Negative Questions MCQs for Sub-Topics of Topic 2: Algebra

Question 1. Which of the following is NOT an algebraic expression?

(A) $3x^2 + 5$

(B) $2(a+b)$

(C) $10$

(D) $x > 5$

Question 2. In the expression $7p - 12$, which of the following is NOT a constant?

(A) $7$

(B) $12$

(C) $-12$

(D) $p$

Question 3. For the term $5xy^2$, which of the following is NOT a factor?

(A) $5$

(B) $x$

(C) $y^2$

(D) $5+x+y^2$

Question 4. In the expression $a^3 + 2a^2 - a + 8$, which of the following is NOT a term?

(A) $a^3$

(B) $2a^2$

(C) $-a$

(D) $8a$

Question 5. Given the expression $4m + 7n$, which of the following describes how it is formed?

(A) Sum of two terms.

(B) Product of four factors.

(C) Each term is a product of a constant and a variable.

(D) It is an algebraic expression.

Which description is NOT correct?

Question 6. Which of the following statements about variables is FALSE?

(A) They are symbols that represent quantities.

(B) They can take on different values.

(C) They are always represented by letters.

(D) They are essential for writing algebraic expressions and equations.

Question 7. In the expression $\frac{x}{3} - 5$, what is the coefficient of $x$?

(A) $1$

(B) $3$

(C) $\frac{1}{3}$

(D) $-5$

Which option is NOT correct?

Question 8. If the value of $y$ is $3$, what is the value of the expression $2y^2 + 5$?

(A) $2(3^2) + 5 = 2(9) + 5 = 18 + 5 = 23$

(B) $2(3)(3) + 5 = 18 + 5 = 23$

(C) $2(3+5)^2 = 2(8)^2 = 2(64) = 128$

(D) The value is $23$.

Which calculation is NOT correct?

Question 9. The perimeter of a rectangle with length $l$ and width $w$ is given by $2(l+w)$. Which of the following is NOT a valid interpretation or application of this expression?

(A) It represents the sum of the lengths of all four sides.

(B) If $l=10$ and $w=5$, the perimeter is $2(10+5) = 30$.

(C) It is equivalent to $2l + 2w$.

(D) It can be used to find the area of the rectangle.

Question 10. Which of the following expressions is NOT formed correctly according to standard algebraic notation?

(A) $x + y$

(B) $3 \times a$ (often written as $3a$)

(C) $p q$ (product of $p$ and $q$)

(D) $xy/$ (division of $x$ by $y$ is $x/y$ or $\frac{x}{y}$)

Question 1. Which of the following additions of algebraic expressions is NOT performed correctly?

(A) $(2x+3) + (4x+1) = 6x+4$

(B) $(5a-2b) + (a+3b) = 6a+b$

(C) $(y^2 + 2y) + (3y^2 - y) = 4y^2 + y$

(D) $(m+n) + (m-n) = m^2 - n^2$

Question 2. Which of the following subtractions of algebraic expressions is NOT performed correctly?

(A) $(7p+5) - (3p+2) = 7p+5-3p-2 = 4p+3$

(B) $(10k - 4) - (5k - 1) = 10k - 4 - 5k + 1 = 5k - 3$

(C) $(r^2 + s^2) - (r^2 - s^2) = r^2 + s^2 - r^2 - s^2 = 0$

(D) $(4y - 3z) - (y + 2z) = 4y - 3z - y - 2z = 3y - 5z$

Question 3. Which of the following multiplications of algebraic expressions is NOT performed correctly?

(A) $3x \times 5y = 15xy$

(B) $(a+1)(a+2) = a^2 + 3a + 2$

(C) $2m(m-4) = 2m^2 - 8m$

(D) $(p-q)^2 = p^2 - q^2$

Question 4. Which of the following divisions of algebraic expressions is NOT performed correctly?

(A) $\frac{6x^2}{2x} = 3x$ (for $x \neq 0$)

(B) $\frac{10a^3b^2}{5ab} = 2a^2b$ (for $a, b \neq 0$)

(C) $\frac{m+n}{m} = 1 + n$ (for $m \neq 0$)

(D) $\frac{y^2 - 4}{y-2} = y+2$ (for $y \neq 2$)

Question 5. To add or subtract algebraic expressions, which of the following steps is NOT necessary?

(A) Identify the like terms.

(B) Group the like terms together.

(C) Add or subtract the coefficients of the like terms.

(D) Multiply the coefficients of the like terms.

Question 6. Which statement about multiplying a binomial by a binomial is FALSE?

(A) The distributive property is used.

(B) Each term of the first binomial is multiplied by each term of the second binomial.

(C) The product is always a trinomial.

(D) The FOIL method (First, Outer, Inner, Last) can be used for $(a+b)(c+d)$.

Question 7. Simplify the expression $5x + 7y - 2x + y$. Which of the following is NOT the correct simplified form?

(A) $(5x - 2x) + (7y + y) = 3x + 8y$

(B) $3x + 8y$

(C) $5x+7y-2x+y = 5x-2x+7y+y = 3x+8y$

(D) $5x+7y-2x+y = (5-2)x + (7+1)y = 3x + 7y$

Question 8. What is the result of multiplying $-4a^2b$ by $3ab^3$?

(A) $-12a^3b^4$

(B) $(-4 \times 3) \times (a^2 \times a) \times (b \times b^3) = -12 \times a^{2+1} \times b^{1+3} = -12a^3b^4$

(C) $-12 a^2 b^3$

(D) The product is $-12a^3b^4$.

Which is NOT correct?

Question 9. Which statement about dividing a polynomial by a monomial is FALSE?

(A) Each term of the polynomial is divided by the monomial.

(B) The result is always a polynomial.

(C) The exponent rules for division are used for the variables.

(D) The coefficients are divided.

Question 10. Consider the expression $(x+y+z)^2$. Which identity is NOT applicable here?

(A) $(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca$

(B) $(A+B)^2 = A^2+2AB+B^2$ (by grouping terms, e.g., $(x+(y+z))^2$)

(C) $(x-y)^2 = x^2 - 2xy + y^2$

(D) The expanded form includes terms like $x^2, y^2, z^2, xy, yz, zx$.

Question 1. Which of the following is NOT a polynomial?

(A) $5x^3 - 7x + 1$

(B) $\frac{1}{2}y^4 + \sqrt{3}y$

(C) $m^{-2} + 2m$

(D) $-9$

Question 2. Which statement about the degree of a polynomial is FALSE?

(A) The degree of a non-zero constant polynomial is $0$.

(B) The degree of the polynomial $P(x) = x^5 - 3x^2 + 10$ is $5$.

(C) The degree of the polynomial $Q(y) = 7y$ is $1$.

(D) The degree of the zero polynomial is $0$.

Question 3. Consider the polynomial $f(x) = x^2 - 4x + 3$. Which of the following is NOT a zero of this polynomial?

(A) $x=1$ (since $f(1) = 1^2 - 4(1) + 3 = 1 - 4 + 3 = 0$)

(B) $x=3$ (since $f(3) = 3^2 - 4(3) + 3 = 9 - 12 + 3 = 0$)

(C) $x=0$ (since $f(0) = 0^2 - 4(0) + 3 = 3 \neq 0$)

(D) The zeroes are the values of $x$ for which $f(x)=0$.

Which option identifies a value that is NOT a zero?

Question 4. What is the value of the polynomial $P(y) = 2y^3 - y + 7$ at $y = -1$?

(A) $P(-1) = 2(-1)^3 - (-1) + 7 = 2(-1) + 1 + 7 = -2 + 1 + 7 = 6$

(B) $P(-1) = 2(1) - (-1) + 7 = 2 + 1 + 7 = 10$

(C) The value is $6$.

(D) $P(-1) = 2(-1) - 1 + 7 = -2 - 1 + 7 = 4$

Which calculation is NOT correct?

Question 5. Which of the following describes a trinomial?

(A) A polynomial with exactly three terms.

(B) A polynomial with degree $3$.

(C) An example is $x^2 + 2x + 1$.

(D) An example is $4a - 5b + 7c$.

Which description is NOT necessarily true for all trinomials?

Question 6. For the quadratic polynomial $ax^2 + bx + c$, the sum of the zeroes is $-b/a$ and the product of the zeroes is $c/a$. Which statement about the relationship between zeroes and coefficients is FALSE?

(A) For $x^2 - 5x + 6$, sum of zeroes $= 5$, product of zeroes $= 6$.

(B) For $2x^2 + 3x - 5$, sum of zeroes $= -3/2$, product of zeroes $= -5/2$.

(C) For $3x^2 + 9x$, sum of zeroes $= -9/3 = -3$, product of zeroes $= 0/3 = 0$.

(D) For $x^2 + 4$, sum of zeroes $= -0/1 = 0$, product of zeroes $= 4/1 = 4$.

Question 7. The graph of a quadratic polynomial $y = ax^2 + bx + c$ is a parabola. Which statement about the graphical representation of the zeroes of a polynomial is FALSE?

(A) The real zeroes of a polynomial are the x-intercepts of its graph.

(B) A quadratic polynomial can have at most two real zeroes.

(C) If the graph of a quadratic polynomial does not intersect the x-axis, it has no real zeroes.

(D) A cubic polynomial always intersects the x-axis at exactly three points.

Question 8. Which of the following is NOT a monomial?

(A) $5x^2$

(B) $-7y$

(C) $9$

(D) $a+b$

Question 9. Which statement about the leading coefficient of a polynomial is FALSE?

(A) It is the coefficient of the term with the highest degree.

(B) For the polynomial $3x^4 - 2x^5 + x$, the leading coefficient is $3$.

(C) For the polynomial $7 - y^2$, the leading coefficient is $-1$.

(D) The leading coefficient can be any real number except $0$ for a non-constant polynomial.

Question 10. Evaluate the polynomial $Q(z) = z^4 - 3z^2 + 2$ at $z = 2$. Which of the following is NOT the correct value?

(A) $Q(2) = 2^4 - 3(2)^2 + 2 = 16 - 3(4) + 2 = 16 - 12 + 2 = 6$

(B) $Q(2) = 16 - 12 + 2 = 6$

(C) The value is $6$.

(D) $Q(2) = (2-3+2)^4 = 1^4 = 1$

Question 1. According to the Remainder Theorem, if a polynomial $P(x)$ is divided by $x-a$, the remainder is $P(a)$. Which of the following applications of the Remainder Theorem is NOT correct?

(A) When $x^2 + 5x + 6$ is divided by $x-1$, the remainder is $1^2 + 5(1) + 6 = 12$.

(B) When $x^3 - 2x + 1$ is divided by $x+1$, the remainder is $(-1)^3 - 2(-1) + 1 = -1 + 2 + 1 = 2$.

(C) When $2x - 3$ is divided by $x-2$, the remainder is $2(2) - 3 = 4 - 3 = 1$.

(D) When $x^2 + 4$ is divided by $x+2$, the remainder is $2^2 + 4 = 4 + 4 = 8$.

Question 2. According to the Factor Theorem, $x-a$ is a factor of a polynomial $P(x)$ if and only if $P(a) = 0$. Which of the following statements based on the Factor Theorem is FALSE?

(A) Since $P(2) = 0$ for $P(x) = x^2 - 4$, $(x-2)$ is a factor of $P(x)$.

(B) Since $Q(-1) = 0$ for $Q(x) = x^3 + 1$, $(x+1)$ is a factor of $Q(x)$.

(C) Since $R(1) = 3$ for $R(x) = x^2+x+1$, $(x-1)$ is a factor of $R(x)$.

(D) If $(x-a)$ is a factor of $P(x)$, then $a$ is a zero of $P(x)$.

Question 3. Which of the following statements about polynomial division is FALSE?

(A) The Division Algorithm for polynomials states that for polynomials $P(x)$ and $D(x)$ (where $D(x) \neq 0$), there exist unique polynomials $Q(x)$ (quotient) and $R(x)$ (remainder) such that $P(x) = D(x)Q(x) + R(x)$, where $R(x)=0$ or degree of $R(x) <$ degree of $D(x)$.

(B) If $P(x)$ is divided by $x-a$ and the remainder is $0$, then $P(a)=0$ and $x-a$ is a factor of $P(x)$.

(C) When dividing a polynomial by a binomial, the remainder is always a constant.

(D) Long division of polynomials is similar to long division of numbers.

Question 4. Divide $x^2 - 5x + 6$ by $x-2$. Which of the following statements about this division is FALSE?

(A) The quotient is $x-3$.

(B) The remainder is $0$.

(C) $(x-2)$ is a factor of $x^2 - 5x + 6$.

(D) The remainder is obtained by evaluating $P(-2)$, where $P(x) = x^2 - 5x + 6$.

