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How many irrational numbers exist between two rational numbers?

Consider a hyperbola $H:x^2-2y^2=4.$ Let the tangent at a point $P(4,\sqrt{6})$ meet the $x-$axis at $Q$ and latus rectum at $R(x_1,y_1),x_1>0.$ If $F$ is a focus of $H$ which is nearer to the point $P,$ then the area of $\Delta QFR$ is equal to:

Which term of the sequence $20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4}$......is the negative term?

In the given figure, O is a point inside a $\Delta PQR$ such that $\angle POR={90}^o,OP=6cm$ and $OR=8cm$. If $PQ=24cmandQR=26cm$, prove that $\Delta PQR$ is right - angled.

1.Two zeroes of the polynomial p(x)=x cube-4x square+x+6 are 2 and -1 of p(x)

A mathematician trying to cross a street happened to witness a bank robbery. When the police questioned him, he stated that the number plate of the van in which the thieves escaped had its last four digits as follows:
The first digit is 5. The last digit is the square of the second digit. The third digit is twice the second digit. Also, he noticed the sum of digits to be “9”. What is the quadratic equation that the police need to frame to find the $2^{nd}$ digit, given that $b$ is the second digit?

If the mean of $x$ and $\frac{1}{x}$ is M, the mean of $x^3$ and $\frac{1}{x^3}$ is _____.

Two circles of radii 8 cm and 5 cm with their centres A and B respectively touch externally as shown in the figure. Calculate the length of direct common tangent PQ

Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

[Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

Pranali and prasad started walking to the East and to North respectively, from the same point and at same speed.After 2 hours distance between them was 15√2km.Find their speed per hour.

In the given figure, $ABCDEF$ is a regular hexagon. $AB,CD$ and $EF$ are the diameters of the semicircles. If $BC=7\text{cm}$, then the area of the shaded region is equal to

Question 34

The circumference of a circle whose area is $81\pi r^2$, is

(a) $9\pi r$
(b) $18\pi r$
(c) $3\pi r$
(d) $81\pi r$

Can π be an exponent for any variable

If $△$ABC and $△$PQR are two similar triangles shown in the figure. AM and AN are the medians on $△$ABC and $△$PQR respectively. The ratio of areas of $△$ABC and $△$PQR is 9:25. If AM = PO = 5 cm. Find the value of 3ON.

If length , breadth and height of cuboid are increased by 20 %,what is the percentage increase in the the total surface area of cuboid?

People of Khejadli village take good care of plants, trees and animals. They say that plants and animals can survive without us, but we cannot survive without them. Inspired by her elders Amrita marked some land for her pets (Camel and Ox) and plants. Find the ratio of the areas kept for animals and plants to the living area.

What is the condition for two concentric circles of radii $r_1$ and $r_2$ to be congruent ?

A copper sphere of diameter 18 cm is drawn into a wire of diameter 4 mm. Find the length of the wire.

What is the value of 200 lb in kilograms?

If the cumulative frequency at a particular class interval 30 -35 is 19 and the cumulative frequency at the next class -interval i.e 35-40 is 27 , the frequency at 35-40 is __________

If sin (A + B) = 1, cos (A – B) = 1 and $0^{\circ }<A+B\le {90}^{\circ }$, find A and B.

The probability of getting a black, face card from a well-shuffled 52 pack card is___

The line $3x-2y=24$ meets $x-$axis at $A$ and $y-$axis at $B$. The perpendicular bisector of $AB$ meets the line through $(0,-1)$ and parallel to $x-$axis at $C$. Then $C$ is

The predecessor of the integer $-1$ is

Question 5
Convert the given frequency distribution into a continuous grouped frequency distribution.

$\begin{array}{cc}\text{Class interval}& \text{Frequency}\\ 150-153& 7\\ 154-157& 7\\ 158-161& 15\\ 162-165& 10\\ 166-169& 5\\ 170-173& 6\end{array}$

In which intervals would 153.5 and 157.5 be included?

If the product of the roots of $x^2–3x+k=10$ is –2, then find the value of k.