Consider : f \left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin x} \over {1 - \sin x}}} } \right),x \in \left( {0,{\pi \over 2}} \right). A normal to $$y=$$ f$$\left(x\right)$$ at x = {\pi \over 6} also passes through the point:

A wire of length $$2$$ units is cut into two parts which are bent respectively to form a square of side $$=x$$ units and a circle of radius $$=r$$ units. If the sum of the areas of the square and the circle so formed is minimum, then:

Let C be a curve given by y(x) = 1 + \sqrt {4x - 3} ,x > {3 \over 4}. If P is a point on C, such that the tangent at P has slope {2 \over 3}, then a point through which the normal at P passes, is :

Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval :

The minimum distance of a point on the curve y = x2−4 from the origin is :

If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3−1, t $$\in$$ R, meets the curve again at a point Q, then the coordinates of Q are :

The area (in sq. units) of the region described by A= {(x, y) $$|$$y$$\ge$$ x2 $$-$$ 5x + 4, x + y $$\ge$$ 1, y $$\le$$ 0} is :

The area (in sq. units) of the region $$\left\{\left(x,y\right):y^2\ge 2xand{x}^2+y^2\le 4x,x\ge 0,y\ge 0\right\}$$ is :

The Boolean expression $$\left(p\wedge \sim q\right)\vee q\vee \left(\sim p\wedge q\right)$$ is equivalent to :

The contrapositive of the following statement, “If the side of a square doubles, then its area increases four times”, is :

Consider the following two statements : P : If 7 is an odd number, then 7 is divisible by 2. Q : If 7 is a prime number, then 7 is an odd number If V1 is the truth value of the contrapositive of P and V2 is the truth value of contrapositive of Q, then the ordered pair (V1 , V2) equals :

A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is 90 s, 91 s, 95 s and 92 s. If the minimum division in the measuring clock is 1 s, then the reported mean time should be :

A screw gauge with a pitch of 0.5 mm and a circular scale with 50 divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the 45th division coincides with the main scale line and that the zero of the main scale is barely visible. What is the thickness of the sheet if the main scale reading is 0.5 mm and the 25th division coincides with the main scale line?

In the following ‘I’ refers to current and other symbols have their usual meaning. Choose the option that corresponds to the dimensions of electrical conductivity :

A, B, C and D are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation AD = C ln (BD) holds true. Then which of the combination is not a meaningful quantity ?

A hemispherical glass body of radius 10 cm and refractive index 1.5 is silvered on its curved surface. A small air bubble is 6 cm below the flat surface inside it along the axis. The position of the image of the air bubble made by the mirror is seen :

A convex lens, of focal length 30 cm, a concave lens of focal length 120 cm, and aplane mirror are arranged as shown. For an object kept at a distance of 60 cm from the convex lens, the final image, formed by the combination, is a real image, at a distance of :

To determine refractive index of glass slab using a travelling microscope, minimum number of readings required are :

To find the focal length of a convex mirror, a student records the following data : .tg {border-collapse:collapse;border-spacing:0;} .tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;} .tg .tg-wrg0{font-size:22px;text-align:center;vertical-align:top} .tg .tg-wv9z{font-size:22px;text-align:center} Object Pin Convex Lens Convex Mirror Image Pin 22.2 cm 32.2 cm 45.8 cm 71.2 cm The focal length of the convex lens is f1 and that of mirror is f2. Then taking index correction to be negligibly small, f1 and f2 are close to :

An observer looks at a distant tree of height $$10$$ $$m$$ with a telescope of magnifying power of $$20.$$ To the observer the tree appears: