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Integrate the following functions.
$\int$ sin (ax+b)cos (ax+b)dx.

$Domianofthefunctionf\left(x\right)=lo{g}_2\left(lo{g}_4\left(lo{g}_2\left(lo{g}_3(x^2+4x-23)\right)\right)\right)is$

If fifth term of a G.P. is $2,$ then the product of its first $9$ terms is

If the function $f\left(x\right)=\{\begin{array}{ccc}{k}_1{(x-\pi )}^2-1,& x\le \pi & \\ k_2\cos x,& x>\pi & \end{array}$ is twice differentiable, then the ordered pair $(k_1,k_2)$ is equal to:

The distance of the point (1, 0, 2) from the point of intersection of the line
$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane x - y + z = 16 is

The value of ${\sin }^{-1}\left(\sin \frac{5\pi }{6}\right)$ is

The total number of four-digit numbers $xyzt$ such that $x<y=z>t$ is

Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this.
(v) On $Z^{+}$, defined $\ast$ by $a\ast b=a$

Differentiate the
function with respect to x.

Let $\alpha$ and $\beta$ be the roots of equation $p{x}^2+qx+r=0,p\ne 0.$ If $p,q,r$ are in A.P. and $\frac{1}{\alpha }+\frac{1}{\beta }=4,$ then the value of $|\alpha -\beta |$ is

In SHM, which equation represent general equation, equation from mean position, equation from extreme position respectively.

$\left(P\right)y=A\sin (wt+\alpha )$

$\left(Q\right)y=A\sin \left(wt\right)$

$\left(R\right)y=A\cos \left(wt\right)$

$A.R,Q,P$

$B.P,R,Q$

$C.P,Q,R$

$D.Q,R,P$

If $\alpha$ is the greater root of $x^2-5x+4=0$ and $\alpha +m=2$, then the value of $m$ is

The equation of the circle which is touched by $y=x$, has its centre on the positive direction of the $x$-axis and cuts off a chord of length $2$ units along the line $\sqrt{3}y-x=0$, is

Find X if $2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]and3X+2Y=\left[\begin{array}{cc}7& -2\\ -1& 5\end{array}\right]$

1. 3.5

If ${\int }_{sinx}^1{t}^2.f\left(t\right)dt=1-sinx,\forall xϵ(0,\frac{\pi }{2})$ then the value of $f\left(\frac{1}{\sqrt{3}}\right)$ is

If the roots of $4x^2-(5k+1)x+5k=0$ differ by unity then the sum of all the possible values of $k$ is

lf the projections of the line segment $AB$ on the $YZ$-plane, $ZX$-plane, $XY$-plane are $\sqrt{160},\sqrt{153},5$ respectively, then the projection of $AB$ on the $z$-axis is

In
triangle ABC which of the following is not true:

A.

B.

C.

D.

If $\sin A=\frac{12}{13},\cos B=-\frac{3}{5},0<A<\frac{\pi }{2},$ $\pi <B<\frac{3\pi }{2}$, then the value of $\sin (A+B)$ is

The number of roots of $e^x+0.5x^2-2=0$ in the range $[-5,5]$ is

Suppose $x$ and $y$ are natural numbers, then the number of ordered pairs $(x,y)$ which satisfy $x+y\le 5$ is

Let $A$ be a vector parallel to line of intersection of planes $P_1:x=0$ and $P_2:x-y-z=0$ through origin. The acute angle between $A$ and $2\hat{i}+\hat{j}-2\hat{k}$ is

Let $A$ and $B$ be any two events such that $P\left(A\right)=\frac{1}{2}\text{and}P\left(B\right)=\frac{1}{3}$. Then the value of $P(A^{\prime }\cap B^{\prime }{)}^{\prime }+P(A^{\prime }\cup B^{\prime }{)}^{\prime }$ is

Using the method of integration,find the area of the triangle ABC,coordinates of whose vertices are A(4,1),B(6,6) and C(8,4).