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$Thedomainof\sqrt{sin\left(cosx\right)}$

The value of $\left|\begin{array}{ccc}(b+c{)}^2& ba& ac\\ ba& (c+a{)}^2& cb\\ ca& cb& (a+b{)}^2\end{array}\right|$ is:

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right].$ $U_1,U_2,U_3$ are column matrices satisfying $A{U}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],$ $A{U}_2=\left[\begin{array}{c}2\\ 3\\ 0\end{array}\right]$ and $A{U}_3=\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right].$ If $U$ is $3\times 3$ matrix whose columns are $U_1,U_2$ and $U_3,$ then $det\left(U\right)$ equals

For the differential equation given in the question find a particular solution satisfying the given condition.

$\frac{dy}{dx}+2ytanx=sinx,y=0whenx=\frac{\pi }{3}.$

A tangent to the parabola $y^2=8x$ makes an angle of ${45}^{\circ }$ with the straight line $y=3x+5$. Then one of the points of contact has the coordinates

If $f\left(x\right)=x+{\int }_0^1(x{y}^2+x^{2}y)f\left(y\right)dy$, and $f\left(x\right)=\frac{A{x}^2+Bx}{119}$ then $\left[\frac{A+B}{100}\right]$ = (where [.] is G.I.F) ___

Solution set of the inequality $lo{g}_{0.8}\left(lo{g}_6\frac{x^2+x}{x+4}\right)<0$

Find the
derivative of the following functions (it is to be understood that a,
b, c, d, p, q, r and s are
fixed non-zero constants and m and n are integers):

If $y\left(\alpha \right)=\sqrt{2\left(\frac{\tan \alpha +\cot \alpha }{1+{\tan }^2\alpha }\right)+\frac{1}{{\sin }^2\alpha }}$ where $\alpha \in (\frac{3\pi }{4},\pi )$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is

Let $f:R\to (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

If X and Y are two sets such that n (X) = 17, n (Y) = 23 and $n(X\cup Y)=38,$ find $n(X\cap Y).$

If $|x-2|\le 1,$ then

The value of $\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ab& b^2+1& bc\\ ac& bc& c^2+1\end{array}\right|$ is:

The value(s) of ${}^{\prime }{m}^{\prime }$ for which the area of the region bounded by the curve $y=x–x^2$ and the line $y=mx\left(\frac{9}{2}\right)$ is sq.units, is

How much money does 4 ten-rupee notes and 3 five-rupees notes together make? What about 7 ten rupee notes and 4 five-rupees notes?

Let us denote the number of ten rupee notes by t, five rupees notes by f and the total amount by m. What is the relation involving t, f, m?

Distance between two parallel planes $2x+y+2z=8$ and $4x+2y+4z+5=0$ is :

The remainder when $7^{103}$ is divided by 25 is:

If $C_1$ and $C_2$ are circles whose equations are $x^2+y^2-20x+64=0$ and $x^2+y^2+30x+144=0$, then the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$ is

The least value of $a$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=a$ has atleast one solution in the interval $(0,\frac{\pi }{2})$ is

If the parabola $x^2=ay$ makes an intercept of length $\sqrt{40}$ units on the line $y-2x=1$, then $a$ is equal to

A function $f$ is such that $f\left(h\right)=-1,f(-h)=1$ as $h\to 0^{+}$ has a maximum at $x=0.$ Then the set of value(s) of $f\left(0\right)$ can be

If $(1,-2)$ is a pole of the circle $x^2+y^2-10x-10y+25=0$, then the equation of the diameter which bisects the polar is

Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}$ equals

If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three unit vectors equally inclined to each, then the maximum angle between vectors is

If f(x) = $\{\begin{array}{cc}\sin x& x\ne n\pi ,n=0,\pm 1,\pm \mathrm{2...}\\ 2,& otherwise\end{array}$ and g(x) = $\{\begin{array}{cc}{x}^2+1,& x\ne 0,2\\ 4,& x=0\\ 5,& x=2\end{array}$, then $\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}$ is

If the circle $x^2+y^2+2\lambda x=0,\lambda \in R$ touches the parabola $y^2=4x$ externally, then

The set of all points, where the function $f\left(x\right)=\frac{x}{1+|x|}$ is differentiable, is

Question 61

State whether the following statement is true or false:

Ratio of area of a circle to the area of a square whose side equals radius of circle, is $1:\pi$

Given $f\left(x\right)=|x-1|+|x+1|$. Then f(x) is

A plane which is tangent to the sphere $(x-a{)}^2+(y-a{)}^2+(z-a{)}^2=2a^2$

a normal is drawn to the sphere which makes equal intercepts to the coordinate axis. Let the the equation of the line of the intersection is $x-2a=\frac{y-a}{-2}=z-0$

Then the equation of the tangent plane is

Let $y=y\left(x\right)$ be the solution of the differential equaiton ${\text{cosec}}^{2}xdy+2dx=(1+y\cos 2x){\text{cosec}}^{2}xdx$, with $y\left(\frac{\pi }{4}\right)=0.$ Then, the value of $\left(y\right(0)+1{)}^2$ is equal to: