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Equation of the line of shortest distace between the lines $\frac{x}{2}=\frac{y}{-3}=\frac{z}{1}$ and $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{5}$ is

Let f(x) be a function satisfying f ’(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = $x^2$. Then, the value of the integral ${\int }_0^{1}f\left(x\right)g\left(x\right)dx,$ is

The circle $x^2+y^2-4x-4y+4=0$ is inscribed in a triangle which has two of its sides along the coordinate axes. If the locus of the circumcenter of the triangle is $x+y-xy+k\sqrt{x^2+y^2}=0$, then the value of $k$ is

If $(1+x{)}^n=C_0+C_{1}x+\dots ..+C_n{x}^n$, then the value of $\underset{0\le r<s\le n}{\sum \sum }{C}_r{C}_s$ is equal to

The number of values of x where the function f(x) = 2 (cos 3x + cos $\sqrt{3x}$ attains its maximum, is

In the parabola $y^2+4=4x$, a chord passing through point $(2,0)$ cuts the parabola at $P$ and $Q$. If coordinates of $P$ are $(5,4)$ and the tangents at $P$ and $Q$ meet at $R$, then



$\begin{array}{cccc}& \text{List-I}& & \text{List-II}\\ \text{(I)}& \text{The focus of the parabola is}& \text{(P)}& (0,\frac{3}{2})\\ \text{(II)}& \text{The centroid of}△PQR\text{is}& \text{(Q)}& (4,0)\\ \text{(III)}& \text{The circumcentre of}△PQR\text{is}& \text{(R)}& (\frac{25}{12},\frac{3}{2})\\ \text{(IV)}& \text{The orthocentre of}△PQR\text{is}& \text{(S)}& (\frac{25}{8},\frac{3}{2})\\ & & \text{(T)}& (\frac{25}{4},\frac{3}{2})\\ & & \text{(U)}& (2,0)\end{array}$



Which of the following is the only CORRECT combination?

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces 20% and plant $T_2$ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P(computer turns out to be defective, given that it is produced in plant $T_1)$ = 10P (computer turns out to be defective, given that it is produced in plant $T_2$), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then, the probability that it is produced in plant $T_2$, is ?

Let $PQ$ be a focal chord of $y^2=4ax$. The tangents to parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a(a>0)$. If the chord $PQ$ subtends an angle $\theta$ at the vertex of prabola then $\tan \theta =$

In the polynomial (x - 1)(x - 2)(x - 3)............... .........(x - 100), the coefficient of $x^{99}$ is

A circle is inscribed in an equilateral triangle of side a, the area of any square inscribed in the circle is

If |x| < 1, then the sum of the series 1 + 2x + $3x^2$ + $4x^3$ + ....... $\infty$ will be

Value of $i^n+i^{n+1}+i^{n+2}+i^{n+3}$, when $n\in I$ is equal to

The number of real solution(s) of $||x-2|-2|-2|x|=|x-3|$ is

If the $(r+1{)}^{th}$ term in the expansion of ${(\sqrt[3]{3\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{3a}}})}^{21}$ has the same power of a and b, then value of r is

If $\left|\begin{array}{ccc}a& b-c& c+b\\ a+c& b& c-a\\ a-b& a+b& c\end{array}\right|=0,$ then the line $ax+by+c=0$ passes through the fixed point which is

What is the inverse of the function { (1,2),(2,3), (3,1) } from the set A ={1,2,3} to itself?

Find the parametric and symmetric equation of the line passing through the point $(2,3,4)$ and perpendicular to the plane $5x+6y-7z=20$ ?

$3[si{n}^4(\frac{3\pi }{2}-\alpha )+si{n}^4(3\pi +\alpha \left)\right]-2[si{n}^6(\frac{\pi }{2}+\alpha )+si{n}^6(5\pi -\alpha \left)\right]=$

If $I_n=\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}{\cot }^{n}xdx$, then

Let $\varphi (x,y)=0$ be the equation of a circle, If $\varphi (0,\lambda )=0$ has equal roots $\lambda =2,2$ and $\varphi (\lambda ,0)=0$ has roots $\lambda =\frac{4}{5},5,$ then the roots of the circle is

Let $\lambda$ be a real number for which the system of linear equations

$x+y+z=6$

$4x+\lambda y-\lambda z=\lambda -2$

$3x+2y-4z=-5$

has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation :

