Let OA, OB, OC and OD are rays in the anticlockwise direction such that ∠ AOB = ∠COD = 100°, ∠BOC = 82° and ∠AOD = 78°. Is it true to say that AOC and BOD are lines?
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The linear equation 2x – 5y = 7 has (A) A unique solution (B) Two solutions (C) Infinitely many solutions (D) No solution
The equation 2x + 5y = 7 has a unique solution, if x, y are: (A) Natural numbers (B) Positive real numbers (C) Real numbers (D) Rational numbers
If (2, 0) is a solution of the linear equation 2x + 3y = k, then the value of k is (A) 4 (B) 6 (C) 5 (D) 2
Any solution of the linear equation 2x + 0y + 9 = 0 in two variables is of the form (A) (– 9/2, m) (B) (n, – 9/2) (C) (0, – 9/2) (D) (– 9, 0)
The graph of the linear equation 2x + 3y = 6 cuts the y – axis at the point (A) (2, 0) (B) (0, 3) (C) (3, 0) (D) (0, 2)
The equation x = 7, in two variables, can be written as (A) 1. x + 1. y = 7 (B) 1. x + 0. y = 7 (C) 0. x + 1. y = 7 (D) 0. x + 0. y = 7
Any point on the x – axis is of the form (A) (x, y) (B) (0, y) (C) (x, 0) (D) (x, x)
Any point on the line y = x is of the form (A) (a, a) (B) (0, a) (C) (a, 0) (D) (a, – a)
The equation of x – axis is of the form
(A) x = 0 (B) y = 0 (C) x + y = 0 (D) x = y
The graph of y = 6 is a line
(A) parallel to x – axis at a distance 6 units from the origin. (B) parallel to y – axis at a distance 6 units from the origin. (C) making an intercept 6 on the x – axis. (D) making an intercept 6 on both the axis.
x = 5, y = 2 is a solution of the linear equation
(A) x + 2 y = 7 (B) 5x + 2y = 7 (C) x + y = 7 (D) 5 x + y = 7
If a linear equation has solutions (–2, 2), (0, 0) and (2, – 2), then it is of the form
(A) y – x = 0 (B) x + y = 0 (C) –2x + y = 0 (D) –x + 2y = 0
The positive solutions of the equation ax + by + c = 0 always lie in the
(A) 1st quadrant (B) 2nd quadrant (C) 3rd quadrant (D) 4th quadrant
The graph of the linear equation 2x + 3y = 6 is a line which meets the x-axis at the point
(A) (0, 2) (B) (2, 0) (C) (3, 0) (D) (0, 3)
If we multiply or divide both sides of a linear equation with a non-zero number, then the solution of the linear equation:
(A) Changes (B) Remains the same (C) Changes in case of multiplication only (D) Changes in case of division only
How many linear equations in x and y can be satisfied by x = 1 and y = 2?
(A) Only one (B) Two (C) Infinitely many (D) Three
The point of the form (a, a) always lies on: (A) x – axis (B) y – axis (C) On the line y = x (D) On the line x + y = 0
The point of the form (a, – a) always lies on the line (A) x = a (B) y = – a (C) y = x (D) x + y = 0
The point (0, 3) lies on the graph of the linear equation 3x + 4y = 12.
The graph of the linear equation x + 2y = 7 passes through the point (0, 7).
Every point on the graph of a linear equation in two variables does not represent a solution of the linear equation.
The graph of every linear equation in two variables need not be a line.
Read the following statement: An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are 60° each. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in a equilateral triangle.
Question 2: Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two intersecting lines l and m and the line n in the above statement.]
Read the following statements which are taken as axioms: (i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal. (ii) If a transversal intersect two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent? Justify your answer
Read the following two statements which are taken as axioms: (i) If two lines intersect each other, then the vertically opposite angles are not equal. (ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°. Is this system of axioms consistent? Justify your answer.
Read the following axioms: (i) Things which are equal to the same thing are equal to one another. (ii) If equals are added to equals, the wholes are equal. (iii) Things which are double of the same thing are equal to one another. Check whether the given system of axioms is consistent or inconsistent.
Two salesmen make equal sales during the month of August. In September, each salesman doubles his sale of the month of August. Compare their sales in September.
It is known that x + y = 10 and that x = z. Show that z + y = 10?
Euclidean geometry is valid only for curved surfaces.
The boundaries of the solids are curves
The edges of a surface are curves.
The things which are double of the same thing are equal to one another
If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C.
The statements that are proved are called axioms.
“For every line l and for every point P not lying on a given line l, there exists a unique line m passing through P and parallel to l ” is known as Playfair’s axiom
Two distinct intersecting lines cannot be parallel to the same line
Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.