Question 5. Use the Remainder Theorem to find the remainder when $P(x) = x^3 - 3x^2 + 4x - 5$ is divided by $x-3$. Which calculation is NOT correct?

(A) Remainder is $P(3)$.

(B) $P(3) = 3^3 - 3(3)^2 + 4(3) - 5 = 27 - 3(9) + 12 - 5 = 27 - 27 + 12 - 5 = 7$.

(C) The remainder is $7$.

(D) The remainder is $P(-3) = (-3)^3 - 3(-3)^2 + 4(-3) - 5 = -27 - 3(9) - 12 - 5 = -27 - 27 - 12 - 5 = -71$.

Question 6. Is $(x+2)$ a factor of $P(x) = x^3 + 8$? Use the Factor Theorem to determine this. Which statement is FALSE?

(A) According to the Factor Theorem, if $P(-2) = 0$, then $(x+2)$ is a factor.

(B) $P(-2) = (-2)^3 + 8 = -8 + 8 = 0$.

(C) Since $P(-2) = 0$, $(x+2)$ is a factor of $P(x)$.

(D) Since $P(2) = 2^3 + 8 = 16 \neq 0$, $(x-2)$ is a factor of $P(x)$.

Question 7. Which of the following is NOT a correct application of the Division Algorithm for polynomials?

(A) $x^2 - 9 = (x-3)(x+3) + 0$

(B) $x^2 + 1 = x(x) + 1$ (Divisor $D(x)=x$, Quotient $Q(x)=x$, Remainder $R(x)=1$. Degree of $R(x)$ is $0$, which is less than degree of $D(x)$ which is $1$).

(C) $x^3 + 2x = x^2(x) + 2x$ (Divisor $D(x)=x^2$, Quotient $Q(x)=x$, Remainder $R(x)=2x$. Degree of $R(x)$ is $1$, which is less than degree of $D(x)$ which is $2$).

(D) $x^2 + 5x + 7 = (x+2)(x+3) + 1$ (Quotient $x+3$, Remainder $1$ when dividing by $x+2$)

Question 8. When a polynomial $P(x)$ is divided by a linear polynomial $ax+b$, the remainder is $P(-b/a)$. Which application of this is NOT correct?

(A) When $2x+1$ is divided by $2x-1$, remainder is $P(1/2) = 2(1/2) + 1 = 1+1 = 2$.

(B) When $3x^2 - 5x + 4$ is divided by $x+2$, remainder is $P(-2) = 3(-2)^2 - 5(-2) + 4 = 3(4) + 10 + 4 = 12 + 10 + 4 = 26$.

(C) When $4x - 8$ is divided by $2x-4$, remainder is $P(2) = 4(2) - 8 = 8 - 8 = 0$.

(D) When $x^2 + x + 1$ is divided by $2x$, remainder is $P(0) = 0^2 + 0 + 1 = 1$.

Question 9. Which statement about synthetic division is FALSE?

(A) It is a shortcut method for polynomial division when the divisor is a linear factor of the form $x-a$ or $ax-b$.

(B) It only uses the coefficients of the polynomial.

(C) It can be used to divide a polynomial by any polynomial, regardless of its degree.

(D) The last number in the result of synthetic division is the remainder.

Question 10. If $P(x) = (x-a)Q(x) + R$, where $R$ is the remainder when $P(x)$ is divided by $x-a$. Which of the following relationships is NOT correct?

(A) $P(a) = (a-a)Q(a) + R = 0 \times Q(a) + R = R$

(B) The remainder is equal to $P(a)$.

(C) If $R=0$, then $P(a) \neq 0$.

(D) If $R=0$, then $x-a$ is a factor of $P(x)$.

Question 1. Which of the following is NOT an algebraic identity?

(A) $(a+b)^2 = a^2 + 2ab + b^2$

(B) $(x-y)^2 = x^2 - 2xy + y^2$

(C) $(p+q)(p-q) = p^2 - q^2$

(D) $(m+1)^2 = m^2 + 1$

Question 2. Consider the expansion of $(2x+3)^2$. Which of the following is NOT part of the correct expansion using the identity $(a+b)^2 = a^2 + 2ab + b^2$?

(A) $(2x)^2 = 4x^2$

(B) $2(2x)(3) = 12x$

(C) $3^2 = 9$

(D) The sum of the terms is $4x^2 + 9$.

Question 3. Simplify the expression $(5a-2b)^2$ using the identity $(x-y)^2 = x^2 - 2xy + y^2$. Which of the following is NOT the correct result?

(A) $(5a)^2 - 2(5a)(2b) + (2b)^2 = 25a^2 - 20ab + 4b^2$

(B) $25a^2 - 20ab + 4b^2$

(C) $25a^2 + 4b^2$

(D) $(5a-2b)(5a-2b) = 25a^2 - 10ab - 10ab + 4b^2 = 25a^2 - 20ab + 4b^2$

Question 4. Use the identity $(a+b)(a-b) = a^2 - b^2$ to simplify $103 \times 97$. Which of the following is NOT the correct approach or result?

(A) Write $103$ as $100+3$ and $97$ as $100-3$.

(B) $(100+3)(100-3) = 100^2 - 3^2 = 10000 - 9 = 9991$.

(C) The result is $9991$.

(D) $103 \times 97 = (100)(90) + (100)(7) + (3)(90) + (3)(7) = 9000 + 700 + 270 + 21 = 9991$.

Question 5. Consider the identity $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. Which statement about this identity is FALSE?

(A) It can be written as $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$.

(B) It is obtained by multiplying $(a+b)$ by $(a+b)^2$.

(C) The expanded form is $a^3 + b^3 + 3a^2b^2$.

(D) If $a=1$ and $b=1$, $(1+1)^3 = 2^3 = 8$, and $1^3 + 3(1)^2(1) + 3(1)(1)^2 + 1^3 = 1 + 3 + 3 + 1 = 8$.

Question 6. Expand $(x-2y)^3$ using the identity $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$. Which of the following terms is NOT part of the correct expansion?

(A) $x^3$

(B) $-3x^2(2y) = -6x^2y$

(C) $3x(2y)^2 = 3x(4y^2) = 12xy^2$

(D) $(2y)^3 = 8y^3$

Question 7. Which of the following is NOT a standard algebraic identity?

(A) $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

(B) $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$

(C) $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

(D) $a^2 + b^2 = (a+b)^2 - 2ab$

Question 8. Simplify the expression $(p+q)^2 - (p-q)^2$. Which of the following is NOT a correct step or the final simplified form?

(A) $(p^2 + 2pq + q^2) - (p^2 - 2pq + q^2)$

(B) $p^2 + 2pq + q^2 - p^2 + 2pq - q^2$

(C) $(p^2 - p^2) + (2pq + 2pq) + (q^2 - q^2) = 0 + 4pq + 0 = 4pq$

(D) The simplified form is $2q^2$.

Question 9. Evaluate $99^2$ using an identity. Which of the following is NOT a correct approach or result?

(A) $99^2 = (100-1)^2 = 100^2 - 2(100)(1) + 1^2 = 10000 - 200 + 1 = 9801$.

(B) $99^2 = (90+9)^2 = 90^2 + 2(90)(9) + 9^2 = 8100 + 1620 + 81 = 9801$.

(C) The value is $9801$.

(D) $99^2 = (100-1)^2 = 100^2 - 1^2 = 10000 - 1 = 9999$.

Question 10. The sum of the cubes of three numbers $a, b, c$ minus $3abc$ is given by the identity $a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$. Which statement derived from this identity is FALSE?

(A) If $a+b+c = 0$, then $a^3+b^3+c^3 = 3abc$.

(B) If $a=b=c$, then $a^3+b^3+c^3-3abc = 3a^3 - 3a^3 = 0$, and $(a+a+a)(a^2+a^2+a^2-a^2-a^2-a^2) = 3a(0) = 0$.

(C) If $a+b+c \neq 0$, then $a^3+b^3+c^3-3abc \neq 0$.

(D) The identity is only true for specific values of $a, b, c$.

Question 1. Which of the following expressions is NOT factorised correctly?

(A) $4x + 8y = 4(x+2y)$

(B) $ab - ac = a(b-c)$

(C) $p^2 + pq = p(p+q)$

(D) $12m - 18n = 6(2m - 3n)$

Question 2. Factorise the expression $ay + by - az - bz$ by grouping terms. Which of the following is NOT a correct step or the final factorisation?

(A) Group terms: $(ay + by) - (az + bz)$

(B) Factor common factor from each group: $y(a+b) - z(a+b)$

(C) Factor out the common binomial: $(a+b)(y-z)$

(D) The factorisation is $(a+b)(y+z)$.

Question 3. Factorise the expression $25a^2 - 36b^2$ using an identity. Which of the following is NOT the correct factorisation?

(A) $(5a)^2 - (6b)^2$

(B) Using $x^2 - y^2 = (x-y)(x+y)$, with $x=5a$ and $y=6b$.

(C) $(5a - 6b)(5a + 6b)$

(D) $(5a - 6b)(5a - 6b)$

Question 4. Factorise the trinomial $x^2 + 8x + 15$. Which of the following pairs of factors of $15$ sum to $8$?

(A) $1$ and $15$

(B) $3$ and $5$

(C) $-3$ and $-5$

(D) $-1$ and $-15$

Which pair does NOT sum to $8$?

Question 5. Factorise the quadratic polynomial $2y^2 - 5y - 3$. Which of the following is NOT a correct step in the process?

(A) Find two numbers whose product is $2 \times (-3) = -6$ and whose sum is $-5$. The numbers are $-6$ and $1$.

(B) Rewrite the middle term: $2y^2 - 6y + y - 3$.

(C) Group terms and factor: $(2y^2 - 6y) + (y - 3) = 2y(y-3) + 1(y-3) = (y-3)(2y+1)$.

(D) The factorisation is $(2y-1)(y+3)$.

Question 6. Which of the following cubic polynomials has $(x-1)$ as a factor?

(A) $x^3 - 1$ (since $1^3-1=0$)

(B) $x^3 + 1$ (since $1^3+1=2 \neq 0$)

(C) $x^3 - 3x^2 + 3x - 1 = (x-1)^3$ (since $(1-1)^3 = 0$)

(D) $2x^3 - x^2 - 2x + 1$ (since $2(1)^3 - 1^2 - 2(1) + 1 = 2 - 1 - 2 + 1 = 0$)

Which polynomial does NOT have $(x-1)$ as a factor?

Question 7. Factorise $a^3 + 8$. Which identity is used and what is the correct factorisation?

(A) Identity: $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$. Factorisation: $(a+2)(a^2 - 2a + 4)$.

(B) Identity: $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$. Factorisation: $(a+2)(a^2 - 4a + 4)$.

(C) $a^3+8 = a^3+2^3$. Identity for sum of cubes is applicable.

(D) The factorisation is $(a+2)(a^2 - 2a + 4)$.

Which statement is FALSE?

Question 8. Factorise the expression $5x(a+b) - 3y(a+b)$. Which of the following is NOT the correct factorisation?

(A) $(a+b)(5x - 3y)$

(B) $(5x-3y)(a+b)$

(C) $5xa + 5xb - 3ya - 3yb$

(D) $15xy(a+b)^2$

Question 9. Factorise $y^2 - 10y + 25$. Which identity is applicable?

(A) $(a+b)^2 = a^2 + 2ab + b^2$

(B) $(a-b)^2 = a^2 - 2ab + b^2$

(C) $a^2 - b^2 = (a-b)(a+b)$

(D) $y^2 - 10y + 25$ is not factorisable.

Which statement is FALSE?

Question 10. Which statement about the factors of an algebraic expression is FALSE?

(A) Factors are expressions that multiply together to give the original expression.

(B) A constant term can be a factor.

(C) A variable can be a factor.

(D) The factors of $2x+4$ are $2$ and $x+4$.

Question 1. Which of the following is NOT a linear equation in one variable?

(A) $2x + 5 = 11$

(B) $3y - 4 = 0$

(C) $z^2 + 2z - 3 = 0$

(D) $7(p-1) = 14$

Question 2. The solution or root of a linear equation in one variable is the value of the variable that makes the equation true. Which statement about the equation $3x - 6 = 0$ is FALSE?

(A) The equation is linear.

(B) The variable is $x$.

(C) The root of the equation is $x = -2$.

(D) The root satisfies the equation: $3(2) - 6 = 6 - 6 = 0$.

Question 3. Solve the equation $5m + 10 = 30$. Which step in solving this equation using the rule of transposition is NOT correct?

(A) Transpose $10$ to the RHS: $5m = 30 - 10$.

(B) Simplify the RHS: $5m = 20$.

(C) Divide both sides by $5$: $m = 20/5$.

(D) The solution is $m = 100$.

Question 4. A number is increased by $7$ and the result is $25$. Let the number be $n$. Which equation correctly represents this word problem?

