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Determine whether the given planes are parallel or perpendicular and in case they are neither, find the angle between them.

4x+8y+z-8=0 and y+z-4=0

Consider $f:R_{+}\to [4,\infty )$ given by $f\left(x\right)=x^2+4$. Show that f is ivnertible with the inverse $f^{-1}$ given by $f^{-1}\left(y\right)=\sqrt{y-4}$. where $R_{+}$ is the set of all non-negative real numbers.

How many different words with or without meaning can be formed from-

Find the sum to
indicated number of terms in each of the geometric progressions in
Exercise 7 to 10:

If sin x – sin y = $\frac{1}{2}$ and cos x – cos y = 1 then tan (x + y)=

Let $f\left(x\right)=\frac{1}{2}{a}_0+{\sum }_{i-1}^n{a}_{i}cos\left(ix\right)+{\sum }_{j-1}^n{b}_{j}sin\left(jx\right),then{\int }_{-\pi }^{\pi }f\left(x\right)coskxdx$ then is equal to

If A is a square matrix of order 2 such that $A^2=0$, then

If two sides of a triangle are roots of the equation $x^2-7x+8=0$ and the angle between these sides is ${60}^{\circ },$ then the product of in-radius and circum-radius of triangle is

$\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.......+\frac{1}{n(n+1)}$ equals

Choose the correct answer in the following questions

Smaller area enclosed by the circle $x^2+y^2=4$ and the line x+y=2 is

(a) $2(\pi -2)$ (b) $(\pi -2)$

(c) $(2\pi -1)$ (d) $2(\pi -2)$

The value of $\cot \frac{\pi }{24}$ is

The equivalent resistance of the circuit between the points $A$ and $B$ as shown in figure is

(Given shape contains segments of equal length and resistance $1$ $\Omega$ each)

For
each binary operation * defined
below, determine whether * is
commutative or associative.

(i) On
Z,
define a * b
= a
− b

(ii) On
Q,
define a * b
= ab
+ 1

(iii) On
Q,
define a * b

(iv) On
Z⁺,
define a * b
= 2^ab

(v) On
Z⁺,
define a * b
= a^b

(vi) On
R −
{−1}, define

If $A=\left[\begin{array}{cc}1& 2\\ 4& 2\end{array}\right]$, then show that $|2A|=4|A|$

If two tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ make angles $\alpha$ and $\beta$ with the major axis such that $\tan \alpha +\tan \beta =\lambda ,$ then the locus of their point of intersection is

How many 3-digit even number can be made by using the digit 1,2,3,4,6,7, if no digit is repeated?

If $f\left(x\right)=\frac{4^x}{4^x+2}\mathrm{for}\mathrm{all}x\in R.$

Which of the following is/are true?

The quadratic equation whose roots are $5+\sqrt{11}$ and $5-\sqrt{11}$ is,

If $m$ is the slope of a tangent to the curve $e^y=1+x^2,$ then

If $(x_r,y_r):r=1,2,3,4$ be the points of intersection of the parabola $y^2=4ax$ and the circle $x^2+y^2+2gx+2fy+c=0,$ then

The foot of the perpendicular drawn from the (- 1, - 3, - 5) to a plane is (2, 4, 6). The equation of the plane is:

Which among the following is a two step equation?

Let from any point $P$ on the line $y=x,$ two tangents are drawn to the circle $(x-2{)}^2+y^2=1.$ Then the chord of contact of $P$ with respect to given circle always passes through a fixed point, whose coordinates are given by

From $P(–4,0)$, tangents $PA$ and $P{A}^{\prime }$ are drawn to a circle, $x^2+y^2=4$. Where $A$ and $A^{\prime }$ is point of contact and $A$ lies above x-axis. Rhombus $PA{P}^{\prime }{A}^{\prime }$ is completed.

$\begin{array}{cccccc}& \text{Column-I}& & \text{Column-II}& & \text{Column-III}\\ \left(I\right)& A\equiv (-1,\sqrt{3})& \left(i\right)& PA=2\sqrt{3}& \left(P\right)& P^{\prime }\text{lies on the circle},x^2+y^2=4\\ \left(II\right)& A^{\prime }\equiv (-1,-\sqrt{3})& \left(ii\right)& \text{Area of}\Delta PA{A}^{\prime }=3\sqrt{3}\text{sq.units}& \left(Q\right)& \Delta PA{A}^{\prime }\text{is equilateral}\\ \left(III\right)& P^{\prime }=(4,0)& \left(iii\right)& P{P}^{\prime }=6& \left(R\right)& P^{\prime }\text{lies outside the circle},x^2+y^2=4\\ \left(IV\right)& P^{\prime }=(2,0)& \left(iv\right)& \text{Area of}\Delta PA{A}^{\prime }=4\sqrt{3}\text{sq.units}& \left(S\right)& P^{\prime }\text{lies inside circle},x^2+y^2=4\end{array}$

Which one of the following is correct?

$A_3{B}_2$ is a sparingly soluble salt of molar mass M $\left(gmo{l}^{-1}\right)$ and solubility x g $L^{-1}$. The solubility product satisfies $K_{SP}=a{\left(\frac{x}{M}\right)}^5$. The vlaue of a is ______(Integer answer).

The number of 4 digited numbers that can be formed using the digits 2, 4, 5, 7,8 when repetition is not allowed

If A + B + C = $\pi$ and cos A = cos B cos C, then tan B tan C is equal to

Find the integrating factor for the following differential equation: $xlogx\frac{dy}{dx}+y=2logx$