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11: Chapter 12 - Mathematics

11: Chapter 12 - Mathematics



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NCERT Books for Class 11 Maths PDF Download

NCERT Books Class 11 Maths: The National Council of Educational Research and Training (NCERT) publishes Maths textbooks for Class 11. The NCERT Class 11th Maths textbooks are well known for it’s updated and thoroughly revised syllabus. The NCERT Maths Books are based on the latest exam pattern and CBSE syllabus.

NCERT keeps on updating the Maths books with the help of the latest question papers of each year. The Class 11 Maths books of NCERT are very well known for its presentation. The use of NCERT Books Class 11 Maths is not only suitable for studying the regular syllabus of various boards but it can also be useful for the candidates appearing for various competitive exams, Engineering Entrance Exams, and Olympiads.


Introduction to Three Dimensional Geometry Class 11 MCQs Questions with Answers

Question 1.
The cartesian equation of the line is 3x + 1 = 6y – 2 = 1 – z then its direction ratio are
(a) 1/3, 1/6, 1
(b) -1/3, 1/6, 1
(c) 1/3, -1/6, 1
(d) 1/3, 1/6, -1

Answer: (a) 1/3, 1/6, 1
Hint:
Given 3x + 1 = 6y – 2 = 1 – z
= (3x + 1)/1 = (6y – 2)/1 = (1 – z)/1
= (x + 1/3)/(1/3) = (y – 2/6)/(1/6) = (1 – z)/1
= (x + 1/3)/(1/3) = (y – 1/3)/(1/6) = (1 – z)/1
Now, the direction ratios are: 1/3, 1/6, 1

Question 2.
The image of the point P(1, 3, 4) in the plane 2x – y + z = 0 is
(a) (-3, 5, 2)
(b) (3, 5, 2)
(c) (3, -5, 2)
(d) (3, 5, -2)

Answer: (a) (-3, 5, 2)
Hint:
Let image of the point P(1, 3, 4) is Q in the given plane.
The equation of the line through P and normal to the given plane is
(x-1)/2 = (y-3)/-1 = (z-4)/1
Since the line passes through Q, so let the coordinate of Q are (2r + 1, -r + 3, r + 4)
Now, the coordinate of the mid-point of PQ is
(r + 1, -r/2 + 3, r/2 + 4)
Now, this point lies in the given plane.
2(r + 1) – (-r/2 + 3) + (r/2 + 4) + 3 = 0
⇒ 2r + 2 + r/2 – 3 + r/2 + 4 + 3 = 0
⇒ 3r + 6 = 0
⇒ r = -2
Hence, the coordinate of Q is (2r + 1, -r + 3, r + 4) = (-4 + 1, 2 + 3, -2 + 4)
= (-3, 5, 2)

Question 3.
Three planes x + y = 0, y + z = 0, and x + z = 0
(a) none of these
(b) meet in a line
(c) meet in a unique point
(d) meet taken two at a time in parallel lines

Answer: (c) meet in a unique point
Hint:
Given, three planes are
x + y = 0 …….. 1
y + z = 0 …….. 2
and x + z = 0 ……… 3
add these planes, we get
2(x + y + z) = 0
⇒ x + y + z = 0 ……… 4
From equation 1
0 + z = 0
⇒ z = 0
From equation 2
x + 0 = 0
⇒ x = 0
From equation 3
y + 0 = 0
⇒ y = 0
So, (x, y, z) = (0, 0, 0)
Hence, the three planes meet in a unique point.

Question 4.
The coordinate of foot of perpendicular drawn from the point A(1, 0, 3) to the join of the point B(4, 7, 1) and C(3, 5, 3) are
(a) (5/3, 7/3, 17/3)
(b) (5, 7, 17)
(c) (5/3, -7/3, 17/3)
(d) (5/7, -7/3, -17/3)

Answer: (a) (5/3, 7/3, 17/3)
Hint:
Let D be the foot of perpendicular and let it divide BC in the ration m : 1
Then the coordinates of D are <(3m + 4)/(m + 1), (5m + 7)/(m + 1), (3m + 1)/(m + 1)>
Now, AD ⊥ BC
⇒ AD . BC = 0
⇒ -(2m + 3) – 2(5m + 7) – 4 = 0
⇒ m = -7/4
So, the coordinate of D are (5/3, 7/3, 17/3)

