Algebra Class 10 - Linear Equations and Quadratics
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Questions and Answers

Which condition indicates that a pair of linear equations is consistent and has exactly one solution?

  • a1/a2 = b1/b2 = c1/c2
  • a1/a2 = b1/b2 ≠ c1/c2
  • a1/a2 ≠ b1/b2 ≠ c1/c2
  • a1/a2 ≠ b1/b2 (correct)

    • Which method is NOT a valid algebraic method for solving a pair of linear equations?

  • Cross Multiplication
  • Elimination Method
  • Graphical Interpretation (correct)
  • Substitution Method

    • If the discriminant D of a quadratic equation ax² + bx + c = 0 is equal to 0, what is the nature of the roots?

  • Two equal roots (correct)
  • Two distinct real roots
  • Two irrational roots
  • No real roots

    • In the quadratic formula, what does D represent?

    <p>The discriminant</p> Signup and view all the answers

    Which of the following statements about coincident lines is TRUE?

    <p>They have infinitely many solutions.</p> Signup and view all the answers

    When will a quadratic equation have no real roots?

    <p>If D &lt; 0</p> Signup and view all the answers

    Which case would result in a system of equations being inconsistent?

    <p>a1/a2 = b1/b2 ≠ c1/c2</p> Signup and view all the answers

    What is the first step in solving a quadratic equation by factorization?

    <p>Rewrite in standard form</p> Signup and view all the answers

    What can be inferred about the relationship between the tangents drawn from an external point to a circle and the angle subtended at the center?

    <p>The angle at the center is double the angle between the tangents.</p> Signup and view all the answers

    In a quadrilateral circumscribing a circle, which relationship holds true?

    <p>The sum of one pair of opposite sides equals the sum of the other pair.</p> Signup and view all the answers

    What must be true for a parallelogram to be classified as a rhombus?

    <p>All sides must be congruent.</p> Signup and view all the answers

    When dividing a line segment in a given ratio, which method can be employed?

    <p>Using parallel lines to create segments of equal lengths.</p> Signup and view all the answers

    What does the bisecting of chord AP in the larger circle indicate about the distances from point P to points A and B?

    <p>AP = BP.</p> Signup and view all the answers

    What is the formula for the sum of the first 'n' even natural numbers?

    <p>$n(2 + 2n)/2$</p> Signup and view all the answers

    Which of the following is true regarding congruent shapes?

    <p>They have same shape and same size.</p> Signup and view all the answers

    What does the SAS similarity criterion imply for two triangles?

    <p>The sides opposite the equal angles are proportional.</p> Signup and view all the answers

    If the square of one side of a triangle is equal to the sum of the squares of the other two sides, what can be concluded about the triangle?

    <p>It is a right-angled triangle.</p> Signup and view all the answers

    For two triangles to be similar by the SSS criterion, what condition must be fulfilled?

    <p>Their sides must be in the same ratio.</p> Signup and view all the answers

    How is the area of two similar triangles related to their corresponding sides?

    <p>The areas are proportional to the squares of their corresponding sides.</p> Signup and view all the answers

    Which of the following statements about arithmetic mean (A.M.) in an A.P. is correct?

    <p>If a, b, c are in A.P., then b = (a + c)/2.</p> Signup and view all the answers

    What are the coordinates of a point located on the x-axis?

    <p>The coordinates are (x, 0).</p> Signup and view all the answers

    If the last term of an arithmetic progression is represented as l, which equation correctly defines l?

    <p>l = a + (n-1)d</p> Signup and view all the answers

    In the context of triangle similarity, which of the following criteria is sufficient to prove similarity?

    <p>At least two angles are equal.</p> Signup and view all the answers

    In a rectangular coordinate system, what is the nature of the y-coordinate for points on the x-axis?

    <p>It is always zero.</p> Signup and view all the answers

    What represents the sum of the first 'n' natural numbers?

    <p>$n(n + 1)/2$</p> Signup and view all the answers

    Which of the following statements about the origin in a coordinate system is true?

    <p>The origin exists at (0,0).</p> Signup and view all the answers

    Which term describes the distance of a point from the y-axis?

    <p>Abscissa</p> Signup and view all the answers

    Which of the following best describes similar figures?

    <p>Same shape but different size.</p> Signup and view all the answers

    What is one of the main divisions created by the coordinate axes in the plane?

    <p>Quadrants</p> Signup and view all the answers

    What happens to the angle of elevation as an object moves towards the right of the observer's line of sight?

    <p>It increases.</p> Signup and view all the answers

    How many tangents can be drawn from a point inside a circle?

    <p>No tangents</p> Signup and view all the answers

    If a point lies outside a circle, how many tangents can be drawn from that point to the circle?

    <p>Two</p> Signup and view all the answers

    Which trigonometric ratio is equal to $ an 60^ ext{°}$?

    <p>$ ext{√3}$</p> Signup and view all the answers

    What is the angle of depression if the angle of elevation from point P to point O is $ ext{45°}$?

    <p>$ ext{45°}$</p> Signup and view all the answers

    According to the properties of tangents, what is true about the lengths of tangents drawn from an external point to a circle?

    <p>They are equal.</p> Signup and view all the answers

    What can be said about the tangent at any point on a circle in relation to the radius at that point?

    <p>It is perpendicular to the radius.</p> Signup and view all the answers

    What characteristic defines a circle?

    <p>A collection of points at a constant distance from a fixed point.</p> Signup and view all the answers

    What is the formula for the area of a segment of a circle defined by angle θ?

    <p>θ/360° × πr² - area of ΔOAB</p> Signup and view all the answers

    How is the slant height of a right circular cone calculated?

    <p>l = √(r² + h²)</p> Signup and view all the answers

    What is the formula for the total surface area of a hollow cylinder?

    <p>2π(Rh + rh + R² - r²)</p> Signup and view all the answers

    Calculate the volume of a cylinder with radius r and height h.

    <p>πr²h</p> Signup and view all the answers

    What is the formula for the longest diagonal of a cuboid?

    <p>√(l² + b² + h²)</p> Signup and view all the answers

    What formula represents the curved surface area of a hollow hemisphere?

    <p>2πR² + 2πr²</p> Signup and view all the answers

    What is the formula used to calculate the area of a sector of a circle with angle θ?

    <p>θ/360° × πr²</p> Signup and view all the answers

    How is the cross-sectional area of a hollow cylinder calculated?

    <p>2πRh + 2πrh</p> Signup and view all the answers

    Study Notes

    Numbers

    • Real numbers include rational and irrational numbers
    • Rational numbers can be expressed as p/q, where p and q are integers and q is not zero.
    • Rational numbers include fractions, terminating decimals and non-terminating repeating decimals
    • Irrational numbers cannot be expressed in the form p/q
    • Examples of irrational numbers include √2, √3, √5, √7, π

    Real Numbers

    • Rational numbers and irrational numbers together form the real numbers
    • A real number can be either rational or irrational

    Fundamental Theorem of Arithmetic

    • Every composite number can be expressed as a product of primes.
    • This factorization is unique, apart from the order of the prime factors.

    Polynomials

    • Polynomials are algebraic expressions made up of one or more terms.
    • The standard form of a polynomial has the terms written in descending order of their degree.
    • Coefficients are real numbers
    • The leading coefficient is the coefficient of the highest degree term.
    • A real number a is a zero of the polynomial p(x) if p(x) = 0

    Nature of Roots of a Quadratic Equation

    • If the discriminant (D) of a quadratic equation is positive and a perfect square, the roots are rational and unequal.
    • If the discriminant (D) is positive but not a perfect square the roots are irrational and unequal
    • If D=0, the roots are equal and real.
    • If D is negative, there are no real roots.

    Sequence and Progression

    • A systematic arrangement of numbers according to a rule is a sequence
    • Sequences of numbers that follow specific patterns are called progressions.
    • Arithmetic Progression (AP): The difference between consecutive terms is constant.
    • Finite Arithmetic Progression (AP): Finite number of terms
    • Infinite Arithmetic Progression (AP): Infinite number of terms
    • The nth term of an AP can be expressed as Tn = a + (n-1)d
    • The sum of n terms of an AP can be calculated using the formula Sn= n/2 [2a +(n-1)d], where a is the first term and d is the common difference.

    Similar Triangles

    • Two triangles are similar if their corresponding angles are equal. The corresponding sides are in the same ratio.

    Coordinate System

    • Coordinate System is a system of assigning addresses to positions in a plane (2D) or space(3D).
    • A point on a plane( or space) is defined by an ordered pair/tuple of numbers representing its position
    • Rectangular Coordinate System: A grid with x-axis and y-axis.
    • Points are represented by ordered pairs (x, y).
    • Abscissa (x-coordinate): The distance from the y-axis.
    • Ordinate (y-coordinate):The distance from the x-axis.
    • Origin: The point of intersection of the x and y axes (0,0)
    • Quadrants: The coordinate axes divide the plane into four quadrants, the signs of x and y coordinates vary in each quadrants

    Distance Formula

    • In a coordinate plane with two points A (x1, y1) and B (x2, y2), the distance between them is given by the formula AB = √(x2 - x1)² + (y2 - y1)²

    Section Formula

    • The coordinates of a point dividing a line segment in a given ratio can be derived.
    • The x-coordinate and y-coordinate has to be calculated individually to find the total coordinates.
    • The length of an arc is calculated based on the angle of the sector
    • The area of a sector and segment of a circle depend on radii and sector angles.

    Surface Areas and Volumes

    • Surface areas and volumes of geometric solids such as cubes, cuboids, cylinders, cones, spheres and hemispheres are derived using appropriate formulas.

    Trigonometric Ratios

    • Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles.
    • Trigonometric ratios connect angles to the ratios of side lengths in right triangles.

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    Description

    Test your knowledge of linear equations and quadratic equations with this quiz designed for 10th class mathematics. Questions cover concepts like consistency of equations, discriminants, and factorization methods. Challenge yourself to see how well you understand the key principles of these important algebra topics.

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