Physics

Work, Energy and Power
Motion of System of Particles and Rigid Body
Gravitation
Behaviour of Perfect Gas and Kinetic Theory
Current Electricity
Magnetic Effects of Current and Magnetism
Electromagnetic Induction and Alternating Currents
Electromagnetic Waves
Optics
Uncategorized
Physical World and Measurement
Kinematics
Vectors
Laws of Motion
Properties of Bulk Matter
Thermodynamics
Oscillations and Waves
Electrostatics
Dual Nature of Matter and Radiation
Atoms and Nuclei
Electronic Devices & Semiconductor
Communication System

Chemistry

Solutions
Electrochemistry
Chemical Kinetics
Surface Chemistry
General Principles and Processes of Isolation of Elements
Some Basic Concepts of Chemistry
Structure of Atom
Classification of Elements and Periodicity in Properties
Chemical Bonding and Molecular Structure
States of Matter: Gases and Liquids
Equilibrium
Redox Reactions
Hydrogen
s-Block Element (Alkali and Alkaline earth metals)
Some p-Block Elements
Organic Chemistry- Some Basic Principles and Techniques
Hydrocarbons
Environmental Chemistry
Solid State
p-Block Elements
d and f Block Elements
Coordination Compounds
Haloalkanes and Haloarenes
Alcohols, Phenols and Ethers
Aldehydes, Ketones and Carboxylic Acids
Organic Compounds Containing Nitrogen
Amines
Biomolecules
Polymers
Chemistry in Everyday Life
Thermodynamics
Uncategorized
Nuclear Chemistry

Mathematics

Coordinate Geometry
Circle and System of Circles
Calculus
Mathematical Reasoning
Statistics and Probability
Properties of Triangle
Vectors and Three-Dimensional Geometry
Trigonometric Ratios & Identities
Uncategorized
Sets, Relations and Functions
Algebra
Permutations and Combinations
Linear Programming
Matrices and Determinants
Logarithm, Indices, Surds and Partial Fraction
Progressions
Trigonometric Equation & Inequalities, Solution Of Triangles
Correlation and Regression
Trigonometry
Principle of Mathematical Induction
Complex Numbers and Quadratic Equations
Linear Inequalities
Binomial Theorem
Sequence and Series
Conic Sections
Differential Calculus
Limit, Continuity and Differentiability
Integral Calculus
Differential Equations

Q 1.

Correct4

Incorrect-1

The period of the function is

Period of

Period of

Q 2.

Correct4

Incorrect-1

The period of the function is

Period of and period of

L.C.M. of and

Q 3.

Correct4

Incorrect-1

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle ABC (R being the radius of the circum-circle)

and . So two sides and two angles are known. So is known. Therefore, two sides and included angle is known. So, is uniquely known in case .

If a, b, c are known the is uniquely known in case . So, sides a, b and angle A, B are known. So is known. Therefore two sides and included angle is known. So, is uniquely known in case .

So, only a side and an angle is known. So, is not uniquely known in case

Q 4.

Correct4

Incorrect-1

The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is

and

⇒ .

Q 5.

Correct4

Incorrect-1

If the roots of the equation x^{2} - 2ax + a^{2} + a - 3 = 0 are real less than 3, then

a< 2

2 ≤ a ≤ 3

3 < a ≤ 4

a > 4

s

f (x) = x^{2} - 2ax + a^{2} + a - 3 = 0, f (3) /> 0, α + β < 6, Δ ≥ 0.

⇒ a^{2} - 5a + 6 /> 0, a < 3, - 4a + 12 /> 0 ⇒ a < 2 or a /> 3, a < 3,

a < 3 ⇒ a < 2

Q 6.

Correct4

Incorrect-1

If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation

0 ≤ M ≤ 1

1 ≤ M ≤ 2

2 ≤ M ≤ 3

3 ≤ M ≤ 4

3 ≤ M ≤ 4

As A.M. ≥ G.M. for positive real numbers, we get

⇒ M ≤ I

(Putting values)

Also(a + b) (c + d) /> 0

[^{.}.^{.} a, b, c, d /> 0]

0 ≤ M ≤ 1

Q 7.