Question 5.
The locus of a point which moves so that the difference of the squares of its distances from two given points is constant, is a
(a) Straight line
(b) Plane
(c) Sphere
(d) None of these

Answer: (b) Plane
Hint:
Let the position vectors of the given points A and B be a and b respectively and that of the variable point be r.
Now, given that
PA² – PB² = k (constant)
⇒ |AP|² – |BP|² = k
⇒ |r – a|² – |r – b|² = k
⇒ (|r|² + |a|² – 2r.a) – (|r|² + |b|² – 2r.b) = k
⇒ 2r.(b – a) = k + |b|² – |a|²
⇒ r.(b – a) = (k + |b|² – |a|²)/2
⇒ r.(b – a) = C where C = (k + |b|² – |a|²)/2 = constant
So, it represents the equation of a plane.

Question 6.
The equation of the set of point P, the sum of whose distance from A(4, 0, 0) and B(-4, 0, 0) is equal to 10 is
(a) 9x² + 25y² + 25z² + 225 = 0
(b) 9x² + 25y² + 25z² – 225 = 0
(c) 9x² + 25y² – 25z² – 225 = 0
(d) 9x² – 25y² – 25z² – 225 = 0

Answer: (b) 9x² + 25y² + 25z² – 225 = 0
Hint:
Let the point P is (x, y, z)
Now given that
PA + PB = 10
⇒ √ <(x-4)² + y² + z²>+ √ <(x+4)² + y² + z²>= 10
⇒ √ <(x-4)² + y² + z²>= 10 – √<(x+4)² + y² + z²>
Now square both side
[√<(x-4)² + y² + z²>]² = (10)² + [<(x+4)² + y² + z²>]² – 2 ×10×√<(x+4)² + y² + z²>
⇒ <(x-4)² + y² + z²>= 100 + <(x+4)² + y² + z²>– 20×√<(x+4)² + y² + z²>
⇒ x² + 16 – 8x + y² + z² = 100 + x² + 16 + 8x + y² + z² – 20×√<(x+4)² + y² + z²>
⇒ – 8x = 100 + 8x – 20×√<(x+4)² + y² + z²>
⇒ -8x -8x – 100 = – 20×√<(x+4)² + y² + z²>
⇒ -16x -100 = – 20×√<(x+4)² + y² + z²>
⇒ 4x + 25 = 5×√<(x+4)² + y² + z²>
Again square both side,
(4x + 25)² = 25 ×[√<(x+4)² + y² + z²>]²
⇒ 16x² + 625 + 200x = 25×<(x+4)² + y² + z²>
⇒ 16x² + 625 + 200x = 25×(x² + 16 + 8x + y² + z²)
⇒ 16x² + 625 + 200x = 25x² + 400 + 200x + 25y² + 25z²
⇒ 25x² + 400 + 200x + 25y² + 25z² – 16x² – 625 – 200x = 0
⇒ 9x² + 25y² + 25z² – 225 = 0

Question 7.
The maximum distance between points (3sin θ, 0, 0) and (4cos θ, 0, 0) is
(a) 3
(b) 4
(c) 5
(d) Can not be find

Answer: (c) 5
Hint:
Given two points are (3sin θ, 0, 0) and (4cos θ, 0, 0)
Now distance = √<(4cos θ – 3sin θ)² + (0 – 0)² + (0 – 0)²>
⇒ distance = √<(4cos θ – 3sin θ)²>
⇒ distance = 4cos θ – 3sin θ ……………. 1
Now, maximum value of 4cos θ – 3sin θ = √<(4² + (-3)²>
= √(16 + 9)
= √25
= 5
From equation 1, we get
distance = 5
So, the maximum distance between points (3sin θ, 0, 0) and (4cos θ, 0, 0) is 5

Question 8.
A vector r is equally inclined with the coordinate axes. If the tip of r is in the positive octant and |r| = 6, then r is
(a) 2√3(i – j + k)
(b) 2√3(-i + j + k)
(c) 2√3(i + j – k)
(d) 2√3(i + j + k)

Answer: (d) 2√3(i + j + k)
Hint:
Let l, m, n are DCs of r.
Given, l = m = n
⇒ l² + m² + n² = 1
⇒ 3l² = 1
⇒ l² = 1/3
⇒ l = m = n = 1/√3
So, r = |r|(li + mj + nk)
⇒ r = 6(i/√3 + j/√3 + k/√3)
⇒ r = 2√3(i + j + k)

