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The coefficient of $x^r$ in the expansion of (1+3x+6$x^2+10x^3$+......) is :

If the value of the integral $\int \frac{x}{(x+1)(x+2)}dx=\ln \left|\frac{f\left(x\right)}{g\left(x\right)}\right|+C,$ then which of the following statement(s) is(are) correct?

(where $f\left(x\right),g\left(x\right)$ are real valued function and $C$ is integration constant)

The number of points with non-negative integral coordinates that lie in the interior of the region common to the circle $x^2+y^2=16$ and the parabola $y^2=4x,$ is

The area of a square whose two sides lie on lines x+y=1 and x+y+2=0 will be .

If the curves $y=\frac{\ln x}{x}$ and $y=\lambda x^2$ (where $\lambda$ is constant) touch each other, then $\lambda$ is -

How many different boat parties of 8, consisting of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls ?

A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges of typing one English and one Hindi page separtely. However typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

A code word of 4 letters consists of two distinct consonants from the English alphabets followed by two digits from 1 to 9, with repetition allowed in digits. If the number of code words so formed ending with an even digit is $432\times k$, then $k$ is equal to

The value of $\sin \left(\frac{25\pi }{3}\right)+\sec \left(\frac{41\pi }{4}\right)+\tan (-\frac{16\pi }{3})-\text{cosec}(-\frac{33\pi }{4})$ is

A quadratic equation with rational coefficients if one of its roots is ${\cot }^2{18}^{\circ }$ is

If the line y=mx+1 touches the hyperbola $\frac{x^z}{9}-\frac{y^z}{2}$ = 1 then m =

1) What is the derivative of e^{​​​​​​y} with respect to x?

2) if cos y =xcos(a+y), with cos a not equal to +_1. Prove that dy/dx = cos^{​​​​2}(a+y)/sin a.

Solution of differential equation $\frac{dy}{dx}+\frac{\tan y}{x}=x{e}^x\sec y$ is

The differential equation for the line y=mx+c is (where c is arbitrary constant)

Write minors and cofactors of elements of following determinants.

$\left|\begin{array}{cc}2& -4\\ 0& 3\end{array}\right|$

$\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|$

The minimum value of the expression is $y=|x+1|+|x-3|$ is

If $f\left(x\right)=\{\begin{array}{cc}\left[x\right],& -2\le x\le -\frac{1}{2}\\ 2x^2-1,& -\frac{1}{2}<x\le 2\end{array};$ where $[.]$ represents the greatest integer function, then the function $f(x-1)$ is discontinous at the points

Choose the correct answer.
$\int \frac{e^x(1+x)}{co{s}^2\left(e^{x}x\right)}dx$ is equal to
$\left(a\right)-cot\left(e^{x}x\right)+C\left(b\right)tan\left(x{e}^x\right)+C\left(c\right)tan\left(e^x\right)+C\left(d\right)cot\left(e^x\right)+C$

Find the point on the curve $y=x^3-11x+5$ at which the tangent is y=x-11

Find the number of solutions for cos $y=cosx$.

Two projectiles are projected at angles $\left(\theta \right)$ and $(\frac{\pi }{2}-\theta )$ to the horizontal respectively with same speed $20\text{m/s}$. One of them rises $10\text{m}$ higher than the other. Find the angle of projection $\theta$.

The equation of the chord joining two points $(x_1,y_1)$ and $(x_2,y_2)$ on the rectangular hyperbola $xy=c^2$ is

In $△ABC,$ sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively. Then the value of $a\cos A+b\cos B+c\cos C=$

Given an example of a map
(ii) which is not one-one but onto.

The obtuse angular bisector between the lines $L_1:\frac{x}{-2}=\frac{y-2}{3}=\frac{z-1}{2}$ and $L_2:\frac{x}{-3}=\frac{y-2}{2}=\frac{z-1}{-2}$ is $\frac{x}{a}=\frac{y-2}{3}=\frac{z-1}{b},(a,b\in R).$ Then the correct option(s) among the following statement(s) is/are:

One card is drawn at random from a well- shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
E: the card drawn is a spade, F: the card drawn is an ace

E : the card drawn is black, F: the card drawn is a king

E: the card drawn is a king or queen
F: the card drawn is a queen or jack

The value of $k\in R,$ for which the following system of linear equations

$3x–y+4z=3,$

$x+2y–3z=–2,$

$6x+5y+kz=–3,$

has infinitely many solutions, is:

If $A=\left[\begin{array}{cc}x& y\end{array}\right],B=\left[\begin{array}{cc}a& h\\ h& b\end{array}\right],C=\left[\begin{array}{c}x\\ y\end{array}\right]$, then $ABC=$

The equation of the curve passing through the origin if the middle point of the segment of its normal form any point of the curve to the x-axis, lies on the parabola $2y^2=x$

If $A$ and $B$ are independent events of a sample space such that $P\left(A\right)=0.2,P\left(B\right)=0.5,$ then which of the following(s) is(are) correct?