ICSE Class 10

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

Objective

1. 1.
   Find the co-ordinates of the point C which divides the join of A $(4, -3)$ and B $(9, 7)$ in the ratio $3:2$.

   1. A. $(7, 3)$
   2. B. $(6.2, 1.8)$
   3. C. $(5.5, -0.2)$
   4. D. $(8, 5)$
   [1]
2. 2.
   The line segment joining A $(-3, 1)$ and B $(5, -4)$ is a diameter of a circle whose centre is C, as shown in the diagram below. Which labelled point corresponds to the centre C of the circle?

   1. A. (-3, 1)
   2. B. (1, -1.5)
   3. C. (5, -4)
   4. D. (2, -2)
   [1]
3. 3.
   Find the distance between the point A (-3, 5) and the point B on the x-axis with 9 as its abscissa.

   1. A. 5 units
   2. B. 12 units
   3. C. 13 units
   4. D. 15 units
   [1]
4. 4.
   Find the co-ordinates of the mid-point of the line segment joining the points P (4, -6) and Q (-2, 4).

   1. A. (1, -1)
   2. B. (3, -1)
   3. C. (1, -5)
   4. D. (-1, 1)
   [1]
5. 5.
   Find the co-ordinates of the point P which divides the line segment joining A (-4, 6) and B (3, -5) in the ratio 3 : 2.

   1. A. P = (-1, 1)
   2. B. P = (1, -1)
   3. C. P = (-6/5, 8/5)
   4. D. P = (0, 0)
   [1]
6. 6.
   In the given triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), identify the correct label for the centroid G in the diagram .

   1. A. G = (x₁ + x₂ + x₃, y₁ + y₂ + y₃)
   2. B. G = ( (x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3 )
   3. C. G = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
   4. D. G = ( (x₂ + x₃)/2, (y₂ + y₃)/2 )
   [1]
7. 7.
   In what ratio is the line joining P(5,3) and Q(-5,3) divided by the Y-axis, and what are the coordinates of the point of intersection?

   1. A. Ratio 2:1, coordinates (0, 3)
   2. B. Ratio 1:1, coordinates (0, 3)
   3. C. Ratio 1:2, coordinates (0, 4)
   4. D. Ratio 3:1, coordinates (0, 2)
   [1]
8. 8.
   Find a point P on the line segment joining A $(14, -5)$ and B $(-4, 4)$ which is twice as far from A as from B. What are the coordinates of P?

   1. A. $(2, 1)$
   2. B. $(8, -2)$
   3. C. $(6, -1)$
   4. D. $(-2, 3)$
   [1]
9. 9.
   Construct a triangle ABC, in which AB = 4·2 cm, BC = 6·3 cm and AC = 5 cm. Draw the perpendicular bisector of BC which meets AC at point D. Why is D equidistant from B and C?

   1. A. D lies on the perpendicular bisector of BC, so it is equidistant from B and C.
   2. B. D divides AC into two equal parts, making it equidistant from B and C.
   3. C. D is the midpoint of BC, so it must be equidistant from B and C.
   4. D. D is closer to C because AC is shorter than AB, making distances unequal.
   [1]
10. 10.
    Three consecutive vertices of a parallelogram ABCD are A(10, -6), B(2, -6) and C(-4, -2). Find the coordinates of the fourth vertex D.

    1. A. (4, -2)
    2. B. (-2, 4)
    3. C. (2, -4)
    4. D. (-4, 2)
    [1]
11. 11.
    Show that P (3, m-5) is a point of trisection of the line segment joining the points A (4, -2) and B (1, 4). Which of the following values of 'm' satisfies this condition?

    1. A. m = 1
    2. B. m = 5
    3. C. m = -1
    4. D. m = 7
    [1]
12. 12.
    In the given diagram of triangle ABC, P (-2, 5) is the mid-point of AB, Q (2, 4) is the mid-point of BC, and R (-1, 2) is the mid-point of AC. Which of the following correctly identifies the coordinates of vertex B?

    1. A. (-5, 3)
    2. B. (1, 7)
    3. C. (3, 1)
    4. D. (-1, 2)
    [1]
13. 13.
    $x$-axis divides the line segment joining $A(2, -3)$ and $B(5, 6)$ in the ratio:

    1. A. 2:3
    2. B. 3:5
    3. C. 1:2
    4. D. 2:1
    [1]
14. 14.
    In the given figure, `` the line segment AB meets the $x$-axis at A and the $y$-axis at B. The point P $(-3, 1)$ on AB divides it in the ratio $2:3$. Which labelled point corresponds to the coordinates of A?

    1. A. (-5, 0)
    2. B. (-3, 0)
    3. C. (0, 0)
    4. D. (3, 0)
    [1]
15. 15.
    Calculate the ratio in which the line joining the points (-3, -1) and (5, 7) is divided by the line x = 2. Also, determine the co-ordinates of the point of intersection. Which of the following options correctly represents the ratio and the point of intersection?

    1. A. Ratio = 3 : 5, Point of intersection = (2, 4)
    2. B. Ratio = 5 : 3, Point of intersection = (2, 4)
    3. C. Ratio = 2 : 1, Point of intersection = (2, 3)
    4. D. Ratio = 5 : 2, Point of intersection = (2, 5)
    [1]
16. 16.
    Find the ratio in which the line $2x + y = 4$ divides the line segment joining the points P(2, -2) and Q(3, 7).

    1. A. 2 : 9
    2. B. 9 : 2
    3. C. 3 : 7
    4. D. 1 : 4
    [1]
17. 17.
    The points $(2, -1)$, $(-1, 4)$ and $(-2, 2)$ are mid-points of the sides of a triangle. What are the coordinates of its vertices?

    1. A. $(3, 5)$, $(1, -3)$, and $(-5, 1)$
    2. B. $(1, 1)$, $(-3, 7)$, and $(-4, 0)$
    3. C. $(2, 3)$, $(-1, -3)$, and $(-2, 5)$
    4. D. $(4, -2)$, $(0, 6)$, and $(-3, 3)$
    [1]
18. 18.
    Point $C(-1, 2)$ divides internally the line segment joining $A(2, 5)$ and $B(x, y)$ in the ratio $3:4$. What is the value of $x^2 + y^2$?

    1. A. 25
    2. B. 29
    3. C. 13
    4. D. 34
    [1]
19. 19.
    The line joining points (a + b, b + a) and (a - b, b - a) is divided by the point (a, b). What is the ratio of division?

    1. A. 1:2
    2. B. 2:1
    3. C. 1:1
    4. D. 3:1
    [1]
20. 20.
    Determine the ratio in which the line 3x + y - 9 = 0 divides the segment joining the points (1, 3) and (2, 7).

    1. A. 1:2 internally
    2. B. 2:3 internally
    3. C. 3:4 internally
    4. D. 4:5 internally
    [1]