Question 9.
The plane 2x – (1 + a)y + 3az = 0 passes through the intersection of the planes
2x – y = 0 and y + 3z = 0
2x – y = 0 and y – 3z = 0
2x + 3z = 0 and y = 0
2x – 3z = 0 and y = 0

Answer: (d) A
Hint:
Given, equation of plane is:
2x – (1 + a)y + 3az = 0
=> (2x – y) + a(-y + 3z) = 0
which is passing through the intersection of the planes
2x – y = 0 and -y + 3z = 0
2x – y = 0 and y – 3z = 0

Question 10.
If the end points of a diagonal of a square are (1, -2, 3) and (2, -3, 5) then the length of the side of square is
(a) √3 unit
(b) 2√3 unit
(c) 3√3 unit
(d) 4√3 unit

Answer: (a) √3 unit
Hint:
Let a is the length of the side of a square.
Given, the diagonal of a square are (1,–2,3) and (2, -3, 5)
Now, length of the diagonal of square = √<(1 – 2)² + (-2 + 3)² + (3 – 5)²>
= √<1 + 1 + 4>
= √6
Again length of the diagonal of square is √2 times the length of side of the square.
⇒ a√2 = √6
⇒ a√2 = √3×√2
⇒ a = √3
So, the length of side of square is √3 unit

Question 11.
The coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ plane is
(a) (0, 17/2, 13/2)
(b) (0, -17/2, -13/2)
(c) (0, 17/2, -13/2)
(d) None of these

Answer: (c) (0, 17/2, -13/2)
Hint:
The line passing through the points (5, 1, 6) and (3, 4, 1) is given as
(x-5)/(3-5) = (y-1)/(4-1) = (z-6)/(1-6)
⇒ (x-5)/(-2) = (y-1)/3 = (z-6)/(-5) = k(say)
⇒ (x-5)/(-2) = k
⇒ x – 5 = -2k
⇒ x = 5 – 2k
(y-1)/3 = k
⇒ y – 1 = 3k
⇒ y = 3k + 1
and (z-6)/(-5) = k
⇒ z – 6 = -5k
⇒ z = 6 – 5k
Now, any point on the line is of the form (5 – 2k, 3k + 1, 6 – 5k)
The equation of YZ-plane is x = 0
Since the line passes through YZ-plane
So, 5 – 2k = 0
⇒ k = 5/2
Now, 3k + 1 = 3 × 5/2 + 1 = 15/2 + 1 = 17/2
and 6 – 5k = 6 – 5×5/2 = 6 – 25/2 = -13/2
Hence, the required point is (0, 17/2, -13/2)

Question 12.
The angle between the vectors with direction ratios are 4, -3, 5 and 3, 4, 5 is
(a) π/2
(b) π/3
(c) π/4
(d) π/6

Answer: (b) π/3
Hint:
Let a is a vector parallel to the vector having direction ratio is 4, -3, 5
⇒ a = 4i – 3j + 5k
Let b is a vector parallel to the vector having direction ratio is 3 ,4, 5
⇒ b = 3i + 4j + 5k
Let θ be the angle between the given vectors.
Now, cos θ = (a . b)/(|a|×|b|)
⇒ cos θ = (12 – 12 + 25)/<√(16 + 9 + 25)×√(9 + 16 + 25)>
⇒ cos θ = 25/<√(50)×√(50)>
⇒ cos θ = 25/50
⇒ cos θ = 1/2
⇒ cos θ = π/3
⇒ θ = π/3
So, the angle between the vectors with direction ratios are 4, -3, 5 and 3, 4, 5 is π/3

Question 13.
The equation of plane passing through the point i + j + k and parallel to the plane r . (2i – j + 2k) = 5 is
(a) r . (2i – j + 2k) = 2
(b) r . (2i – j + 2k) = 3
(c) r . (2i – j + 2k) = 4
(d) r . (2i – j + 2k) = 5

Answer: (b) r . (2i – j + 2k) = 3
Hint:
The equation of plane parallel to the plane r . (2i – j + 2k) = 5 is
r . (2i – j + 2k) = d
Since it passes through the point i + j + k, therefore
(i + j + k) . (2i – j + 2k) = d
⇒ d = 2 – 1 + 2
⇒ d = 3
So, the required equation of the plane is
r . (2i – j + 2k) = 3

