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The lengths and bearings of a closed traverse PQRSP are given below.

If ${\log }_3(2x^2+6x-5)>1$, then the number of integral values of $x$ which do not satisfy the inequality, is

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X=1) + P(X=2) equals:

If z is a complex number of unit modulus, the value of $\frac{2+2z}{3+3\overline{z}}$ is

If $-7\le x^2+8x+5\le 14$, then the interval(s) in which $x$ lies is/are

Find the area of the
triangle formed by the lines joining the vertex of the parabola x²
= 12y to the ends of its latus rectum.

$\underset{x\to \infty }{lim}\frac{(2x+1{)}^{40}(4x-1{)}^5}{(2x+3{)}^{45}}$

If $\underset{0}{\overset{1}{\int }}\frac{e^x}{1+x}dx=k,$ then $\underset{0}{\overset{1}{\int }}\frac{e^x}{(1+x{)}^2}dx$ is equal to

The domain of $\sqrt{x+2}+\frac{1}{{\log }_{10}(x+1)}$ is

If the normals at $(x_i,y_i)$, i = 1, 2, 3, 4 to the rectangular hyperbola xy = 2 meet at the point (3, 4), then

A radioactive nucleus $A$ with a half life $T$, decays into a nucleus $B$. At $t=0$, there is no nucleus $B$. At sometime $t$, the ratio of the number of $B$ to that of $A$ is $0.3$. Then, $t$ is given by

Which of the following is not a tautology ?

If the extremities of a diagonal of a square are $(1,-2,3)$ and $(2,-3,5)$, then the length of its side $\left(\text{in units}\right)$ is:

The number of positive integral solutions of $\frac{x^2(3x-4{)}^3(x-2{)}^4}{(x-5{)}^5(2x-7{)}^6}\le 0$ is

Equation of the common tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and the circle $x^2+y^2=12$ is

Two radioactive materials $X_1$ and $X_2$ contain same number of nuclei. If $6\lambda {\text{sec}}^{-1}$ and $4\lambda {\text{sec}}^{-1}$ are the decay constants of $X_1$ and $X_2$ respectively, then the ratio of number of nuclei undecayed of $X_1$ to that of $X_2$ will be $1/e$ after a time:

If $\alpha$ and $\beta$ are the roots of the equation $x^2+5x-7=0$. Then a equation with roots
$\frac{1}{\alpha }and\frac{1}{\beta }$
is .

If the direction cosines of a variable line in two adjacent positions be l, m, n and l + a, m + b, n + c and the small angle between the two positions be $\theta$, then :

The Equation of the directrix to parabola $y^2=8x$ is _____

For $x\in (0,\frac{\pi }{2})$, which of the following is/are correct

Pick out
the solution from the values given in the bracket next to each
equation. Show that the other values do not satisfy the equation.

(a) 5m
= 60 (10, 5, 12, 15)

(b) n
+ 12 = 20 (12, 8, 20, 0)

(c) p
− 5 = 5 (0, 10, 5 − 5)

(d) (7,
2, 10, 14)

(e) r
− 4 = 0 (4, − 4, 8, 0)

(f) x +
4 = 2 (− 2, 0, 2, 4)

If the ratio of the roots of the equation $a{x}^2+bx+c=0$ be $p:q$, then

Divide the largest 6 digit number by the largest 2 digit number.

Let $\alpha ,\beta ,\gamma ,\delta$ be real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2\ne 0$ and $\alpha +\gamma =1.$ Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta .$ Then which of the following statements is/are TRUE?

The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar if

If points $z_1=a+i,z_2=1+ib$ and $z_0=0+i0$ form an equilateral triangle where $(a,b)\in (0,1),$ then