(A) $n + 7 = 25$

(B) $n - 7 = 25$

(C) $7n = 25$

(D) $n = 25 - 7$

Which equation is NOT correct?

Question 5. Solve the equation $\frac{y}{4} - 2 = 3$. Which is NOT a correct step or the solution?

(A) Add $2$ to both sides: $\frac{y}{4} = 3 + 2 = 5$.

(B) Multiply both sides by $4$: $y = 5 \times 4$.

(C) The solution is $y = 20$.

(D) Multiply both sides by $4$: $y - 2 \times 4 = 3 \times 4 \implies y - 8 = 12 \implies y = 20$.

Question 6. The sum of three consecutive integers is $51$. Let the smallest integer be $x$. The next two consecutive integers are $x+1$ and $x+2$. The equation is $x + (x+1) + (x+2) = 51$. Which statement is FALSE?

(A) The equation simplifies to $3x + 3 = 51$.

(B) Solving the equation gives $3x = 48$, so $x = 16$.

(C) The three consecutive integers are $16, 17$, and $18$.

(D) The sum of $16, 17$, and $18$ is $16+17+18 = 50$.

Question 7. Which statement about solving linear equations in one variable is FALSE?

(A) The goal is to isolate the variable on one side of the equation.

(B) The same operation must be performed on both sides of the equation to maintain equality.

(C) Multiplying or dividing both sides by zero is allowed.

(D) Transposition involves moving a term from one side to the other with a change in sign.

Question 8. A shopkeeper sells an item for $\textsf{₹} 600$ and makes a profit of $\textsf{₹} 150$. Let the cost price be $\textsf{₹} c$. The selling price is the cost price plus the profit, i.e., $600 = c + 150$. What is the cost price of the item?

(A) The equation is $600 = c + 150$.

(B) Solving for $c$: $c = 600 - 150 = 450$.

(C) The cost price is $\textsf{₹} 450$.

(D) The cost price is $\textsf{₹} 750$.

Which statement is FALSE?

Question 9. Solve the equation $2(x-3) = 10$. Which is NOT a correct step or the solution?

(A) Divide both sides by $2$: $x-3 = 10/2 = 5$.

(B) Add $3$ to both sides: $x = 5 + 3 = 8$.

(C) The solution is $x=8$.

(D) Distribute the $2$: $2x - 6 = 10$. Transpose $-6$: $2x = 10 - 6 = 4$. Divide by $2$: $x=2$.

Question 10. Which statement about the equation $0 \times x = 5$ is TRUE?

(A) The solution is $x = 5/0 = \text{undefined}$.

(B) The solution is $x=5$.

(C) The equation has no solution.

(D) The equation has infinitely many solutions.

Which statement is FALSE?

Question 1. Which of the following is NOT a linear equation in two variables?

(A) $x + y = 10$

(B) $2x - 3y = 5$

(C) $4a = 7b$

(D) $xy = 10$

Question 2. A solution to a linear equation in two variables is a pair of values $(x, y)$ that satisfies the equation. For the equation $2x + y = 7$, which of the following is NOT a solution?

(A) $(1, 5)$ (since $2(1) + 5 = 2 + 5 = 7$)

(B) $(0, 7)$ (since $2(0) + 7 = 0 + 7 = 7$)

(C) $(3, 1)$ (since $2(3) + 1 = 6 + 1 = 7$)

(D) $(2, 4)$ (since $2(2) + 4 = 4 + 4 = 8 \neq 7$)

Question 3. Which statement about the graph of a linear equation in two variables is FALSE?

(A) The graph is always a straight line.

(B) Every point on the line is a solution to the equation.

(C) A linear equation in two variables always has infinitely many solutions.

(D) The graph of $x+y=5$ is a parabola.

Question 4. What is the equation of a line parallel to the x-axis and passing through the point $(3, 5)$?

(A) $x = 3$

(B) $y = 5$

(C) $x = 5$

(D) $y = 3$

Which equation is NOT correct?

Question 5. What is the equation of a line parallel to the y-axis and passing through the point $(-2, 4)$?

(A) $x = -2$

(B) $y = 4$

(C) $x = 4$

(D) $y = -2$

Which equation is NOT correct?

Question 6. Consider the equation $x + y = 0$. Which statement is FALSE?

(A) The origin $(0, 0)$ is a solution.

(B) The line passes through the origin.

(C) The line is parallel to the x-axis.

(D) The line represents $y = -x$.

Question 7. The standard form of a linear equation in two variables is $ax + by + c = 0$, where $a, b, c$ are real numbers, and $a$ and $b$ are not both zero. Which equation is NOT in the standard form?

(A) $2x + 3y - 6 = 0$

(B) $y = 5x - 1$

(C) $4x = 7$

(D) $xy + y = 2$

Question 8. Which of the following points lies on the graph of the equation $3x - 2y = 12$?

(A) $(4, 0)$ (since $3(4) - 2(0) = 12 - 0 = 12$)

(B) $(0, -6)$ (since $3(0) - 2(-6) = 0 + 12 = 12$)

(C) $(2, -3)$ (since $3(2) - 2(-3) = 6 + 6 = 12$)

(D) $(6, 3)$ (since $3(6) - 2(3) = 18 - 6 = 12$)

Which point does NOT lie on the graph?

Question 9. The equation $y=3$ represents a line. Which statement about this line is FALSE?

(A) It is a horizontal line.

(B) It is parallel to the x-axis.

(C) It passes through the point $(0, 3)$.

(D) It is parallel to the y-axis.

Question 10. To graph a linear equation in two variables, you can find at least two solutions and draw a line through them. For the equation $x - y = 5$, which pair of points, when plotted, would NOT help you draw the graph?

(A) $(5, 0)$ and $(0, -5)$

(B) $(6, 1)$ and $(4, -1)$

(C) $(10, 5)$ and $(0, 5)$

(D) $(7, 2)$ and $(3, -2)$

Question 1. A system of linear equations can have a unique solution, no solution, or infinitely many solutions. Which type of system is NOT possible?

(A) Consistent system with a unique solution.

(B) Consistent system with infinitely many solutions.

(C) Inconsistent system with no solution.

(D) System with exactly two solutions.

Question 2. Consider the system of equations: $x+y=5$ and $x-y=1$. Which statement about this system is FALSE?

(A) The lines intersect at a single point.

(B) The system is consistent.

(C) The solution is $(3, 2)$.

(D) The lines are parallel and distinct.

Question 3. Consider the system: $2x + 3y = 7$ and $4x + 6y = 14$. Which statement is FALSE?

(A) The ratio of coefficients of $x$ is $2/4 = 1/2$.

(B) The ratio of coefficients of $y$ is $3/6 = 1/2$.

(C) The ratio of constant terms is $7/14 = 1/2$.

(D) The lines are intersecting at a unique point.

Question 4. Which method is NOT typically used for algebraically solving a pair of linear equations in two variables?

(A) Substitution method

(B) Elimination method

(C) Cross-multiplication method

(D) Graphical method

Question 5. Consider the system: $x - 2y = 4$ and $2x - 4y = 10$. Which statement is TRUE?

(A) The lines are identical.

(B) The lines are parallel and distinct.

(C) The lines intersect at a unique point.

(D) The system has infinitely many solutions.

Which statement is FALSE?

Question 6. Solve the system using the substitution method: $y = x + 1$ and $2x + y = 7$. Which is NOT a correct step or the solution?

(A) Substitute the first equation into the second: $2x + (x+1) = 7$.

(B) Simplify and solve for $x$: $3x + 1 = 7 \implies 3x = 6 \implies x = 2$.

(C) Substitute $x=2$ back into the first equation: $y = 2 + 1 = 3$.

(D) The solution is $(3, 2)$.

Question 7. Solve the system using the elimination method: $2x + y = 8$ and $x - y = 1$. Which is NOT a correct step or the solution?

(A) Add the two equations to eliminate $y$: $(2x+y) + (x-y) = 8+1 \implies 3x = 9 \implies x=3$.

(B) Substitute $x=3$ into the second equation: $3 - y = 1 \implies y = 3 - 1 = 2$.

(C) The solution is $(3, 2)$.

(D) Subtract the second equation from the first: $(2x+y) - (x-y) = 8-1 \implies x = 7$.

Question 8. Consider the system $ax+by=c_1$ and $dx+ey=c_2$. This system has no solution if $\frac{a}{d} = \frac{b}{e} \neq \frac{c_1}{c_2}$. Which statement is FALSE?

(A) The lines are parallel.

(B) The lines are distinct.

(C) The system is consistent.

(D) There is no point that lies on both lines.

Question 9. Which type of system of linear equations corresponds to the case where the graphs of the two equations are the same line?

(A) Inconsistent system

(B) Consistent system with a unique solution

(C) Consistent system with infinitely many solutions

(D) System with no solution

Which option is NOT correct?

Question 10. Which of the following systems of equations is NOT reducible to a pair of linear equations in two variables using suitable substitutions?

(A) $\frac{1}{x} + \frac{1}{y} = 5$, $\frac{2}{x} - \frac{3}{y} = 4$ (Substitute $u=1/x, v=1/y$)

(B) $\frac{x+y}{xy} = 2$, $\frac{x-y}{xy} = 1$ (Divide by $xy$ to get $\frac{1}{y}+\frac{1}{x}=2$, $\frac{1}{y}-\frac{1}{x}=1$, then substitute $u=1/x, v=1/y$)

(C) $\frac{x}{y} = 5$, $x+y = 6$ (This is already a linear system)

(D) $\frac{1}{x+y} + \frac{1}{x-y} = 3$, $\frac{2}{x+y} - \frac{3}{x-y} = 1$ (Substitute $u=1/(x+y), v=1/(x-y)$)

Question 1. Which of the following is NOT a quadratic equation?

(A) $x^2 + 5x + 6 = 0$

(B) $2y^2 - 3y = 0$

(C) $z + \frac{1}{z} = 2$ (becomes $z^2 - 2z + 1 = 0$)

(D) $m^3 - 4m + 1 = 0$

Question 2. The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. Which statement is FALSE?

(A) In $x^2 - 7x + 10 = 0$, $a=1, b=-7, c=10$.

(B) In $4y^2 - 9 = 0$, $a=4, b=0, c=-9$.

(C) In $3z = 5z^2 - 2$, the standard form is $5z^2 - 3z - 2 = 0$.

(D) In $m^2 = m^2 + 2m + 1$, the equation is quadratic with $a=1, b=2, c=1$.

Question 3. Which method is NOT a standard method for solving a quadratic equation?

(A) Factorisation

(B) Completing the square

(C) Using the quadratic formula

(D) Graphical addition

Question 4. Solve the quadratic equation $x^2 - 7x + 10 = 0$ by factorisation. Which of the following is NOT a correct step or the solution?

(A) Find two numbers that multiply to $10$ and add up to $-7$. The numbers are $-2$ and $-5$.

(B) Rewrite the equation as $(x-2)(x-5) = 0$.

(C) Set each factor to zero: $x-2 = 0$ or $x-5 = 0$.

(D) The solutions are $x=2$ and $x=-5$.

Question 5. Which statement about the discriminant $\Delta = b^2 - 4ac$ of a quadratic equation $ax^2 + bx + c = 0$ is FALSE?

(A) If $\Delta > 0$, the equation has two distinct real roots.

(B) If $\Delta = 0$, the equation has two equal real roots.

(C) If $\Delta < 0$, the equation has no real roots (complex roots).

(D) The discriminant determines the sum of the roots.

Question 6. Solve the equation $2y^2 + y - 3 = 0$ using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Which calculation or step is NOT correct?

(A) $a=2, b=1, c=-3$.

(B) $\Delta = b^2 - 4ac = 1^2 - 4(2)(-3) = 1 + 24 = 25$.

(C) $y = \frac{-1 \pm \sqrt{25}}{2(2)} = \frac{-1 \pm 5}{4}$.

(D) The solutions are $y = \frac{-1+5}{4} = \frac{4}{4} = 1$ and $y = \frac{-1-5}{4} = \frac{-6}{4} = -3/2$. The solutions are $y=1$ and $y=3/2$.

Question 7. Which quadratic equation has roots $2$ and $3$?

(A) $(x-2)(x-3) = 0 \implies x^2 - 5x + 6 = 0$

(B) $(x+2)(x+3) = 0 \implies x^2 + 5x + 6 = 0$

(C) $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \implies x^2 - (2+3)x + (2 \times 3) = 0 \implies x^2 - 5x + 6 = 0$

(D) $x^2 - 6x + 5 = 0$

Which option is NOT the correct quadratic equation with roots $2$ and $3$?

Question 8. Complete the square for the expression $x^2 + 6x$. Which is NOT a correct step?

(A) Take half of the coefficient of $x$: $6/2 = 3$.

