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If $\int \frac{\sin x+\cos x}{9+16\sin 2x}dx=A\ln \left|\frac{4\sin x-4\cos x+B}{4\cos x-4\sin x+P}\right|+C,$ then which of the following is/are true?

(where $A,B,P$ are fixed constants and $C$ is integration constant)

If z = (1 + i $\sqrt{3}$).Find the value of ${log}_e$z.

If the curve $y=a{x}^{\frac{1}{2}}+bx$ passes through the point $(1,2)$ and $y\ge 0$ for $0\le x\le 9$ and the area enclosed by the curve, the $x-$ axis and the line $x=4$ is $8$ sq. units, then which of the following is/are true

Consider a branch of the hyperbola $x^2-2y^2-2\sqrt{2}x-4\sqrt{2}y-6=0$ with vertex at the point A. Let B be one of the end points of its latusrectum. If C is the focus of the hyperbola nearest to the point A, then the area of the $\Delta ABC$ is

The half-life of $T{e}^{99}$ is $6h$. The activity of $T{e}^{99}$ in a patient, $60h$ after receiving an injection containing this radioisotope is at least $0.125\mu Ci$. What was the minimum activity (in $\mu Ci$) of the sample injected?

Find the equation of the parabola with vertex at the origin, passing through the point $P(3,-4)$ and symmetric about the y-axis.

Out of 200 students who are trying to improve their vocabulary, 120 students read newspaper H, 50 read newspaper T and 30 read both newspaper H and T. Find the number of students

i) who read H but not T

ii) who read T but not H

iii) who don't read any newspaper

Equation of the ellipse with foci ($\pm$5,0) and length of major axis 26 is

If $5x-3\ge 3x-5$ and $x\in R^{-}$, then $x\in$

If $f\left(x\right)$ satisfies the relation $f\left(x\right)-\lambda \underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin x(\cos t\cdot f(t\left)\right)dt=\sin x$ and $f\left(x\right)=2$ has at least one real root, then

There are three bags $B_1,B_2,\text{and}{B}_3$ containing $2$ red & $3$ white, $5$ red & $5$ white, $3$ red & $2$ white balls respectively. A ball is drawn from bag $B_1$ and placed in $B_2$. Then a ball is drawn from bag $B_2$ and placed in $B_3$. Then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed, if same colour balls are used in the first and the second transfers is (assuming all balls are different)

If $a\in R-\left\{0\right\}$ and $\left|\begin{array}{ccc}x+1& x& x\\ x& x+a& x\\ x& x& x+a^2\end{array}\right|=0,$ then $x$ belongs to

The complete set of values of $a$ for which the inequality $a{x}^2-(3+2a)x+6>0,a\ne 0$ holds good for exactly three integral values of $x$ 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

Match the following by appropriately matching the lists based on the information given in $\text{Column I}$ and $\text{Column II}$.

$\begin{array}{cc}ColumnI& ColumnII\left(Typeof△ABC\right)\\ \text{a.}\cot \frac{A}{2}=\frac{b+c}{a}& \text{p. always right angled}\\ \text{b.}a\tan A+b\tan B=(a+b)\tan \frac{A+B}{2}& \text{q. always isosceles}\\ \text{c.}a\cos A=b\cos B& \text{r. may be right angled}\\ \text{d.}\cos A=\frac{\sin B}{2\sin C}& \text{s. may be right angled isosceles}\end{array}$

The sum of the infinite G.P. a + a $r^2$ + a $r^3$ . . . . . . a is finite. Then,

Evaluate $\Delta =\left|\begin{array}{ccc}0& \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0& \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$

The total number of triangles in the given figure is

If $a={\sin }^4(\frac{3\pi }{2}-\alpha )+{\sin }^4(3\pi +\alpha )$ and $b={\sin }^6(\frac{\pi }{2}+\alpha )+{\sin }^6(5\pi -\alpha )$, then the value of $3a-2b$ is

The value of $x$ satisfying the equation ${\sin }^4\left(\frac{x}{3}\right)+{\cos }^4\left(\frac{x}{3}\right)>\frac{1}{2}$ is (where $n\in Z$)

Show that the pooints (5, 5), (6, 4), (-2, 4) and (7, 1 ) all lie on a circle, and find its equation, centre and radius.

