Completing Statements MCQs for Sub-Topics of Topic 2: Algebra

Question 1. In the expression $7y - 15$, the letter $y$ is referred to as a____

(A) constant.

(B) coefficient.

(C) term.

(D) variable.

Question 2. A fixed numerical value in an algebraic expression is called a____

(A) variable.

(B) coefficient.

(C) constant.

(D) term.

Question 3. A product of factors, which can be numerical or algebraic, is called a____

(A) sum.

(B) difference.

(C) term.

(D) equation.

Question 4. In the term $12m$, the numerical factor 12 is known as the____

(A) variable.

(B) constant.

(C) coefficient.

(D) exponent.

Question 5. A combination of constants and variables, connected by fundamental mathematical operations, is called an____

(A) equation.

(B) identity.

(C) algebraic expression.

(D) inequality.

Question 6. To find the value of an algebraic expression, we substitute the given numerical values for the____

(A) constants.

(B) coefficients.

(C) terms.

(D) variables.

Question 7. The expression $\textsf{₹} 10p + \textsf{₹} 20n$ can represent the total cost of buying $p$ pens at $\textsf{₹} 10$ each and $n$ notebooks at $\textsf{₹} 20$ each, illustrating the use of variables to represent____

(A) fixed values.

(B) unknown or changing quantities.

(C) mathematical operations.

(D) types of items.

Question 8. In the expression $5x^2 - 3x + 9$, the terms are separated by____

(A) multiplication signs.

(B) equality signs.

(C) addition or subtraction signs.

(D) coefficients.

Question 9. The expression for the area of a square with side $s$ is $s^2$. If $s = 6$ cm, the value of the expression is $6^2 = 36$ cm$^2$. This demonstrates evaluating an expression for a specific____

(A) variable.

(B) coefficient.

(C) context.

(D) term.

Question 10. Variables are widely used in algebra to formulate general rules, laws, and formulas, providing a way to represent relationships between quantities in a concise and____

(A) limited manner.

(B) specific instance.

(C) symbolic manner.

(D) graphical manner.

Question 1. To add or subtract algebraic expressions, we combine terms that have the same variables raised to the same powers. These are called____

(A) like terms.

(B) unlike terms.

(C) constant terms.

(D) coefficients.

Question 2. The sum of the expressions $5a + 3b$ and $2a - b$ is obtained by adding the coefficients of the like terms, which gives____

(A) $7a + 4b$.

(B) $7a + 2b$.

(C) $3a + 2b$.

(D) $7a - 2b$.

Question 3. When subtracting one algebraic expression from another, like $(8x + 5) - (3x - 2)$, we change the sign of each term in the second expression and then____

(A) multiply the resulting expressions.

(B) add the resulting expressions.

(C) subtract the resulting expressions.

(D) divide the resulting expressions.

Question 4. The product of two monomials like $4xy$ and $3x^2$ is found by multiplying the coefficients and adding the exponents of the same variables, resulting in____

(A) $12x^2y$.

(B) $12x^3y$.

(C) $7x^3y$.

(D) $12x^3y^2$.

Question 5. To multiply a polynomial by a monomial, we use the distributive property to multiply the monomial by each term of the polynomial, for example, $2x(x^2 + 3) = 2x(x^2) + 2x(3)$, which simplifies to____

(A) $2x^3 + 6$.

(B) $2x^3 + 6x$.

(C) $3x^3 + 6x$.

(D) $2x^3 + 3$.

Question 6. When multiplying two binomials like $(x+a)$ and $(x+b)$, we multiply each term in the first binomial by each term in the second binomial and then combine like terms, resulting in $x^2 + (a+b)x + ab$. This method is often remembered by the acronym____

(A) PEMDAS.

(B) BODMAS.

(C) FOIL.

(D) LCM.

Question 7. To divide a polynomial by a monomial, we divide each term of the polynomial by the monomial separately. For example, $\frac{6a^3 + 9a^2}{3a}$ (for $a \neq 0$) is equal to $\frac{6a^3}{3a} + \frac{9a^2}{3a}$, which simplifies to____

(A) $2a^2 + 3a$.

(B) $2a^2 + 3$.

(C) $2a + 3a$.

(D) $2a^2 + 3a^2$.

Question 8. The sum of three expressions $x+y$, $y+z$, and $z+x$ is found by adding all terms and combining like terms, which results in____

(A) $x+y+z$.

(B) $2x+2y+2z$.

(C) $xy+yz+zx$.

(D) $x^2+y^2+z^2$.

Question 9. Subtracting $4p^2 - 3q^2$ from $7p^2 + 2q^2$ means calculating $(7p^2 + 2q^2) - (4p^2 - 3q^2)$, which simplifies to____

(A) $3p^2 - q^2$.

(B) $3p^2 + 5q^2$.

(C) $11p^2 - q^2$.

(D) $11p^2 + 5q^2$.

Question 10. When performing operations on algebraic expressions, it is crucial to follow the order of operations (like BODMAS/PEMDAS) and combine only____

(A) constant terms.

(B) variable terms.

(C) like terms.

(D) coefficients.

Question 1. An algebraic expression in which the variables have only non-negative integer powers is called a____

(A) rational expression.

(B) polynomial.

(C) linear equation.

(D) quadratic equation.

Question 2. The highest power of the variable in a polynomial is called its____

(A) coefficient.

(B) term.

(C) degree.

(D) zero.

Question 3. A polynomial with degree 1 is called a____

(A) constant polynomial.

(B) linear polynomial.

(C) quadratic polynomial.

(D) cubic polynomial.

Question 4. A polynomial with exactly three terms is called a____

(A) monomial.

(B) binomial.

(C) trinomial.

(D) multinomial.

Question 5. For a polynomial $P(x)$, if $P(a) = 0$, then 'a' is called a____ of the polynomial.

(A) term.

(B) coefficient.

(C) factor.

(D) zero.

Question 6. The graph of a quadratic polynomial $y = ax^2 + bx + c$ is a parabola, and its real zeroes are the points where the graph intersects the____

(A) y-axis.

(B) x-axis.

(C) origin.

(D) vertex.

Question 7. For a quadratic polynomial $ax^2 + bx + c$, the sum of its zeroes is given by the formula____

(A) $c/a$.

(B) $-b/a$.

(C) $b/a$.

(D) $-c/a$.

Question 8. A polynomial with degree 3 is called a____

(A) linear polynomial.

(B) quadratic polynomial.

(C) cubic polynomial.

(D) quartic polynomial.

Question 9. The leading coefficient of a polynomial is the coefficient of the term with the____

(A) lowest degree.

(B) highest degree.

(C) constant term.

(D) zero coefficient.

Question 10. Evaluating a polynomial $P(x)$ at a specific value, say $x=k$, means substituting $k$ for $x$ and simplifying the expression to find the____ of the polynomial at $x=k$.

(A) zero.

(B) degree.

(C) value.

(D) coefficient.

Question 1. The Division Algorithm for polynomials states that when a polynomial $P(x)$ is divided by a non-zero polynomial $D(x)$, 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 the degree of $R(x)$ is strictly less than the degree of____

(A) $P(x)$.

(B) $Q(x)$.

(C) $D(x)$.

(D) $R(x)$.

Question 2. According to the Remainder Theorem, if a polynomial $P(x)$ is divided by a linear polynomial $x-a$, the remainder is given by____

(A) $Q(a)$.

(B) $P(a)$.

(C) $P(-a)$.

(D) 0.

Question 3. The Factor Theorem is a special case of the Remainder Theorem; it states that $x-a$ is a factor of the polynomial $P(x)$ if and only if $P(a)$ is equal to____

(A) a constant value.

(B) the quotient.

(C) the remainder.

(D) zero.

Question 4. When a polynomial $P(x)$ is divided by a linear polynomial $ax+b$, the remainder can be found by evaluating $P(x)$ at $x = -b/a$, which is a direct application of the____

(A) Factor Theorem.

(B) Division Algorithm.

(C) Remainder Theorem.

(D) Identity Theorem.

Question 5. If the remainder when $P(x)$ is divided by $x-5$ is 0, then according to the Factor Theorem, $x-5$ is a____ of $P(x)$.

