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The figure shows a parabola whose equation is $x=\frac{y^2}{8}$. For very small aperture, the power of the convex mirror made from this parabola will be

[Assume $x$ and $y$ are in metres]

Find the derivative of f(x) = 3x at x = 2

Without expanding the determinant, prove that

The value of the integral $\int \frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}dx$ is

(where $C$ is integration constant)

If $l_i^2+m_i^2+n_i^2=1$ for $i=1,2,3$ & $l_i{l}_j+m_i{m}_j+n_i{n}_j=0$ for $i,j\in \{1,2,3\}$ and $i\ne j$ and $\Delta =\left|\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right|,$ then

The difference between the greatest and least values of the function $f\left(x\right)=\sin 2x-x,$ on $[-\frac{\pi }{2},\frac{\pi }{2}]$ is

If a, b, c are in G.P., prove that log a, log b, log c are in A.P.

The values of $\alpha$ for which the point $(\alpha -1,\alpha +1)$ lies in the larger segment of the circle $x^2+y^2-x-y-6=0$ made by the chord whose equation is $x+y-2=0$ is

If the function $f\left(x\right)=\{\begin{array}{cc}a|\pi -x|+1,& x\le 5\\ & \\ b|x-\pi |+3,& x>5\end{array}$

is continuous at $x=5$, then the value of $a-b$ is



[1 mark]

Find the probability distribution of

number of heads in two tosses of a coin

number of tails in the simultaneous tosses of three coins

number of heads in four tosses of a coin.

If
is
the A.M. between a and b, then find the value of n.

Let $A=\left[\begin{array}{ccc}2& b& 1\\ b& b^2+1& b\\ 1& b& 2\end{array}\right]$ where $b>0$. Then the minimum value of $\frac{det\left(A\right)}{b}$ is :

If $f\left(x\right)=\sin x+{\int }_0^x{f}^{\prime }\left(t\right)(2\sin t-{\sin }^{2}t)dt$ then $f\left(x\right)$ is

The angles of a triangle are in A.P. and the number of degrees in the least angle is to the number of degrees in the mean angle as 1 : 120. Find the angles in radians.

Let $P$ be the point of intersection of the common tangents to the parabola $y^2=12x$ and the hyperbola $8x^2-y^2=8.$ If $S$ and $S^{\prime }$ denote the foci of the hyperbola where $S$ lies on the positive $x-$axis then $P$ divides $S{S}^{\prime }$ in a ratio

A debate club consists of $6$ girls and $4$ boys. A team of $4$ members is to be selected from this club including a selection of a captain (from among these four members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

If matrix $A=\left[\begin{array}{cc}2& y\\ 2y& 8\end{array}\right]$ is non-singular, then the value(s) of $y$ can be

The minimum distance between the curves $x^2=4y$ and $x^2+y^2+18x+12y+81=0$

${lim}_{x\to 1}\frac{1-\frac{1}{x}}{sin\pi (x-1)}$

For the given differential equation find the general solution.

$(x+3y^2)\frac{dy}{dx}=y(y>0).$

If $\alpha$ and $\beta$ are the roots of the equation $x^2+3x+5=0$, then ${\alpha }^2+{\beta }^2$ is

Find the sum to
indicated number of terms in each of the geometric progressions in
Exercise 7 to 10:

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis,

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+2x^2-3x+1=0$, then

A vertical pole subtends an angle $ta{n}^{-1}\left(1/2\right)$ at a point P on the ground. The angle subtended by the upper half of the pole at the point P is

Show that if A and B are square matrices such that AB = BA, then $(A+B{)}^2=A^2+2AB+B^2$.

A variable line has intercepts $e$ and $e^{\prime }$ on the coordinate axes, where $\frac{e}{2}$ and $\frac{e^{\prime }}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola respectively. The value of $r$ for which the line always touches the circle $x^2+y^2=r^2$ is

If $sinx+cosecx=2$, then $si{n}^{n}x+cose{c}^{n}x$ is equal to

Let chords of the circle $x^2+y^2=a^2$ touch the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ Then their middle points lie on the curve