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Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), x-axis and the ordinates x = π/4 and x = β > π/4 is (β sin β + (π/4) cos β + √2β). Then f(π/2) is

Solution: Given ∫π/4β f(x) dx = (β sin β + (π/4) cos β + √2β Differentiate both sides w.r.t.β... View Article

The angle bisector of a triangle divides the opposite side into two parts proportional to the other two sides of the triangle. Prove it.

Solution: Let ABC be the triangle. AD is the internal bisector of ∠BAC which meets BC at D. We have to prove BD/DC = AB/AC Draw CE parallel... View Article

How to convert an improper fraction into a mixed fraction?

Solution: To convert an improper fraction to a mixed fraction, we should divide the numerator by the denominator. Then write down the quotient... View Article

What is central angle?

Solution: A central angle is an angle whose vertex is the centre of a circle and whose arms are radii intersecting the circle at two distinct... View Article

How to calculate quartiles?

Solution: We can calculate the quartiles using the formulas given below. First quartile, Q1 = (n + 1)/4th term Second quartile, Q2 =... View Article

How to calculate coefficient of correlation?

Solution: We can calculate Pearson’s coefficient of correlation using the following formula. ρ(X,Y) = cov (X,Y) /... View Article

A ball is projected horizontally. After 3 second from projection its velocity becomes 1.25 times of the velocity of projection. Its velocity of projection is:-

a) 10 m/s b) 20 m/s c) 30 m/s d) 40 m/s Answer: d) 40 m/s Solution: vy = uy + ayt vy = 0 + 9.8 x 3 = 29. 4 m/s v'... View Article

The shape of drops and bubbles are spherical due to its

a) thrust on drop b) surface tension c) viscosity d) residual pressure Answer: b) surface tension Solution: A liquid surface... View Article

The solenoid has a core with a material of relative permeability of 500. The windings of the solenoid are insulated from the core and carry a current of 1A. If the number of turns is 500 per meter, then magnetisation is?

a) 2.5 x 103 A/m b) 2.5 x 105 A/m c) 2.0 x 103 A/m d) 2.0 x 103 A/m Answer: b) 2.5 x 105 A/m Solution: Here, n = 500... View Article

Explain surface tension on the basis of the molecular theory.

All the molecules in the liquid are pulled by other water molecules due to intermolecular force. The forces on a molecule are balanced out by... View Article

In the circuit shown here, the point ‘C’ is kept connected to point ‘A’ till the current flowing through the circuit becomes constant. Afterward, suddenly, point ‘C’ is disconnected from point ‘A’ and connected to point ‘B’ at time t=0. Ratio of the voltage across resistance and the inductor at t =L/R will be equal to:

a) 0 b) 1 c) -1 d) infinity Answer: c) -1 Solution:' Let the emf of the battery be E.From the question, the point C of the circuit is... View Article

A 4 cm tall object is placed perpendicular to the principal axis of a convex lens of focal length 24 cm. The distance of the object from the lens is 16 cm. Find the position, size and nature of the image formed, using the lens formula.

Solution: For convex lens f=+24cm,u=−16cm Using lens formula (1/f)=(1/v)−(1/u) (1/24)=(1/v)−(1/−16)... View Article

A block of weight W is suspended by a string of fixed length. The ends of the string are held at various positions as shown in the figures below. In which case, if any, is the magnitude of the tension along the string largest?

a. b. c. d. Answer: a Solution: From the free body diagrams 2Tcosθ = W Hence, T = W/2cosθ. If the θ increases then cosθ will decrease... View Article

A train is moving due east and a car is moving due north with equal speeds. A passenger in the train finds that the car is moving towards

a) North - East b) North - West c) South - West d) South - East Answer: b) North - West Solution: Let be the velocity of the train. We... View Article

The block of mass M moving on the frictionless horizontal surface collides with the spring of spring constant K and compresses it by length L. The maximum momentum of the block after collision is?

a) Zero b) ML2/K c) √MKL d) KL2/2M Answer: c) √MKL Solution: (1/2)mv2 = (1/2)kL2 v = L√(K/m) Maximum momentum, mv = mL√(K/m) = L√KM... View Article

A body is acted upon by force which is inversely proportional to the distance covered. The work done will be proportional to:

a) s b) s2 c) √s d) none of the above Answer: d) none of the above Solution: F ∝ (1/s) ⇒ F = k/s... View Article

A bullet of mass 20g has an initial speed of 1ms⁻¹, just before it starts penetrating a mud wall of thickness 20 cm. if the wall offers a mean resistance of 2.5×10⁻² N, the speed of the bullet after emerging from the other side of the wall is close to:

a) 0.4 m/s b) 0.1 m/s c) 0.3 m/s d) 0.7 m/s Answer: d) 0.7 m/s Solution: mass, m = 20 g Initial speed, u = 1m/s The... View Article

A small body of mass m falls to the earth from an infinite distance away. What will be its velocity on reaching the earth? (Radius of the earth = R, acceleration due to gravity on the surface of the earth is g):

a) gR b)2gR c) √gR d) √2gR Answer: d) √2gR Solution: change in potential energy = change in kinetic energy... View Article

Name the different types of polyhedral solids.

Below are the names of different types of polyhedral solids: Tetrahedron (or pyramid) Cube Octahedron Dodecahedron Icosahedron Cuboid Rectangular... View Article

What is a ratio?

Ration means the quantitative relation between two quantities representing the number of times one quantity contains or is contained within the... View Article

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