The three steps from solids to points are: (A) Solids - surfaces - lines - points (B) Solids - lines - surfaces - points (C) Lines - points - surfaces - solids (D) Lines - surfaces - points – solids
The number of dimensions, a surface has: (A) 1 (B) 2 (C) 3 (D) 0
The number of dimension, a point has: (A) 0 (B) 1 (C) 2 (D) 3
Euclid divided his famous treatise “The Elements” into: (A) 13 chapters (B) 12 chapters (C) 11 chapters (D) 9 chapters
The total number of propositions in the Elements are: (A) 465 (B) 460 (C) 13 (D) 55
Boundaries of solids are: (A) surfaces (B) curves (C) lines (D) points
Boundaries of surfaces are: (A) surfaces (B) curves (C) lines (D) points
In Indus Valley Civilization (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio (A) 1 : 3 : 4 (B) 4 : 2 : 1 (C) 4 : 4 : 1 (D) 4 : 3 : 2
A pyramid is a solid figure, the base of which is (A) only a triangle (B) only a square (C) only a rectangle (D) any polygon
The side faces of a pyramid are: (A) Triangles (B) Squares (C) Polygons (D) Trapeziums
It is known that if x + y = 10 then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is: (A) First Axiom (B) Second Axiom (C) Third Axiom (D) Fourth Axiom
In ancient India, the shapes of altars used for house hold rituals were: (A) Squares and circles (B) Triangles and rectangles (C) Trapeziums and pyramids (D) Rectangles and squares
The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is: (A) Seven (B) Eight (C) Nine (D) Eleven
In Ancient India, Altars with combination of shapes like rectangles, triangles and trapeziums were used for: (A) Public worship (B) Household rituals (C) Both A and B (D) None of A, B, C
Euclid belongs to the country: (A) Babylonia (B) Egypt (C) Greece (D) India
Thales belongs to the country: (A) Babylonia (B) Egypt (C) Greece (D) Rome
Pythagoras was a student of: (A) Thales (B) Euclid (C) Both A and B (D) Archimedes
Which of the following needs a proof? (A) Theorem (B) Axiom (C) Definition (D) Postulate
‘Lines are parallel if they do not intersect’ is stated in the form of (A) an axiom (B) a definition (C) a postulate (D) a proof
Angles of a triangle are in the ratio 2 : 4 : 3. The smallest angle of the triangle is(A) 60° (B) 40° (C) 80° (D) 20°
If one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles can be(A) 50° (B) 65° (C) 145° (D) 155°
The angles of a triangle are in the ratio 5: 3: 7. The triangle is(A) an acute angled triangle (B) an obtuse angled triangle(C) a right triangle (D) an isosceles triangle
If one angle of a triangle is equal to the sum of the other two angles, then the triangle is(A) an isosceles triangle(B) an obtuse triangle(C) an equilateral triangle(D) a right triangle
A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel
If two lines intersect, prove that the vertically opposite angles are equal.
The angles of a triangle are in the ratio 2 : 3 : 4. Find the angles of the triangle.
A triangle ABC is right angled at A. L is a point on BC such that AL ⊥ BC. Prove that ∠BAL = ∠ACB.
Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.
In the given figure, the side QR of ΔPQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR=∠QPR.
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In the given figure, if PQ ⊥ PS, PQ || SR, ∠ SQR = 28º and ∠ QRT = 65º, then find the values of x and y.
In the given figure, if lines PQ and RS intersect at point T, such that ∠ PRT = 40º, ∠ RPT = 95º and ∠ TSQ = 75º, find ∠ SQT.
In the given figure, if AB || DE, ∠ BAC = 35º and ∠ CDE = 53º, find ∠ DCE.
In the given figure, ∠ X = 62º, ∠ XYZ = 54º. If YO and ZO are the bisectors of ∠ XYZ and ∠ XZY respectively of ΔXYZ, find ∠ OZY and ∠ YOZ.
In the given figure, sides QP and RQ of ΔPQR are produced to points S and T respectively. If ∠ SPR = 135º and ∠ PQT = 110º, find ∠ PRQ.
In the given figure, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.
In the given figure, if AB || CD, ∠ APQ = 50º and ∠ PRD = 127º, find x and y.
In the given figure, if PQ || ST, ∠ PQR = 110º and ∠ RST = 130º, find ∠ QRS.
[Hint: Draw a line parallel to ST through point R.]
In the given figure, If AB || CD, EF ⊥ CD and ∠ GED = 126º, find ∠ AGE, ∠ GEF and ∠ FGE.
In the given figure, if AB || CD, CD || EF and y: z = 3: 7, find x.
In the given figure, find the values of x and y and then show that AB || CD.
It is given that ∠XYZ = and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP.
In the given figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that
In the given figure, if then prove that AOB is a line.
In the given figure, ∠ PQR = ∠ PRQ, then prove that ∠ PQS = ∠ PRT.
the given figure, lines XY and MN intersect at O. If ∠POY = and a:b = 2 : 3, find c.
In the given figure, lines AB and CD intersect at O. If and find ∠BOE and reflex ∠COE.
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