Correct4

Incorrect-1

If b > a, then the equation (x - a) (x - b) - 1 = 0 has

Both roots in (a, b)

Both roots in (-∞, a),

Both roots in (b, + ∞)

One root in (-∞, a) and the other in (b, +∞)

The given equation is (x - a)(x - b) - 1 = 0, b /> a

Or x^{2} - (a + b) x + ab - 1 = 0

Since coeff. of x^{2} i.e. 1 /> 0 it represents upward parabola, intersecting x axis at two points. (corresponding to two real roots, D being +ve)

Also f = f = - 1 ⇒ curve is below x-axis at a and b ⇒ a

and b both lie between the roots.

Thus the graph of given eq^{n} is as shown.

From graph it is clear that one root of the equation lies in

(- ∞, a) and other in (b, ∞).

.

Q 8.

Correct4

Incorrect-1

For the equation 3x^{2} + px + 3 = 0, p > 0, if one of the root is square of the other, then p is equal to

1/3

1

3

2/3

Let α, α^{2} be the roots of 3x^{2} + px + 3 = 0

Now S = α + α^{2} = - p/3, p = α^{3} = 1 ⇒ α = 1, ω, ω^{2};

α + α^{2} = - p/3 ⇒ ω + ω^{2} = -p/3 ⇒ - 1= -p/3 ⇒ p = 3

Q 9.

Correct4

Incorrect-1

For all 'x', x^{2} + 2ax + 10 - 3a > 0, then the interval in which 'a' lies is

a< - 5

- 1 < a < 2

a> 5

2 < a < 5

x^{2} + 2ax + 20 - 3x /> 0 x

⇒ D < 0 ⇒ 4a^{2} - 4(10 - 3a) < 0 ⇒ a^{2} + 3a - 10 < 0

⇒ (a + 5) (a - 2) < 0 ⇒ a ε (- 5, 2)

Q 10.

Correct4

Incorrect-1

If in ΔABC, ∠A = 90^{0} and c, sin B, cos B are rational numbers, then

A is rational and b is irrational

A is irrational and b is rational

A and b both are rational

None of these

Here C = 90^{0} - B and sin C = cos B.

Also a =

⇒ a, b are rational

Q 11.

Correct4

Incorrect-1

If A is the area and 2s the sum of the sides of a triangle, then

A ≤

A ≤

A />

None of these

We have 2s = a + b + c, A^{2} = s(s - a) (s - b) (s - c).

Now A.M ≥ G.M

⇒ ≥ [s(s - a) (s - b) (s - c)]^{1/4}

⇒ ≥ [A^{2}]^{1/4 }⇒ ≥ A^{1/2 }⇒ A ≤.

Also ≥ [(s - a) (s - b) (s - c)]^{1/3}

oror ⇒ A ≤

Q 12.

Correct4

Incorrect-1

ABC is a triangle whose medians AD and BE are perpendicular to each other. If AD = p and BE = q then area

of ΔABC is-

AM = , MD = , BM =

BM^{2} = BM^{2} + DM^{2}

Q 13.

Correct4

Incorrect-1

In a ΔABC if ∠A = 60^{0}, then ∠B - ∠C has value equal to

15^{0 }^{ }

30^{0}^{ }

22.5^{0 }^{ }

45^{0}

(given)

⇒

Q tan

tancot 30^{0}

tan

⇒ = 15^{0}

⇒ B - C = 30^{0}

Q 14.

Correct4

Incorrect-1

If D is the mid-point of side BC of a triangle ABC and AD is perpendicular to AC, then

3b^{2} = a^{2} - c^{2 }^{ }

3a^{2} = b^{2} - 3c^{2}^{}

b^{2} = a^{2} - c^{2 }^{ }

a^{2}+ b^{2}= 5c^{2}

From the right-angled triangle CAD, we have

cos C =

⇒

⇒a^{2} + b^{2}_{ }- c^{2} = 4b^{2}

⇒ a^{2} - c^{2}_{ }= 3b^{2}

Q 15.

Correct4

Incorrect-1

In a triangle, if the sum of two sides is x and their product is y such that (x + z) (x - z) = y where z is the third side of the triangle then the triangle is

Equilateral

Tight angled

Obtuse angled

None of these

Let ABC be the triangle with b + c = x and

bc = y then a = z, and from the given relations we have

(b + c + a) (b + c - a) = bc

⇒ b^{2} + c^{2}_{ }- a^{2} = - bc

⇒

⇒cos A = - = cos 120^{0}

⇒ A = 120^{0} and the triangle is obtuse angled.

