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If the letters of the word $SACHIN$ are arranged in all possible ways and these words(with or without meaning) are written out as in dictionary, then the position of the word $SACHIN$ will be

If $z$ is a complex number, then the conjugate of $z+2\overline{z}$ is

Given that $A=\{1,2,3\}$, $B=\{3,4\}$ and $C=\{4,5,6\}$, then $n(A\cup (B\cap C\left)\right)=$

Equation of the conjugate axis of the hyperbola $5x^2-4y^2-30x-8y-39$ = 0

Minimum z = x + 2y subject to x ≤ 20, y ≥ 10, x ≥ 0, y ≥ 0

The cubic polynomial $p\left(x\right)$ satisfies $p\left(0\right)=1,p\left(1\right)=20,p\left(2\right)=40,p\left(3\right)=60.$ Then the value of $p\left(4\right)$ is

Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.

Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:
$X=\{x:f(x)=0\},Y=\{x:f^{\prime }(x)=0\},$
$Z=\{x:g(x)=0\},W=\{x:g^{\prime }(x)=0\},$.
$List-I$ contains the sets $X,Y,Z$ and $W$. $List-II$ contains some information regarding these sets.

$\begin{array}{cccc}& ListI& & ListII\\ \left(I\right)& X& \left(P\right)& \supseteq \{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \}\\ \left(II\right)& Y& \left(Q\right)& \text{an arithmetic progression}\\ \left(III\right)& Z& \left(R\right)& \text{NOT an arithmetic progression}\\ \left(IV\right)& Z& \left(S\right)& \supseteq \{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\}\\ & & \left(T\right)& \supseteq \{\frac{\pi }{3},\frac{2\pi }{3},\pi \}\\ & & \left(U\right)& \supseteq \{\frac{\pi }{6},\frac{3\pi }{4}\}\end{array}$
Q.15 which of the following is the only CORRECT combination?

If in the expansion of $(a-2b{)}^n$, the sum of $5^{\text{th}}$ and $6^{\text{th}}$ terms is $0$, then the value of $\frac{a}{b}$ is equal to

If $\cos \left(\log i^{4i}\right)=a+bi$, then

If set $A=\left\{\right(r,s)|r<s;r,s\in W\}$, then the number of element(s) in set $A$ such that ${}^7{C}_r+{}^7{C}_{r-1}={}^8{C}_s$ is

The argument of $\frac{1-i}{1+i}$ is

Prove that the lines $y=\sqrt{3}x+1,y=4andy=-\sqrt{3}x+2$ form an equilateral triangle.

The number of possible straight lines, passing through (2, 3) and forming a triangle with coordinate axes, whose area is 12 sq. units is

The sum of the series
$\frac{2}{1\times 2}+\frac{5}{2\times 3}\times 2+\frac{10}{3\times 4}\times 2^2+\frac{17}{4\times 5}\times 2^3+----$
$--$ upto $9$ terms is given as

A variable plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ at a unit distance from origin cuts the coordinate axes at A, B and C. Centroid (x,y,z) satisfies the equation $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=K$ is

If one vertex of a chord of parabola $y^2=8ax$ is at $(0,0),$ then the locus of a point which divides the chord in the ratio $1:2$ is

Find the absolute maximum and minimum values of the function f given by $f\left(x\right)=co{s}^{2}x+sinx,xϵ[0,\pi ]$

The range of values of $\theta$ in the interval $(0,\pi )$ such that the points $(3,5)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y-1=0$, is

Find the rate of change of the area of a circle with respect to its radius r when

r = 4 cm

In a multiple choice question there are four alternative answers of which one or more than one is correct. A candidate will get marks on the question only if he ticks all the correct answers. The candidate decides to tick answers at random. If he is allowed upto three chances to answer the question, the probability that he will get marks on it is given by:

Find the points of local maxima, local minima and the points of inflection of the function f(x)= $x^5-5x^4+5x^3-1.$ Also find the corresponding local maximum and local minimum values.

$In(0,\pi ),\text{the number of solutions of the equation}tan\theta +tan2\theta +tan3\theta =tan\theta tan2\theta tan3\theta is$

Differentiate the following functions with respect to x :

$lo{g}_{3}x+3lo{g}_{3}x+2tanx$

Let $y=f\left(x\right)$ is a parabola of the form $y=x^2+ax+1$ and tangent to the parabola at the point of intersection with $y-$axis also touches the circle $x^2+y^2=r^2$.If it is known that no point of the parabola is below $x-$axis then the radius of circle when $a$ attains its maximum value in units is

The middle term of the binomial expansion of $(a+b{)}^n$ is the $6^{th}$ term. Find the possible values of n, if n is odd.

If $a_n=\sum _{r=0}^n\frac{1}{{}^n{C}_r}$ , then $\sum _{r=0}^n\frac{r}{{}^n{C}_r}$ equals:

What is the value of determinant given if x, y, z are in AP with common difference d.

$\left|\begin{array}{ccc}x+y& y& 1\\ y+z& z& 1\\ z+x& x& 1\end{array}\right|$

If $P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3)$ and $S(x_4,y_4)$ are four concyclic points on the rectangular hyperbola $xy=c^2$, then coordinates of the orthocenter of the $△PQR$ is

$Ifcos\alpha +cos\beta =0=sin\alpha +sin\beta ,thencos2\alpha +cos2\beta isequalto$

If ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z=\pi \left({\sec }^2\right(u)+{\sec }^4(v)+{\sec }^6(w\left)\right)$, where $u,v,w$ are least non-negative angles such that $u<v<w$, then the value of $x^{2000}+y^{2002}+z^{2004}+\frac{36\pi }{u+v+w}$ is

Find the equation of the ellipse whose centre lies at the origin, major axis lies on the x-axis, the eccentricity is $\frac{2}{3}$ and the length of the latus rectum is 5 units.

The mirror image of the directrix of the parabola
$y=x^2+5x+7$ in the line mirror $x+y=1$ is

The solution of the differential equation $ydx-xdy=y^2\tan \left(\frac{x}{y}\right)dx$ is
( $C$ is constant of integration)

If the tangents at P(t1) and Q(t2) in the parabola $y^2=4ax$ meet in T, then find the coordinates of point T.

sin 20 (4+sec20 ) is

If $f\left(x\right)$ is defined on $(0,1)$, then the domain of $g\left(x\right)=f\left(e^x\right)+f\left({\log }_e|x|\right)$ is

Find the $6^{\text{th}}$ term of the expansion $(y^{1/2}+x^{1/3}{)}^n,$ if the value of coefficient of $3^{\text{rd}}$ term from the end is $45$

Evaluate ${\sum }_{k=1}^{11}(2+3^k)$