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The maximum value of q until which the approximation $sinq\approx q$ holds to within 10%

Evaluate $\int (\ln x{)}^x\left(\ln \right(\ln x)+\frac{1}{\ln x})dx$

(where $C$ is constant of integration)

If the angle between $\overrightarrow{a}=2x^2\hat{i}+4x\hat{j}+\hat{k}$ and $\overrightarrow{b}=7\hat{i}-2\hat{j}+x\hat{k}$ is obtuse and the angle between $\overrightarrow{b}$ and $Z-$ axis is acute and less than $\frac{\pi }{6},$ then

Find the least positive value of $x$ such that $147$ $\equiv$ $(x+5)\left(\text{mod}7\right)$.

Let $a,r,s$ and $t$ be non –zero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)$ and $S(a{s}^2,2as)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is point $(2a,0)$.

The value of $r$ is

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg \left(\frac{1+z}{1+\overline{z}}\right)$ equals :

If f (x) = a log |x| + bx² + x has extreme values at x = –1 and at x = 2, then values of a and b are

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle ${45}^{\circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha ,\beta ),(0,\beta )$and $(0,0)$ is equal to:

Evaluate tan^-1(a - b)/(1+ab)+ tan^-1(b - c)/(1+bc)+ tan^-1(c - a)/(1+cb) ; ab,bc,ac>-1

Prove
that: cos 6x = 32 cos⁶ x – 48 cos⁴
x + 18 cos² x – 1

If the area bounded by the curve $4x^2-8x+y^2-2y+1=0$ is divided into two parts by straight line $2x+y=5,$ then the ratio of smaller to the larger area is

If sum of
the perpendicular distances of a variable point P (x, y)
from the lines x + y – 5 = 0 and 3x –
2y + 7 = 0 is always 10. Show that P must move on a line.

If G be the geometric mean of $x$ and $y$, then $\frac{1}{G^2-x^2}+\frac{1}{G^2-y^2}=$

If a circle whose one end of the diameter is focus of the parabola $y^2=4x$ and other end is a point on parabola, then which of the following line will always be a tangent to the given circle?

The variance of data $1001,1003,1006,1007,1009,1010$ is

How many terms of the sequence $\sqrt{3},3,3\sqrt{3},....$ must be taken to make the sum $39+13\sqrt{3}$ ?

The domain of the function $\frac{1}{\sqrt{[x{]}^2-5[x]+6}}$ is
(where [.] denotes the greatest integer function)

Which of the following is true for $y\left(x\right)$ that satisfies the differential equation $\frac{dy}{dx}=xy-1+x-y;y\left(0\right)=0$

Find the number of diagonals of (i) a hexagon (ii) a polygon of 16 sides.

If the function $f\left(x\right)=\{\begin{array}{cc}x+a^2\sqrt{2}sinx,& 0\le x<\frac{\pi }{4}\\ xcotx+b,& \frac{\pi }{4}\le x\le \frac{\pi }{2}\\ bsin2x-acos2x,& \frac{\pi }{2}<x\le \pi \end{array}$ is continuous in the interval $[0,\pi ]$ Then the values of (a, b) are
I. (-1, -1)
II. (0, 0)
III. (-1, 1)
IV. (1, 1)

If a circle and rectangular hyperbola $xy=c^2$ meets in 4 points $P,Q,RandS$ then $O{P}^2+O{Q}^2+O{R}^2+O{S}^2=$______ where r is the radius of the circle. O is the origin.

If the value of determinant $\Delta =\left|\begin{array}{ccc}2+x^2& x& x^2\\ 2+x& \sqrt{x}& x\\ 2+x^4& x^2& x^4\end{array}\right|$ is $6$. Then which of the following statement is correct?

If $\int \frac{(x-1{)}^2}{x^4+x^2+1}dx=\frac{1}{\sqrt{a}}{\tan }^{-1}\left(\frac{x^2-1}{x\sqrt{3}}\right)-\frac{b}{\sqrt{a}}{\tan }^{-1}\left(\frac{2x^2+1}{\sqrt{3}}\right)+C,$ then which of the following is/are CORRECT?

(where $a,b$ are fixed constants and $C$ is integration constant)

The values of $a$ for which the number $6$ lies in between the roots of the equation $x^2+2(a-3)x+9=0,$ belong to

Let f:
X → Y be an invertible function. Show that f
has unique inverse.

(Hint:
suppose g₁ and g₂ are two
inverses of f. Then for all y ∈
Y,

fog₁(y)
= I_Y(y) = fog₂(y).
Use one-one ness of f).

For any two non-empty sets $A$ and $B$, $(A\cup B{)}^C\cap (A^C\cap B)$ is equal to