(B) Square the result: $3^2 = 9$.

(C) Add and subtract this value: $x^2 + 6x + 9 - 9$.

(D) Factor the first three terms: $(x-3)^2 - 9$.

Question 9. Which statement about the relationship between the roots $\alpha, \beta$ and coefficients of $ax^2 + bx + c = 0$ is FALSE?

(A) $\alpha + \beta = -b/a$

(B) $\alpha \beta = c/a$

(C) $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-b/a}{c/a} = -b/c$ (if $c \neq 0$)

(D) $\alpha^2 + \beta^2 = (\alpha + \beta)^2 = (-b/a)^2 = b^2/a^2$

Question 10. Which quadratic equation has no real roots?

(A) $x^2 + x + 1 = 0$ ($\Delta = 1^2 - 4(1)(1) = 1 - 4 = -3 < 0$)

(B) $x^2 - 4x + 4 = 0$ ($\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0$)

(C) $x^2 + 5 = 0$ ($\Delta = 0^2 - 4(1)(5) = -20 < 0$)

(D) $x^2 - 3x + 2 = 0$ ($\Delta = (-3)^2 - 4(1)(2) = 9 - 8 = 1 > 0$)

Which equation does NOT have no real roots?

Question 1. Which of the following is NOT a complex number?

(A) $3 + 4i$

(B) $5$ (can be written as $5 + 0i$)

(C) $2i$ (can be written as $0 + 2i$)

(D) $\sqrt{-9}$ (This is $3i$, which is a complex number)

Which option represents something that is NOT a complex number (as defined in Intro and Algebra)?

Question 2. The imaginary unit $i$ is defined as $i = \sqrt{-1}$. Which statement about the powers of $i$ is FALSE?

(A) $i^2 = -1$

(B) $i^3 = -i$

(C) $i^4 = 1$

(D) $i^5 = -1$

Question 3. Given two complex numbers $Z_1 = 2+3i$ and $Z_2 = 4-i$. Which of the following calculations is NOT correct?

(A) $Z_1 + Z_2 = (2+4) + (3i-i) = 6 + 2i$

(B) $Z_1 - Z_2 = (2-4) + (3i-(-i)) = -2 + (3i+i) = -2 + 4i$

(C) $Z_1 \times Z_2 = (2+3i)(4-i) = 2(4) + 2(-i) + 3i(4) + 3i(-i) = 8 - 2i + 12i - 3i^2 = 8 + 10i - 3(-1) = 8 + 10i + 3 = 11 + 10i$

(D) $\frac{Z_1}{Z_2} = \frac{2+3i}{4-i} = \frac{2+3i}{4-i} \times \frac{4+i}{4+i} = \frac{(2+3i)(4+i)}{4^2 - i^2} = \frac{8 + 2i + 12i + 3i^2}{16 - (-1)} = \frac{8 + 14i - 3}{17} = \frac{5 + 14i}{17} = \frac{5}{17} + \frac{14}{17}i$. The calculation for division result is incorrect.

Question 4. Which of the following identities for complex numbers is FALSE?

(A) $Z + \bar{Z} = 2 \times \text{Re}(Z)$

(B) $Z \times \bar{Z} = |Z|^2$

(C) $(Z_1 + Z_2)^2 = Z_1^2 + 2Z_1Z_2 + Z_2^2$

(D) $(Z_1 - Z_2)^2 = Z_1^2 - Z_2^2$

Question 5. Simplify $i^{19}$. Which calculation is NOT correct?

(A) Divide $19$ by $4$: $19 = 4 \times 4 + 3$.

(B) $i^{19} = i^{4 \times 4 + 3} = (i^4)^4 \times i^3$.

(C) Since $i^4=1$ and $i^3 = -i$, $i^{19} = 1^4 \times (-i) = -i$.

(D) $i^{19} = i^{16} \times i^3 = (i^4)^4 \times i^3 = (-1)^4 \times i^3 = 1 \times i^3 = i^3 = -i$.

Question 6. What is the real part of the complex number $Z = \frac{1}{1+i}$?

(A) $\frac{1}{1+i} = \frac{1-i}{(1+i)(1-i)} = \frac{1-i}{1^2 - i^2} = \frac{1-i}{1 - (-1)} = \frac{1-i}{2} = \frac{1}{2} - \frac{1}{2}i$. Real part is $1/2$.

(B) The real part is $1/2$.

(C) The real part is $1$.

(D) $\frac{1}{1+i} = \frac{1}{\sqrt{2}(\cos(\pi/4) + i\sin(\pi/4))} = \frac{1}{\sqrt{2}}(\cos(-\pi/4) + i\sin(-\pi/4)) = \frac{1}{\sqrt{2}}(\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}}) = \frac{1}{2} - \frac{1}{2}i$. Real part is $1/2$.

Which statement about the real part is FALSE?

Question 7. Which statement about the imaginary unit $i$ is FALSE?

(A) $i$ is a real number.

(B) $i^2 = -1$.

(C) $i$ is part of the set of complex numbers.

(D) The real part of $i$ is $0$.

Question 8. Given $Z = a+bi$. Which of the following expressions does NOT simplify to a real number?

(A) $Z + \bar{Z} = (a+bi) + (a-bi) = 2a$

(B) $Z - \bar{Z} = (a+bi) - (a-bi) = 2bi$

(C) $Z \times \bar{Z} = (a+bi)(a-bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2$

(D) $\text{Re}(Z)$

Question 9. What is the imaginary part of the complex number $Z = \frac{3+2i}{1-i}$?

(A) $\frac{3+2i}{1-i} = \frac{(3+2i)(1+i)}{(1-i)(1+i)} = \frac{3 + 3i + 2i + 2i^2}{1^2 - i^2} = \frac{3 + 5i - 2}{1 - (-1)} = \frac{1 + 5i}{2} = \frac{1}{2} + \frac{5}{2}i$. Imaginary part is $5/2$.

(B) The imaginary part is $5/2$.

(C) The imaginary part is $5$.

(D) The imaginary part is $5i/2$.

Which statement about the imaginary part is FALSE?

Question 10. Which identity is NOT correct for complex numbers $Z_1$ and $Z_2$?

(A) $\overline{Z_1 + Z_2} = \bar{Z_1} + \bar{Z_2}$

(B) $\overline{Z_1 Z_2} = \bar{Z_1} \bar{Z_2}$

(C) $\overline{\left(\frac{Z_1}{Z_2}\right)} = \frac{\bar{Z_1}}{\bar{Z_2}}$ (for $Z_2 \neq 0$)

(D) $|Z_1 + Z_2| = |Z_1| + |Z_2|$

Question 1. Which statement about the Argand plane is FALSE?

(A) The horizontal axis represents the real part of a complex number.

(B) The vertical axis represents the imaginary part of a complex number.

(C) A complex number $a+bi$ is represented by the point $(a, b)$.

(D) The origin represents the complex number $1+i$.

Question 2. What is the modulus of the complex number $Z = -3 - 4i$?

(A) $|Z| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

(B) $|Z| = -3 - 4 = -7$

(C) $|Z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$

(D) The modulus is $5$.

Which calculation is NOT correct?

Question 3. The conjugate of a complex number $Z = a+bi$ is $\bar{Z} = a-bi$. Which statement about conjugates is FALSE?

(A) The conjugate of $2+5i$ is $2-5i$.

(B) The conjugate of $-4+i$ is $4-i$.

(C) If $Z$ is a real number, then $Z = \bar{Z}$.

(D) The conjugate of $6i$ is $-6i$.

Question 4. Which of the following is NOT a property of the modulus of complex numbers?

(A) $|Z| \geq 0$

(B) $|Z| = |\bar{Z}|$

(C) $|Z_1 Z_2| = |Z_1| |Z_2|$

(D) $|Z_1 + Z_2| = |Z_1| + |Z_2|$ (Triangle inequality gives $\leq$, not necessarily $=$)

Question 5. Which statement about the polar representation $Z = r(\cos\theta + i\sin\theta)$ of a complex number $Z = a+bi$ is FALSE?

(A) $r$ is the modulus of $Z$, $r = |Z| = \sqrt{a^2+b^2}$.

(B) $\theta$ is the argument or amplitude of $Z$, where $\tan\theta = b/a$ (if $a \neq 0$).

(C) $\theta$ is always in the range $[0, 2\pi)$.

(D) $a = r\cos\theta$ and $b = r\sin\theta$.

Question 6. What is the modulus of the complex number $Z = 5i$?

(A) $|Z| = \sqrt{0^2 + 5^2} = \sqrt{25} = 5$

(B) $|Z| = 5$

(C) $|Z| = \sqrt{5^2} = 5$

(D) $|Z| = -5$

Which calculation is NOT correct?

Question 7. Which of the following is NOT a property of complex conjugates?

(A) $\overline{(\bar{Z})} = Z$

(B) $Z$ is purely imaginary if and only if $Z = -\bar{Z}$.

(C) $\text{Re}(Z) = \frac{Z+\bar{Z}}{2}$

(D) $\text{Im}(Z) = \frac{Z+\bar{Z}}{2i}$

Question 8. Convert the complex number $Z = \sqrt{3} + i$ to polar form. Which step or result is FALSE?

(A) Modulus $r = |\sqrt{3} + i| = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3+1} = \sqrt{4} = 2$.

(B) Argument $\theta$: $\tan\theta = 1/\sqrt{3}$. Since $\sqrt{3}+i$ is in the first quadrant, $\theta = \pi/6$ or $30^\circ$.

(C) Polar form is $2(\cos(\pi/6) + i\sin(\pi/6))$.

(D) Polar form is $2(\cos(30^\circ) + i\sin(60^\circ))$.

Question 9. Which statement about finding the square root of a complex number is FALSE?

(A) Every non-zero complex number has exactly two square roots.

(B) The two square roots are always negatives of each other.

(C) The square roots of a positive real number are real.

(D) The square roots of a negative real number are real.

Question 10. Which statement about representing complex numbers is FALSE?

(A) The Argand plane is used for geometric representation.

(B) The horizontal axis is called the imaginary axis.

(C) The complex number $0$ is represented by the origin.

(D) The polar form gives the distance from the origin and the angle with the positive real axis.

Question 1. Consider the quadratic equation $ax^2 + bx + c = 0$ where $a, b, c$ are real coefficients. Which condition guarantees that the equation has complex roots?

(A) The discriminant $\Delta = b^2 - 4ac > 0$.

(B) The discriminant $\Delta = b^2 - 4ac = 0$.

(C) The discriminant $\Delta = b^2 - 4ac < 0$.

(D) The leading coefficient $a = 0$.

Which condition does NOT guarantee complex roots?

Question 2. Solve the quadratic equation $x^2 + 2x + 2 = 0$. Which step or result is FALSE?

(A) Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, with $a=1, b=2, c=2$.

(B) $\Delta = b^2 - 4ac = 2^2 - 4(1)(2) = 4 - 8 = -4$.

(C) $x = \frac{-2 \pm \sqrt{-4}}{2(1)} = \frac{-2 \pm 2i}{2} = -1 \pm i$.

(D) The roots are real and distinct.

Question 3. If a quadratic equation with real coefficients has one complex root $2+3i$, what must be the other root?

(A) $2-3i$ (complex conjugate)

(B) $-2-3i$

(C) $-2+3i$

(D) Any other complex number.

Which option is NOT correct?

Question 4. Which quadratic equation with real coefficients has roots $1+i$ and $1-i$?

(A) Sum of roots $= (1+i) + (1-i) = 2$. Product of roots $= (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2$. Equation is $x^2 - (\text{sum})x + (\text{product}) = 0 \implies x^2 - 2x + 2 = 0$.

(B) $x^2 - 2x + 2 = 0$.

(C) $(x-(1+i))(x-(1-i)) = 0 \implies ((x-1)-i)((x-1)+i) = (x-1)^2 - i^2 = x^2 - 2x + 1 - (-1) = x^2 - 2x + 2 = 0$.

(D) $x^2 + 2x - 2 = 0$.

Which option is NOT the correct equation?

Question 5. Consider the equation $x^2 + bx + c = 0$. If the roots are complex conjugates, $p \pm qi$ (where $q \neq 0$), which statement about the coefficients $b$ and $c$ is FALSE?

(A) $b = -(\text{sum of roots}) = -((p+qi) + (p-qi)) = -(2p) = -2p$. So $b$ is real.

(B) $c = \text{product of roots} = (p+qi)(p-qi) = p^2 - (qi)^2 = p^2 - q^2i^2 = p^2 + q^2$. So $c$ is real.

(C) Both $b$ and $c$ must be real numbers.

(D) If the roots are complex, the coefficients $b$ and $c$ must be complex.

Question 6. Solve the equation $x^2 - 6x + 10 = 0$. Which step or result is FALSE?