Question 14.
A vector r is equally inclined with the coordinate axes. If the tip of r is in the positive octant and |r| = 6, then r is
(a) 2√3(i – j + k)
(b) 2√3(-i + j + k)
(c) 2√3(i + j – k)
(d) 2√3(i + j + k)

Answer: (d) 2√3(i + j + k)
Hint:
Let l, m, n are DCs of r.
Given, l = m = n
⇒ l² + m² + n² = 1
⇒ 3l² = 1
⇒ l² = 1/3
⇒ l = m = n = 1/√3
So, r = |r|(li + mj + nk)
⇒ r = 6(i/√3 + j/√3 + k/√3)
⇒ r = 2√3(i + j + k)

Question 15.
The maximum distance between points (3sin θ, 0, 0) and (4cos θ, 0, 0) is
(a) 3
(b) 4
(c) 5
(d) Can not be find

Answer: (c) 5
Hint:
Given two points are (3sin θ, 0, 0) and (4cos θ, 0, 0)
Now distance = √<(4cos θ – 3sin θ)² + (0 – 0)² + (0 – 0)²>
⇒ distance = √<(4cos θ – 3sin θ)²>
⇒ distance = 4cos θ – 3sin θ …………….1
Now, maximum value of 4cos θ – 3sin θ = √<(4² + (-3)²>
= √(16 + 9)
= √25
= 5
From equation 1, we get
distance = 5
So, the maximum distance between points (3sin θ, 0, 0) and (4cos θ, 0, 0) is 5

Question 16.
The image of the point P(1, 3, 4) in the plane 2x – y + z = 0 is
(a) (-3, 5, 2)
(b) (3, 5, 2)
(c) (3, -5, 2)
(d) (3, 5, -2)

Answer: (a) (-3, 5, 2)
Hint:
Let image of the point P(1, 3, 4) is Q in the given plane.
The equation of the line through P and normal to the given plane is
(x-1)/2 = (y-3)/-1 = (z-4)/1
Since the line passes through Q, so let the coordinate of Q are (2r + 1, -r + 3, r + 4)
Now, the coordinate of the mid-point of PQ is
(r + 1, -r/2 + 3, r/2 + 4)
Now, this point lies in the given plane.
2(r + 1) – (-r/2 + 3) + (r/2 + 4) + 3 = 0
⇒ 2r + 2 + r/2 – 3 + r/2 + 4 + 3 = 0
⇒ 3r + 6 = 0
⇒ r = -2
Hence, the coordinate of Q is (2r + 1, -r + 3, r + 4) = (-4 + 1, 2 + 3, -2 + 4)
= (-3, 5, 2)

Question 17.
The points on the y- axis which are at a distance of 3 units from the point (2, 3, -1) is
(a) either (0, -1, 0) or (0, -7, 0)
(b) either (0, 1, 0) or (0, 7, 0)
(c) either (0, 1, 0) or (0, -7, 0)
(d) either (0, -1, 0) or (0, 7, 0)

Answer: (d) either (0, -1, 0) or (0, 7, 0)
Hint:
Let the point on y-axis is O(0, y, 0)
Given point is A(2, 3, -1)
Given OA = 3
⇒ OA² = 9
⇒ (2 – 0)² + (3 – y)² + (-1 – 0)² = 9
⇒ 4 + (3 – y)² + 1 = 9
⇒ 5 + (3 – y)² = 9
⇒ (3 – y)² = 9 – 5
⇒ (3 – y)² = 4
⇒ 3 – y = √4
⇒ 3 – y = ±4
⇒ 3 – y = 4 and 3 – y = -4
⇒ y = -1, 7
So, the point is either (0, -1, 0) or (0, 7, 0)

Question 18.
If α, β, γ are the angles made by a half ray of a line respectively with positive directions of X-axis Y-axis and Z-axis, then sin² α + sin² β + sin² γ =
(a) 1
(b) 0
(c) -1
(d) None of these

Answer: (d) None of these
Hint:
Let l, m, n be the direction cosines of the given vector.
Then, α, β, γ
l = cos α
m = cos β
n = cos γ
Now, l² + m² + n² = 1
⇒ cos² α + cos² β + cos² γ = 1
⇒ 1 – sin² α + 1 – sin² β + 1 – sin² γ = 1
⇒ 3 – sin² α – sin² β – sin² γ = 1
⇒ 3 – 1 = sin² α + sin² β + sin² γ
⇒ sin² α + sin² β + sin² γ = 2

