The value of cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right) is :

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

In a shop there are five types of ice-cream available. A child buys six ice-cream. Statement - 1: The number of different ways the child can buy the six ice-cream is $${}^{10}{C}_5$$. Statement - 2: The number of different ways the child can buy the six ice-cream is equal to the number of different ways of arranging 6 A and 4 B's in a row.

STATEMENT - 1 : For every natural number $$n\ge 2,$$ ${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n .$ STATEMENT - 2 : For every natural number $$n\ge 2,$$, $\sqrt {n\left( {n + 1} \right)} < n + 1.$

The quadratic equations $$x^2-6x+a=0$$ and $$x^2-cx+6=0$$ have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

If the straight lines {{x - 1} \over k} = {{y - 2} \over 2} = {{z - 3} \over 3} and {{x - 2} \over 3} = {{y - 3} \over k} = {{z - 1} \over 2} intersects at a point, then the integer $$k$$ is equal to

The line passing through the points $$(5,1,a)$$ and $$(3,b,1)$$ crosses the $$yz$$-plane at the point \left( {0,{{17} \over 2}, - {{ - 13} \over 2}} \right) . Then

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

Suppose the cubic $$x^3-px+q$$ has three distinct real roots where $$p>0$$ and $$q>0$$. Then which one of the following holds?

How many real solutions does the equation $$x^7+14x^5+16x^3+30x-560=0$$ have?

The area of the plane region bounded by the curves $$x+2y^2=0$$ and $$x+3y^2=1$$ is equal to :

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number iff y is a transcendental number”. Statement –1: r is equivalent to either q or p. Statement –2: r is equivalent to $$\sim \left(p\leftrightarrow \sim q\right)$$

The statement $$p\to \left(q\to p\right)$$ is equivalent to

The dimension of magnetic field in M, L, T and C (coulomb) is given as

A body of mass m = 3.513 kg is moving along the x-axis with a speed of 5.00 ms−1. The magnitude of its momentum is recorded as

Two full turns of the circular scale of a screw gauge cover a distance of 1 mm on its main scale. The total number of divisions on the circular scale is 50. Further, it is found that the screw gauge has a zero error of − 0.03 mm while measuring the diameter of a thin wire, a student notes the main scale reading of 3 mm and the number of circular scale divisions in line with the main scale as 35. The diameter of the wire is

An experiment is performed to find the refractive index of glass using a travelling microscope. In this experiment distances are measured by

A student measures the focal length of a convex lens by putting an object pin at a distance $${}^{\prime }{u}^{\prime }$$ from the lens and measuring the distance $${}^{\prime }{v}^{\prime }$$ of the image pin. The graph between $${}^{\prime }{u}^{\prime }$$ and $${}^{\prime }{v}^{\prime }$$ plotted by the student should look like

Shown in the figure below is a meter-bridge set up with null deflection in the galvanometer. The value of the unknown resister $$R$$ is