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If $|x|<2$, what are the range of values for x?

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{4}}(2se{c}^{2}x+x^3+2)dx$

Let f:
R → R be defined as f(x) = 3x.
Choose the correct answer.

(A) f is one-one
onto (B) f is many-one onto

(C) f is one-one but not
onto (D) f is neither one-one nor onto

If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point $P(6,-2)$ to the ellipse $4x^2+9y^2=36,$ then $m_1^2+m_2^2$ is equal to

Let P($E_1,E_2$) denote the event that exactly one out of $E_{1}and{E}_2$ occurs. For three events A, B, C; it is known that P(A, B) = P(B, C) = P(C, A) = p and $P(A\cap B\cap C)=p^2$ where 0 < p < $\frac{1}{2}$. Then probability that at least one of A, B, C occur is

Let $S_n=\frac{1}{n}\sum _{r=0}^{n-1}f\left(\frac{r}{n}\right),T_n=\frac{1}{n}\sum _{r=1}^{n}f\left(\frac{r}{n}\right)$ and $f$ is strictly decreasing function, then which of the following option(s) is/are correct?

The sum of intercepts of any tangent on the curve $\sqrt{x}+\sqrt{y}=2$ is

Form the
differential equation representing the family of curves given by
where
a is an arbitrary constant.

The value of $\underset{0}{\overset{\frac{1}{\sqrt{2}}}{\int }}{({\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2)}^{\frac{1}{2}}dx$ is

Find the value of

The value of $\frac{1}{2}{\cos }^{-1}\left(\cos 4\right)$ is equal to



[2 marks]

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$are three vectors such that $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=1$ and $|\overrightarrow{c}|=2$ and $|\overrightarrow{b}\times \overrightarrow{c}|=\sqrt{3}$ and $\overrightarrow{b}-3\overrightarrow{c}=\lambda \overrightarrow{a}$, then the possible value of $\left[\lambda \right]$ is (where $[.]$ denotes greatest integer function)

Any point on the parabola whose focus is (0, 1) and the directrix is x+2=0 is given by

If equation of a plane is given $4x+2y+12z=7$ then $x,y\&z$ intercepts will be

The domain of definition of the function
$f\left(x\right)=x\cdot \frac{1+2(x+4{)}^{-0.5}}{2-(x+6{)}^{0.5}}+(x+5{)}^{0.5}+4(x+10{)}^{-0.5}$ is

The plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1$ cuts the axes in $A,B,C$ then the area of the $△ABC$ (in $\text{sq.units}$) is

Select the correct graph of $f\left(x\right)=|\cos |x-\frac{\pi }{4}||.$

The normal duration of I.Sc. Physics practical period in Indian colleges is 100 minutes. Express this period in micro centuries, 1 micro century $={10}^{-6}\times 100$ years.

The point on the parabola $y=x^2+7x+2$ which is closest to the line $y=3x-3$ is

If rolle's theorem is applicable on $f\left(x\right)=x^{\alpha }\tan x$ in $[-\frac{\pi }{4},\frac{\pi }{4}],$ then the value of $\alpha +1$ can be

Electric field at a point due to an electric dipole on an axis inclined at an angle theta <90° to the dipole axis is perpendicular to the dipole axis if the angle theta is

The general solution of differential equation $\frac{dy}{dx}+ysecx=tanx(0<x<\frac{\pi }{2})$ is

A quadratic polynomial $p\left(x\right)$ has $1+\sqrt{5}$ and $1-\sqrt{5}$ as its zeros and $p\left(1\right)=2$. Then the value of $p\left(0\right)$ is

An angle of intersection of the curves, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $x^2+y^2=ab,$ $a>b$, is

Let $z_1$ and $z_2$ be two complex numbers such that $|z_1|=|z_2|=1.$ If $C={\left[\begin{array}{cc}{\overline{z}}_1& -z_2\\ {\overline{z}}_2& z_1\end{array}\right]}^{-1}{\left[\begin{array}{cc}{z}_1& z_2\\ -{\overline{z}}_2& {\overline{z}}_1\end{array}\right]}^{-1},$ then the sum of principal diagonal entries of $C$ is

The equation of the circle in diameter form with centre $(4,–2)$ and passing through the point $(2,-2)$ is

Write the value of $\frac{d}{dx}\left(log|x|\right).$

The radius of the circle $x^2+y^2+2x+8y+8=0$ is

Out of 3n consecutive integers, three are selected at random. The probability that their sum is divisible by 3 is