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A circle passes through the point $(3,\sqrt{\frac{7}{2}})$ and touches the line pair $x^2-y^2-2x+1=0$. The coordinates of the centre of the circle are

The scientists at D.O.S.A records the following observations:



The observations can be reprsented as .

A five digit number is formed with digits 0. 1. 2. 3. 4 without repetition. A number is selected at random, then the probability that it is divisible by 4 is

The probabilities that a student passes in mathematics, physics and chemistry are $m,p$ and $c$ respectively. Of these subjects, the students have a $75\%$, chance of passing in atleast one, a $50\%$ chance of passing in at least two and a $40\%$ chance of passing in exactly two. Which of the following relation(s) is/are true?

If $(x^2+x+1)+(x^2+2x+3)+(x^2+3x+5)+\cdots +(x^2+20x+39)=4500,$ then $x$ is equal to

Let $f:R\to R$ and $g:C\to C$ be two functions defined as $f\left(x\right)=x^2$ and $g\left(x\right)=x^2$. Are they equal functions?

Evaluate the determinant.
$\left|\begin{array}{ccc}3& -4& 5\\ 1& 1& -2\\ 2& 3& 1\end{array}\right|$

If $a+b\sqrt{7}=\frac{4+\sqrt{7}}{3-\sqrt{7}}$, then find the value of $a+b$.

Find the ${20}^{th}$ term and sum of the 20 term of the series 1, 4, 7, 10....

A random
variable X has the following probability distribution.

Determine

(i) k

(ii) P
(X < 3)

(iii) P
(X > 6)

(iv) P
(0 < X < 3)

Let $A$ be a non-empty set such that $A\times A$ has $9$ elements among which $(-2,0),(0,2)$ are found. Then $A$ is equal to

The smallest positive value of $\theta$' satisfying the equation $\sqrt{3}(cot\theta +tan\theta )=4$ is

If $4x^2+4y^2=a^2,$ then which of the following options is CORRECT?



[1 mark]

The value of $\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=0}^{2n-1}\frac{n^2}{n^2+4r^2}$ is

Show that the given differential equation is homogeneous and then solve it.

(x-y)dy-(x+y)dx=0.

There is a discount of $\$\frac{1}{6}$ on each type of book.



After discount, the combined cost of both the books is .

If the ratio of sum of $n$ terms of $2$ different $A.P.$ is $\frac{2n-1}{5n+10}$, then the ratio of their ${15}^{th}$ term is

Find the equation of the circle which passes through the origin and has its center on the line $x+y+4=0$ and cuts the circle $x^2+y^2-4x+2y+4=0$ orthogonally.

Find the derivative of the following function

f(x) = $\frac{3^x\cot 45°}{2x}$

The differential equation of all the straight lines which are at a constant distance of ${}^{\prime }{a}^{\prime }$ from the origin is

The volume of a cube is increasing at a rate of 7 cubic cm per second. The rate of change of its surface area when the length of an edge is 12 cm is

In a triangle $ABC,$ right angled at the vertex $A,$ if the position vectors of $A,B$ and $C$ are respectively $3\hat{i}+\hat{j}–\hat{k},–\hat{i}+3\hat{j}+p\hat{k}$ and $5\hat{i}+q\hat{j}–4\hat{k}$ then the point $(p,q)$ lies on a line

Find the equations to the diagonals of the rectangle the equations of whose sides are x = a, x = a', y = b and y = b'.

If the letters of the word 'MISSISSIPPI' are written down at random in a row, what is the probability that four S's come together.

Let P, Q, R and S be the points on the plane with position vectors
$-2\hat{i}-\hat{j},4\hat{i},3\hat{i}+3\hat{j}and-3\hat{i}+2\hat{j}$, respectively. The quadrilateral PQRS must be a

The equation of the circle passing through the points (4,1),(6,5) whose centre lies on the 4x+y-16=0 is

If $|a^x-2|+|8-a^x|<5$ , then $x\in$____ , where $a\in (1,\infty )$

Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}. An element (a, b) of their Cartesian product $A\times B$ is chosen at random. The probability that a + b = 9, is

The value of $\sin \left(3{\sin }^{-1}\left(\frac{1}{5}\right)\right)$ is

For any quadratic expression $f\left(x\right)=a{x}^2+bx+c,a<0$ and if $(v,f(v\left)\right)$ is it's vertex, then $y\in$