The function f(x) defined as f(x) = $\sqrt{(x-4{)}^2}$

Consider a triangular plot $ABC$ with sides $AB=7m,BC=5m$ and $CA=6m$. A vertical lamp-post at the mid point $D$ of $AC$ subtends an angle ${30}^{\circ }$ at $B$. The height (in $m$) of the lamp-post is :

Let the two vertices of a triangle are $(2,-1)$ and $(3,2)$ and third vertex lies on the line $x+y=5$. If the area of triangle is $4\text{sq. units}$, then the coordinates of the third vertex is/are

The area of the director circle of the ellipse $\frac{x^2}{5}+\frac{y^2}{4}=1$ is

If ${\log }_{10}7=0.8451$ and ${\log }_{10}11=1.0414$, then the number of digits in ${77}^{100}$ is

In a ${\Delta }^{le}ABC,tan\frac{A}{2},Tan\frac{B}{2}Tan\frac{C}{2}$ are in H.P. then the value of cot $\left(\frac{A}{2}\right).cot\left(\frac{C}{2}\right)=$

If $a_1,a_2,\cdot \cdot \cdot ,a_{15}$ are in A.P. and $a_1+a_8+a_{15}=15$, then $a_2+a_3+a_8+a_{13}+a_{14}$ equals

Sum of values of $x$, satisfying the equation $\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}=4-2x-x^2,$ is

If $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3x^2-10x-25=0,$ then the value of $3{\sin }^2(A+B)-10\sin (A+B)\cdot \cos (A+B)-25{\cos }^2(A+B)$ is :

Let $A=\{x_1,x_2,x_3,\dots ,x_8\},B=\{y_1,y_2,y_3\}$ then the total number of functions from $A$ to $B$ such that all the elements of $B$ has atleast one pre image and there are exactly four elements in $A$ having image as $y_3,$ are

In the expansion of ${(\frac{x}{\cos \theta }+\frac{1}{x\sin \theta })}^{16}$, if $l_1$ is the least value of the term independent of $x$ when $\frac{\pi }{8}\le \theta \le \frac{\pi }{4}$ and $l_2$ is the least value of the term independent of $x$ when $\frac{\pi }{16}\le \theta \le \frac{\pi }{8},$ then the ratio $l_2:l_1$ is equal to :

If z lies on the circle $|z-1|=1$, then $\frac{z-2}{z}$ equals

The sum of real roots of the equation $\left|\begin{array}{ccc}x& -6& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0,$ is equal to :

If the equation $x^4-(k-1)x^2+(2-k)=0$ has three distinct real roots, then the possible value(s) of $k$ is/are

In a right angled triangle, medians drawn from the acute angles make an angle of $\theta$ which each other and L is the length of the hypotenuse. Then the area of the triangle is equal to:

If $a{x}^2+2bx+3c=0,a\ne 0,c>0$ does not have any real roots, then which of the following is/are true?

$\Delta \left(r\right)=\left|\begin{array}{ccc}r& r^2& r^3\\ 1& 2& 4\\ 2& 1& 3\end{array}\right|$ then ${\sum }_{r=1}^5\Delta \left(r\right)$ will be _____

Let $p,q,r\in R^{+}$ such that $27pqr\ge (p+q+r{)}^3$ and $3p+4q+5r=12.$ Then the value of $p+q+r$ is

Let $A$ and $B$ be two events such that $P\left(A\right)=\frac{3}{8},P\left(B\right)=\frac{1}{2}$ and $P(A\cup B)=\frac{5}{8}$. Then which of the following do/does hold good?

If the roots of $x^2$ - (a - 3)x + a = 0 are such that atleast ont of the roots is greater than 2, then a$ϵ$

If $n\left(U\right)=60,n\left(A\right)=21,n\left(B\right)=43,$ then minimum and maximum value of $n(A\cup B)$ is

If the curve $\frac{x^2}{4}+y^2=1$ and $\frac{x^2}{a^2}+y^2=1$ for suitable values of ${}^{\prime }{a}^{\prime }$ cut on four concyclic points, the equation of the circle passing through these points is

The number of real roots of the equation $x^2-12|x|+20=0$ is $P.$ Then the values of $a$ for which the equation $||x-2|+a|=P$ can have four distinct solutions, is

If $tan\theta =\sqrt{\frac{3}{2}}$, the sum of the infinite series 1 + 2 $(1-cos\theta )+3(1-cos\theta {)}^2+4(1-cos\theta {)}^3+....\infty$ is