⇒ A is an obtuse angle.

Q 16.

Correct4

Incorrect-1

In ΔABC, a cos (B - C) + b cos (C - A) + c cos (A - B) (where a, b, c are sides of Δ) equals

None

a = 2R sin A

b = 2R sin B

& c = 2R sin C

Q 17.

Correct4

Incorrect-1

In a triangle ABC, ∠A = 60^{0}, a = 4 and b = 3, then c is a root

of the equation

c^{2} + 3c + 7 = 0

c^{2} -3c + 7 = 0

c^{2} + 3c -7 = 0

c^{2} -3c -7 = 0

cos 60^{0} = ⇒ c^{2} -3c - 7 = 0

Q 18.

Correct4

Incorrect-1

In a triangle ABC, if + = then C is equal to -

30^{0 }^{ }

60^{0 }

75^{0 }^{ }

90^{0}^{ }

The given relation can be written as

=

⇒ (a + b + 2c) (a + b + c) = 3(a + c) (b + c)

⇒ (a + b)^{2} + 3c(a + b) + 2c^{2} = 3(ab + ac + bc+ c^{2})

⇒ a^{2} + b^{2} - c^{2} = ab

cos C =

= =

⇒ C = 60^{0}.

Q 19.

Correct4

Incorrect-1

If 4 cos A cos B + sin 2A + sin 2B + sin 2C = 4 then ΔABC is

Right angle

Isosceles

Right angled isosceles

None of these

Q 4 cos A cos B + 4 sin A sin B sin C = 4

sin C = ≤1 ... (1)

⇒ cos (A - B) ≥ 1

⇒ cos (A - B) = 1

True if A = B ... (2)

Put (2) in (1)

sin C = 1

C = 90^{0}

Q 20.

Correct4

Incorrect-1

Thesumof all solutions of the equation is

None of these

Here, or

or

where ;

The required sum = 30π.

Q 21.

Correct4

Incorrect-1

In any triangle ABC if , then the triangle isC

Right angled

Equilateral

Isosceles

None of these

⇒

⇒

⇒= sinA

⇒

⇒

⇒

⇒

Triangle is isosceles.

Q 22.

Correct4

Incorrect-1

If in a ΔABC, the sides b, a, c are in A.P. then-

acos2+ c cos2=

c cos2 + b cos2=

bcos2 + a cos2=

2 sin= cos

b + c = 2a

Option B :

⇒ c cos2 B/2 + b cos2 C/2 = 3a/2

⇒ 2c cos2 B/2 + 2b cos2 C/2 = 3a

⇒ c (1 + cos B) + b(1 + cos C) = 3a

⇒ b + c + c cos B + b cos C = 3a

⇒ b + c + a = 3a

⇒b + c = 2a which is in A.P.

Q 23.

Correct4

Incorrect-1

If p_{1}, p_{2}, p_{3} are respectively the perpendiculars from the vertices of a Δ to the opposite sides, then -

++= r

p_{1}p_{2}p_{3}=

++=

++=

p1= , p2=, p3=

⇒++=

⇒ == ...(i)

p1p2p3 === ......(ii)

Also ++=

(a cos A + b cos B + c cos C)

=(4 sinA sin B sin C) ==.......(iii)

and ++

=×+×+×

= 2Δ= 2Δ

= 2(a^{2} + b^{2} + c^{2}). =

Q 24.

Correct4

Incorrect-1

In a right angled triangle acute angle α, β satisfy tan α + tan β + tan^{2}α + tan^{2}β = 4, if hypotenuse is of length d then area of Δ is-

d^{2 }^{ }

d^{2}^{ }

α + β = 90^{0}

tan α + cot α + tan^{2} β + cot^{2} β = 4

Let tan α = t

t ++ t^{2 }+ = 4

t = 1 satisfy

α = β = 45^{0}

So area is

Q 25.

Correct4

Incorrect-1

If cos A + cos B+ cos C = , then (ratio of inradius (r) and R is circumradius) is a root of the equation -

- 2 + = 0

- 2 - 1 = 0

+ 2 - 1 = 0

+ 3 - 1 = 0

cos A + cos B + cos C =

⇒ 1 + 4 sinA/2 sin B/2 sinC/2 =

⇒1 + = ⇒ = - 1

Your request has been placed successfully.