Question 19.
If P(x, y, z) is a point on the line segment joining Q(2, 2, 4) and R(3, 5, 6) such that the projections of OP on the axes are 13/5, 19/5, 26/5 respectively, then P divides QR in the ration
(a) 1 : 2
(b) 3 : 2
(c) 2 : 3
(d) 1 : 3

Answer: (b) 3 : 2
Hint:
Since OP has projections 13/5, 19/5 and 26/5 on the coordinate axes, therefore
OP = 13i/5 + 19j/5 + 26/5k
Let P divides the join of Q(2, 2, 4) and R(3, 5, 6) in the ratio m : 1
Then the position vector of P is
<(3m + 2)/(m + 1), (5m + 2)/(m + 1), (6m + 4)/(m + 1)>
So, 13i/5 + 19j/5 + 26/5k = (3m + 2)/(m + 1)+ (5m + 2)/(m + 1)+ (6m + 4)/(m + 1)
⇒ (3m + 2)/(m + 1) = 13/5
⇒ 2m = 3
⇒ m = 3/2
⇒ m : 1 = 3 : 2
Hence, P divides QR in the ration 3 : 2

Question 20.
In a three dimensional space, the equation 3x – 4y = 0 represents
(a) a plane containing Y axis
(b) none of these
(c) a plane containing Z axis
(d) a plane containing X axis

Answer: (c) a plane containing Z axis
Hint:
Given, equation is 3x – 4y = 0
Here z = 0
So, the given equation 3x – 4y = 0 represents a plane containing Z axis.

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Frequently Asked Questions: FAQs

Here we have provided the most frequently asked questions (FAQs) related to NCERT Solutions for Class 11 Maths Chapter 12, Three Dimensional Geometry:

Ans: In the NCERT Class 11 Chapter 12, Three Dimensional Geometry, students are introduced to a new segment of mathematics where the students view Geometry in a new light, in the form of a three-dimensional figure. Through this chapter, students get to learn about the representation of points and different three-dimensional figures, salient features of three-dimensional shapes, and their characteristics. Students get to learn about various topics such as the representation of a point in three-dimension, cube, cuboid, cone, volume and area of three-dimensional shapes, and much more.

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NCERT Solutions for class 11 Maths

Download NCERT Solutions for class 11 Maths in PDF form – all chapters. Along with the NCERT book, a revision book is also given, based on CBSE syllabus, providing ample practice of questions.

11th Maths – NCERT Solutions

Chapter 1: Sets

In this chapter logical approach to set theory is discussed. The term set falls in the category of undefined terms in mathematics. Also to be an element of a set is also undefined term. However a set means a well-defined collection of objects.

Chapter 2: Relations and Functions

The concept of functions is very fundamental in modern mathematics. French mathematician Descartes used the word FUNCTION in the year 1637 and James Gregory gave the definition of a function in 1667.

Chapter 3: Trigonometric Functions

The study of trigonometry initially started in India. The ancient Indian mathematicians Aryabhatta (476 AD), Brahmagupta (598 AD), Bhaskara I (600 AD) and Bhaskara II (1114 AD) got important results.

Chapter 4: Principle of Mathematical Induction

The early traces of mathematical induction can be found in Euclid’s proof that number of primes is infinite. Bhaskara II’s cyclic method (Chakravala) also introduces mathematical induction.

Chapter 5: Complex Numbers and Quadratic Equations

To allow the square root of negative numbers, the real number system is extended to complex numbers. In fact, Greeks were the first to recognize the fact that square root of a negative number does not exist in the real number system. It is also mentioned in ‘GANITASARA SANGRAHA’ by Indian mathematician Mahaviracharya (850 AD).

Chapter 6: Linear Inequalities

In this chapter, we will study how the inequalities arise in day to day practice. Whenever we compare two quantities, they are more likely to be unequal than equal.

Chapter 7: Permutations and Combinations

Permutation – A permutation is an arrangement in a definite order of a number of distinct of n different objects taking r at a time. Combinations – The number of ways of selecting r things out of n different things is called r combination number of n things.

Chapter 8: Binomial Theorem

It is believed that in the eleventh century, Persian poet and mathematician Omar Khayyam gave the general formula for (a + b)^n, where n is a positive integer. This formula or expansion is called Binomial theorem.

Chapter 9: Sequences and Series

Historically, Aryabhata was the first mathematician to give the formula for the sum of the square of the first n natural numbers, the sum of cubes of first n natural numbers, etc. This is given in his work ARYABHATIYAM.

