CLASS X MATHEMATICS (51)
Paper Pattern:
There will be one paper of two and a half hours duration carrying 80 marks and Internal Assessment of 20 marks.
Certain questions may require the use of Mathematical tables (Logarithmic and Trigonometric tables).
1. Commercial Mathematics
- (i) Goods and Services Tax (GST):
Computation of tax including problems involving discounts, list-price, profit, loss, basic/cost price including inverse cases.
Candidates are also expected to find price paid by the consumer after paying State Goods and Service Tax (SGST) and Central Goods and Service Tax (CGST). Different rates as in vogue on different types of items will be provided. Problems based on corresponding inverse cases are also included. - (ii) Banking:
Recurring Deposit Accounts: computation of interest and maturity value using the formula:
\[ I = \frac{P \times n(n+1)r}{2 \times 12 \times 100} \]
\[ MV = P \times n + I \] - (iii) Shares and Dividends:
(a) Face/Nominal Value, Market Value, Dividend, Rate of Dividend, Premium.
(b) Formulae:
\[ \text{Income} = \text{Number of Shares} \times \text{Rate of Dividend} \times \text{Face Value} \] \[ \text{Return} = \left( \frac{\text{Income}}{\text{Investment}} \right) \times 100 \] Note: Brokerage and fractional shares not included.
2. Algebra
- (i) Linear Inequations:
Linear Inequations in one unknown for \( x \in \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{R} \). Solving:
• Algebraically and writing the solution in set notation form.
• Representation of solution on the number line. - (ii) Quadratic Equations in one variable:
(a) Nature of roots:
• Two distinct real roots if \( b^2 - 4ac > 0 \)
• Two equal real roots if \( b^2 - 4ac = 0 \)
• No real roots if \( b^2 - 4ac < 0 \)
(b) Solving Quadratic equations by:
• Factorisation
• Using the quadratic formula.
(c) Solving simple quadratic equation problems. - (iii) Ratio and Proportion:
(a) Proportion, Continued proportion, Mean proportion
(b) Componendo, Dividendo, Alternendo, Invertendo properties and their combinations.
(c) Direct simple applications on proportions only. - (iv) Factorisation of Polynomials:
(a) Factor Theorem.
(b) Remainder Theorem.
(c) Factorising a polynomial completely after obtaining one factor using Factor Theorem.
Note: \( f(x) \) not to exceed degree 3. - (v) Matrices:
(a) Order of a matrix, Row and Column matrices.
(b) Compatibility for addition and multiplication.
(c) Null and Identity matrices.
(d) Addition and subtraction of \( 2 \times 2 \) matrices.
(e) Multiplication of a \( 2 \times 2 \) matrix by:
• a non-zero rational number
• another matrix. - (vi) Arithmetic and Geometric Progressions:
• Finding their General Term.
• Finding the Sum of their first ‘n’ terms.
• Simple Applications. - (vii) Coordinate Geometry:
(a) Reflection:
• Reflection of a point in the lines \( x=0 \), \( y=0 \), \( x=a \), \( y=a \), and the origin.
• Invariant points.
(b) Co-ordinates expressed as \((x,y)\), Section formula, Midpoint formula, Concept of slope, Equation of a line.
• Section and Mid-point formula (Internal section only, coordinates of the centroid of a triangle included).
• Equation of a line:
\[ \text{Slope-intercept form: } y = mx + c \] \[ \text{Two-point form: } (y-y_1) = m(x-x_1) \] • Geometric understanding of slope \(m\) as \( \tan \theta \) where \( \theta \) is the angle the line makes with the positive x-axis.
• Geometric understanding of \(c\) as the y-intercept.
• Conditions for two lines to be parallel or perpendicular.
• Simple applications.
3. Geometry
- (a) Similarity:
• Similarity as a size transformation.
• Comparison with Congruency; keyword: proportionality.
• Three conditions: SSS, SAS, AA.
• Applications of Basic Proportionality Theorem.
• Areas of similar triangles are proportional to the squares of corresponding sides.
• Direct applications including maps and models. - (b) Loci:
• The locus of a point at a fixed distance from a fixed point is a circle.
• The locus of a point equidistant from two intersecting lines is the angle bisector.
• The locus of a point equidistant from two given points is the perpendicular bisector.
Note: Proofs not required. - (c) Circles:
(i) Angle Properties:
• Angle subtended by an arc at the centre is double that at any point on the circle.
• Angles in the same segment are equal.
• Angle in a semi-circle is a right angle.
(ii) Cyclic Properties:
• Opposite angles of a cyclic quadrilateral are supplementary.
• The exterior angle of a cyclic quadrilateral equals the opposite interior angle.
(iii) Tangent and Secant Properties:
• The tangent at any point of a circle is perpendicular to the radius at that point.
• If two circles touch, their point of contact lies on the straight line joining their centres.
• Two tangents drawn from an external point are equal in length.
• If two chords intersect internally or externally, the product of the lengths of their segments is equal.
• Tangent-secant and tangent-chord angle properties.
Note: Proofs required unless specified otherwise.
(iv) Constructions:
• Construction of tangents to a circle from an external point.
• Circumscribing and inscribing a circle about a triangle and a regular hexagon.
4. Mensuration
-
Area and Volume of solids: Cylinder, Cone, and Sphere.
• Total surface and curved surface area.
• Volume.
• Cost-related problems.
• Inner and Outer volume, melting and recasting method.
• Combination of solids included.
Note: Problems on Frustum are not included.
5. Trigonometry
- (a) Identities:
• Solving/proving simple algebraic trigonometric expressions using:
\[ \sin^2 A + \cos^2 A = 1 \] \[ 1 + \tan^2 A = \sec^2 A \] \[ 1 + \cot^2 A = \csc^2 A \quad (0^\circ \leq A \leq 90^\circ) \] - (b) Heights and Distances:
• Solving 2-D problems involving angles of elevation and depression using trigonometric tables.
Note: Cases involving more than two right-angled triangles are excluded.
6. Statistics
-
• Basic concepts: Mean, Median, Mode.
• Computation of Measures of Central Tendency:
(Mean, Median, Mode for raw and grouped data).
• Graphical Representation: Histograms and Less than Ogive.
• Calculation of Upper Quartile, Lower Quartile, Median, Interquartile Range.
Mean by three methods:
• Direct method:
\[ \bar{x} = \frac{\Sigma fx}{\Sigma f} \] • Short-cut method:
\[ \bar{x} = A + \frac{\Sigma fd}{\Sigma f} \] • Step-deviation method:
\[ \bar{x} = A + \left( \frac{\Sigma ft}{\Sigma f} \times i \right) \]
7. Probability
-
• Random experiments, Sample space, Events.
• Definition of probability:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \] • Simple problems on single events.
Aims:
- To acquire knowledge and understanding of mathematical terms, symbols, concepts, principles, processes, and proofs.
- To develop an understanding of mathematical concepts and their application to further studies.
- To develop skills to apply mathematical knowledge to real-life problems.
- To develop necessary skills to work with technological devices like calculators and computers.
- To develop drawing skills, skills of reading tables, charts, and graphs.
- To develop an interest in mathematics.
Tags:
ICSE