CLAT UG Quantitative Techniques Syllabus

How is loss percentage calculated?

Loss% = (Loss / CP) × 100, calculated on the Cost Price.

What is the formula for SP in terms of CP and profit%?

SP = CP × (100 + Profit%)/100. For a loss: SP = CP × (100 − Loss%)/100.

What are Marked Price (MP) and Discount?

Marked Price (list price) is the price tagged on an item before discount; Discount = MP − SP, and Discount% = (Discount / MP) × 100 (calculated on MP).

A shopkeeper sells two items at the same SP, one at x% profit and the other at x% loss. What is the net result?

There is always a net loss of (x²/100)%.

What is a linear equation in one variable, and how many solutions does it have?

An equation of the form ax + b = 0 (a ≠ 0), where the variable has degree 1. It has exactly one solution: x = −b/a.

What is the general form of a linear equation in two variables?

ax + by + c = 0, where a and b are not both zero; its graph is a straight line.

What does it mean when a pair of linear equations has a unique solution (graphically)?

The two lines intersect at exactly one point; condition: a₁/a₂ ≠ b₁/b₂.

What condition makes a pair of linear equations have no solution?

The lines are parallel: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (the system is inconsistent).

What condition makes a pair of linear equations have infinitely many solutions?

The lines coincide: a₁/a₂ = b₁/b₂ = c₁/c₂ (the system is dependent/consistent).

Name two algebraic methods to solve a pair of linear equations.

Substitution method and elimination method (also cross-multiplication).

What is the standard form of a quadratic equation?

ax² + bx + c = 0, where a ≠ 0 (the highest power of the variable is 2).

State the quadratic formula.

x = [−b ± √(b² − 4ac)] / (2a), for ax² + bx + c = 0.

What is the discriminant of a quadratic equation and its symbol?

The discriminant is D = b² − 4ac; it determines the nature of the roots.

How does the discriminant determine the nature of the roots?

If D > 0: two distinct real roots; if D = 0: two equal real roots; if D < 0: no real roots (roots are complex).

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

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

How do you form a quadratic equation given its roots α and β?

x² − (α + β)x + αβ = 0, i.e., x² − (sum of roots)x + (product of roots) = 0.

What is a polynomial?

An algebraic expression of the form aₙxⁿ + ... + a₁x + a₀ where exponents are whole numbers and coefficients are real, e.g., 3x² − 2x + 5.

What is the degree of a polynomial?

The highest power of the variable in the polynomial, e.g., 4x³ + x − 7 has degree 3.

Name polynomials of degree 1, 2, and 3.

Degree 1 = linear; degree 2 = quadratic; degree 3 = cubic.

What is a zero (root) of a polynomial?

A value of the variable for which the polynomial equals 0, i.e., p(k) = 0 means k is a zero of p(x).

State the Remainder Theorem.

If a polynomial p(x) is divided by (x − a), the remainder is p(a).

State the Factor Theorem.

(x − a) is a factor of polynomial p(x) if and only if p(a) = 0.

How many zeroes can a polynomial of degree n have at most?

At most n zeroes (a degree-n polynomial has at most n real roots).

What is a bar graph used for?

To compare discrete categories of data using rectangular bars of equal width whose lengths/heights are proportional to the values they represent.

What is the difference between a bar graph and a histogram?

A bar graph shows discrete/categorical data with gaps between bars; a histogram shows continuous data in class intervals with no gaps between bars.

What does a pie chart represent and what is its total?

A pie chart shows parts of a whole as sectors of a circle; the whole circle represents 100% (or 360°).

How do you convert a percentage into the central angle of a pie chart?

Central angle = (Percentage / 100) × 360°, i.e., each 1% corresponds to 3.6°.

In a pie chart, what central angle represents 25% of the data?

90°, since 25% × 360° = 90° (a quarter of the circle).

What is a data table (tabulation) used for in quantitative analysis?

To organize numerical data in rows and columns for easy reading, comparison, and computation of totals, averages, ratios, and trends.

When comparing data, which is generally better for showing proportions vs. comparing magnitudes: a pie chart or a bar graph?

A pie chart is better for showing proportions/share of a whole; a bar graph is better for comparing absolute magnitudes across categories.