(A) coefficient.

(B) multiple.

(C) factor.

(D) root.

Question 6. Polynomial long division is a method used to divide a polynomial by another polynomial of equal or lower degree, yielding a quotient and a____

(A) factor.

(B) zero.

(C) constant.

(D) remainder.

Question 7. If the remainder is 0 when $P(x)$ is divided by $D(x)$, it means $D(x)$ is a____ of $P(x)$.

(A) multiple.

(B) factor.

(C) quotient.

(D) remainder.

Question 8. Synthetic division is a streamlined method for dividing polynomials by linear binomials of the form $x-a$, making the division process faster than traditional long division, but it is limited to this specific form of____

(A) polynomial.

(B) quotient.

(C) dividend.

(D) divisor.

Question 9. The Division Algorithm guarantees that the remainder in polynomial division is either the zero polynomial or a polynomial whose degree is strictly less than the degree of the____

(A) dividend.

(B) divisor.

(C) quotient.

(D) remainder itself.

Question 10. If we want to check if a specific value $a$ is a zero of a polynomial $P(x)$, we can use the Remainder Theorem by calculating $P(a)$; if $P(a)$ equals zero, then $a$ is a zero and $x-a$ is a____

(A) multiple.

(B) term.

(C) zero.

(D) factor.

Question 1. An equation that is true for every value of the variable(s) is called an algebraic____

(A) equation.

(B) inequality.

(C) expression.

(D) identity.

Question 2. The identity $(a+b)^2 = a^2 + 2ab + b^2$ is used to find the square of a binomial sum, meaning it provides a formula for $(a+b) \times$____?

(A) $(a-b)$.

(B) $(a+b)$.

(C) $a^2 + b^2$.

(D) $2ab$.

Question 3. The identity $(x-y)^2 = x^2 - 2xy + y^2$ is used to expand the square of a binomial difference, demonstrating that the square of $(x-y)$ is equal to $x^2$ minus twice the product of $x$ and $y$, plus the square of____

(A) $x$.

(B) $-y$.

(C) $y$.

(D) $x-y$.

Question 4. The identity $a^2 - b^2 = (a-b)(a+b)$ is used to factorise the difference of two squares, showing that the difference of squares of $a$ and $b$ is equal to the product of their difference and their____

(A) difference.

(B) sum.

(C) product.

(D) quotient.

Question 5. The identity $(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$ is used to expand the cube of a binomial sum, providing a formula for the expansion of $(x+y) \times$____?

(A) $(x+y)$.

(B) $(x+y)^2$.

(C) $(x-y)$.

(D) $x^3 + y^3$.

Question 6. Algebraic identities provide shortcuts for expanding and factorising algebraic expressions, simplifying calculations and operations compared to direct multiplication or factoring by____

(A) grouping.

(B) inspection.

(C) trial and error.

(D) formula.

Question 7. The identity $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ is used to factorise the sum of two____

(A) squares.

(B) cubes.

(C) binomials.

(D) trinomials.

Question 8. Evaluating $52 \times 48$ can be simplified using the identity $(a+b)(a-b) = a^2 - b^2$ by writing $52$ as $50+2$ and $48$ as $50-2$, giving the result as $50^2 - 2^2 = 2500 - 4 = 2496$. This illustrates using identities for simplifying numerical____

(A) problems.

(B) operations.

(C) expressions.

(D) variables.

Question 9. The identity $(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca$ is an extension of the binomial square identity to a trinomial sum, indicating the square of the sum of three terms includes the sum of their squares plus twice the sum of the products of the terms taken____ at a time.

(A) one.

(B) two.

(C) three.

(D) any number.

Question 10. Using the identity $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ can help factorise expressions like $x^3 - 27$, where $a=x$ and $b=3$, resulting in $(x-3)(x^2 + 3x + 9)$. This shows how identities are used in____ of algebraic expressions.

(A) expansion.

(B) simplification.

(C) evaluation.

(D) factorisation.

Question 1. Factorisation is the process of expressing an algebraic expression as a product of two or more expressions, called its____

(A) terms.

(B) coefficients.

(C) factors.

(D) variables.

