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Question 1 (i)

Form the pair of linear equations in the following problems, and find their solutions graphically.

10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

. If a, b, c are three unit vectors such that $a\times (b\times c)=\frac{1}{2}$b then the angles between a, b and a, c are

If $x=e^{y+e^{y+e^{y+e^{y+\cdots \infty }}}}$

The integrating factor of the differential equation $x\frac{dy}{dx}+xy\cot x=0$ is $(x\ne 0)$

If $3{\tan }^{-1}x+{\cot }^{-1}x=\pi$ then $x$ equal to

Find the values of the following expressions :

$\left(i\right)i^{49}+i^{68}+i^{89}+i^{110}\left(ii\right)i^{30}+i^{80}+i^{120}\left(iii\right)i+i^2+i^3+i^4\left(iv\right)i^5+i^{10}+i^{15}\left(v\right)\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}\left(vi\right)1+i^2+i^4+i^6+i^8+...+i^{20}$

The values of a function f(x) are tabulaed below

If the mean of the set of number $x_1,x_2...x_n$ is $\overline{x}$ ,then the mean of the number $x_i+2i,1\le \le n$ is

Using the word '$\text{AEROPLANE}$', different words are formed and are put in the dictionary order, then what will be the ${50}^{\text{th}}$ word?

For real values of x, cos(x) can be written in one of the forms of a convergent series given below:

If the mean deviation about the median of the numbers $a,2a,......,50a$ is $50$, then $|a|$ equals

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{1}=1$ and the circle $x^2+y^2=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to :

If cos A + sin A = √2 sinA , Then, Find sin A - cos A

In a swimming race $3$ swimmers compete . The probability of $A$ and $B$ winning is same and twice that of $C$.What is the probability that $B$ or $C$ wins. Assuming no two finish the race at the same time.

The two curves $x^3-3x{y}^2+2=0$ and $3x^{2}y-y^3=2$

The area of the region bounded by the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}(x,y>0)$ and the coordinate axes is

If equations $x^2-3x+4=0$ and $4x^2-2[3a+b]x+b=0(a,b\in R)$ have a common root, then the complete set of values of $a$ is

$($Here, $\left[K\right]$ denotes the greatest integer less than or equal to $K.)$

Integrate the following functions.
$\int \frac{4x+1}{\sqrt{2x^2+x-3}}dx.$

The degree of the differential equation ${[1+{\left(\frac{dy}{dx}\right)}^2]}^2=\frac{d^{2}y}{d{x}^2}$ is

Show that $\left(ii\right)\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]\left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\ne \left[\begin{array}{ccc}-1& 1& 0\\ 0& -1& 1\\ 2& 3& 4\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 0\\ 1& 1& 0\end{array}\right]$

The general solution of the differential equation $\frac{dy}{dx}=\frac{x^2+xy+y^2}{x^2}$ is

(where $c$ is constant of integration)

If $2+i\sqrt{3}$ is a root of the equation $x^2+px+q=0$, where p and q are real, then (p, q) =

If a, b, c are in A.P.; b, c, d are in G.P. and $\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P. prove that a, c, e are in G.P.

Number of solutions of $8\cos x=x$ will be

For the differential equation $ydx+y^{2}dy=xdy,x\in R,y\left(1\right)=1,$ then the value of $y(-3)$ is:

1. 220

The value of $\underset{n\to \infty }{lim}\left\{\frac{\sqrt{n+1}+\sqrt{n+2}+...+\sqrt{2n-1}}{n^{3/2}}\right\}$ is