(A) $\Delta = (-6)^2 - 4(1)(10) = 36 - 40 = -4$.

(B) The roots are real and equal.

(C) $x = \frac{-(-6) \pm \sqrt{-4}}{2(1)} = \frac{6 \pm 2i}{2} = 3 \pm i$.

(D) The roots are $3+i$ and $3-i$.

Question 7. Which of the following quadratic equations with real coefficients does NOT have complex roots?

(A) $x^2 + 1 = 0$ ($\Delta = 0^2 - 4(1)(1) = -4 < 0$)

(B) $x^2 - 2x + 3 = 0$ ($\Delta = (-2)^2 - 4(1)(3) = 4 - 12 = -8 < 0$)

(C) $3x^2 - x + 1 = 0$ ($\Delta = (-1)^2 - 4(3)(1) = 1 - 12 = -11 < 0$)

(D) $x^2 + 4x + 3 = 0$ ($\Delta = 4^2 - 4(1)(3) = 16 - 12 = 4 > 0$)

Question 8. If the roots of a quadratic equation are $p \pm qi$ (with $q \neq 0$), which statement is FALSE?

(A) The coefficients of the quadratic equation must be real.

(B) The discriminant of the quadratic equation must be negative.

(C) The quadratic equation can be formed as $x^2 - (\text{sum})x + (\text{product}) = 0$.

(D) The quadratic equation can be $x^2 + (p \pm qi)x + (p \pm qi) = 0$.

Question 9. Solve $x^2 = -9$. Which step or result is FALSE?

(A) Take the square root of both sides: $x = \pm \sqrt{-9}$.

(B) $\sqrt{-9} = \sqrt{9 \times -1} = \sqrt{9} \times \sqrt{-1} = 3i$.

(C) The solutions are $x = 3$ and $x = -3$.

(D) The solutions are $x = 3i$ and $x = -3i$.

Question 10. Which statement about quadratic equations with complex roots is TRUE?

(A) The discriminant is always positive.

(B) The roots are always real and equal.

(C) If the coefficients are real, the complex roots always appear in conjugate pairs.

(D) If the coefficients are complex, the complex roots always appear in conjugate pairs.

Which statement is FALSE?

Question 1. Which of the following is NOT a linear inequality in one variable?

(A) $2x + 5 \geq 10$

(B) $3y < 7$

(C) $z^2 - 4 \leq 0$

(D) $5(p-1) > p$

Question 2. The solution to a linear inequality in one variable can be represented on a number line. Which statement about solving inequalities is FALSE?

(A) Adding or subtracting the same number on both sides does not change the inequality sign.

(B) Multiplying or dividing both sides by a positive number does not change the inequality sign.

(C) Multiplying or dividing both sides by a negative number requires reversing the inequality sign.

(D) Multiplying or dividing both sides by zero is allowed and results in $0 > 0$ or $0 < 0$.

Question 3. Solve the inequality $3x - 4 < 5$. Which step or result is FALSE?

(A) Add $4$ to both sides: $3x < 5 + 4 = 9$.

(B) Divide both sides by $3$: $x < 9/3 = 3$.

(C) The solution set is $x < 3$.

(D) The solution can be graphed on a number line with a closed circle at $3$ and shading to the left.

Question 4. Consider the inequality $-2y + 6 \geq 10$. Which step or result is FALSE?

(A) Subtract $6$ from both sides: $-2y \geq 10 - 6 = 4$.

(B) Divide both sides by $-2$ and reverse the inequality sign: $y \leq 4/(-2) = -2$.

(C) The solution set is $y \leq -2$.

(D) The solution can be graphed on a number line with an open circle at $-2$ and shading to the left.

Question 5. Which of the following inequalities is NOT a linear inequality in two variables?

(A) $x + y \leq 5$

(B) $2x - 3y > 0$

(C) $y \geq 4x + 1$

(D) $x^2 + y^2 < 9$

Question 6. Which statement about the graphical solution of a linear inequality in two variables (like $ax+by > c$) is FALSE?

(A) The boundary line is obtained by replacing the inequality sign with an equality sign ($ax+by=c$).

(B) The boundary line is solid if the inequality includes "equal to" ($\leq$ or $\geq$) and dashed if it does not ($<$ or $>$).

(C) The solution region is one of the half-planes formed by the boundary line.

(D) To determine which half-plane is the solution, you always shade the region containing the origin $(0,0)$.

Question 7. Consider the system of linear inequalities: $x \geq 0$, $y \geq 0$, $x+y \leq 5$. Which point is NOT in the solution region of this system?

(A) $(0, 0)$ (since $0 \geq 0$, $0 \geq 0$, $0+0 \leq 5$ is true)

(B) $(2, 2)$ (since $2 \geq 0$, $2 \geq 0$, $2+2 \leq 5$ is true)

(C) $(5, 0)$ (since $5 \geq 0$, $0 \geq 0$, $5+0 \leq 5$ is true)

(D) $(3, 3)$ (since $3 \geq 0$, $3 \geq 0$, $3+3 = 6 \leq 5$ is false)

Question 8. Which of the following represents a numerical inequality?

(A) $5 > 3$

(B) $x + 2 < 7$

(C) $-1 \leq 0$

(D) $10 \neq 12$

Which option is NOT a numerical inequality?

Question 9. The solution to a system of linear inequalities in two variables is the region where the solution regions of all inequalities overlap. Which statement is FALSE?

(A) The solution region is the intersection of the individual solution regions.

(B) The solution region is always a bounded region.

(C) The corners or vertices of the solution region are found by solving pairs of equations corresponding to the boundary lines.

(D) A system of inequalities might have no solution region if the individual regions do not overlap.

Question 10. Consider the inequality $y > 2x - 1$. To graph this, we first graph the line $y = 2x - 1$. Which point would you NOT use as a test point to determine which side to shade?

(A) $(0, 0)$

(B) $(2, 0)$

(C) $(-1, -1)$

(D) A point that lies on the line $y = 2x - 1$.

Question 1. Which of the following is NOT a sequence?

(A) $2, 4, 6, 8, ...$ (Arithmetic Progression)

(B) $1, 2, 4, 8, ...$ (Geometric Progression)

(C) $1, 1, 2, 3, 5, 8, ...$ (Fibonacci sequence)

(D) $2 + 4 + 6 + 8 + \dots$ (This is a series)

Question 2. An Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant. Which statement about an AP is FALSE?

(A) The general term is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.

(B) The sequence $5, 10, 15, 20$ is an AP with common difference $5$.

(C) The sequence $10, 7, 4, 1$ is an AP with common difference $3$.

(D) The sum of the first $n$ terms of an AP is $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$.

Question 3. The sum of the first $n$ terms of an AP is given by $S_n = \frac{n}{2}(2a_1 + (n-1)d)$. Which application of this formula is NOT correct?

(A) Sum of the first $10$ terms of AP: $2, 4, 6, ...$ ($a_1=2, d=2$): $S_{10} = \frac{10}{2}(2 \times 2 + (10-1) \times 2) = 5(4 + 9 \times 2) = 5(4+18) = 5 \times 22 = 110$.

(B) Sum of the first $5$ terms of AP: $1, 5, 9, ...$ ($a_1=1, d=4$): $S_5 = \frac{5}{2}(2 \times 1 + (5-1) \times 4) = \frac{5}{2}(2 + 4 \times 4) = \frac{5}{2}(2+16) = \frac{5}{2}(18) = 45$.

(C) Sum of the first $n$ terms of AP: $3, 3, 3, ...$ ($a_1=3, d=0$): $S_n = \frac{n}{2}(2 \times 3 + (n-1) \times 0) = \frac{n}{2}(6) = 3n$.

(D) Sum of the first $3$ terms of AP: $-1, 0, 1, ...$ ($a_1=-1, d=1$): $S_3 = 3 \times (-1) + (3-1) \times 1 = -3 + 2 = -1$. (This is calculating the third term, not the sum).

Question 4. A Geometric Progression (GP) is a sequence where the ratio between consecutive terms is constant. Which statement about a GP is FALSE?

(A) The general term is $a_n = a_1 r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.

(B) The sequence $3, 6, 12, 24$ is a GP with common ratio $2$.

(C) The sequence $10, 5, 2.5, 1.25$ is a GP with common ratio $0.5$.

(D) The sum of the first $n$ terms of a GP is $S_n = \frac{a_1(r^n-1)}{r+1}$ (for $r \neq -1$).

Question 5. The Arithmetic Mean (AM) of two positive numbers $a$ and $b$ is $\frac{a+b}{2}$, and the Geometric Mean (GM) is $\sqrt{ab}$. Which statement about AM and GM is FALSE?

(A) AM $\geq$ GM always holds for positive numbers.

(B) AM = GM if and only if $a=b$.

(C) For numbers $2$ and $8$, AM $= \frac{2+8}{2} = 5$, GM $= \sqrt{2 \times 8} = \sqrt{16} = 4$. AM $\geq$ GM holds.

(D) For numbers $-4$ and $-9$, AM $= \frac{-4-9}{2} = -6.5$, GM $= \sqrt{(-4)(-9)} = \sqrt{36} = 6$. AM $\geq$ GM holds.

Question 6. Which of the following is NOT a standard type of sequence or series studied in basic algebra?

(A) Arithmetic Progression (AP)

(B) Geometric Progression (GP)

(C) Harmonic Progression (HP)

(D) Fibonacci Sequence

Question 7. The sum of an infinite Geometric Progression with first term $a_1$ and common ratio $r$ ($|r|<1$) is given by $S_\infty = \frac{a_1}{1-r}$. Which application of this formula is FALSE?

(A) Sum of $1, 1/2, 1/4, ...$ ($a_1=1, r=1/2$): $S_\infty = \frac{1}{1-1/2} = \frac{1}{1/2} = 2$.

(B) Sum of $3, -1, 1/3, ...$ ($a_1=3, r=-1/3$): $S_\infty = \frac{3}{1-(-1/3)} = \frac{3}{1+1/3} = \frac{3}{4/3} = 3 \times \frac{3}{4} = 9/4$.

(C) Sum of $2, 4, 8, ...$ ($a_1=2, r=2$): $S_\infty = \frac{2}{1-2} = \frac{2}{-1} = -2$. (Formula only applies if $|r|<1$)

(D) The sum of $0.3, 0.03, 0.003, ...$ ($a_1=0.3, r=0.1$): $S_\infty = \frac{0.3}{1-0.1} = \frac{0.3}{0.9} = \frac{1}{3}$.

Question 8. Which statement about the $n$-th term of a sequence is FALSE?

(A) It is a formula that gives the value of any term in the sequence based on its position $n$.

(B) For the sequence $2, 4, 6, 8, ...$ the $n$-th term is $a_n = 2n$.

(C) For the sequence $1, 4, 9, 16, ...$ the $n$-th term is $a_n = n^2$.

(D) The $n$-th term can only be expressed using $n$ itself, not previous terms.

Question 9. Which statement about the sum of a series is FALSE?

(A) A finite series has a definite sum.

(B) An infinite arithmetic series always diverges (its sum does not approach a finite value) unless the common difference is $0$.

(C) An infinite geometric series always converges (its sum approaches a finite value).

(D) The sum of a series is the result of adding the terms of a sequence.

Question 10. Which application scenario does NOT typically involve arithmetic or geometric progressions?

(A) Calculating compound interest on an investment over time.

(B) Calculating the total distance traveled by a ball bouncing to decreasing heights.

(C) Predicting the temperature fluctuations throughout a day.

(D) Calculating the total amount repaid on a loan with fixed monthly installments (principal + interest).

Question 1. The Principle of Mathematical Induction (PMI) is a method used to prove statements about positive integers. Which statement about PMI is FALSE?

(A) It consists of a base case and an inductive step.

(B) The base case involves proving the statement for $n=1$ (or the smallest relevant integer).

(C) The inductive step involves proving the statement for $n=k+1$ assuming it is true for $n=k$ (the inductive hypothesis).

(D) If the statement is true for $n=k$, it is automatically true for all $n > k$.

Question 2. Suppose you are trying to prove the statement $P(n): 1+3+5+\dots+(2n-1) = n^2$ for all positive integers $n$. Which of the following is NOT a correct part of the inductive step?

(A) Assume $P(k)$ is true: $1+3+5+\dots+(2k-1) = k^2$ for some positive integer $k$.

(B) Consider the sum of the first $k+1$ terms: $1+3+5+\dots+(2k-1) + (2(k+1)-1)$.

(C) Replace $1+3+5+\dots+(2k-1)$ with $k^2$: $k^2 + (2(k+1)-1)$.

(D) Prove that $k^2 + (2k+1) = (k+1)^2$. (This is a correct step in proving $P(k+1)$)

Question 3. Which statement accurately describes the inductive hypothesis in a proof by PMI?

