Indian Army GD / Technical Recruitment Mathematics Syllabus

What is the simple interest formula?

Simple Interest = (P x R x T)/100, where P is principal, R is annual rate percent, and T is time in years.

What is the compound interest formula for amount compounded annually?

Amount = P(1 + R/100)^T, and Compound Interest = Amount - P.

What is the basic formula linking distance, speed, and time?

Distance = Speed x Time, so Speed = Distance/Time and Time = Distance/Speed.

How do you convert speed from km/h to m/s?

Multiply the km/h value by 5/18 to get m/s (and multiply m/s by 18/5 to get km/h).

How is combined work rate calculated when A finishes in x days and B in y days?

Together they do (1/x + 1/y) of the work per day, so the time taken together is xy/(x+y) days.

What are like terms and unlike terms in algebra?

Like terms have the same variables raised to the same powers (e.g. 3x and 5x) and can be combined; unlike terms differ in variables or powers and cannot be combined.

State the algebraic identity for (a + b)^2 and (a - b)^2.

(a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2.

State the identity for (a + b)(a - b) and for a^3 + b^3.

(a + b)(a - b) = a^2 - b^2, and a^3 + b^3 = (a + b)(a^2 - ab + b^2).

State the identity for (a + b + c)^2.

(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.

How do you solve a pair of linear equations by elimination?

Make the coefficient of one variable equal in both equations, add or subtract to eliminate it, solve for the remaining variable, then back-substitute to find the other.

What is the standard form of a quadratic equation and the quadratic formula?

Standard form is ax^2 + bx + c = 0 (a not 0); the roots are x = [-b +/- sqrt(b^2 - 4ac)]/(2a).

What does the discriminant of a quadratic equation tell you?

For D = b^2 - 4ac: D > 0 gives two distinct real roots, D = 0 gives two equal real roots, and D < 0 gives no real (complex) roots.

What is the sum and product of the roots of ax^2 + bx + c = 0?

Sum of roots = -b/a and product of roots = c/a.

What is the HCF and LCM of two polynomials?

The HCF is the product of the common factors of highest degree present in both polynomials; the LCM is the product of factors of highest power occurring in either polynomial.

State the relationship between HCF and LCM of two polynomials.

For two polynomials, HCF x LCM = product of the two polynomials.

What are complementary, supplementary, and vertically opposite angles?

Complementary angles sum to 90 degrees, supplementary angles sum to 180 degrees, and vertically opposite angles (formed by two intersecting lines) are equal.

State the angle sum property and the exterior angle property of a triangle.

The three interior angles of a triangle sum to 180 degrees, and an exterior angle equals the sum of the two opposite interior angles.

How are triangles classified by their sides and by their angles?

By sides: scalene (all unequal), isosceles (two equal), equilateral (all equal). By angles: acute (all < 90), right (one = 90), obtuse (one > 90).

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: hypotenuse^2 = base^2 + perpendicular^2.

What is the relationship between a central angle and an inscribed angle subtending the same arc of a circle?

The central angle is twice the inscribed angle subtending the same arc; angles in the same segment are equal, and the angle in a semicircle is 90 degrees.

What is the property of a tangent to a circle at the point of contact?

A tangent is perpendicular to the radius drawn to the point of contact, and tangents drawn from an external point are equal in length.

What is the formula for the sum of interior angles of a polygon with n sides?

Sum of interior angles = (n - 2) x 180 degrees; each interior angle of a regular polygon = (n - 2) x 180/n.

What are the three criteria commonly used to prove two triangles congruent?

SSS (three sides equal), SAS (two sides and included angle equal), and ASA (two angles and included side equal); RHS applies to right triangles.

What is the basic proportionality (Thales) theorem?

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those two sides in the same ratio.

State the area and perimeter formulas for a rectangle and a circle.

Rectangle: area = length x breadth, perimeter = 2(length + breadth). Circle: area = pi r^2, circumference = 2 pi r.

What are the area formulas for a triangle and a trapezium?

Triangle area = (1/2) x base x height; trapezium area = (1/2) x (sum of parallel sides) x height between them.

State the total surface area and volume of a cube and a cuboid.

Cube (side a): TSA = 6a^2, volume = a^3. Cuboid (l, b, h): TSA = 2(lb + bh + hl), volume = l x b x h.

State the curved surface area, total surface area, and volume of a cylinder.

For radius r and height h: CSA = 2 pi r h, TSA = 2 pi r(r + h), volume = pi r^2 h.

State the volume and surface area of a sphere and a cone.

Sphere (radius r): volume = (4/3) pi r^3, surface area = 4 pi r^2. Cone (radius r, height h, slant l): volume = (1/3) pi r^2 h, CSA = pi r l.

How do you find the volume of a combined solid such as a cylinder topped by a hemisphere?

Add the volumes of the individual solids (e.g. pi r^2 h + (2/3) pi r^3), and for surface area add only the exposed surfaces, excluding shared interfaces.

Define the six trigonometric ratios in a right triangle.

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent, and cosec, sec, cot are their respective reciprocals.

State the three fundamental trigonometric identities.

sin^2 A + cos^2 A = 1, 1 + tan^2 A = sec^2 A, and 1 + cot^2 A = cosec^2 A.

Give the values of sin, cos, and tan for 0, 30, 45, 60, and 90 degrees (sin row).

sin 0 = 0, sin 30 = 1/2, sin 45 = 1/sqrt2, sin 60 = sqrt3/2, sin 90 = 1; cos values are the reverse, and tan = sin/cos.

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

The angle of elevation is the angle above the horizontal to an object higher than the observer; the angle of depression is the angle below the horizontal to an object lower than the observer.

How do you find the height of a tower using the angle of elevation?

If the angle of elevation is theta from a point at horizontal distance d, then height = d x tan(theta).

What is the difference between mean, median, and mode?

Mean is the arithmetic average, median is the middle value of ordered data, and mode is the most frequently occurring value.

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

Mode = 3 x Median - 2 x Mean (approximately, for a moderately skewed distribution).

What are the common ways to represent statistical data graphically?

Bar graphs, histograms, pie charts, frequency polygons, and ogives (cumulative frequency curves).