Agniveer (Agnipath Scheme) Quantitative Aptitude and Mathematics Syllabus

How do you adjust the compound interest formula when interest is compounded half-yearly?

Use rate R/2 and time 2T: A = P(1 + R/200)^(2T). For quarterly: A = P(1 + R/400)^(4T).

What is the formula for the average (arithmetic mean) of n numbers?

Average = (Sum of all observations) / (Number of observations). So Sum = Average × n.

What is the average of the first n natural numbers?

Average = (n + 1)/2. (Sum of first n natural numbers = n(n+1)/2.)

In simple partnership, how is profit divided among partners?

Profit is shared in the ratio of the capitals invested (when invested for the same time). If A invests Ca and B invests Cb, profit ratio = Ca : Cb.

In partnership with different time periods, how is profit shared?

Profit is divided in the ratio of (Capital × Time) for each partner: A:B = (Ca × Ta) : (Cb × Tb).

What does the rule of alligation give you, and state the formula?

Alligation finds the ratio in which two ingredients at different prices are mixed to get a desired mean price. Ratio = (Dearer − Mean) : (Mean − Cheaper).

State the fundamental relationship between speed, distance, and time.

Speed = Distance / Time; Distance = Speed × Time; Time = Distance / Speed.

How do you convert km/hr to m/s and m/s to km/hr?

km/hr to m/s: multiply by 5/18. m/s to km/hr: multiply by 18/5.

What are relative speeds for two objects moving in the same vs opposite directions?

Same direction: relative speed = difference of speeds (|s1 − s2|). Opposite direction: relative speed = sum of speeds (s1 + s2).

How long does a train of length L moving at speed S take to cross a platform of length P?

Time = (L + P)/S. To cross a pole or point object, time = L/S (only the train's own length).

If A can do a work in x days and B in y days, how long do they take working together?

Together they take xy/(x + y) days. Their combined one-day work = 1/x + 1/y.

In time and work, what is the relationship between efficiency and time taken?

Efficiency is inversely proportional to time. If A is twice as efficient as B, A takes half the time. Ratio of efficiencies = inverse ratio of times.

State the pipes-and-cisterns rule for an inlet and an outlet pipe working together.

If an inlet fills in x hours and an outlet empties in y hours, net filling per hour = 1/x − 1/y; tank fills in xy/(y − x) hours (when x < y).

If the present ages of A and B are in ratio a:b and after n years the ratio becomes c:d, how do you set up the equations?

Let ages be ax and bx. Then (ax + n)/(bx + n) = c/d. Solve for x, then present ages are ax and bx.

State the algebraic identities for (a+b)², (a−b)², and a²−b².

(a+b)² = a² + 2ab + b²; (a−b)² = a² − 2ab + b²; a² − b² = (a+b)(a−b).

State the identities for (a+b)³ and a³+b³.

(a+b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a+b). a³ + b³ = (a+b)(a² − ab + b²).

What is the identity for a³ + b³ + c³ when a + b + c = 0?

If a + b + c = 0, then a³ + b³ + c³ = 3abc.

How do you solve a linear equation in one variable like 3x + 5 = 20?

Isolate the variable: move constants to the other side and divide by the coefficient. 3x = 15, so x = 5.

What are the methods to solve a pair of linear equations in two variables?

Substitution, elimination, cross-multiplication (algebraic) and graphical method. The solution is the point of intersection of the two lines.

For a pair of linear equations a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, when is the system consistent or inconsistent?

Unique solution (consistent): a₁/a₂ ≠ b₁/b₂. Infinitely many (consistent, dependent): a₁/a₂ = b₁/b₂ = c₁/c₂. No solution (inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

State the quadratic formula for ax² + bx + c = 0.

x = [−b ± √(b² − 4ac)] / (2a), provided a ≠ 0.

What does the discriminant of a quadratic equation tell you?

D = b² − 4ac. If D > 0: two distinct real roots. If D = 0: two equal real roots. If D < 0: no real roots (complex conjugates).

For ax² + bx + c = 0, what are the sum and product of the roots?

Sum of roots = −b/a; Product of roots = c/a.

What are complementary, supplementary, and vertically opposite angles?

Complementary: two angles summing to 90°. Supplementary: two angles summing to 180°. Vertically opposite angles (formed by two intersecting lines) are equal.

State the angle sum properties of a triangle and the exterior angle theorem.

Sum of interior angles of a triangle = 180°. Exterior angle = sum of the two interior opposite (remote) angles.

State the Pythagoras theorem.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: hypotenuse² = base² + perpendicular².

What is the sum of interior angles of a polygon with n sides, and the measure of each interior angle of a regular polygon?

Sum of interior angles = (n − 2) × 180°. Each interior angle of a regular polygon = (n − 2) × 180° / n. Sum of exterior angles = 360°.

State the formulas for circumference and area of a circle.

Circumference = 2πr (or πd); Area = πr², where r = radius, d = diameter.

State the area and perimeter formulas for a rectangle, square, triangle, and circle.

Rectangle: Area = l×b, Perimeter = 2(l+b). Square: Area = a², Perimeter = 4a. Triangle: Area = ½×base×height. Circle: Area = πr², Circumference = 2πr.

State Heron's formula for the area of a triangle with sides a, b, c.

Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter.

State the surface area and volume formulas for a cube and a cuboid.

Cube (side a): Volume = a³, Total Surface Area = 6a². Cuboid (l,b,h): Volume = l×b×h, TSA = 2(lb + bh + hl).

State the volume and surface area formulas for a cylinder and a sphere.

Cylinder (r,h): Volume = πr²h, Curved SA = 2πrh, TSA = 2πr(r+h). Sphere (r): Volume = (4/3)πr³, Surface Area = 4πr².

State the volume and surface area formulas for a cone.

Cone (radius r, height h, slant height l): Volume = (1/3)πr²h, Curved SA = πrl, TSA = πr(r+l), where l = √(r²+h²).

Define the six trigonometric ratios in a right triangle.

sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent, cosecθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ.

State the fundamental Pythagorean trigonometric identities.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.

Give the values of sin, cos, and tan at 0°, 30°, 45°, 60°, and 90°.

sin: 0, 1/2, 1/√2, √3/2, 1. cos: 1, √3/2, 1/√2, 1/2, 0. tan: 0, 1/√3, 1, √3, undefined.

In heights and distances, what are the angle of elevation and angle of depression?

Angle of elevation: the angle above the horizontal when looking up at an object. Angle of depression: the angle below the horizontal when looking down at an object. Both are measured from the horizontal line of sight.

How do you find the height of an object using the angle of elevation θ from distance d?

Height = d × tanθ (object height above eye/base level), since tanθ = height/distance. Add observer's eye height if measured from eye level.

What is data interpretation and what are common chart types used?

Data interpretation is analyzing given data (tables, charts) to answer questions. Common forms: tables, bar graphs, line graphs, pie charts, and histograms.

How do you read a pie chart to find a quantity, and how many degrees equal the whole?

The whole circle = 360° (or 100%). A sector's value = (sector angle/360°) × total, or (sector percent/100) × total.

Define mean, median, and mode.

Mean = sum of observations / number of observations. Median = the middle value when data is arranged in order. Mode = the most frequently occurring value.

How do you find the median when the number of observations is even vs odd?

Arrange data in ascending order. Odd n: median = the ((n+1)/2)th value. Even n: median = average of the (n/2)th and (n/2 + 1)th values.

State the empirical relationship between mean, median, and mode.

Mode = 3 × Median − 2 × Mean. (Equivalently: Mean − Mode = 3(Mean − Median).)