Chapter 10: Straight Lines

French mathematician Rene Descartes was the first mathematician who used algebra for the study of geometry. Using Cartesian coordinates, he represented lines and curves by algebraic equation.

Chapter 11: Conic Sections

The special curves like circles, ellipse, parabolas and hyperbolas are called conic sections or more commonly conics. The names PARABOLA and HYPERBOLA are given by Apollonius (262 BC – 19 BC).

Chapter 12: Introduction to Three Dimensional Geometry

Earlier the concepts of plane coordinate geometry were initiated by French mathematician Rene Descartes and also by Fermat in the beginning of 17th century. In this chapter we will study the coordinate geometry in the 3 – D space.

Chapter 13: Limits and Derivatives

BRAHMAGUPTA’S YUKTIBHASHA is considered to be the first book on calculus. BHASKAR’S work on calculus precedes much before the time of LEIBJITZ and NEWTON. BHASKARA – II used principles of differential calculus in problems on Astronomy.

Chapter 14: Mathematical Reasoning

In mathematics, mainly two kinds of reasoning occur. One is inductive reasoning which is studied in chapter 4 – mathematical induction and the other is deductive reasoning which we intend to study in this chapter.

Chapter 15: Statistics

We know that the statistics deals with data collection for specific purposes. We will do the next level of statistics in chapter than whatever we have studied in classes 8, 9 and 10.

Chapter 16: Probability

Probability is the word we use calculating the degree of the certainty of events in ideal conditions. An experiment means an operation which can produce some well-defined outcomes. The classical approach is given by Blaise Pascal and the axiomatic approach is given by a Russian mathematician A Kolmogorov in 1937.


  1. Do your calculations faster with maths formulas.
  2. Solving questions are always easy if you know the maths formulas.
  3. Application in maths numerical are easy.
  4. Enhance your score in maths subject with maths formula.
  5. With entrancei maths formula pdf you can revise all maths formula at a time which help in many entrance exam.

Apart from above mentioned points Math formula will always helpful in many areas of subjects and can be applied in several topics,these formulas are useful in all most entrance exam just after class 10 or 12.

Maths is one of the important subjects of student&rsquos life.The team at Entrancei is dedicatedly involved in developing the most advanced Maths formulas list. Command over the Maths formulas can offer an extra cutting edge to the students in their exams. The Maths formulas are useful in covering the syllabus effectively. The students can plan their study effectively with this study material. Students can score effective marks with this study material. Since the syllabus is very large it is very crucial to study each and every aspect. With this Maths formulas list, students can complete their syllabus with a time span.

Why Maths formulas are important?

Since mathematics is one such subject that needs a lot amount of formulae, so this study material helps the students to memorize them effectively. We have provided a time-syllabus dependent strategy which helps students to progress accordingly. Since the time the most lacked for students preparing for their examinations, we have made efforts in our Maths formulas list to minimize it. An appropriate study of Maths formulas helps the students to gauge their strength and weaknesses. The strengths can be overall maintained where-as weaknesses could be minimized with optimum efforts.

Students looking forward to preparing for their preparation in competitive examinations can use Maths formulas.We have provided extra coverage to typical Maths formulas requiring extra efforts with detailed explanations. Since remembering formulas can be tricky to a certain extent, it is believed that preparing with the right study material can be useful. We have devised certain strategies that could be helpful in preparing off the students. The faculty preparing the study material are subject experts, with years of experience. Students looking forward to preparing for the examination can go through our website. The Maths formulas list we have provided is free of cost and can be accessed by students. The study material can be shared among friends.

How to Study Maths formulas effectively?

Since the Maths formulas list is prepared in Pdf format it can be accessed on multiple devices. The team at Entrancei has curated the most important Maths formulas list of students.The faculties at Entrancei are highly qualified professor&rsquos experts from prominent IITs.The Maths formulas list has been divided into various segments. All the study material is devised adhering to the latest syllabus only. Since we truly believe in equal education to all the students, we have provided complete study material for free.This material could be accessed free of cost. Since the time has arrived where finding the appropriate teachers is a difficult task.

Why Entrancei is best for formulas?

