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If $a=-\hat{i}+\hat{j}+2\hat{k}$, $b=2\hat{i}-\hat{j}-\hat{k}$ then $c=-2\hat{i}+\hat{j}+3\hat{k}$, then the angle between 2a-c and a+b is:

The distance between the foci of the ellipse x=5cos$\theta$, y=4sin$\theta$ is

The value of ${\cos }^3({60}^{\circ }-A)-{\cos }^3({60}^{\circ }+A)$ is

Let $f\left(x\right)=({\lambda }^2+\lambda -2)x^2+(\lambda +2)x$ be a quadratic polynomial. The sum of all integral values of $\lambda$ for which $f\left(x\right)<1\forall x\in R$, is

Equation of a straight line passing through the point $(4,5)$ and equally inclined to the lines $3x=4y+7$ and $5y=12x+6$ is

The solution set of $x^4-8x^2-9\le 0$ is

If z is a complex number, then z.$\overline{z}$ = 0 if and only if

The area of an equilateral triangle inscribed in the circle $x^2+y^2+2gx+2fy+c=0$ is

If $a$ is a complex number such that $|a|=1$ and $a{z}^2+z+1=0$ has one purely imaginary root, then $\cos \left(\arg \right(a\left)\right)$ is

$(2+2\lambda +\mu )x^2+(1+2\lambda +2\mu )y^2+(3+5\lambda +3\mu )xy$

$-(16+14\lambda +11\mu )x-(10+13\lambda +17\mu )+(24+20\lambda +30\mu )=0$ is an equation of circle only when

If two tangents drawn from the point $(\alpha ,\beta )$ to the parabola $y^2=4x$ such that the slope of one tangent is double of the other, then

The vertex of the parabola $x^2+8x+12y+4=0$ is

The interval(s) of $x$ which satisfy the inequality $\frac{(1-3x{)}^7(1+5x{)}^4(x+5{)}^6}{(x^2-1)(x-6{)}^3(1-2x{)}^4}>0$ is/are

If the area of the quadrilateral formed by the tangents from the origin to the circle $x^2+y^2+6x-10y+c=0$ and radii corresponding to the point of contact is $15$ sq. units, then value(s) of $c$ can be

If in a $△ABC$, $\tan \frac{A}{2},\tan \frac{B}{2},\tan \frac{C}{2}$ are in H.P., then the minimum value of $\cot \frac{B}{2}$ is

If $A={45}^{\circ },$ then the value of ${\cos }^{2}B+{\sin }^2(A+B)+2\sin A\sin ({180}^{\circ }+B)\cos ({360}^{\circ }+A+B)$ is

The equation of the hyperbola whose asymptotes are $2x-y=3$ and $3x+y=7$ and passing though the point $(1,1)$ is

The line $4x+3y-4=0$ divides the circumference of the circle centred at $(5,3),$ in the ratio $1:2.$ Then the equation of the circle is

The possible values of $x^2+4$ lie in the interval

Two parabolas with a common vertex and with axes along $x-$ axis and $y-$ axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $3$, then the equation of the common tangent to the two parabolas is :

What is the number of principle diagonal elements in a square matrix of order n $\times$ n.

Let $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ be three non-coplanar unit vectors equally inclined to each other at an angle of $\frac{\pi }{3}$. Then the value of $|\overrightarrow{p}\times (\overrightarrow{q}\times \overrightarrow{r})|$ is

If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the from $7^m+7^n$ is divisible by 5, equals

Derivative of $log|x|$ w.r.t. $|x|$ is

Let $f:R\to R$ be a function defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2+2mx-1,& x\le 0\\ mx-1,& x>0\end{array}.$ If $f$ is one-one, then $m$ can be

The probabilities of different faces of a biased dice to appear are as follows

$\begin{array}{ccccccc}Facenumber& 1& 2& 3& 4& 5& 6\\ Probability& 0.1& 0.32& 0.21& 0.15& 0.05& 0.17\end{array}$

The dice is thrown and it is known that either the face number 1 or 2 will appear. Then, the probability of the face number 1 to appear is

If ${\log }_{x+3}(x^2-x)<1,$ then $x\in$

The set of values of $a$ for which the equation $\sqrt{a}\cos x-2\sin x=\sqrt{2}+\sqrt{2-a}$ possesses a solution, is

An integrating factor for the differential equation

$1+x^2)dx-(ta{n}^{-1}y-x)dy=0$

[MP PET 1993]

Mean of five observations is $4.4$ and variance is $8.24$. If three of the observations are $1,2$ and $6$, then the other two observations are

Find the radius of the circle $x^2+y^2-2x+4y-11=0$

If f(x) = $\sqrt{x}$ and $g\left(x\right)=x$ , then $\left(\frac{f}{g}\right)\left(x\right)$ equals to $(x\ne 0)$

General solution of the equation $\sin \theta \sin ({60}^{\circ }+\theta )\sin ({60}^{\circ }-\theta )=\frac{\sqrt{5}-1}{16}$ is

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is :

The set of real values of $t\in [-\frac{\pi }{2},\frac{\pi }{2}]$ satisfying $2\sin t=\frac{1-2x+5x^2}{3x^2-2x-1}$, $\forall x\in R-\{-\frac{1}{3},1\}$ lies in the interval

If $a^2(a+p)=b^2(b+p)=c^2(c+p),$ where $a,b,c,p\in R,$ then value of $ab+bc+ca$ is

If a function f(x )is continuous in [2,5] , differentiable in (2,5) and f(2) = f(5) then how many value x can have where f'(x) nullifies for sure?

If x satisfies $\left(x-1\right|+\left(x-2\right|+\left(x-3\right|\ge 6$, then

If $a,b,c\in R^{+}$ are such that $2a,b,4c$ are in A.P. and $c,a,\text{and}b$ are in G.P., then

If $\cot \theta =\frac{1}{2}$ and $\sec \varphi =-\frac{5}{3}$, where $\theta \in (\pi ,\frac{3\pi }{2})$ and $\varphi \in (\frac{\pi }{2},\pi )$, then the value of $\cot (\theta -\varphi )$ is

General values of $x$ for which $\sin 2x+\cos x=0$ is/are :

A tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at any point $P$ meets the line $x=0$ at a point $Q.$ Let $R$ be the image of $Q$ in the line $y=x$, then the circle whose extremities of a diameter are $Q$ and $R$ passes through a fixed point. The fixed point is

The value of ${lim}_{x\to \infty }x[ta{n}^{-1}\left(\frac{x+1}{x+2}\right)-ta{n}^{-1}\left(\frac{x}{x+2}\right)]is$

If $x\in (\frac{\pi }{2},\pi ),$ then $\sqrt{\frac{1-\sin x}{1+\sin x}}$ is equal to

If two vertices of a triangle are (6,4), (2,6) and its centroid is (4, 6), then the third vertex is