ICSE Class 10

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

Objective

1. 1.
   What is the equation of the line parallel to $3x + 2y = 8$ and passing through the point $(0,1)$?

   1. A. $3x + 2y = 8$
   2. B. $3x + 2y = 2$
   3. C. $2x + 3y = 3$
   4. D. $3x - 2y = 1$
   [1]
2. 2.
   Which of the following is the slope of the line passing through the points $(7, 7)$ and $(5, 1)$?

   1. A. 2
   2. B. 3
   3. C. 4
   4. D. -3
   [1]
3. 3.
   Find the equation of a line whose y-intercept is -1 and inclination is 45°.

   1. A. y = x - 1
   2. B. y = -x - 1
   3. C. y = x + 1
   4. D. y = -x + 1
   [1]
4. 4.
   Find the slope of the line passing through the points A $(-2, 1)$ and B $(3, -4)$.

   1. A. -1
   2. B. 1
   3. C. -5
   4. D. 5
   [1]
5. 5.
   What is the slope of a line whose inclination is $60^{\circ}$?

   1. A. $\sqrt{3}$
   2. B. 1
   3. C. $\frac{1}{\sqrt{3}}$
   4. D. $\frac{1}{2}$
   [1]
6. 6.
   A straight line passes through the points $P(2, -5)$ and $Q(4, 3)$. What is the slope of the line $PQ$?

   1. A. 2
   2. B. 3
   3. C. 4
   4. D. -3
   [1]
7. 7.
   Select the equation of a straight line with slope $-2$ and which intersects the $x$-axis at a distance of 3 units to the left of the origin.

   1. A. $2x - y - 6 = 0$
   2. B. $2x + y + 6 = 0$
   3. C. $2x - y - 3 = 0$
   4. D. $2x - y + 3 = 0$
   [1]
8. 8.
   Which of the following is the slope of a line perpendicular to $PQ$ if $P(5, -3)$ and $Q(7, 3)$?

   1. A. $-\frac{7}{8}$
   2. B. $-\frac{5}{3}$
   3. C. $\frac{3}{4}$
   4. D. $\frac{3}{2}$
   [1]
9. 9.
   Write down the equation of the line whose gradient is $-\frac{2}{5}$ and which passes through point P, where P divides the line segment joining A (4, -8) and B (12, 0) in the ratio 3 : 1. Which of the following is the correct equation of the line?

   1. A. $2x + 5y + 20 = 0$
   2. B. $2x + 5y - 20 = 0$
   3. C. $5x + 2y + 10 = 0$
   4. D. $2x - 5y + 20 = 0$
   [1]
10. 10.
    Lines $2x - by + 5 = 0$ and $ax + 3y = 2$ are parallel. Which of the following relations correctly connects $a$ and $b$?

    1. A. $ab = 6$
    2. B. $ab = -6$
    3. C. $a + b = -1$
    4. D. $2a = 3b$
    [1]
11. 11.
    The line through $A(-2,3)$ and $B(4,b)$ is perpendicular to the line $2x - 4y = 5$. What is the value of $b$?

    1. A. -3
    2. B. -9
    3. C. 3
    4. D. 9
    [1]
12. 12.
    In triangle ABC, the co-ordinates of vertices A, B and C are (4, 7), (-2, 3) and (0, 1) respectively. Which of the following represents the equation of the median through vertex A and the equation of the line through vertex B parallel to AC?

    1. A. Median: 3x - 4y + 16 = 0; Parallel line: 3x - 2y + 12 = 0
    2. B. Median: 4x - 3y + 5 = 0; Parallel line: 2x - 3y + 10 = 0
    3. C. Median: x - y + 3 = 0; Parallel line: 3x + 2y - 5 = 0
    4. D. Median: 3x + 4y - 20 = 0; Parallel line: 2x + 3y - 12 = 0
    [1]
13. 13.
    In $\Delta ABC$, $A = (3, 5)$, $B = (7, 8)$ and $C = (1, -10)$. Find the equation of the median through A.

    1. A. $13x - 11y + 16 = 0$
    2. B. $6x + y - 23 = 0$
    3. C. $11x - 13y + 32 = 0$
    4. D. $x - 6y + 27 = 0$
    [1]
14. 14.
    Find the value of $p$ for which the lines $2x + 3y - 7 = 0$ and $4y - px - 12 = 0$ are perpendicular to each other.

    1. A. $p = 3$
    2. B. $p = 6$
    3. C. $p = -6$
    4. D. $p = 8$
    [1]
15. 15.
    Consider the line passing through the points $(-2, 3)$ and $(4, 1)$ and the line given by the equation $3x = y + 1$. Which of the following statements is correct regarding these lines?

    1. A. The lines are perpendicular, and the line $3x = y + 1$ bisects the join of $(-2, 3)$ and $(4, 1)$.
    2. B. The lines are parallel, and the line $3x = y + 1$ does not bisect the join of $(-2, 3)$ and $(4, 1)$.
    3. C. The lines are perpendicular, but the line $3x = y + 1$ does not bisect the join of $(-2, 3)$ and $(4, 1)$.
    4. D. The lines are neither perpendicular nor parallel, and the line $3x = y + 1$ bisects the join of $(-2, 3)$ and $(4, 1)$.
    [1]
16. 16.
    Find the equation of the line that passes through the point $P(-2, 1)$ and is parallel to the line joining the points $A(4, -3)$ and $B(-1, 5)$. What is the equation of this line?

    1. A. $8x + 5y + 11 = 0$
    2. B. $5x + 8y - 2 = 0$
    3. C. $8x - 5y + 21 = 0$
    4. D. $5x - 8y + 18 = 0$
    [1]
17. 17.
    In the given rhombus ABCD, the coordinates of vertices A and C are (3, 6) and (-1, 2) respectively. The diagram below shows the rhombus with diagonal AC and diagonal BD intersecting at right angles. Which of the following is the equation of diagonal BD?

    1. A. x - y + 3 = 0
    2. B. x + y - 5 = 0
    3. C. x - y - 1 = 0
    4. D. x + y - 1 = 0
    [1]
18. 18.
    The gradient of the line joining $P(6, k)$ and $Q(1 - 3k, 3)$ is $\frac{1}{7}$. What is the value of $k$?

    1. A. -11
    2. B. 7
    3. C. 7
    4. D. -3
    [1]
19. 19.
    Find the equation of the perpendicular from the point P(1, -2) on the line 4x - 3y - 5 = 0. Also, which of the following correctly represents the equation of the perpendicular and the co-ordinates of the foot of the perpendicular?

    1. A. Equation: 3x + 4y + 5 = 0; Foot: (-1, -2)
    2. B. Equation: 4x - 3y + 10 = 0; Foot: (1, 1)
    3. C. Equation: 3x - 4y - 5 = 0; Foot: (2, -3)
    4. D. Equation: 4x + 3y + 2 = 0; Foot: (-2, 1)
    [1]
20. 20.
    Find the equation of the line that passes through the point of intersection of the lines $5x - 8y + 23 = 0$ and $7x + 6y - 71 = 0$ and is perpendicular to the line $4x - 2y = 3$. Which of the following is the correct equation?

    1. A. $x + 2y = 17$
    2. B. $2x + y = 16$
    3. C. $x - 2y = -7$
    4. D. $4x - 2y = 8$
    [1]