Question 2. One method of factorisation is taking out the common factors present in all the terms of the expression. For example, $5xy + 10x$ can be factorised as $5x(y+2)$, where $5x$ is the greatest____ factor.

(A) variable.

(B) numerical.

(C) algebraic.

(D) common.

Question 3. Factorising by grouping involves arranging terms in groups such that each group has a common factor, and then taking out the common factor from each group to reveal a common____

(A) monomial.

(B) binomial.

(C) trinomial.

(D) polynomial.

Question 4. Factorisation can be done using algebraic identities, such as factorising the difference of two squares $p^2 - q^2$ as $(p-q)(p+q)$, which is an application of the identity for the difference of two____

(A) terms.

(B) squares.

(C) cubes.

(D) products.

Question 5. To factorise a quadratic trinomial of the form $ax^2 + bx + c$, one method is to find two numbers whose product is $ac$ and whose sum is $b$, and then rewrite the middle term $bx$ using these numbers to factor by____

(A) identities.

(B) common factors.

(C) grouping.

(D) long division.

Question 6. Factorising a cubic polynomial, like $x^3 - 6x^2 + 11x - 6$, often starts by finding one factor using the Factor Theorem (checking if $P(a)=0$ for values of $a$ that divide the constant term), and then using polynomial division to find the remaining____

(A) terms.

(B) coefficients.

(C) zeroes.

(D) quadratic factor.

Question 7. The expression $y^2 - 10y + 25$ is a perfect square trinomial because it can be factorised as $(y-5)^2$, which is an application of the identity for the square of a binomial____

(A) sum.

(B) difference.

(C) product.

(D) quotient.

Question 8. Factorising by taking out common factors simplifies expressions and helps in solving equations or simplifying fractions involving algebraic expressions by revealing the common multiplicative____

(A) terms.

(B) sums.

(C) differences.

(D) components.

Question 9. The expression $a^3 + b^3$ can be factorised using the sum of cubes identity as $(a+b)(a^2 - ab + b^2)$, which provides a specific pattern for factorising binomials that are sums of two quantities raised to the power of____

(A) 2.

(B) 3.

(C) 4.

(D) any integer.

Question 10. Factorisation helps in solving quadratic equations by setting each linear factor equal to zero, since the product of factors is zero if and only if at least one of the factors is____

(A) one.

(B) negative.

(C) positive.

(D) zero.

Question 1. An equation is a statement that two mathematical expressions are equal, indicated by the use of an____

(A) inequality sign.

(B) equality sign.

(C) algebraic expression.

(D) operation sign.

Question 2. A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are real numbers and $a$ is not equal to____

(A) 0.

(B) 1.

(C) $-1$.

(D) $b$.

Question 3. The value of the variable that satisfies a linear equation in one variable, making the left side equal to the right side, is called the____ or root of the equation.

(A) coefficient.

(B) term.

(C) solution.

(D) constant.

Question 4. The rule of transposition is a shortcut for performing the same operation on both sides of an equation; it allows moving a term from one side to the other by changing its____

(A) coefficient.

(B) variable.

(C) sign.

(D) position.

Question 5. Solving a linear equation in one variable involves isolating the variable on one side of the equation by applying inverse operations, such as using subtraction to undo addition or division to undo____

(A) subtraction.

(B) multiplication.

(C) addition.

(D) division.

Question 6. Word problems based on linear equations in one variable require translating the given information into a mathematical equation, solving the equation, and then checking the solution in the context of the original____

(A) variable.

(B) equation.

(C) problem.

(D) method.

Question 7. If an equation simplifies to $0 \times x = c$, where $c$ is a non-zero constant, the equation has____

(A) a unique solution.

(B) no solution.

(C) infinitely many solutions.

(D) two solutions.

Question 8. The cost price of an item plus the profit is equal to the selling price. If the selling price is $\textsf{₹} 500$ and the profit is $\textsf{₹} 100$, letting the cost price be $c$, the equation is $c + 100 = 500$. Solving this gives $c = 400$, which represents the____ in $\textsf{₹}$.

(A) selling price.

(B) profit.

(C) cost price.

(D) variable.

