Explore topic-wise InterviewSolutions in .

Here we can use the compound interest based formula,

Population after n years = P*[1 - (r/100)]ⁿ

Population after 3 years = 8000,000*[1-(8/100)]³

Calculating we get,

Population after 3 years = 6229504 

Considering the given series

16, 8, 8, 12, 30, 60

The logic of the given series can be explained as

16 × 0.5 = 8

8 × 1 = 8

8 × 1.5 = 12

12 × 2 = 24

24 × 2.5 = 60

∴ Wrong term in given number series is 30.

Using the formula,

⇒ Volume = Height × Length × Breadth

Substitute the values,

⇒ Volume = (h) × (7h) × (7h - 2)

Remove the brackets,

⇒ Volume = h × 7h × 7h - 2

Take variable and numbers seperately,

⇒ Volume = (h × 7h × 7h) - 2

Multiply and solve,

⇒ Volume = 49h - 2 cm³

∴ Thus, the volume of the rectangular box 49h - 2 cm³.

Correct option is: (A) cos²B – cos²A

Correct option is: (A) cos²A

Correct answer is (D) tan A

Correct answer is (B) (X + 2)² = 3(x + 4)

Correct option is: (A) 2cos\(\frac{C+D}{2}\).sin\(\frac{C-D}{2}\)

Correct option is: (A) \(\frac{sin^2\theta}{cos\theta}\)

Correct answer is (B) 2

Correct option is: (B) secθ

Correct option is: (A) 2 - √3

Correct option is: (A) sinθ

Correct option is: (D) cot2A

Correct option is: (B) \(2\sqrt{p^2+Q^2}\)

Correct answer is (C) \(\frac{x^2 + 1}{2x}\)

Correct answer is (C) 1

Correct option is: (c) 3 : 4

Given: A(−8,13),B(x,7) are points.
Midpoint:- P(4,10)
By using mid-point formula to calculate the x-coordinate of the midpoint,<span style="margin: 0px; padding: 0px; border: 0px; outline-color: initial; outline-style: initial; background: tra

Correct option is: (C) (0, 0)

Correct option is: (A) (4, 2)

Correct option is: (C) (3, 5)

Correct answer is (c) \(\left(\frac{-9}{2}, \frac{5}{2}\right)\)

Let required point on x- axis is (x, 0).

\(\therefore\) Distance between points (x, 0) and (-2, 3) is 5√3.

\(\therefore\) \(\sqrt{(-2-x)^2+(3-0)^2}\) = 5√3

⇒ (2 + x)² + 9 = 25 x 3 (By squaring both sides)

⇒ (x + 2)² = 75 - 9 = 66

⇒ (x + 2) = \(\pm\sqrt{66}\)

⇒ x = -2 \(\pm\sqrt{66}\)

Hence required point is (-2\(+\sqrt{66}\), 0) or (-2,\(-\sqrt{66}\), 0)

Let direction cosines of line are l, m and n.

\(\therefore\) l + m + n = 0

4l + m - 2n = 0

(\(\because\) Line is present in both planes, therefore line is perpendicular to normals of both planes)

⇒ \(\frac{l}{-2-1}=\frac{m}{4+2}=\frac{n}{1-4}\) = k (let)

⇒ l = -3k, m = 6k, n = -3k

But l, m, n are direction cosines.

l = \(\frac{-3k}{\sqrt{(-3k)^2+(6k)^2+(-3k)^2}}\)

\(=\frac{-3k}{\sqrt{9k^2+36k^2+9k^2}}\) 

\(=\frac{-3k}{\sqrt{54k^2}}=\frac{-3}{3\sqrt6}=\frac{-1}{\sqrt 6}\)

m = \(\frac{6k}{\sqrt{(-3k)^2+(6k)^2+(-3k)^2}}\) = \(\frac{6k}{3\sqrt6}\) = \(\frac{2}{\sqrt6}\) 

n = \(\frac{-3k}{\sqrt{(-3k)^2+(6k)^2+(-3k)^2}}\) = \(\frac{-3k}{3\sqrt6}=\frac{-1}{\sqrt6}\)

\(\therefore\) Direction cosines of line are \((\mp\frac1{\sqrt6},\pm\frac{2}{\sqrt6},\mp\frac1{\sqrt6}).\)

Laplace transformation plays a major role in control system engineering. To analyze the control system, Laplace transforms of different functions have to be carried out. Both the properties of the Laplace transform and the inverse Laplace transformation are used in analyzing the dynamic control system. 

In the development of a particular branch of mathematics, the mathematician is chiefly concerned with a logical rigorous treatment of the subject matter, whereas the teacher is usually concerned with its psychological organization and presentation. It is the curriculum organizer who is called upon to integrate the two approaches.

Main branches of Pure mathematics:

• Algebra: Algebra is first accepted as a branch of mathematics. It is a kind of arithmetic where we use unknown quantities along with numbers. These unknown quantities are represented by letters of the English alphabet such as X, Y, A, B, etc. or symbols. The use of letters helps us to generalize the formulas and rules that you write and also helps you to find the unknown missing values in the algebraic expressions and equations.
• Geometry: It is the most practical branch of mathematics that deals with shapes and sizes of figures and their properties. The basic elements of geometry are points, lines, angles, surfaces, and solids.
• Trigonometry: Derived from Greek trigōnon, "triangle" and metron, "measure", it is a branch of mathematics that studies relationships between side lengths and angles of triangles.
• Calculus: It is a branch of mathematics concerned with instantaneous rates of change and the summation of infinitely many small factors.
• Statistics and Probability: The branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation, and display of numerical data. 

Also Note:

• Arithmetic: It is the oldest and the most elementary among other branches of mathematics. It deals with numbers and the basic operations- addition, subtraction, multiplication, and division, between them.
• Analysis: The analysis is a branch of mathematics that studies continuous changes and includes the theories of integration, differentiation, measure, limits, analytic functions, and infinite series.

Hence, we conclude that Geometry teaches Logical thinking and natural design.

A) always true

Correct option is  B) False

Mathematics not only is "number work‟ or "computation‟, but also is more about forming generalizations, seeing relationships and developing logical thinking and reasoning. Mathematics should be visualized as the vehicle to train a child to think, reason, analyze and to articulate logically. Mathematics should be shown as a way of thinking, an art forms a beauty and as human achievement. 

Note that:

• Locke stated, “Mathematics is a way to settle in the mind a habit of reasoning.” A more comprehensive definition of mathematics was given by Courant and Robin when they defined mathematics in the following way. “Mathematics is an expression of the human mind which reflects the active will, the contemplative reason and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.”
• The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics; theorems and theories; are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty.

Hence, we conclude that the above statement is of Locke.

Correct option is  D) 11

C) Hypothesis

A) Inductive reasoning

Correct option is  A) Axiom

Correct option is  C) 4

B) conjecture

Correct option is  C) contradiction

A) Mathematical proof

D) Not possible

1. The maturity of a bill of exchange  refers to the date on which a bill of exchange or a promissory note becomes due for payment. 

2. Dishonour of a bill, happens when the acceptor of the bill fails to make the payment on the date of maturity of the bill. Hence, liability of the acceptor is restored. 

3. Noting of the bill is recording the facts of its dishonour by a Notary public. 

4. 

5. 

B) inconsistent

Given that a cosθ + bsinθ = m. … (1) 

And a sinθ – b cosθ = n. … (2) 

Now, squaring equation (2) and (3), we get 

m² = (a cosθ + sinθ)² 

= a² co²θ + b² sin²θ + 2absinθcosθ. … (3) 

And n² = (a sinθ − b cosθ)² = a² sin² + b² cos² – 2absinθcosθ. … (4) 

Now, adding equations (3) & (4), we get 

m² + n² = a² (cos²θ + sin²θ) + b² (sin²θ + cos²θ) + 2ab sinθcosθ – 2ab sinθcosθ 

⇒ m²+ n² = a² + b² . (∵ sin²θ + cos²θ = 1) 

Hence Proved.

Let one son is x years old and other son is y years old, 

Given that man’s age is 3 times the sum of ages of his two sons. 

Therefore, the man’s age = 3(x + y).   ... (1) 

After 5 years the man’s age is 3(x + y) + 5. 

But given that after 5 years, the man’s age will be twice of the ages of his two sons. After 5 years the age of first son is x + 5 and the age of second son is y + 5. 

Therefore, 3(x + y) + 5 = 2((x+5) + (y+5)) 

⇒ 3(x + y) + 5 = 2(x + y + 10) = 2(x + y) +20 

⇒ x + y = 20 – 5 = 15. 

⇒ x + y = 15.

Therefore, the man’s age = 3 (x + y) = 3 × 15 = 45 years. [From equation (1)]

cot A : cot B : cot C = 1 : 4 : 15

Let cot A = x, cot B = 4x and cot C = 15x

⇒ A = cot^-1x, B = cot^-14x + cot^-115x = 180°

(\(\because\) cot^-1 x + cot^-1 y = cot^-1(\(\frac{xy-1}{x+y}\)))

⇒ cot^-1\(\left(\cfrac{\frac{4x^2-1}{5x}\times15-1}{\frac{4x^2-1}{5x}+15x}\right)\)= 180°

⇒  cot^-1\(\left(\cfrac{\frac{15x(4x^2-1)-5x}{5x}}{\frac{4x^2-1+75x^2}{5x}}\right)\)= 180°

⇒ \(\frac{15x(4x^2-1)-5x}{79x^2-1}\) = cot 180° = \(\frac{cos180^{\circ}}{sin180^{\circ}}\) = \(\frac{-1}0\)

\(\therefore\) 79x² - 1 = 0

⇒ x = \(\frac{1}{\sqrt{79}}\)

Now, A = cot^-1x = cot^-1\((\frac1{\sqrt{79}})\)

B = cot^-14x = cot^-1\((\frac{4}{\sqrt{79}})\) 

C = cot^-115x = cot^-1\((\frac{15}{\sqrt{79}})\)

A > B > C 

(\(\because\) cot^-1 x is a decreasing function and x < 4x < 15x)

Greatest angle is A.

Answer is (c) 2x + 5y - 6z + k = 0

Answer is (a) x/√(1 + x²)

correct option:

(C) z = 0 

correct option:

(B) 12

Answer is (b) -1/5

Answer is (d) 0