Let $f:R\overrightarrow{0,1}$ be a continuous function. Then, which of the following function (s) has (have) the value zero at some point in the interval (0,1)? $e^x-{\int }_0^{x}f\left(t\right)\sin tdt$ (b) $f\left(x\right)+{\int }_0^{\frac{\pi }{2}}f\left(t\right)\sin tdt$(c)$x-{\int }_0^{\frac{\pi }{2}-x}f\left(t\right)\cos tdt$ (d) $x^9-f\left(x\right)$

Let $a,b,xandy$ be real numbers such that $a-b=1andy\ne 0.$ If the complex number $z=x+iy$ satisfies $Im\left(\frac{az+b}{z+1}\right)=y$ , then which of the following is (are) possible value9s) of x? (a)$-1-\sqrt{1-y^2}$ (b) $1+\sqrt{1+y^2}$(c)$-1+\sqrt{1-y^2}$ (d) $-1-\sqrt{1+y^2}$

Word of length 10 are formed using the letters A,B,C,D,E,F,G,H,I,J. Let $x$be the number of such words where no letter is repeated; and let $y$be the number of such words where exactly one letter is repeated twice and no other letter is repeated. The, $\frac{y}{9x}=$

Which of the following is (are) NOT the square of a $3\times 3$ matrix with real entries? (a)$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$ (b) $\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$ (c)$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ (d) $\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$

For how many values, of p, the circle $x^2+y^2+2x+4y-p=0$and the coordinate axes have exactly three common points?

Let $f:R\vec{R}$be a differentiable function such that $f\left(0\right),f\left(\frac{\pi }{2}\right)=3and{f}^{prime}\left(0\right)=1.$If $g\left(x\right)={\int }_x^{\frac{\pi }{2}}\left[f^{prime}\left(t\right)\cos ect-\cot t\cos ectf\left(t\right)\right]dtf\text{or}x\left(0,\frac{\pi }{2}\right],$then ${\left(\lim \right)}_{x\vec{0}}g\left(x\right)=$

If $2x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$ then which of the following CANNOT be sides of a right angled triangle? (a)$a,4,2$ (b) $a,4,1$(c)$2a,4,1$ (d) $2a,8,1$

Let $\left[x\right]$ be the greatest integer less than or equal to $x\dot{}$ Then, at which of the following point (s) function $f\left(x\right)=x\cos \left(\pi \left(x+\left[x\right]\right)\right)$ is discontinuous? (a)$x=1$ (b) $x=-1$ (c) $x=0$ (d) $x=2$

Let $XandY$be two events that $P\left(X\right)=\frac{1}{3},P\left(X\text{|}Y\right)=\frac{1}{2}andP\left(Y\text{|}X\right)=\frac{2}{5}$then:$P\left(Y\right)=\frac{4}{15}$ (b) $P\left(X\cup Y\right)=\frac{2}{5}$$P\left(X^{prime}\text{|}Y\right)=\frac{1}{2}$ (d) $P\left(X\cap Y\right)=\frac{1}{5}$

If a chord, which is not a tangent of the parabola $y^2=16x$has the equation $2x+y=p,$and midpoint $\left(h,k\right),$then which of the following is(are) possible values (s) of $p,handk?$$p=-1,h=1,k=-3$ $p=2,h=3,k=-4$ $p=-2,h=2,k=-4$ $p=5,h=4,k=-3$