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ABC is an equilateral triangle of side length 20 cm. The minimum distance of vertex A from its opposite side is _____.

Show that

$\frac{1\times 2^2+2\times 3^2+\dots \dots +n\times (n+1{)}^2}{1^2\times 2+2^2\times 3+\dots \dots +n^2\times (n+1)}=\frac{3n+5}{3n+1}$

Question 57

In the following question, fill in the blanks to make the statements true.

The usual form fo $2.3\times {10}^{10}$ is ___ .

Question 3(iv)
Find the value of:

$\left(3^{-1}+4^{-1}+5^{-1}\right)^0$

What is the simplest rationalising factor of cube root of 500

The co-ordinate of point A in number line is 5.Find the co-ordinate of points on the same number line which are 13 units away from point A

How to find irrational number between any two numbers

Choose the correct statement

Choose the correct order for the sequence.

The young two American students

Find coordinates of the points of trisection of the line segment joining the point.

A (-4,2) and B (-4, 9)

In the adjoining figure, ∠a = (120 - x)^o and ∠b = 5x^o, the measures of ∠a and ∠b.

In the figure, ABCD is a trapezium in which AB || DC and its diagonals AC and BD intersect at O. Then ar(∆AOD) is equal to

In parallelogram ABCD, AB = 16 cm. The altitudes corresponding to sides AB and AD are respectively 8 cm and 10 cm. Then AD is equal to

If the base of a parallelogram is 8 cm and its altitude is 5 cm, then its area is equal to

The value of m, if a = 3 and b = 1 is the solution of equation 2a + 3b = m is:

Express $\frac{1}{32}$ in exponential form

The Pythagorean theorem describes the relationship between three sides of a

The value of cot $\theta$ in the given right triangle is

In the given figure, ABCDE is a pentagon. EG // DA meets BA produced at G and CF // DB meets AB produced at F. The ratio of area of pentagon ABCDE : area of $\Delta GDF$ is

Two notebooks and one pencil cost ₹ 35, while 3 notebooks and four pencils cost ₹ 65. Calculate the cost of one notebook and one pencil.

In the given figure, PA, QB and RC are perpendiculars to AC.

Determine the value of $\frac{1}{x}+\frac{1}{z}$.

In a right triangle ABC, the right angle is at B. Which of the following is true about the other two angles A and C?

The largest cube is carved out of a sphere of diameter $10\sqrt{3}cm$. Find the volume of the cube.

Given the area of rectangle is $A=25a^2-45a+18$. The length is given as ($5a-3$). Therefore, the width is:

The value of polynomial p(x) = x² – a² at x = -a

Two parallel lines $l$ and $m$ are intersected by a transversal $t$. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

Factorise
$27a^3+64b^3$

In the figure, M, N and P are the mid points of AB, AC and BC respectively. If MN = 3 cm, NP = 3.5 cm and MP = 2.5 cm, calculate BC, AB and AC.

The mean of a,b,c,d and e is 28. If the mean of a,c, and e is 24, what is the mean of b and d?

Describe some fundamental characteristics of statistics.

P and Q are points on opposite sides AD and BC of a parallelogram ABCD such that PQ passes through the point of intersection O of its diagonals AC and BD. Then, point O ___.

The table given below shows the months of birth of 36 students of a class:

$\begin{array}{ccccccccccccc}\text{Month of birth}& \text{Jan.}& \text{Feb.}& \text{March}& \text{April}& \text{May}& \text{June}& \text{July}& \text{Aug.}& \text{Sept.}& \text{Oct.}& \text{Nov.}& \text{Dec.}\\ \text{No. of students}& 4& 3& 5& 0& 1& 6& 1& 3& 4& 3& 4& 2\end{array}$

A student is chosen at random from the class. What is the probability that the chosen student was born in October?

(a) $\frac{1}{3}$

(b) $\frac{2}{3}$

(c) $\frac{1}{4}$

(d) $\frac{1}{12}$

Three angles of a quadrilateral are respectively equal to ${110}^{\circ },{50}^{\circ }$ and ${40}^{\circ }$. Find its fourth angle.

In the figure, write all pairs of adjacent angles and all the linear pairs.

Inradius of a regular hexagon of sides 6 cm =

The given graph represents the equation _____.

Prove that each angle of an equilateral triangle is ${60}^{\circ }$

If $(x+y{)}^3-(x-y{)}^3-6y(x^2-y^2)=k{y}^2$ , then k =

For what value of m, the system of equations $mx+3y=m–3,and12x+my=m$ will have no solution.

The distance of the point P (4,3) from the origin is

Question 154

Write the following in expanded form.

(a) 74836 (b) 574021 (c) 8907010

Two circles with centres, A and B and of radii 5 cm and 3 cm respectively touch each other internally. If the perpendicular bisector of segment AB meets the bigger circle in P and Q. Find the length of PQ.