We have compiled all the important Maths formulas list from the subject experts.This study material could be accessed by just making an account on our website. This Maths formulas list provides students with hands-on experience to practice more and more sums.This type of study material is crucial for faster completion of the syllabus.The rapid revision could be initiated with our Maths formulas list. Students effectively prepare for their board examination as well as national exams.This study material acts as pre-hand assistance to tackle mathematics effectively. Our experts have provided top-notch study material which can be to clear all the doubts.

How Maths formulas Helps you in your final Revision

Maths is subject of application of concepts and formulas every numerical you are going to solve use a maths formula .Speed and accuracy of solving maths questions also depends how fast you apply concept in the numerical and how fast you concludes what is asked in the questions for all these you need good memories of all maths formulas used in that chapter. It is observed that because of multiple concepts and & formulas some time students forget concept or formulas because of which he or she did mistake, and each error bring your down by -1 marks .So what should you do to avoid such error ? answering to this questions revision of all important maths formulas for all class students is must . Chapter wise maths formulas are highly helpful for your final revision of the chapter. We highly recommend NCERT text books for maths and use as a reference NCERT solutions prepared by entrancei.

Best approach to use formulas

Maths formula are helpful for all class students, students must have interest in maths subjects and in general all students love to solve math questions if he understand the concepts .One must start maths from the best text book in which the theory part is well explained with the solved questions .Do read the theory at first and try to build your own approach write down the important rules and theory which are important .After developing clear concept now it&rsquos time to learn the application of theory.

Maths formula act as a bridge between your theory and its application once you have completed the theory read the maths formula from the pdf of entrancei.Write down all important maths formula in your note book refer NCERT.Once you know all formula now it&rsquos time to apply these maths formula in questions. Start solving from the solved questions of your text book and try to understand the application of concept during the numerical solving time make your own solutions which different form the solution is given in your text book , check the application of concept and maths formula. Solve as many as Maths Questions as you can.

Entrancei Maths formula Benefits

  1. Entrancei Maths formula pdf Hand book will help you to score good marks in your school as well as entrance exam .
  2. These maths formula pdf will help you to revise entire maths of your syllabus in just few hours and empower you to retain all formula during your exam time.
  3. Entrancei maths formula is highly useful for the students who are preparing for entrance exam like NTSE, JEE Olympiad& RMO.
  4. These maths formula pdf are extremely helpful for quick revision of entire chapter.

FAQ For Maths Formulas

Q-1. Who can use Entrancei Maths Formulas ?

Ans- Academic team of Entrancei prepared Maths formulas for students who are in class 6 to 12. We have uploaded chapter wise formulas sheet for effective revision and can be freely downloaded from Entrancei.

Q-2. How to use Maths formula sheet ?

Ans- The best way to use formula sheet is to start the chapter form your textbook and read the theory given in the text book try to build conceptual clarity in the chapter and with the help of solved examples. Once you understand the chapter very well before going to solve the exercise given in the text book try to revise all Maths formulas given in the sheet. This will help you to remember the formulas which are used in the questions.

Q-3. Are these Maths formulas are free to use ?

Ans- Yes Academic team of entrancei prepared chapter wise Maths formulas for class 7 to 12 and available free to download alone with Maths questions which is posted separately chapter wise.

Q-4. Are these Maths formulas are helpful for competitive entrance exam ?

Ans- Yes, all the chapter wise sheet of formulas is prepared such a way that it consists of all-important formulas asked in board school or competitive entrance exam like Olympiad , CBSE, NTSE, JEE and JEE advance. Students are recommended to check out your text book try to prepare your own notes of derivation and application of formulas build your conceptual clarity on the chapter try to resolve the examples which give you more clarity on the concepts. Maths formulas sheet must be use for reference not to be mugged up.

Q-5.What do you mean by Maths formulas?

Ans-The formula may be a fact or rule written with Maths symbols. It usually connects two or more with an equal sign. Once you know the value of one quantity, you can use the formula to find the value of the other.

Q-6.Is it necessary to know how does a Maths formula work?

Ans-Indeed, it is necessary to understand and solve equations either when you want to work as a mathematician or in another field that uses Maths, or when you want to become a Maths teacher or a teacher in an area that uses Maths.

Q-7.What are the uses of the Maths formulas?

Ans-Mathematics is one of the most important subjects in a student's life. Mastering the Maths formulas can give students an extra edge on their exams. The Maths formulas are useful to cover the curriculum effectively. With this learning material, students can plan their studies effectively and achieve effective grades. Since the curriculum is very large, it is very important to study every aspect. This mathematical formula allows students to complete their curriculum at a specified time.