Question 9. If multiplying both sides of a linear equation by the same non-zero number results in an equivalent equation, it demonstrates the multiplication property of____

(A) inequality.

(B) expressions.

(C) equality.

(D) variables.

Question 10. A linear equation in one variable has at most____ solution.

(A) zero.

(B) one.

(C) two.

(D) infinitely many.

Question 1. A linear equation in two variables is an equation that can be written in the form $ax + by + c = 0$, where $a, b, c$ are real numbers and $a$ and $b$ are not both equal to____

(A) 1.

(B) -1.

(C) 0.

(D) c.

Question 2. The graph of a linear equation in two variables is always a____

(A) curve.

(B) point.

(C) straight line.

(D) parabola.

Question 3. Every point that lies on the graph of a linear equation in two variables is a____ to the equation.

(A) variable.

(B) coefficient.

(C) term.

(D) solution.

Question 4. A linear equation in two variables typically has____ solutions.

(A) a unique.

(B) no.

(C) exactly two.

(D) infinitely many.

Question 5. The equation of a line parallel to the x-axis has the form $y = k$, where $k$ is a constant, indicating that the y-coordinate of all points on the line is____

(A) zero.

(B) varying.

(C) constant.

(D) twice the x-coordinate.

Question 6. The equation of a line parallel to the y-axis has the form $x = k$, where $k$ is a constant, indicating that the x-coordinate of all points on the line is____

(A) zero.

(B) varying.

(C) constant.

(D) half the y-coordinate.

Question 7. The point where the graph of a linear equation in two variables intersects the x-axis is found by setting the____ coordinate to zero and solving for the other coordinate.

(A) x.

(B) y.

(C) constant.

(D) coefficient.

Question 8. To graph a linear equation in two variables, it is sufficient to find at least____ solutions and draw the straight line passing through them.

(A) one.

(B) two.

(C) three.

(D) four.

Question 9. The equation $y = 0$ represents the graph of the____

(A) y-axis.

(B) x-axis.

(C) origin.

(D) line parallel to x-axis.

Question 10. The equation $x = 0$ represents the graph of the____

(A) y-axis.

(B) x-axis.

(C) origin.

(D) line parallel to y-axis.

Question 1. A set of two or more linear equations involving the same variables is called a system of linear equations, and the solution to the system is the pair of values that satisfies____ equations simultaneously.

(A) the first.

(B) the second.

(C) at least one.

(D) all.

Question 2. Graphically, the solution to a pair of linear equations in two variables corresponds to the point(s) where the graphs of the two equations____

(A) are parallel.

(B) coincide.

(C) intersect.

(D) are perpendicular.

Question 3. A pair of linear equations is called consistent if it has at least one solution, meaning the graphs of the equations either intersect at a unique point or are____

(A) parallel and distinct.

(B) intersecting at multiple points.

(C) coincident lines.

(D) perpendicular.

Question 4. An inconsistent pair of linear equations has no solution, which means their graphs are parallel and____

(A) intersecting.

(B) coincident.

(C) distinct.

(D) perpendicular.

Question 5. The graphical method of solving a pair of linear equations involves plotting the graph of each equation on the same coordinate plane and finding the coordinates of the point of____

(A) origin.

(B) intersection.

(C) midpoint.

(D) endpoint.

Question 6. The substitution method for solving a system of linear equations involves solving one equation for one variable and substituting that expression into the____

(A) first equation.

(B) original equation.

(C) other equation.

(D) solution.

Question 7. The elimination method for solving a system of linear equations involves multiplying the equations by suitable numbers such that the coefficients of one variable become equal or opposite, and then adding or subtracting the equations to____ that variable.

(A) substitute.

(B) isolate.

(C) eliminate.

(D) combine.

Question 8. The cross-multiplication method is a formula-based approach for solving a system of two linear equations in two variables, derived from the elimination method, providing a direct way to find the values of the____

(A) coefficients.

(B) constants.

(C) variables.

(D) determinants.

Question 9. Some pairs of equations that are not linear can be reduced to linear form by making suitable substitutions, like substituting $u = \frac{1}{x}$ and $v = \frac{1}{y}$ for equations involving terms like $\frac{1}{x}$ and $\frac{1}{y}$. This transforms the original equations into a system of linear equations in terms of the new____

(A) variables.

