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If $\sin \theta =-\frac{4}{5}$ and $\pi <\theta <\frac{3\pi }{2}$, then $\tan \theta +\cos \theta =$

The equation of the line parallel to the line joining $(4,2)$ and $(2,4)$ and whose $y$-intercept is $4$ units along positive $y$- axis is

If the system of equations $ax+y+z=0;x+by+z=0$ and $x+y+cz=0$ has a non-trivial solution, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

A tetrahedron has vertices $P(1,2,1),Q(2,1,3),R(-1,1,2)$ and $O(0,0,0).$ The angle between the faces $OPQ$ and $PQR$ is:

Which of the following sets can be the subset of the general solution of the equation $1+\cos 3x=2\cos 2x$ ?

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar unit vectors such that $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\frac{(\overrightarrow{b}+\overrightarrow{c})}{\sqrt{2}},$ then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

If $\tan A,\tan B$ are the roots the quadratic equation $\sqrt{3}{x}^2-2x-\sqrt{3}=0$, $0<A+B<\pi$, then $A+B$ is equal to

If both roots of the equation $x^2+ax+2=0$ lie in the interval $(0,3),$ then the exhaustive range of values of $a$ is

On a set {a,b,c}, we define an equivalence relation 'R'. If this relation is {(a,a),(b,b),(c,c),(a,b),(b,a)}. How many equivalence classes will be formed from this relation?

Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6i{z}_0^{81}-3i{z}_0^{93}$, then $\arg z$ is equal to :

The number of non-negative integral solutions of the equation $x+y+z+5t=15$ is

The perpendicular distance of a line from origin is 2 units and the perpendicular makes an angle α with X-axis such that sin α=13. The equation of line is .

The value of $\frac{3\times 1^3}{1^2}+\frac{5\times (1^3+2^3)}{1^2+2^2}+\frac{7\times (1^3+2^3+3^3)}{1^2+2^2+3^2}+\cdots \text{upto}10\text{terms}$, is

$(\overrightarrow{a}.\hat{i})(\overrightarrow{a}\times \hat{i})+(\overrightarrow{a}.\hat{j})(\overrightarrow{a}\times \hat{j})+(\overrightarrow{a}.\hat{k})(\overrightarrow{a}\times \hat{k})$ is always equal to

Three distinct points $P(3u^2,2u^3);Q(3v^2,2v^3)andR(3w^2,2w^3)$ are collinear and equation $a{x}^3+b{x}^2+cx+d=0$ has roots u, v and w, then which of the following is true

Let $y=y\left(x\right)$ be the solution curve of the differential equation, $(y^2-x)\frac{dy}{dx}=1$, satisfying $y\left(0\right)=1$. This curve intersects the $x$-axis at a point whose abscissa is :

The point of intersection of the lines $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$ is

Six points in a plane be joining in all possible way by staright lines, and if no two of them be coincident or parallel, and no three pass through the same point (with the exception of the original $6$ points). The number of distinct points of intersection is equal to

The value of $(1+\cos \theta +i\sin \theta {)}^n+(1+\cos \theta -i\sin \theta {)}^n$ is

A parabolic curve is described parametrically by $x-3=t^2,$ $y=4t.$ Then equation of the parabola is

If the tangent and normal to a rectangular hyporbola $xy=c^2$ at a point cuts off intercepts $a_1$ and $a_2$ on $x-$axis and $b_1$, and $b_2$ on the $y-$axis, then $a_1{a}_2+b_1{b}_2=$

The angle (in degree) between the hour hand and the minute hand in a circular clock at $03:25$ hours is

Equation of a common tangent to the circle, $x^2+y^2-6x=0$ and the parabola, $y^2=4x$, is

The sets of real values of x for which

$lo{g}_{2x+3}{x}^2<lo{g}_{2x+3}(2x+3)$

includes

Equation of normal to the curve at y = $4x^2$ at (a,b) is x+16y-258 = 0 . Find the value of (a+b)

All the values of $x$ for which $x^2-5x+6$ is non-negative are

For $a>b>c>0$, the distance between (1,1) and the point of intersection of the lines $ax+by+c=0$ and $bx+ay+c=0$ is less then $2\sqrt{2}$. Then

A five letter word is to be formed such that the letters appearing in the odd positions are taken from the unrepeated letters of the word MATHEMATICS whereas the letters which occupy even places are taken from amongst the repeated letters.___

Which of the following statements is/are correct about identity function?

The set of values of $k$ for which the equation $x^4+(k-1)x^3+x^2+(k-1)x+1=0$ has $2$ positive and $2$ negative roots is

If $\int (2-3si{n}^{2}x)\sqrt{secx}dx=2f\left(x\right)\sqrt{g\left(x\right)}+c$ and f(x) is non constant function then

If the equation $x^2+9y^2-4x+3=0$ is satisfied for all real values of $x$ and $y,$ then

If $\int \frac{dx}{x\sqrt{1-x^3}}=alog\left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right|+C$ then a =

If $|z|=1$ , then $\arg \left(\sqrt{\frac{(1+z)}{(1-z)}}\right)$ will be

The term independent of x in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$ is

The $6^{\text{th}}$ (from the beginning) term in the expansion of the ${[\sqrt{2^{\log (10-3^x)}}+\sqrt[5]{52^{(x-2)\log 3}}]}^m$

is equal to $21$. It is known that the binomial coefficient of the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ term in the expansion are in an $A.P.$, then the value of $x$ is/are

You are asked to construct a distance chart representing distances between cities given in a set with all the cities of the same set. What kind of matrix will come out of it?

A curve is represented parametrically by the equations $x=f\left(t\right)=a^{\ln \left(b^t\right)}$ and $y=g\left(t\right)=b^{-\ln \left(a^t\right)};a,b>0$ and $a\ne 1,b\ne 1$ where $t\in R$.



Which of the following is not a correct expression for $\frac{dy}{dx}?$

The value of $i^{57}+\frac{1}{i^{125}}$ is :

Let $n\left(U\right)=700$, $n\left(A\right)=200$, $n\left(B\right)=300$ and $n(A\cap B)=100$, Then $n(A^C\cap B^C)=$

Term independent of $x$ in the expansion of ${(x-\frac{1}{x^2})}^{3n}$ (where $n$ is even natural number)

Find the value of x, if $5{\tan }^{-1}x+2co{t}^{-1}x=2\pi .$

From the vertex of the parabola $y^2=4ax$ pair of perpendicular chords are drawn. If this chords are adjacent sides of a rectangle, then the locus of the vertex of the rectangle diagonally opposite to the vertex of parabola is

If "$x$" is an element of the set "$Y$", then we write it as