1.Make your calculations faster with Maths formulas.

2.Solving questions is always easy if we know the formulas.

3.Applications in Maths numbers are simple.

4.Improve your Maths score with Maths formulas.

Q-8.Why Maths formulas are important?

Ans-Mathematics is one such subject that requires a lot of formulas. An Appropriate study of mathematical formulas will help students assess their strengths and weaknesses. Students who want to prepare for competitive exams can use Maths formulas. Memorizing formulas can be difficult to some extent. It is believed that preparation with the right study materials can be helpful.

Q-9.How to remember Maths formulas?

Ans- To remember Maths formulas use the following tips.

1.Use and write formulas first when solving related questions.

2.Call up formulas regularly and visualize them.

3.Learn memory and apply creative memory links to memorize long-term formulas.

4.Take a test with formulas in 2 to 3 weeks and write down all formulas.

5.Make it more interesting and with the help of a like-minded friend, play a game by asking for formulas at random. The winner can throw a party.

6.Understand the logic behind the formula and learn how the formula is derived.

Q-10.Why Entrancei is best for Maths formulas?

Ans-We have compiled the full list of key Maths formulas from the subject matter experts. This study material can be assessed simply by creating an account on our website. This list of Maths formulas provides students with hands-on experience to practice more and more sums. This type of study material is critical to completing the curriculum faster. The quick check could start with our list of Maths formulas. Students prepare effectively for their board exams and national exams. This learning material serves as a preliminary aid to tackle Mathematics effectively. Our experts have top quality study materials that can dispel doubts.

About Entrancei Maths Quiz

Entrancei uploaded chapter wise Maths quiz for students who are in class 7, 8 , 9, 10 and 11 do appear for this.


  • 10th Maths - Chapter 1 - Relations & Functions
  • 12th Maths - Chapter 2 - Complex Numbers
  • 12th Maths - Chapter 4 - Inverse Trigonometry Functions
  • Chapter 4 - 12th Business Maths - Differential Equation
  • Class 10
  • Class 10 Algebra
  • Class 10 Chapter 2 Sequences and Series
  • Class 10 Physics
  • Class 10 Question Papers
  • Class 11
  • Class 11 Sets Relations and Functions
  • Class 11 Binomial Theorem Sequences And Series
  • Class 11 Business Maths
  • Class 11 CHAPTER 12 Introduction to Probability Theory
  • Class 11 CHAPTER 7 Matrices and Determinants
  • Class 11 CHAPTER 8 Vector Algebra-I
  • Class 11 Combinatorics and Mathematical Induction
  • Class 11 Maths CHAPTER 10 Differentiability and Methods of Differentiation
  • Class 11 Maths CHAPTER 11 Integral Calculus
  • Class 11 Maths CHAPTER 9 Differential Calculus - Limits and Continuity
  • Class 11 Maths Full Solution
  • Class 11 Maths One mark
  • Class 11 Maths Syllabus
  • Class 11 Theorem
  • Class 11 Trigonometry
  • Class 11 Two Dimensional Analytical Geometry
  • Class 12 Applications of Matrices and Determinants
  • Class 12 Business Maths
  • Class 12 Business Maths - Differential Equations -Chapter 4
  • Class 12 Maths Full
  • Class 12 Maths-Two Dimensional Analytical Geometry II
  • Class 12 Physics
  • Class 12 Quarterly Exam Question paper
  • Class 8 Science
  • Class 9 Question Papers
  • Class12- Business Maths- Chapter 2-Integral Calculus I-தொகை நுண்கணிதம் I
  • Formula
  • How to Prepare Class 11 Maths for Board Exam
  • I Mid -Term Question Paper
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  • Quarterly Exam Portions Tamilnadu Syllabus
  • Question Papers

NCERT Solutions for Class 12 Maths

All solutions are easily accessible for mobile users, as these are being download from google drive. If desktop use face any difficulty in download these PDF solutions, use Crome browser to open files.

Class 12 Maths NCERT Solutions

Solutions of all chapters are given below. To explore these chapters, click the links given at the end of each chapter. In the explanation of each chapter of class 12 maths, historical facts about chapters, main points of the chapter, assignments, chapter test, previous years questions, etc. are given to under stand the facts related to the chapters more easily.


Watch the video: Μαθηματικά - Αναλογίες - ΣΤ Δημοτικού Επ. 65 (August 2022).