(B) constants.

(C) terms.

(D) coefficients.

Question 10. A pair of linear equations has a unique solution if the ratio of coefficients of $x$ is not equal to the ratio of coefficients of $y$, i.e., $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$. Graphically, this corresponds to the lines being____

(A) parallel.

(B) coincident.

(C) intersecting.

(D) perpendicular.

Question 1. A quadratic equation in one variable is an equation that can be written in the standard form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a$ is not equal to____

(A) 0.

(B) 1.

(C) -1.

(D) c.

Question 2. The highest power of the variable in a quadratic equation is____

(A) 0.

(B) 1.

(C) 2.

(D) 3.

Question 3. The solutions to a quadratic equation are also called its____ or roots.

(A) coefficients.

(B) terms.

(C) zeroes.

(D) exponents.

Question 4. For a quadratic equation $ax^2 + bx + c = 0$, the sum of its roots $\alpha$ and $\beta$ is given by $\alpha + \beta = -b/a$, and the product is given by $\alpha \beta = c/a$. This establishes the relationship between the roots and the____

(A) variables.

(B) constants.

(C) coefficients.

(D) discriminant.

Question 5. One method of solving a quadratic equation is by factorisation, which involves expressing the quadratic polynomial as a product of two linear factors and setting each factor equal to____

(A) one.

(B) negative one.

(C) zero.

(D) the other factor.

Question 6. Completing the square is a method to solve a quadratic equation by transforming it into the form $(x+k)^2 = m$, where the left side is a perfect square trinomial, allowing us to take the square root of both____

(A) coefficients.

(B) sides.

(C) terms.

(D) variables.

Question 7. The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, provides a direct method to find the roots of any quadratic equation $ax^2 + bx + c = 0$, provided $a$ is not____

(A) 1.

(B) -1.

(C) 0.

(D) greater than $b$.

Question 8. The expression $b^2 - 4ac$ within the quadratic formula is called the discriminant ($\Delta$), and it determines the nature of the____ of the quadratic equation.

(A) coefficients.

(B) terms.

(C) variables.

(D) roots.

Question 9. If the discriminant ($\Delta$) of a quadratic equation is positive ($\Delta > 0$), the equation has two distinct____ roots.

(A) real.

(B) complex.

(C) equal.

(D) irrational.

Question 10. If the discriminant ($\Delta$) of a quadratic equation is zero ($\Delta = 0$), the equation has two equal____ roots.

(A) real.

(B) complex.

(C) distinct.

(D) irrational.

Question 1. A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $i^2$ equals____

(A) 1.

(B) -1.

(C) $i$.

(D) $-i$.

Question 2. In the complex number $a+bi$, the real number $a$ is called the real part, and the real number $b$ is called the____ part.

(A) complex.

(B) imaginary.

(C) pure.

(D) conjugate.

Question 3. To add two complex numbers $(a+bi)$ and $(c+di)$, we add their real parts and imaginary parts separately: $(a+bi) + (c+di) = (a+c) + (b+d)i$. This is similar to adding binomials by combining____

(A) constants.

(B) variables.

(C) like terms.

(D) coefficients.

Question 4. To multiply two complex numbers $(a+bi)$ and $(c+di)$, we use the distributive property and the fact that $i^2 = -1$, resulting in $(ac - bd) + (ad + bc)i$. This operation is part of the____ of complex numbers.

(A) addition.

(B) subtraction.

(C) algebra.

(D) representation.

Question 5. The powers of the imaginary unit $i$ follow a cycle of four values: $i^1=i$, $i^2=-1$, $i^3=-i$, and $i^4=1$. To simplify $i^n$ for any integer $n$, we divide $n$ by 4 and use the remainder to determine the equivalent power of____

(A) $-1$.

(B) $i$.

(C) 1.

(D) $-i$.

Question 6. The identity $(Z_1 + Z_2)^2 = Z_1^2 + 2Z_1Z_2 + Z_2^2$ holds for complex numbers, similar to real numbers, demonstrating that algebraic identities involving sums and differences raised to powers also apply in the domain of____ numbers.

