SSC CHSL (Combined Higher Secondary Level) Tier II - Module-Specific Topics Syllabus

What is the slope-intercept form of a straight line and what does each symbol mean?

$y = mx + c$, where $m$ is the slope (gradient) and $c$ is the $y$-intercept.

How do you find the slope of a line passing through points $(x_1,y_1)$ and $(x_2,y_2)$?

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

What is the condition on slopes for two lines to be (a) parallel and (b) perpendicular?

Parallel: $m_1 = m_2$. Perpendicular: $m_1 \cdot m_2 = -1$.

State the formula for the area of a triangle with base $b$ and height $h$, and Heron's formula.

Area $= \frac{1}{2}bh$. Heron's formula: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$.

Give the curved surface area and total surface area of a cylinder of radius $r$ and height $h$.

CSA $= 2\pi r h$; TSA $= 2\pi r(h + r)$; Volume $= \pi r^{2} h$.

What are the volume and total surface area of a sphere of radius $r$?

Volume $= \frac{4}{3}\pi r^{3}$; Surface area $= 4\pi r^{2}$.

State the volume and curved surface area of a cone with radius $r$, height $h$, slant height $l$.

Volume $= \frac{1}{3}\pi r^{2} h$; CSA $= \pi r l$, where $l = \sqrt{r^{2} + h^{2}}$.

State the Pythagorean theorem for a right triangle.

In a right triangle, $\text{hypotenuse}^{2} = \text{base}^{2} + \text{height}^{2}$, i.e. $c^{2} = a^{2} + b^{2}$.

Define $\sin\theta$, $\cos\theta$ and $\tan\theta$ in a right triangle.

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$.

State the fundamental Pythagorean trigonometric identity.

$\sin^{2}\theta + \cos^{2}\theta = 1$.

Give the values of $\sin\theta$ for $\theta = 0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}$.

$0,\ \frac{1}{2},\ \frac{1}{\sqrt{2}},\ \frac{\sqrt{3}}{2},\ 1$ respectively.

What is the sum of the interior angles of a triangle, and of the exterior angle property?

Interior angles sum to $180^{\circ}$. An exterior angle equals the sum of the two opposite interior angles.

State the angle sum of a polygon with $n$ sides.

Sum of interior angles $= (n-2) \times 180^{\circ}$.

What is the relationship between a tangent to a circle and the radius at the point of contact?

The tangent is perpendicular to the radius drawn to the point of contact (angle $= 90^{\circ}$).

State the theorem about the angle subtended by a diameter on the circle.

The angle in a semicircle (subtended by a diameter at any point on the circle) is a right angle, $90^{\circ}$.

What is the tangent-secant length property for two tangents drawn from an external point?

The lengths of two tangents drawn from an external point to a circle are equal.

In a heights and distances problem, define angle of elevation and angle of depression.

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

If a tower of height $h$ subtends an angle of elevation $\theta$ from a point at distance $d$, what is the relation?

$\tan\theta = \frac{h}{d}$, so $h = d\tan\theta$.

Define mean (arithmetic average) for a set of $n$ values.

$$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$$ — the sum of all values divided by the number of values.

How do you find the median of a data set?

Arrange data in ascending order; the median is the middle value. For $n$ odd it is the $\frac{n+1}{2}$th term; for $n$ even it is the average of the $\frac{n}{2}$th and $\left(\frac{n}{2}+1\right)$th terms.

What is the mode of a data set?

The mode is the value that occurs most frequently in the data set. A set may have no mode, one mode, or several modes.

State the empirical relationship between mean, median and mode.

$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$.

In a frequency distribution, what is the class mark (mid-value) of a class interval?

$\text{Class mark} = \frac{\text{lower limit} + \text{upper limit}}{2}$.

What is the classical (theoretical) definition of probability of an event?

$$P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$$ where $0 \leq P(E) \leq 1$.

State the probability of the complement of an event $E$.

$P(\text{not } E) = 1 - P(E)$.

In reasoning, what is a semantic analogy and how does it differ from a number/figural analogy?

A semantic analogy relates words by meaning (e.g. doctor : hospital). A number analogy relates by numerical pattern, and a figural analogy relates by shape/position patterns rather than meaning.

In symbolic operations reasoning, if symbols like $@, \#, \$$ replace $+, -, \times$, what is the first step to solve such problems?

Substitute each given symbol with its defined arithmetic operation, then evaluate using the BODMAS/order-of-operations rule.

What is coding-decoding in reasoning, and name a common type based on letter positions?

Coding-decoding is encrypting a word/number by a rule and decoding it back. A common type is letter-shifting, e.g. each letter moved forward by a fixed number of positions in the alphabet (Caesar-style shift).

Identify the rule and next term of the series $2, 6, 12, 20, 30, \ldots$

Differences increase by $2$: $4,6,8,10,\ldots$ (i.e. $n(n+1)$). The next term is $30 + 12 = 42$.

In problem solving and critical thinking, what does it mean to 'draw an inference' from given statements?

To draw an inference is to derive a logically valid conclusion that necessarily follows from the given premises, without adding outside assumptions or information not stated.