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In the picture below, ΔPQR is right angled. Also, ∠A = ∠P and BC = QR.

Prove that the diameter of the circumcircle of ΔABC is equal to the length of PQ.

Two dice are thrown at the same time. Find the probability of getting different values on both.

Which of the following equations has a solution x = 7, y = 3?

If bi-quadratic equation $x^4-2x^3+4x^2+6x-21=0$ can also be written as $(x^2+p)(x^2-2x+q)=0$. Find the sum of p + q.

The angle of elevation of the top Q of a verticle tower PQ from a point X on the ground is ${60}^{\circ }$. At a point Y, 40 m vertically above X, the angle of elevation is ${45}^{\circ }$. Find the height of tower PQ. [Take $\sqrt{3}=\mathrm{1.732.}$]

Ahmed has a Recurring Deposit Account in a bank. He deposits Rs2,500 per month for 2 years. If he gets Rs 66,250 at the time of maturity, find
(i) The interest paid by the bank
(ii) The rate of interest.

Given that the line y = – & + 5 = 3 are ||, find ‘a’.

Which of the following is the perfect relation between $\sin \theta$ and $\cos \theta$ for any value of $\theta$?

Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. the construct another triangle whose sides are times the corresponding sides of the given triangle. Give the justification of the construction.

The value of $\text{adj}\left(A\right)\cdot \text{adj}\left(A^{-1}\right)$ such that $A=PQR$ and all matrices are non-singular of same order($A^{-1}$ is inverse of matrix $A$ and $I_n$ is an identity matrix), is

How many solutions of the equation 15x - 14y + 11 = 0 are possible?

If 'l', 'b', and 'h' are the length, breadth, and height of the rectangular box respectively, then the product of the areas of all the faces of the box is equal to

If A and B are two square matrices such that $B=-A^{-1}BA$, then $(A+B{)}^2$ is equal to

Question 1 (i)
Without actually performing the long division, state whether $\frac{13}{3125}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

The point of intersection of all the medians of a triangle is called the ___ of the triangle.

If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its vertex.

Which term of the AP : 3, 8, 13, 18, . . . , is 78?

Consider the figure where $\angle AOB={90}^{\circ }$ and $\angle ABC={30}^{\circ }.$ Then, $\angle CAO$ = ___.

The digits of a positive integer whose three digits are in AP and their sum is 21. The number obtained by reversing the digits is 396 less than the original number. Find the number.

Find the equation of the circle with
Centre (-a, -b) and radius $\sqrt{a^2-b^2}$

If two adjacent vertices of a parallelogram are (3,2) and (-1,0) and the diagonals intersect at (2, -5) then find the coordinates of the other two vertices.

Radii of 2 circles are 16 cm and 8 cm. Find the distance between their centres if they touch internally.

Question 5 (ii)
Rationalize the denominator of :
$\frac{1}{(\sqrt{7}-\sqrt{6})}$ .

A single letter is selected at random from the word ‘PROBABILITY’. Find the probability that it is a vowel.

A total of Rs. 10,000 is distributed among 150 persons as gift. A gift is either of Rs. 50 or Rs. 100. The total number of Rs. 50 gifts is _________.

If $S$ is the sum of the first $10$ terms of the series ${\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)+{\tan }^{-1}\left(\frac{1}{13}\right)+{\tan }^{-1}\left(\frac{1}{21}\right)+\cdots ,$ then $\tan \left(S\right)$ is equal to :

The first negative term of the sequence 114, 109, 104 ............ is

The roots of a quadratic equation $x^2-4x-{\log }_{3}a=0$ are real. Then what is the least value of $a$?