The range of the function $f\left(x\right)=\left[\right\{2x+3\left\}\right]$ is

$\left(\right[.]$ represents the greatest integer function and $\left\{x\right\}$ is the fractional part of $x)$

If $0<x<1$ and $y=\frac{1}{2}{x}^2+\frac{2}{3}{x}^3+\frac{3}{4}{x}^4+\cdots ,$ then the value of $e^{1+y}$ at $x=\frac{1}{2}$ is

The dimensions of the area A of a black hole can be written in terms of the universal constant G, its mass M and the speed of light c as $A=G^{\alpha }{M}^{\beta }{c}^{\gamma }$. Here

Column Matching:



$$\begin{array}{cc}\text{Column (I)}& \text{Column (II)}\\ \begin{array}{cc}\text{(A)}& \text{In a triangle}△XYZ,\text{let}a,b\text{and}c\text{be the}\\ & \text{lengths of the sides opposite to the angles}\\ & X,Y\text{and}Z,\text{respectively. If}2(a^2-b^2=c^2\\ & \text{and}\lambda =\frac{\sin (X-Y)}{\sin Z},\text{then possible values}\\ & \text{of}n\text{for which}\cos \left(n\pi \lambda \right)=0\text{is (are)}\end{array}& \text{(P) 1}\\ & \\ \begin{array}{cc}\text{(B)}& \text{In a triangle}△XYZ,\text{let}a,b\text{and}c\text{be}\\ & \text{the lengths of the sides opposite to the}\\ & \text{angles}X,Y\text{and}Z,\text{respectively. If}\\ & 1+\cos 2X-2\cos 2Y=2\sin X\sin Y,\text{then}\\ & \text{possible value(s) of}\frac{a}{b}\text{is (are)}\end{array}& \text{(Q) 2}\\ & \\ \begin{array}{cc}\text{(C)}& \text{In}{R}^2,\text{let}\sqrt{3}\hat{i}+\hat{j},\hat{i}+\sqrt{3}\hat{j}\text{and}\beta \hat{i}+(1-\beta )\hat{j}\\ & \text{be the position vectors of}X,Y\text{and}Z\text{with}\\ & \text{respect to the origin}O,\text{respectively. If the}\\ & \text{distance of}Z\text{from the bisector of the acute}\\ & \text{angle of}\overrightarrow{OX}\text{with}\overrightarrow{OY}\text{is}\frac{3}{\sqrt{2}},\text{then possible}\\ & \text{value(s) of}|\beta |\text{is (are)}\end{array}& \text{(R) 3}\\ & \\ \begin{array}{cc}\text{(D)}& \text{Suppose that}F\left(\alpha \right)\text{denotes the area of the}\\ & \text{region bounded by}x=0,x=2,y^2=4x\\ & \text{and}y=|\alpha x-1|+|\alpha x-2|+\alpha x,\text{where}\\ & \alpha \in \{0,1\}.\text{Then the value(s) of}F\left(\alpha \right)+\frac{8}{3}\sqrt{2}\text{,}\\ & \text{when}\alpha =0\text{and}\alpha =1,\text{is (are)}\end{array}& \text{(S) 5}\\ & \\ & \text{(T) 6}\\ & \end{array}$$

Option $\text{(D)}$ matches with which of the elements of right hand column?

If $A$ and $B$ are non singular matrix of same order such that $|AB|=10,|A|=5.$ Then the value of $(|A|-|B|{)}^2$ is:

If $2^{2x+2}-a\cdot 2^{x+2}+5-4a\ge 0$ has atleast one real solution, Then a $ϵ$

Let X be a family of sets and R be a relation on X defined by 'A is disjoint from B'. Then R is

The derivative of ${\cos }^{-1}(2x^2-1)$ w.r.t
${\cos }^{-1}x$ is