ICSE Class 10

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

Objective

1. 1.
   If $(x - 2)$ is a factor of $2x^3 - x^2 - px - 2$, then the value of $p$ is:

   1. A. 6
   2. B. 4
   3. C. 5
   4. D. 8
   [1]
2. 2.
   If $(x - 3)$ is a factor of $x^2 + x - a$, then the value of $a$ is

   1. A. $-12$
   2. B. 12
   3. C. 6
   4. D. $-6$
   [1]
3. 3.
   Use the factor theorem to determine which of the following statements is true for any odd positive integer $n$.

   1. A. $x + a$ is a factor of $x^n + a^n$
   2. B. $x - a$ is a factor of $x^n + a^n$
   3. C. $x + a$ is a factor of $x^n - a^n$
   4. D. $x - a$ is a factor of $x^n - a^n$
   [1]
4. 4.
   Find the value of $k$ if $(x - 2)$ is a factor of $x^3 + 2x^2 - kx + 10$. Which of the following is correct?

   1. A. $k = 11$, and $(x + 5)$ is not a factor
   2. B. $k = 13$, and $(x + 5)$ is also a factor
   3. C. $k = 15$, and $(x + 5)$ is not a factor
   4. D. $k = 10$, and $(x + 5)$ is also a factor
   [1]
5. 5.
   Find the remainders when the polynomial $3x^{3} - 4x^{2} + 7x - 5$ is divided by $x - 3$ and $x + 3$ respectively.

   1. A. 61 and -143
   2. B. 52 and -134
   3. C. 67 and -150
   4. D. 70 and -140
   [1]
6. 6.
   Determine whether $x - 1$ is a factor of the polynomial $x^{6} - x^{5} + x^{4} + x^{3} - x^{2} - x + 1$ or not.

   1. A. $x - 1$ is a factor because substituting $x = 1$ yields zero.
   2. B. $x - 1$ is not a factor because substituting $x = 1$ does not yield zero.
   3. C. $x - 1$ is a factor because the polynomial has an even degree.
   4. D. $x - 1$ is not a factor because the polynomial has a constant term of 1.
   [1]
7. 7.
   If the sum of remainders obtained on dividing $x^{3} + (kx + 8)x + k$ by $x + 1$ and $x - 2$ is 1, then the value of $k$ is:

   1. A. 2
   2. B. 1
   3. C. -1
   4. D. -2
   [1]
8. 8.
   What is the remainder when $2x^{3} - 7x^{2} + 5x - 9$ is divided by $2x - 3$?

   1. A. $-\frac{21}{2}$
   2. B. $-\frac{21}{4}$
   3. C. $-\frac{129}{4}$
   4. D. $-\frac{129}{2}$
   [1]
9. 9.
   If both $(x - 2)$ and $(2x - 1)$ are factors of $ax^2 + 5x + b$, which of the following must be true?

   1. A. $a + b = 0$
   2. B. $a - b = 0$
   3. C. $a = 2b$
   4. D. $a = -b$
   [1]
10. 10.
    If $x + a$ is a factor of the polynomials $x^2 + px + q$ and $x^2 + mx + n$, which of the following correctly expresses $a$?

    1. A. $a = \frac{q - n}{p - m}$
    2. B. $a = \frac{n - q}{m - p}$
    3. C. $a = \frac{q + n}{p + m}$
    4. D. $a = \frac{n + q}{m + p}$
    [1]
11. 11.
    Show that $(3x - 2)$ is a factor of $(3x^3 + x^2 - 20x + 12)$. Hence, which of the following is the complete factorization of the given expression?

    1. A. $(3x - 2)(x + 3)(x - 2)$
    2. B. $(3x - 2)(x + 2)(x - 3)$
    3. C. $(3x - 2)(x^2 + x - 6)$
    4. D. $(3x + 2)(x - 3)(x + 2)$
    [1]
12. 12.
    Show that $(x - 3)$ is a factor of $(2x^3 - 3x^2 - 11x + 6)$. Given that $(x - 3)$ is a factor, which of the following is the complete factorization of $(2x^3 - 3x^2 - 11x + 6)$?

    1. A. $(x - 3)(x + 2)(2x + 1)$
    2. B. $(x - 3)(x + 2)(2x - 1)$
    3. C. $(x - 3)(2x + 1)(2x - 1)$
    4. D. $(x - 3)(x - 2)(2x - 1)$
    [1]
13. 13.
    When the polynomial $x^3 + 2x^2 - kx + 4$ is divided by $x - 2$, the remainder is $k$. The value of $k$ is:

    1. A. 10
    2. B. $-\frac{26}{3}$
    3. C. $-\frac{20}{3}$
    4. D. 20
    [1]
14. 14.
    If both $(x - 2)$ and $(x + 3)$ are the factors of the expression $x^3 + ax^2 + bx - 12$, then the value of $(a + b)$ is:

    1. A. 14
    2. B. -4
    3. C. 6
    4. D. -1
    [1]
15. 15.
    The polynomials $2x^3 - 7x^2 + ax - 6$ and $x^3 - 8x^2 + (2a + 1)x - 16$ leave the same remainder when divided by $x - 2$. Find the value of $a$.

    1. A. 5
    2. B. 17
    3. C. -17
    4. D. 34
    [1]
16. 16.
    Show that $(3x + 2)$ is a factor of $(6x^3 + 13x^2 - 4x - 12)$. Which of the following represents the complete factorization of $(6x^3 + 13x^2 - 4x - 12)$?

    1. A. $(3x + 2)(x + 2)(2x + 1)$
    2. B. $(3x + 2)(x + 2)(2x - 1)$
    3. C. $(3x + 2)(2x + 3)(x - 2)$
    4. D. $(3x - 2)(x + 2)(2x - 1)$
    [1]
17. 17.
    Using the factor theorem, determine which of the following is a complete factorisation of the polynomial $2x^{3} + 5x^{2} - 11x - 14$, given that $2x + 7$ is one of its factors.

    1. A. $(2x + 7)(x + 2)(x - 1)$
    2. B. $(2x + 7)(x - 2)(x + 1)$
    3. C. $(2x - 7)(x + 2)(x - 1)$
    4. D. $(2x + 7)(x + 2)(x + 1)$
    [1]
18. 18.
    Given that $(x - 2)$ is a factor of $2x^3 + ax^2 + bx - 14$ and dividing the expression by $(x - 3)$ leaves a remainder of 52, what are the values of $a$ and $b$?

    1. A. $a = 5$, $b = -11$
    2. B. $a = -5$, $b = 11$
    3. C. $a = 3$, $b = -7$
    4. D. $a = -3$, $b = 7$
    [1]
19. 19.
    If $x + a$ is a common factor of the expressions $f(x) = x^2 + px + q$ and $g(x) = x^2 + mx + n$, which of the following correctly expresses the value of $a$ in terms of $p$, $q$, $m$, and $n$?

    1. A. $a = \frac{n - q}{m - p}$
    2. B. $a = \frac{q - n}{p - m}$
    3. C. $a = \frac{m - p}{n - q}$
    4. D. $a = \frac{p - m}{q - n}$
    [1]
20. 20.
    Find the value of $a$, if $x - 2$ is a factor of $2x^5 - 6x^4 - 2ax^3 + 6ax^2 + 4ax + 8$.

    1. A. -1
    2. B. 1.5
    3. C. 2
    4. D. -2
    [1]