. Find the difference between the total distance and displacement covered by him if all the distances in the figure are in kilometers? ___

The radius of two similar right circular cones are $2cm$ and $6cm$. The ratio of their volumes is

Which one of the following is not a factor of $x^3$+ 2$x^2$ + x?

Three girls Reshma, Salma, and Mandeep are playing a game by standing on a circle of radius $5m$ drawn in a park. Reshma throws a ball to Salma, Salma to Mandeep, Mandeep to Reshma. If the distance between Reshma and Salma and between Salma and Mandeep is $6m$ each. What is the distance between Reshma and Mandeep?

If $\sqrt{3}tan2\theta =3$ and $0<\theta \le {90}^{\circ }\theta$ then

$\theta$ = __________________

The midpoints D and E of the sides AB and AC of an equilateral triangle are joined together to form a line segment DE. If AD = 4cm, then find DE.

Cartesian plane consists of two fixed perpendicularly intersecting lines to represent the position of a point on a plane. What are the horizontal and vertical lines, respectively known as?

In quadrilateral ABCD, if $\angle A={60}^{\circ }$ and $\angle B:\angle C:\angle D=2:4:6$, then find the measure of $\angle C.$

Is quadrilateral ABCD a parallelogram?

I. Its opposite sides are equal.

II. Its opposite angles are equal.

The correct answer is: (a)/(b)/(c)/(d).

Assertion-and-Reason Type MCQ

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

$In\Delta ABC,\angle A={50}^{\circ }and\angle B={60}^{\circ }.\text{Determine the longest and shortest sides of the triangle.}$

Choose the correct answer in each of the following :

A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to that of the cone is

(a) 3:5 (b) 2:5 (c) 3:1 (d) 1:3

What is the number of sections a perpendicular bisector divides a line segment into?__

In the given pentagon ABCDE, AB = BC = CD = DE = AE. What is the value of x?

P,Q,R,S are respectively the midpoints of the sides AB,BC,CD and DA of ||gm ABCD.Show that PQRS is a parallelogram and also show that $ar\left(||gmPQRS\right)=\frac{1}{2}ar\left(||gmABCD\right)$.

In the given figure,if $\frac{y}{x}=5and\frac{z}{x}=4$ , then the value of x is ${12}^{\circ }$.

Factrorise
$16x^2+4y^2+9z^2-16xy-12yz+24xz$

$\text{If there are four line segments of length3cm, 7cm, 14cm and 21cm. Then how many triangles can be drawn from these line segments?}$

If any point on a perpendicular bisector is used to form a triangle along with the given line segment, what type of triangle is this triangle?__

Factorise:
$x^2-22x+120$

One of the roots of the equation $(x+\frac{1}{a}{)}^2-b^2=0$ is:

Given P = Product of $x^2$ y and $\frac{x}{y}$ and Q = Quotient obtained when $x^2$ is divided by D. If P = Q, what is the value of D?

ABCD is a quadrilateral and AP and DP are bisectors of ∠A and ∠D. The value of x is:

If (3a+2b) : (5a+3b) = 18 : 29, find a : b

Consider the following statements.

1. The Indian Bankruptcy Code has provided a resolution framework that will help corporates clean up their balance sheets.

2. Government strengthened the balance sheets of Public Sector banks through large re-capitalization package.

Factorise:x^{​​​​​​3}​-7x-6

Factorise: 27x (with power 4) - 8x

A reservoir in the form of cuboid is $50m$ long. If $20kl$ water is removed from the reservoir, water level goes down by $10cm$. Find the width of the reservoir.

Pinku was a hard working student who used to mug up without understanding. He was asked to construct a triangle say ABC and was given details like the base length of the triangle BC, one of the base angles say $\angle$ B and the sum of the other two sides (AB+AC). He went about constructing the triangle in the following way:

He drew the base BC with the given dimension, drew the $\angle$ B along the ray BX with the angle known to him already. He then took B as centre and (AB + AC) as radius and cuts an arc on the ray BX intersecting the ray at D.

He then joins D to C. He then draws a perpendicular bisector of the line DC and the perpendicular bisector intersecting on the ray intersects the ray at point A. The teacher then asked him as to why he did what he did, she retracked the steps and asked him as to how the intersection of the ray and the perpendicular bisector gives A.

Which of the following is the reason for him drawing the perpendicular bisector and intersecting it with the ray?

How many m³ of concrete will be required to fill a cylindrical column 10 m long and 7 m in diameter?

A hexagonal field of side 30 m is to be used for playing cricket. As circular fields are preferred for cricket, a circle of radius 30 m is made circumscribing the hexagon so as to make the field circular. Find the increase in the area of the cricket field.

What is the product of $(a+b)$ and $7a^2$$b^2$?

What is the ratio between AE and AF if EF = 1m ?

Which of these would you prefer for comparing two different sets of data of the same nature?

The graph below shows the ages of presidents of a country at the time of inauguration or start of their tenure.

Find the correct option(s) among the following.

In th figure, ABCD is a parallelogram, AE $\perp$ DC and CF $\perp$ AD. IF AB = 16 cm, AE = 8 cm and CF = 10 cm, Find AD.

Which of the following instruments can be used to create a right angle?

The sides of an equilateral triangle are (2a - b + 5), (a + b) and (2b - a + 2). What is the area of the triangle?

The perimeter of triangle is 450. And is sides are in ratio 12:13:5. Find its area

Find the mean of the given data.

$\begin{array}{cc}Classinterval& Frequency\\ 0-100& 7\\ 100-200& 11\\ 200-300& 15\\ 300-400& 9\\ 400-500& 8\end{array}$

The table given below shows the marks obtained by 30 students in a test.

$\begin{array}{cccccc}\text{Marks}& 1-10& 1-20& 21-30& 31-40& 41-50\\ \text{(Class interval)}& & & & & \\ \text{Number of students}& 7& 10& 6& 4& 3\\ \text{(Frequency)}& & & & & \end{array}$

Out of these students, one is chosen at random. What is the probability that the marks of the chosen student

(i) are 30 or less?

(ii) are 31 or more?

(iii) lie in the interval 21-30?

The value of $\left\{\right(23+2^2{)}^{\frac{2}{3}}+(150-29{)}^{\frac{1}{2}}{\}}^2$, is