(A) It is the statement that needs to be proven for all $n$.

(B) It is the statement assumed to be true for $n=1$ (or the base case).

(C) It is the statement assumed to be true for some arbitrary positive integer $k$ (or $k \geq \text{base case}$).

(D) It is the statement that needs to be proven for $n=k+1$.

Which description is FALSE?

Question 4. To prove that $2^n > n$ for all positive integers $n \geq 1$ using PMI, what is the base case?

(A) Show that $2^k > k$ for some $k \geq 1$.

(B) Show that $2^{k+1} > k+1$ assuming $2^k > k$.

(C) Show that $2^1 > 1$. (This is true since $2>1$)

(D) Assume $2^n > n$ is false for some $n$.

Which option is NOT the base case?

Question 5. Which of the following statements cannot be proven using the basic Principle of Mathematical Induction?

(A) The sum of the interior angles of a convex polygon with $n$ sides is $(n-2) \times 180^\circ$ for $n \geq 3$.

(B) $n^3 + 2n$ is divisible by $3$ for all positive integers $n$.

(C) For every integer $n > 1$, $\int_1^n \frac{1}{x} dx = \ln(n)$.

(D) For every positive integer $n$, $1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$.

Question 6. The steps of a proof by PMI are usually: (I) Base Case, (II) Inductive Hypothesis, (III) Inductive Step. What is the purpose of the inductive step (III)?

(A) To verify the statement for a specific small integer.

(B) To assume the statement is true for all integers up to $k$.

(C) To show that if the statement is true for $k$, it must also be true for $k+1$.

(D) To conclude that the statement is true for all positive integers.

Which option is NOT the purpose of the inductive step?

Question 7. To prove that the sum of the first $n$ odd numbers is $n^2$, i.e., $1+3+\dots+(2n-1)=n^2$. If you assume $1+3+\dots+(2k-1)=k^2$ is true, what is the next term to add to the left side to get the sum of the first $k+1$ terms?

(A) $k+1$

(B) $2(k+1)-1 = 2k+1$

(C) $2k$

(D) $(k+1)^2$

Which term is NOT the correct next term?

Question 8. Which statement about the base case in PMI is FALSE?

(A) It establishes the starting point for the induction.

(B) It is usually $n=1$, but can be any integer for which the statement is claimed to hold.

(C) If the base case is not proven, the entire proof by PMI is invalid for all $n$ starting from that base.

(D) Proving the base case is the most difficult part of the induction proof.

Question 9. The principle "If a statement is true for $n=1$, and if its truth for $n=k$ implies its truth for $n=k+1$, then the statement is true for all positive integers $n$" is the core of PMI. Which analogy is NOT helpful in understanding this principle?

(A) Falling dominoes (If the first domino falls, and if the $k$-th domino falling causes the $(k+1)$-th domino to fall, then all dominoes will fall).

(B) Climbing a ladder (If you can get on the first rung, and if you can climb from the $k$-th rung to the $(k+1)$-th rung, then you can climb to any rung).

(C) Proving a statement by checking every single case for $n=1, 2, 3, \dots$ individually.

(D) A chain reaction where the occurrence of one event triggers the next.

Question 10. Which of the following statements about the strength of induction is FALSE?

(A) Weak induction assumes $P(k)$ is true to prove $P(k+1)$.

(B) Strong induction assumes $P(i)$ is true for all $1 \leq i \leq k$ to prove $P(k+1)$.

(C) Strong induction is a more powerful method and can prove statements that weak induction cannot.

(D) If a statement can be proven by strong induction, it can also be proven by weak induction, although it might be more complex.

Question 1. The Fundamental Principle of Counting states that if an event can occur in $m$ ways, and another independent event can occur in $n$ ways, then the two events can occur in $m \times n$ ways. Which application of this principle is NOT correct?

(A) Choosing one shirt from $5$ options and one pair of trousers from $3$ options: $5 \times 3 = 15$ ways.

(B) Rolling a die and flipping a coin: $6 \times 2 = 12$ outcomes.

(C) Choosing a $3$-digit number using digits $1, 2, 3$ with replacement: $3 \times 3 \times 3 = 27$ numbers.

(D) Choosing $3$ distinct numbers from a set of $10$ numbers where the order matters: $10 \times 9 \times 8$ ways.

Question 2. The factorial notation $n!$ represents the product of all positive integers from $1$ to $n$. Which statement about factorials is FALSE?

(A) $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

(B) $0! = 1$.

(C) $n! = n \times (n-1)!$ for $n > 0$.

(D) $(-3)! = -6$.

Question 3. A permutation is an arrangement of objects in a specific order. The number of permutations of $n$ distinct objects taken $r$ at a time is $P(n, r) = \frac{n!}{(n-r)!}$. Which statement about permutations is FALSE?

(A) Arranging $4$ distinct books on a shelf: $P(4, 4) = 4! = 24$ ways.

(B) Selecting a President, Vice-President, and Secretary from a group of $10$ people: $P(10, 3) = \frac{10!}{7!} = 10 \times 9 \times 8 = 720$ ways.

(C) The number of permutations is always less than the number of combinations for the same $n$ and $r$ (when $r>1$).

(D) If all objects are identical, the number of permutations of $n$ objects is $1$.

Question 4. A combination is a selection of objects where the order does not matter. The number of combinations of $n$ distinct objects taken $r$ at a time is $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$. Which statement about combinations is FALSE?

(A) Choosing a committee of $3$ people from a group of $10$ people: $C(10, 3) = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120$ ways.

(B) Choosing $2$ items from a set of $5$ items: $C(5, 2) = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10$ ways.

(C) $C(n, r) = C(n, n-r)$.

(D) $C(n, n) = n$.

Question 5. Which of the following problems does NOT involve combinations?

(A) Selecting $5$ players from a squad of $15$ to form a cricket team.

(B) Choosing $3$ ice cream flavours from a list of $10$ available flavours.

(C) Arranging $6$ people in a row for a photograph.

(D) Selecting $4$ winning lottery tickets from a pool of $50$ tickets.

Question 6. How many distinct words can be formed by rearranging the letters of the word 'INDIA'? (Note: Letters are not all distinct). Which is NOT the correct approach or result?

(A) The word has $5$ letters, with 'I' repeated $2$ times.

(B) The number of distinct permutations is $\frac{5!}{2!} = \frac{120}{2} = 60$.

(C) The result is $60$.

(D) The number of distinct words is $5! = 120$.

Question 7. Which statement about the relationship between permutations and combinations is FALSE?

(A) $P(n, r) = C(n, r) \times r!$

(B) $C(n, r) = \frac{P(n, r)}{r!}$

(C) Permutations count arrangements, while combinations count selections.

(D) $P(n, r)$ and $C(n, r)$ can be calculated for any non-negative integers $n$ and $r$ (where $r \leq n$).

Question 8. A box contains $6$ red balls and $4$ blue balls. In how many ways can $2$ red balls and $3$ blue balls be selected? Which is NOT the correct approach or result?

(A) Number of ways to select $2$ red balls from $6$ is $C(6, 2)$.

(B) Number of ways to select $3$ blue balls from $4$ is $C(4, 3)$.

(C) By the Fundamental Principle of Counting, the total number of ways is $C(6, 2) + C(4, 3)$.

(D) $C(6, 2) = \frac{6 \times 5}{2 \times 1} = 15$. $C(4, 3) = C(4, 1) = 4$. Total ways = $15 \times 4 = 60$.

Question 9. Which of the following represents a scenario where the order of selection matters?

(A) Choosing $3$ members for a committee from a group of $7$.

(B) Selecting a hand of $5$ cards from a deck of $52$.

(C) Awarding gold, silver, and bronze medals to $3$ distinct runners in a race of $10$ runners.

(D) Choosing $4$ fruits from a basket containing different fruits.

Which scenario does NOT represent a case where order matters?

Question 10. How many ways can $10$ distinct items be divided into two groups of $5$ items each? Which statement is FALSE?

(A) We can select $5$ items for the first group in $C(10, 5)$ ways.

(B) Once the first group of $5$ is selected, the remaining $5$ items form the second group in $C(5, 5)$ ways.

(C) The total number of ways to form the two groups is $C(10, 5) \times C(5, 5) = 252 \times 1 = 252$.

(D) Since the two groups are distinguishable, we divide by $2!$: $\frac{C(10, 5)}{2!} = \frac{252}{2} = 126$ ways.

Question 1. Which statement about the Binomial Theorem for a positive integral index $n$ is FALSE?

(A) The expansion of $(a+b)^n$ has $n+1$ terms.

(B) The sum of the powers of $a$ and $b$ in each term of the expansion is always $n$.

(C) The coefficients of the terms are given by the binomial coefficients $\binom{n}{r}$.

(D) The general term in the expansion is $T_{r+1} = \binom{n}{r} a^r b^{n-r}$.

Question 2. Consider the expansion of $(x+y)^5$. Which of the following is NOT a term in this expansion?

(A) $\binom{5}{0} x^5 y^0 = x^5$

(B) $\binom{5}{1} x^4 y^1 = 5x^4y$

(C) $\binom{5}{2} x^3 y^2 = 10x^3y^2$

(D) $\binom{5}{6} x^{-1} y^6$ (Index $r$ cannot be greater than $n$)

Question 3. Find the coefficient of $x^3$ in the expansion of $(2x-1)^4$. Which is NOT the correct approach or result?

(A) The general term is $T_{r+1} = \binom{4}{r} (2x)^{4-r} (-1)^r = \binom{4}{r} 2^{4-r} x^{4-r} (-1)^r$.

(B) We need $4-r = 3$, so $r=1$.

(C) The coefficient is $\binom{4}{1} 2^{4-1} (-1)^1 = 4 \times 2^3 \times (-1) = 4 \times 8 \times (-1) = -32$.

(D) The coefficient is $32$.

Question 4. Which statement about the middle term(s) in the expansion of $(a+b)^n$ is FALSE?

(A) If $n$ is even, there is one middle term, the $(\frac{n}{2}+1)$-th term.

(B) If $n$ is odd, there are two middle terms, the $(\frac{n+1}{2})$-th and $(\frac{n+3}{2})$-th terms.

(C) In the expansion of $(x+y)^6$, the middle term is the $3$rd term.

(D) In the expansion of $(p+q)^7$, the middle terms are the $4$th and $5$th terms.

Question 5. Find the term independent of $x$ in the expansion of $(x^2 + \frac{1}{x})^6$. Which is NOT the correct approach or result?

(A) The general term is $T_{r+1} = \binom{6}{r} (x^2)^{6-r} (\frac{1}{x})^r = \binom{6}{r} x^{2(6-r)} x^{-r} = \binom{6}{r} x^{12-2r-r} = \binom{6}{r} x^{12-3r}$.

(B) For the term independent of $x$, the power of $x$ is $0$, so $12-3r = 0 \implies 3r = 12 \implies r=4$.

(C) The term independent of $x$ is $\binom{6}{4} x^0 = \binom{6}{4} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15$.

(D) The term independent of $x$ is the coefficient of $x^0$, which is $1$.

Question 6. Which statement about binomial coefficients $\binom{n}{r}$ is FALSE?

(A) $\binom{n}{r} = \binom{n}{n-r}$

(B) $\binom{n}{0} = \binom{n}{n} = 1$

(C) $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$ (Pascal's Identity)

(D) $\binom{n}{r} = \frac{n!}{r!}$

Question 7. What is the sum of the binomial coefficients in the expansion of $(1+x)^n$?

(A) Setting $x=1$ in the expansion $(1+x)^n = \binom{n}{0} + \binom{n}{1}x + \dots + \binom{n}{n}x^n$, we get $(1+1)^n = \binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n}$.

(B) The sum is $2^n$.

(C) The sum is $n!$.

(D) For $n=3$, $(1+x)^3 = 1 + 3x + 3x^2 + x^3$. The coefficients are $1, 3, 3, 1$. Sum $= 1+3+3+1 = 8 = 2^3$.

Which statement about the sum of coefficients is FALSE?

Question 8. Consider the expansion of $(3a - \frac{1}{3a})^4$. Which statement is FALSE?

(A) The number of terms in the expansion is $5$.

(B) The general term is $T_{r+1} = \binom{4}{r} (3a)^{4-r} (-\frac{1}{3a})^r = \binom{4}{r} 3^{4-r} a^{4-r} (-1)^r (3a)^{-r} = \binom{4}{r} 3^{4-r} a^{4-r} (-1)^r 3^{-r} a^{-r} = \binom{4}{r} 3^{4-2r} (-1)^r a^{4-2r}$.

(C) The term independent of $a$ occurs when $4-2r = 0$, so $r=2$. The term is $\binom{4}{2} 3^{4-4} (-1)^2 = 6 \times 3^0 \times 1 = 6$.

