Maths

If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $\left(\mathrm{0,2}\pi \right)-\left\{\pi \right\}$ which satisfy the equation, $2{\mathrm{c}\mathrm{o}\mathrm{t}}^2\theta -\frac{5}{\mathrm{s}\mathrm{i}\mathrm{n}\theta }+4=0$ , then ${\int }_{{\theta }_1}^{{\theta }_2}?{\mathrm{c}\mathrm{o}\mathrm{s}}^2?3\theta d\theta$ is equal to:

In a bombing attack, there is 50% chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99% chance of completely destroying the target, is

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ $$ and then reduced by $$ $q$ , where $p=0$ $$ and $q\ne 0$ $$ . If the new mean and new s.d. become half of their original values, then $q$ $$ is equal to:

Let the vectors a,b,c be such that |a|=2, |b|=4 and |c|=4. If the projection of b on a is equal to the projection of c on a and b is perpendicular to c, then the value of |a+b-c| is

If the lines x+y=a and x-y=b touch the curve y=x²-3x+2 at the points where the curve intersects the x-axis, then a/b is equal to

Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points $A$ and $B$ . Then $(AB{)}^2$  is equal to:

The coefficient of x⁴ in the expansion of (1+x+x²+x³)⁶ in powers of x, is

Let A=(a,b,c) and B=(1,2,3,4). Then the number of elements in the set C={f: A→B | 2∈f(A) and f is not one-one} is

The locus of a point which divides the line segment joining the point $(0,-1)$ $$ and a point on the parabola, $x^2=4y$ ${}^{}$ , internally in the ratio $1:2$ $$ , is

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$ . If the probability that at most two machines will be out of service on the same day is ${\left(\frac{3}{4}\right)}^{3}k$ , then $k$  is equal to: