ICSE Class 10

General instructions: Answer all questions. Marks are shown in brackets [ ].

Objective

1. 1.
   In the diagram below, the line `l` represents the locus of a point equidistant from two fixed points A and B. Identify the correct label for the part marked `X` in the diagram ``.

   1. A. The midpoint of segment AB
   2. B. The perpendicular bisector of segment AB
   3. C. The angle bisector of ∠APB
   4. D. The median from P to segment AB
   [1]
2. 2.
   In the given diagram , a point P moves such that it remains equidistant from a fixed point O. Which of the following best describes the path (locus) traced by point P?

   1. A. A straight line passing through O
   2. B. A circle with O as the centre and the constant distance as radius
   3. C. A parabola with O as the focus
   4. D. An ellipse with O as one of its foci
   [1]
3. 3.
   Geometry made by a locus of a swimmer maintaining the same distance from a pole in water is:

   1. A. Circle
   2. B. Triangle
   3. C. Rectangle
   4. D. Square
   [1]
4. 4.
   Locus of a point whose distance from origin is always equal to is:

   1. A. $a^2 + b^2 = 2$
   2. B. $a^2 - b^2 = 4$
   3. C. $a^2 + b^2 = 4$
   4. D. $a^2 - b^2 = 2$
   [1]
5. 5.
   Let a point P move in such a way that its distance from a fixed line AB is always equal to $d$ cm. The diagram below shows the fixed line AB and the path traced by P. Which labelled part represents the locus of the moving point P?

   1. A. A single straight line parallel to AB at a distance of $d$ cm from it.
   2. B. A pair of straight lines CD and EF, each parallel to AB at a distance of $d$ cm from it.
   3. C. A circle centered on AB with radius $d$ cm.
   4. D. A single straight line perpendicular to AB at a distance of $d$ cm from it.
   [1]
6. 6.
   A point $P$ moves two units above the $x$-axis at a constant distance. The locus of point $P$ is:

   1. A. $x = 2$
   2. B. $y = -2$
   3. C. $y = 2$
   4. D. $x = -2$
   [1]
7. 7.
   A point $Q$ is at the distance of 5 units from (3, 1). Equation of locus of point $D$ is:

   1. A. $x^{2} + y^{2} - 6x + 4y - 15 = 0$
   2. B. $x^{2} + 6x + 4y - 15 = 0$
   3. C. $x^{2} + y^{2} - 6x - 2y - 15 = 0$
   4. D. $x^{2} + y^{2} + 6x - 2y - 15 = 0$
   [1]
8. 8.
   On a graph paper, draw the lines $x = 3$ and $y = -5$. Now, on the same graph paper, what is the locus of the point which is equidistant from the given lines?

   1. A. The pair of lines $y = x - 2$ and $y = -x - 2$
   2. B. The single line $y = x + 2$
   3. C. The pair of lines $x = -5$ and $y = 3$
   4. D. The single line $y = -x + 2$
   [1]
9. 9.
   A point $P$ is equidistant from the points $(-2, 0)$ and $(3, 2)$. The locus of point $P$ is:

   1. A. $10x + 4y + 9 = 0$
   2. B. $10x + 4y + 8 = 0$
   3. C. $10x + 4y - 9 = 0$
   4. D. $10x - 4y + 9 = 0$
   [1]
10. 10.
    Given: CP is the bisector of angle C of Δ ABC. Which property explains why point P is equidistant from sides AC and BC?

    1. A. P lies on the angle bisector of ∠C, so it is equidistant from AC and BC.
    2. B. P is the midpoint of side AB, making it equidistant from AC and BC.
    3. C. P divides CP in the ratio of the adjacent sides, so distances are equal.
    4. D. P lies on the perpendicular bisector of AB, ensuring equal distances.
    [1]
11. 11.
    On a graph paper, draw the line $x = 6$. Now, on the same graph paper, draw the locus of the point which moves in such a way that its distance from the given line is always equal to 3 units. What is the resulting locus?

    1. A. A single line $x = 3$ parallel to $x = 6$.
    2. B. The pair of lines $x = 3$ and $x = 9$, each 3 units from $x = 6$.
    3. C. A circle centered at $x = 6$ with radius 3 units.
    4. D. The lines $y = 3$ and $y = -3$ intersecting $x = 6$.
    [1]
12. 12.
    The locus of vertices of all isosceles triangles having a common base will be the ... of the common base of the triangles:

    1. A. Perpendicular
    2. B. Perpendicular Bisector
    3. C. Bisector
    4. D. None of the above
    [1]
13. 13.
    A point P moves such that its distance from a fixed line AB is always the same. Which of the following best describes the path travelled by P?

    1. A. A single line parallel to AB at the fixed distance.
    2. B. Two lines parallel to AB, each at the fixed distance from AB.
    3. C. A circle centered on AB with radius equal to the fixed distance.
    4. D. A parabola with AB as its directrix.
    [1]
14. 14.
    The locus of a point P such that ∠APB = 90° where AB is a fixed line is:

    1. A. Sphere
    2. B. Cube
    3. C. Circle
    4. D. Line
    [1]
15. 15.
    $AB$ is a fixed line. Which of the following is the locus of the point $P$ so that $\angle APB = 90^{\circ}$.

    1. A. Sphere
    2. B. Cube
    3. C. Circle
    4. D. Line
    [1]
16. 16.
    Using ruler and compasses only, the diagram below shows the construction of a rhombus ABCD whose diagonals AC and BD are 8 cm and 6 cm long respectively. A point P is found by construction such that it is equidistant from AB, AD and also equidistant from C and D. Which labelled part in the diagram represents the point P that satisfies the given conditions?

    1. A. The intersection of diagonal AC and the perpendicular bisector of CD
    2. B. The intersection of diagonals AC and BD
    3. C. The midpoint of diagonal AC
    4. D. The intersection of side AB and the angle bisector of angle BAD
    [1]
17. 17.
    Draw a triangle ABC such that BC = 6.3 cm, AB + AC = 10.5 cm, and ∠ABC = 60°. What is the position of point P such that PA = PB = PC?

    1. A. Intersection of the medians of triangle ABC
    2. B. Intersection of the perpendicular bisectors of BC and AC
    3. C. Intersection of the angle bisectors of triangle ABC
    4. D. Intersection of the altitudes of triangle ABC
    [1]
18. 18.
    Using ruler and compass only, construct a quadrilateral ABCD where AB = 6.5 cm, AD = 4 cm, and ∠DAB = 75°. Point C is equidistant from sides AB and AD and also equidistant from points A and B. How is point C located in the construction?

    1. A. At the intersection of AB and AD
    2. B. At the intersection of the angle bisector of ∠DAB and the perpendicular bisector of AB
    3. C. At the midpoint of AB
    4. D. At the intersection of AD and the perpendicular bisector of AB
    [1]
19. 19.
    Two straight lines PQ and PK cross each other at P at an angle of 75°. S is a stone on the road PQ, 800 m from P towards Q. Using a scale of 1 cm = 100 m, where is the flag staff X located if it is equidistant from P and S and also equidistant from the roads PQ and PK?

    1. A. At the intersection of PS and PK
    2. B. At the intersection of the perpendicular bisector of PS and the angle bisector of ∠KPS
    3. C. At the midpoint of PS
    4. D. At the intersection of PQ and the angle bisector of ∠KPS
    [1]
20. 20.
    Draw a Δ PQR such that QR = 7.5 cm, ∠PQR = 60°, and altitude PN = 4 cm. Inside the triangle, a point A is equidistant from PR and QR and also equidistant from P and R. Which construction step identifies point A?

    1. A. Intersection of the angle bisector of ∠P and the perpendicular bisector of QR
    2. B. Intersection of the angle bisector of ∠R and the perpendicular bisector of PR
    3. C. Intersection of the medians of Δ PQR
    4. D. Intersection of the altitudes of Δ PQR
    [1]