(D) The middle term is the $3$rd term (since $n=4$). The $3$rd term is $T_3 = T_{2+1}$, so $r=2$. Term is $\binom{4}{2} (3a)^{4-2} (-\frac{1}{3a})^2 = 6 (3a)^2 (\frac{1}{9a^2}) = 6 (9a^2) (\frac{1}{9a^2}) = 6$.

Question 9. Which statement about the Binomial Theorem is FALSE?

(A) It provides a formula for expanding $(a+b)^n$ for any positive integer $n$.

(B) It can be used to approximate values like $(1.01)^5$.

(C) The coefficients in the expansion can be found using Pascal's triangle.

(D) It can be used to expand $(a+b)^n$ for any real number $n$.

Question 10. Find the coefficient of $x^7y^2$ in the expansion of $(x+y)^9$. Which is NOT the correct approach or result?

(A) The general term is $T_{r+1} = \binom{9}{r} x^{9-r} y^r$.

(B) We need the term where the power of $y$ is 2, so $r=2$. The power of $x$ is $9-2=7$.

(C) The coefficient is $\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36$.

(D) The coefficient is $\binom{9}{7} = \frac{9 \times 8}{2 \times 1} = 36$.

Which step or result is FALSE?

Question 1. Which statement about matrices is FALSE?

(A) A matrix is a rectangular arrangement of numbers or functions.

(B) The numbers or functions are called the elements or entries of the matrix.

(C) The order of a matrix with $m$ rows and $n$ columns is $m \times n$.

(D) A matrix can be added to any other matrix.

Question 2. Consider the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$. What is the order of this matrix?

(A) $2 \times 3$

(B) $3 \times 2$

(C) $6$

(D) $5$

Which option is NOT the order?

Question 3. Which statement about types of matrices is FALSE?

(A) A row matrix has only one row.

(B) A column matrix has only one column.

(C) A square matrix has the number of rows equal to the number of columns.

(D) A diagonal matrix is a square matrix where all diagonal elements are zero.

Question 4. Given $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} p & q \\ r & s \end{pmatrix}$. Which condition is NOT necessary for matrices $A$ and $B$ to be equal ($A=B$)?

(A) They must be of the same order.

(B) Their corresponding elements must be equal ($a=p, b=q, c=r, d=s$).

(C) They must be square matrices.

(D) If $A$ and $B$ are equal, then $a=p$ and $b=q$.

Question 5. Given $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and scalar $k=2$. Calculate $kA$. Which of the following is NOT the correct resulting matrix?

(A) $\begin{pmatrix} 2 \times 5 & 2 \times 1 \\ 2 \times 2 & 2 \times 3 \end{pmatrix} = \begin{pmatrix} 10 & 2 \\ 4 & 6 \end{pmatrix}$

(B) $\begin{pmatrix} 10 & 2 \\ 4 & 6 \end{pmatrix}$

(C) Each element of the matrix is multiplied by the scalar.

(D) $\begin{pmatrix} 5+2 & 1+2 \\ 2+2 & 3+2 \end{pmatrix} = \begin{pmatrix} 7 & 3 \\ 4 & 5 \end{pmatrix}$

Question 6. Given $A = \begin{pmatrix} 5 & 1 \\ 2 & 3 \end{pmatrix}$ and scalar $k=2$. Calculate $kA$. Which of the following is NOT the correct resulting matrix?

(A) $\begin{pmatrix} 2 \times 5 & 2 \times 1 \\ 2 \times 2 & 2 \times 3 \end{pmatrix} = \begin{pmatrix} 10 & 2 \\ 4 & 6 \end{pmatrix}$

(B) $\begin{pmatrix} 10 & 2 \\ 4 & 6 \end{pmatrix}$

(C) Each element of the matrix is multiplied by the scalar.

(D) $\begin{pmatrix} 5+2 & 1+2 \\ 2+2 & 3+2 \end{pmatrix} = \begin{pmatrix} 7 & 3 \\ 4 & 5 \end{pmatrix}$

Question 7. Given $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$. Calculate $AB$. Which step or result is FALSE?

(A) The product $AB$ is defined because the number of columns in $A$ (2) equals the number of rows in $B$ (2).

(B) The order of the resulting matrix $AB$ is $2 \times 2$.

(C) $AB = \begin{pmatrix} 1 \times 5 + 2 \times 7 & 1 \times 6 + 2 \times 8 \\ 3 \times 5 + 4 \times 7 & 3 \times 6 + 4 \times 8 \end{pmatrix} = \begin{pmatrix} 5 + 14 & 6 + 16 \\ 15 + 28 & 18 + 32 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$.

(D) $AB = \begin{pmatrix} 1 \times 5 & 2 \times 6 \\ 3 \times 7 & 4 \times 8 \end{pmatrix} = \begin{pmatrix} 5 & 12 \\ 21 & 32 \end{pmatrix}$.

Question 8. Which statement about matrix multiplication is FALSE?

(A) Matrix multiplication is generally not commutative ($AB \neq BA$ in general).

(B) Matrix multiplication is associative: $(AB)C = A(BC)$.

(C) Matrix multiplication is distributive over addition: $A(B+C) = AB + AC$.

(D) If $AB = 0$, then $A=0$ or $B=0$.

Question 9. What is an identity matrix ($I$)?

(A) A square matrix where all diagonal elements are $1$ and all non-diagonal elements are $0$.

(B) A matrix such that $AI = IA = A$ for any matrix $A$ for which the products are defined.

(C) For a $2 \times 2$ matrix, the identity matrix is $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$.

(D) It is a diagonal matrix where all diagonal elements are equal to $1$.

Which statement is FALSE?

Question 10. Given $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Which type of matrix is A?

(A) Square matrix

(B) Diagonal matrix

(C) Identity matrix

(D) Zero matrix

Which type of matrix is A NOT?

Question 1. The transpose of a matrix $A$, denoted by $A'$ or $A^T$, is obtained by interchanging its rows and columns. Which statement is FALSE?

(A) If $A$ is an $m \times n$ matrix, then $A'$ is an $n \times m$ matrix.

(B) The transpose of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$.

(C) $(A+B)' = A' + B'$.

(D) $(AB)' = A'B'$.

Question 2. A square matrix $A$ is symmetric if $A' = A$. A square matrix $A$ is skew symmetric if $A' = -A$. Which statement is FALSE?

(A) If $A$ is symmetric, then $a_{ij} = a_{ji}$ for all $i, j$.

(B) If $A$ is skew symmetric, then $a_{ij} = -a_{ji}$ for all $i, j$.

(C) All diagonal elements of a skew symmetric matrix must be zero ($a_{ii} = -a_{ii} \implies 2a_{ii}=0 \implies a_{ii}=0$).

(D) The sum of a matrix and its transpose, $A+A'$, is always a skew symmetric matrix.

Question 3. Elementary operations on a matrix (row or column operations) are used to transform a matrix into a simpler form (like row echelon form). Which of the following is NOT an elementary row operation?

(A) Interchanging two rows ($R_i \leftrightarrow R_j$).

(B) Multiplying a row by any real number.

(C) Adding a multiple of one row to another row ($R_i \to R_i + kR_j$).

(D) Multiplying a row by a non-zero scalar ($R_i \to kR_i$, $k \neq 0$).

Question 4. An invertible matrix (or non-singular matrix) is a square matrix $A$ for which there exists a matrix $B$ such that $AB = BA = I$, where $I$ is the identity matrix. $B$ is called the inverse of $A$, denoted by $A^{-1}$. Which statement is FALSE?

(A) Not all square matrices are invertible.

(B) If a matrix has an inverse, it is unique.

(C) A non-square matrix can have an inverse.

(D) If $A$ and $B$ are invertible matrices of the same order, then $(AB)^{-1} = B^{-1}A^{-1}$.

Question 5. Which statement about symmetric and skew symmetric matrices is FALSE?

(A) Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix: $A = \frac{1}{2}(A+A') + \frac{1}{2}(A-A')$.

(B) If $A$ is a symmetric matrix, then $kA$ is also symmetric for any scalar $k$.

(C) If $A$ is a skew symmetric matrix, then $kA$ is also skew symmetric for any scalar $k$.

(D) If $A$ is symmetric and $B$ is symmetric (of the same order), then $AB$ is always symmetric.

Question 6. Consider the matrix $A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$. Which statement is FALSE?

(A) A is a symmetric matrix.

(B) The determinant of A is $1 \times 4 - 2 \times 2 = 0$.

(C) A is an invertible matrix.

(D) Elementary row operations can be used to transform A.

Question 7. Which of the following properties of transpose is NOT correct?

(A) $(A')' = A$

(B) $(kA)' = kA'$ (for scalar $k$)

(C) $(A+B)' = A' + B'$

(D) $(AB)' = A'B'$

Question 8. Elementary row operations correspond to multiplying the original matrix by elementary matrices. Which statement is FALSE?

(A) An elementary matrix is obtained by performing a single elementary row operation on an identity matrix.

(B) Every elementary matrix is invertible.

(C) Applying a sequence of elementary row operations to a matrix $A$ is equivalent to pre-multiplying $A$ by a sequence of elementary matrices.

(D) Applying a sequence of elementary row operations to a matrix $A$ is equivalent to post-multiplying $A$ by a sequence of elementary matrices.

Question 9. Consider the matrix $A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Which statement is FALSE?

(A) A is a square matrix.

(B) A is a symmetric matrix.

(C) A is a skew symmetric matrix ($A' = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = -A$).

(D) All diagonal elements are zero.

Question 10. Which statement about finding the inverse of a matrix using elementary operations is FALSE?

(A) We start with the augmented matrix $[A | I]$.

(B) We perform elementary row operations to transform $A$ into $I$.

(C) The same elementary row operations are applied to $I$ simultaneously.

(D) If $A$ is transformed into $I$, the matrix on the right side is the adjoint of $A$.

Question 1. Which statement about determinants is FALSE?

(A) A determinant is a scalar value associated with a square matrix.

(B) The determinant of a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $ad - bc$.

(C) Determinants are defined for any rectangular matrix.

(D) The determinant of the identity matrix $I$ is $1$.

Question 2. Which of the following properties of determinants is FALSE?

(A) If any two rows (or columns) of a matrix are interchanged, the sign of the determinant changes.

(B) If any two rows (or columns) of a matrix are identical, the determinant is zero.

(C) If a matrix has a row or column of zeros, the determinant is zero.

(D) Multiplying a single row (or column) of a matrix by a scalar $k$ multiplies the determinant by $k^2$.

Question 3. The area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is given by $\frac{1}{2} |\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}|$. Which statement is FALSE?

(A) If the points are collinear, the area of the triangle is zero.

(B) The determinant value can be positive or negative, so the absolute value is taken for area.

(C) The formula can be used for vertices in any order.

(D) The formula gives the perimeter of the triangle.

Question 4. The adjoint of a square matrix $A$, denoted by $\text{adj}(A)$, is the transpose of the cofactor matrix. For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. Which statement is FALSE?

(A) $A(\text{adj}(A)) = (\text{adj}(A))A = \det(A)I$.

(B) If $\det(A) \neq 0$, then $A^{-1} = \frac{1}{\det(A)}\text{adj}(A)$.

(C) The adjoint matrix exists only for invertible matrices.

(D) The adjoint of $\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$ is $\begin{pmatrix} 3 & -1 \\ -4 & 2 \end{pmatrix}$.

Question 5. Evaluate the determinant of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}$. Which calculation is NOT correct?

(A) This is an upper triangular matrix. The determinant is the product of the diagonal elements.

(B) Determinant $= 1 \times 4 \times 6 = 24$.

(C) Expanding along the first column: $1(4 \times 6 - 5 \times 0) - 0(...) + 0(...) = 1(24) = 24$.

(D) Expanding along the first row: $1(4 \times 6 - 5 \times 0) - 2(0 \times 6 - 5 \times 0) + 3(0 \times 0 - 4 \times 0) = 1(24) - 2(0) + 3(0) = 24$.

Question 6. Which statement about the determinant of a matrix is FALSE?

(A) $\det(A') = \det(A)$.

(B) $\det(AB) = \det(A)\det(B)$.

(C) $\det(kA) = k \det(A)$ for an $n \times n$ matrix $A$ and scalar $k$.

(D) $\det(A+B) = \det(A) + \det(B)$.

Question 7. If the determinant of a square matrix $A$ is zero, what does it imply about the matrix $A$?

(A) A is a singular matrix.

(B) A is an invertible matrix.

(C) The system of linear equations $AX=0$ has non-trivial solutions.

(D) The columns of A are linearly dependent.

Which statement is FALSE?

Question 8. For a $3 \times 3$ matrix, the cofactor of an element $a_{ij}$, denoted by $C_{ij}$, is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column. Which statement about cofactors and adjoint is FALSE?

