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The sum of the squares of two consecutive natural numbers is 421. The numbers are

The invariant point under reflection in the x-axis, y-axis and origin is ________________

A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is ${60}^{\circ }$. When he moves 30 metres away from the bank, he finds the angle of elevation to be ${30}^{\circ }$. Find the height of the tree and the width of the river. [Take $\sqrt{3}=1.732$] [4 MARKS]

If matrix $A=\left[\begin{array}{cc}6& 11\\ 2& 4\end{array}\right],$ then the value of determinant of $A^{2015}-6A^{2014}$ is

A container made of a metal sheet open at the top is of the form of frustum of cone, whose height is 16 cm and the radii of its lower and upper circular edges are 8 cm and 20 cm respectively. Find

(i) the cost of metal sheet used to make the container if it costs Rs 10 per 100 $c{m}^2$

(ii) the cost of milk at the rate of Rs 35 per litre which can fill it completely.

In below shown figure, $\frac{PS}{SQ}=\frac{PT}{TR}$ and$\angle PST=\angle PRQ$. Then $\Delta$PQR is an __ triangle.

Three consecutive positive integers are such that the sum of the square of the first and the product of other two is 46, find the integers,

If a circle is drawn with centre 'C' (0,5) and radius 13 units then the point (12, 10) lies

_{^{​​​If the difference of mode and median of a data is 24, then the difference of median and mean is?}}

What is the slope of the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$?

The parabolas $y^2=4x$ and $x^2=4y$ divide the square region bounded by the lines $x=4,y=4$ and the coordinate axes. If $S_1,S_2,S_3$ are respectively the areas of these parts numbered from top to bottom; then $S_1:S_2:S_3$ is

Find the roots of quadratic equation $x^2-5x+4=0$.

In the given figure,ABCD is a rectangle inscribed in a quadrant of a circle of radius 10 cm.If AD=2$\sqrt{5}$ cm then area of the rectangle is

(a) $32c{m}^2$

(b) $40c{m}^2$

(c) $44c{m}^2$

(d) $48c{m}^2$

Which of the following is/are rationalizing factor for $\sqrt{3}$?

Question 40

Fill in the blanks to make the statements true.

The sides of a right-angled triangle whose hypotenuse is 17 cm are ___ and ___.

The areas of two similar triangles are 81 $c{m}^2$ and 49 $c{m}^2$ respectively. If the altitude of the first triangle is 6.3 cm, find the corresponding altitude of the other.

Additional Information:

1. Half of the investment held in the beginning of the year were sold at 10% profit.

2. Depreciation on Land and Building was ₹ 50,000 for the year.

3. Interest received on investments ₹ 75,000.

1. 7

Write the sum of first n even natural numbers.

ABCD is a square with P and Q are the midpoints of BC and CD respectively. A point is selected at random in the square. Calculate the probability that it lies inside the triangle PCQ.

Aparna has a cumulative deposit in Union bank for 4 years at 9% rate of interest per annum. She receives ₹ 51,607.50 at the time of maturity. Find her monthly installment.

If $cosec(A+B)=\frac{2}{\sqrt{3}},sec(A-B)=\frac{2}{\sqrt{3}},0^o\le A+B\le {90}^o$, Find A and B.

What can you say about the product of a rational and an irrational number ?

Trinagles ABC and DEF are similar.
(i) If area $\left(\Delta ABC\right)=16c{m}^2,area\left(\Delta DEF\right)=25c{m}^2$ and BC = 2.3 cm find EF.
(ii) If area $\left(\Delta ABC\right)=9c{m}^2,area\left(\Delta DEF\right)=64c{m}^2$ and DE=5.1 cm, find AB.
(iii) If AC = 19 cm adn DF = 8 cm, find the ratio of the area of two triangles.
(iv) If area $\left(\Delta ABC\right)=36c{m}^2$, are $\left(\Delta DEF\right)=64c{m}^2$ and DE= 6.2 cm, find AB.
(v) If AB = 1.2 cm and DE=1.4 cm, find the ratio of the areas of $\Delta ABC$ and $\Delta DEF$.

If $1,lo{g}_9(3^{1-x}+2)lo{g}_3({4.3}^x-1)$ are in A.P., then x equals

Two circles touch each other externally.

Prove that the tangents drawn to the two circles from any point on the common tangent are equal in length. [2 MARKS]

Question 9

A solid iron cuboidal block of dimensions 4.4 m × 2.6 m × 1m is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.

If $a_1,a_2,a_3,a_4$ are in AP, then