(A) The cofactor matrix is the matrix where each element $a_{ij}$ is replaced by its cofactor $C_{ij}$.

(B) The adjoint matrix is the transpose of the cofactor matrix.

(C) Expanding the determinant along any row or column using cofactors gives the same value.

(D) The adjoint of a diagonal matrix is always a zero matrix.

Question 9. What is the determinant of the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$?

(A) $1 \times 4 - 2 \times 3 = 4 - 6 = -2$

(B) The determinant is $-2$.

(C) The determinant is $2$.

(D) $\det \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = -2$.

Which is NOT the correct value?

Question 10. Which statement about the adjoint of a matrix is FALSE?

(A) The adjoint is used in the formula for the inverse of a matrix.

(B) The adjoint of a $2 \times 2$ matrix is easy to calculate.

(C) The adjoint of a matrix is always invertible.

(D) The adjoint exists for any square matrix.

Question 1. Which statement about the inverse of a matrix $A$ is FALSE?

(A) The inverse $A^{-1}$ exists if and only if $A$ is a non-singular matrix ($\det(A) \neq 0$).

(B) If $A$ is invertible, then $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix of the same order as $A$.

(C) The inverse of a non-square matrix exists.

(D) If $A$ and $B$ are invertible matrices of the same order, then $(AB)^{-1} = B^{-1}A^{-1}$.

Question 2. The inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by $A^{-1} = \frac{1}{\det(A)}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $\det(A) \neq 0$. Find the inverse of the matrix $\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$. Which step or result is FALSE?

(A) Determinant $= 3 \times 2 - 1 \times 5 = 6 - 5 = 1$. Since determinant is non-zero, the inverse exists.

(B) Adjoint matrix is $\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}$.

(C) Inverse is $A^{-1} = \frac{1}{1}\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}$.

(D) The inverse matrix is $\begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}$.

Which calculation is FALSE?

Question 3. A system of linear equations $AX=B$ can be solved using the matrix inverse method as $X = A^{-1}B$, provided $A$ is invertible. Which statement is FALSE?

(A) The matrix $A$ must be a square matrix.

(B) The system must have a unique solution for this method to work.

(C) $X$ is the column matrix of variables.

(D) The method can be used even if $\det(A) = 0$.

Question 4. Cramer's Rule is a method for solving a system of linear equations using determinants. For a system $AX=B$ with a square matrix $A$, if $\det(A) \neq 0$, the unique solution is given by $x_i = \frac{\det(A_i)}{\det(A)}$, where $A_i$ is the matrix obtained by replacing the $i$-th column of $A$ with the column matrix $B$. Which statement about Cramer's Rule is FALSE?

(A) It can be used for any system of linear equations.

(B) It is particularly useful for systems with a unique solution where calculating determinants is feasible.

(C) If $\det(A) = 0$, Cramer's Rule cannot be used to find a unique solution.

(D) For a $2 \times 2$ system, $a_{11}x + a_{12}y = b_1$, $a_{21}x + a_{22}y = b_2$, $x = \frac{\det \begin{pmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{pmatrix}}{\det \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}}$ and $y = \frac{\det \begin{pmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{pmatrix}}{\det \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}}$.

Question 5. Which statement about the relationship between the inverse of a matrix and the solution of a system of linear equations is FALSE?

(A) If a system $AX=B$ has a unique solution, the matrix $A$ must be invertible.

(B) If the matrix $A$ is invertible, the system $AX=B$ always has a unique solution $X=A^{-1}B$.

(C) If $\det(A)=0$, the system $AX=B$ has no solution.

(D) The matrix inverse method requires calculating the inverse of the coefficient matrix.

Question 6. Solve the system $x+y=3$ and $2x-y=0$ using the matrix inverse method. The coefficient matrix is $A = \begin{pmatrix} 1 & 1 \\ 2 & -1 \end{pmatrix}$. Find $A^{-1}$. Which step or result is FALSE?

(A) $\det(A) = 1(-1) - 1(2) = -1 - 2 = -3$. $A$ is invertible.

(B) $\text{adj}(A) = \begin{pmatrix} -1 & -1 \\ -2 & 1 \end{pmatrix}$.

(C) $A^{-1} = \frac{1}{-3}\begin{pmatrix} -1 & -1 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 1/3 & 1/3 \\ 2/3 & -1/3 \end{pmatrix}$.

(D) The solution $X = A^{-1}B = \begin{pmatrix} 1/3 & 1/3 \\ 2/3 & -1/3 \end{pmatrix} \begin{pmatrix} 3 \\ 0 \end{pmatrix} = \begin{pmatrix} (1/3)\times 3 + (1/3)\times 0 \\ (2/3)\times 3 + (-1/3)\times 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$. So $x=1, y=2$. The solution is $(1, 2)$.

Which calculation is FALSE?

Question 7. Which statement about the inverse of a matrix is FALSE?

(A) $(A^{-1})^{-1} = A$.

(B) $(A')^{-1} = (A^{-1})'$.

(C) $(kA)^{-1} = kA^{-1}$ for a non-zero scalar $k$ and invertible matrix $A$.

(D) $I^{-1} = I$.

Question 8. Which system of linear equations CANNOT be solved using Cramer's Rule?

(A) A system where the number of equations equals the number of variables.

(B) A system where the determinant of the coefficient matrix is non-zero.

(C) A system with more equations than variables.

(D) A system that has a unique solution.

Question 9. Consider the system $2x+y=5$, $4x+2y=10$. Which statement is FALSE?

(A) The determinant of the coefficient matrix $\begin{pmatrix} 2 & 1 \\ 4 & 2 \end{pmatrix}$ is $2 \times 2 - 1 \times 4 = 4 - 4 = 0$.

(B) The coefficient matrix is singular.

(C) The system has a unique solution.

(D) The lines are identical or parallel and distinct. In this case, they are identical ($4x+2y=10$ is $2(2x+y)=2(5)$ which is $2x+y=5$).

Which statement is FALSE?

Question 10. To solve the system $AX=B$ using the matrix inverse method, you need to find the inverse $A^{-1}$. Which method is NOT used to find the inverse of a matrix?

(A) Using the formula $A^{-1} = \frac{1}{\det(A)}\text{adj}(A)$.

(B) Using elementary row operations on the augmented matrix $[A | I]$.

(C) Using Cramer's Rule on the matrix $A$ itself.

(D) Solving the equation $AX=I$ for $X$ using other methods.

Question 1. A word problem translates a real-world scenario into mathematical equations or inequalities. Which of the following is NOT a typical step in solving a word problem using algebraic equations?

(A) Read the problem carefully and identify the unknown quantities.

(B) Assign variables to represent the unknown quantities.

(C) Translate the relationships given in the problem into algebraic equations.

(D) Guess the answer and check if it fits the problem description.

Question 2. The sum of two numbers is $30$, and their difference is $10$. Let the two numbers be $x$ and $y$. The system of equations is $x+y=30$ and $x-y=10$. Which statement is FALSE?

(A) Adding the two equations gives $2x = 40$, so $x=20$.

(B) Substituting $x=20$ into the first equation gives $20+y=30$, so $y=10$.

(C) The two numbers are $20$ and $10$.

(D) The product of the two numbers is $20+10=30$.

Question 3. A rectangular field has a perimeter of $40$ metres. Its length is $4$ metres more than its width. Find the dimensions of the field. Let the width be $w$ and the length be $l$. The equations are $2(l+w) = 40$ and $l = w+4$. Which statement is FALSE?

(A) Substituting the second equation into the first gives $2(w+4+w) = 40$.

(B) Simplifying the equation gives $2(2w+4) = 40 \implies 4w + 8 = 40 \implies 4w = 32 \implies w = 8$.

(C) The width is $8$ metres, and the length is $8+4=12$ metres.

(D) The area of the field is $2(12+8) = 40$ square metres.

Question 4. The product of two consecutive positive integers is $156$. Let the integers be $n$ and $n+1$. The equation is $n(n+1) = 156$, which is $n^2 + n - 156 = 0$. Which statement is FALSE?

(A) We need to factorise $n^2 + n - 156 = 0$. Find two numbers that multiply to $-156$ and add to $1$. The numbers are $13$ and $-12$.

(B) The equation can be written as $(n+13)(n-12) = 0$.

(C) The solutions are $n=-13$ or $n=12$. Since the integers are positive, $n=12$.

(D) The two consecutive positive integers are $-13$ and $-12$.

Question 5. A sum of $\textsf{₹} 5000$ is to be divided between two people such that one person gets $\textsf{₹} 1000$ more than the other. Let the amounts be $x$ and $y$. The equations are $x+y=5000$ and $x = y+1000$. Which statement is FALSE?

(A) Substituting the second equation into the first gives $(y+1000) + y = 5000$.

(B) Simplifying gives $2y + 1000 = 5000 \implies 2y = 4000 \implies y = 2000$.

(C) One person gets $\textsf{₹} 2000$ and the other gets $\textsf{₹} 2000 + 1000 = \textsf{₹} 3000$.

(D) The amounts are $\textsf{₹} 2000$ and $\textsf{₹} 4000$.

Question 6. A boat travels upstream at $10$ km/h and downstream at $14$ km/h. Find the speed of the boat in still water and the speed of the stream. Let the speed of the boat be $u$ and the speed of the stream be $v$. Upstream speed is $u-v$, downstream speed is $u+v$. The equations are $u-v=10$ and $u+v=14$. Which statement is FALSE?

(A) Adding the two equations gives $(u-v) + (u+v) = 10 + 14 \implies 2u = 24 \implies u = 12$.

(B) Substituting $u=12$ into $u+v=14$ gives $12+v=14 \implies v=2$.

(C) The speed of the boat in still water is $12$ km/h, and the speed of the stream is $2$ km/h.

(D) The speed of the boat is $10$ km/h and the speed of the stream is $4$ km/h.

Question 7. Two pipes, A and B, can fill a tank in $20$ minutes and $30$ minutes respectively. If both pipes are opened together, how long will it take to fill the tank? Let the time taken together be $t$. The rates are $1/20$ and $1/30$ tanks per minute. The combined rate is $1/20 + 1/30$. The equation is $t \times (\text{combined rate}) = 1$. Which statement is FALSE?

(A) Combined rate = $\frac{1}{20} + \frac{1}{30} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12}$ tanks per minute.

(B) The equation is $t \times \frac{1}{12} = 1$.

(C) Solving for $t$: $t = 12$ minutes.

(D) The time taken together is less than the time taken by the faster pipe.

Question 8. A sum of $\textsf{₹} 10000$ is invested in two parts, one at $8\%$ annual interest and the other at $10\%$ annual interest. The total annual interest earned is $\textsf{₹} 920$. Let the amount invested at $8\%$ be $x$ and the amount invested at $10\%$ be $y$. The equations are $x+y=10000$ and $0.08x + 0.10y = 920$. Which statement is FALSE?

(A) From the first equation, $y = 10000 - x$. Substitute this into the second equation: $0.08x + 0.10(10000-x) = 920$.

(B) Simplify: $0.08x + 1000 - 0.10x = 920 \implies -0.02x = 920 - 1000 = -80$.

(C) Solve for $x$: $x = -80 / (-0.02) = 4000$.

(D) The amount invested at $8\%$ is $\textsf{₹} 6000$ and at $10\%$ is $\textsf{₹} 4000$.

Question 9. The difference between a two-digit number and the number obtained by reversing its digits is $18$. The sum of the digits is $8$. Let the tens digit be $t$ and the units digit be $u$. The number is $10t + u$. The reversed number is $10u + t$. The equations are $(10t + u) - (10u + t) = 18$ and $t+u = 8$. Which statement is FALSE?

(A) The first equation simplifies to $9t - 9u = 18 \implies t - u = 2$.

(B) We have a system: $t+u = 8$ and $t-u = 2$.

(C) Adding the two equations gives $(t+u) + (t-u) = 8 + 2 \implies 2t = 10 \implies t = 5$.

(D) Substituting $t=5$ into $t+u=8$ gives $5+u=3 \implies u=3$. The number is $53$. The reversed number is $35$. The difference is $53-35 = 18$. The sum of digits is $5+3=8$. The calculation for $u$ is $5+u=8 \implies u=3$. The resulting number is $10(5) + 3 = 53$. Option D states the number is $35$ which is the reversed number, and the calculation $5+u=8 \implies u=3$ is correct, leading to the number $53$. The statement in D that the number is 35 is FALSE.

Question 10. Which type of real-world problem is NOT typically solved using algebraic equations?

(A) Calculating unknown values in geometric shapes (e.g., side lengths, angles).

(B) Determining mixtures and concentrations.

(C) Analyzing speed, distance, and time problems.

(D) Predicting weather patterns with high precision over long periods.