(A) real.

(B) rational.

(C) irrational.

(D) complex.

Question 7. To divide complex numbers, we typically multiply the numerator and denominator by the conjugate of the denominator to make the denominator a real number, facilitating the separation of the real and imaginary parts of the____?

(A) dividend.

(B) divisor.

(C) quotient.

(D) remainder.

Question 8. A complex number of the form $0+bi$, where $b \neq 0$, is called a purely____ number.

(A) real.

(B) complex.

(C) imaginary.

(D) conjugate.

Question 9. Every real number is also a complex number with its imaginary part equal to____

(A) 1.

(B) -1.

(C) $i$.

(D) 0.

Question 10. The set of complex numbers is an extension of the set of real numbers, introduced to provide solutions to polynomial equations that have no solutions in the set of____ numbers.

(A) natural.

(B) integer.

(C) rational.

(D) real.

Question 1. The geometric representation of complex numbers is done on a coordinate plane called the Argand plane or complex plane, where the horizontal axis represents the real part and the vertical axis represents the____ part.

(A) principal.

(B) argument.

(C) modulus.

(D) imaginary.

Question 2. The modulus of a complex number $Z = a+bi$, denoted as $|Z|$, represents the distance of the point $(a, b)$ from the origin on the Argand plane, and is calculated as $\sqrt{a^2 + b^2}$, which is always a non-negative____ number.

(A) complex.

(B) imaginary.

(C) real.

(D) integer.

Question 3. The conjugate of a complex number $Z = a+bi$ is denoted by $\bar{Z}$ or $Z^*$ and is defined as $a-bi$. Geometrically, the conjugate of a complex number is its reflection across the____ axis on the Argand plane.

(A) imaginary.

(B) real.

(C) vertical.

(D) origin.

Question 4. The polar representation of a complex number $Z = a+bi$ is given by $Z = r(\cos\theta + i\sin\theta)$, where $r = |Z|$ is the modulus and $\theta$ is the argument or amplitude of $Z$, representing the angle made by the line segment from the origin to $(a, b)$ with the positive____ axis.

(A) imaginary.

(B) vertical.

(C) real.

(D) x-axis.

Question 5. Finding the square root of a complex number $Z$ involves finding complex numbers $W$ such that $W^2 = Z$. Every non-zero complex number has exactly two square roots, which are negatives of each other, represented by points equidistant from the origin and symmetric with respect to the____

(A) real axis.

(B) imaginary axis.

(C) origin.

(D) unit circle.

Question 6. One important property of the modulus is the Triangle Inequality: $|Z_1 + Z_2| \leq |Z_1| + |Z_2|$, which states that the modulus of the sum of two complex numbers is less than or equal to the sum of their____

(A) arguments.

(B) conjugates.

(C) real parts.

(D) moduli.

Question 7. The product of a complex number and its conjugate, $Z \bar{Z}$, is always a non-negative real number equal to the square of the____ of $Z$.

(A) argument.

(B) real part.

(C) imaginary part.

(D) modulus.

Question 8. The argument of a complex number $Z$ is usually taken in the interval $(-\pi, \pi]$ or $[0, 2\pi)$, and it is not uniquely defined for the complex number____

(A) 1.

(B) $i$.

(C) 0.

(D) $-1$.

Question 9. Properties of conjugates include $\overline{Z_1 + Z_2} = \bar{Z_1} + \bar{Z_2}$ and $\overline{Z_1 Z_2} = \bar{Z_1} \bar{Z_2}$, meaning the conjugate of a sum or product is the sum or product of the____

(A) moduli.

(B) arguments.

(C) conjugates.

(D) real parts.

Question 10. Complex numbers can be represented using ordered pairs $(a,b)$ or vectors on the Argand plane, providing a visual way to understand operations like addition (vector addition) and multiplication (scaling and rotation), linking algebra with____

(A) arithmetic.

(B) trigonometry.

(C) geometry.

(D) calculus.

Question 1. A quadratic equation with real coefficients may have complex roots if its discriminant ($\Delta = b^2 - 4ac$) is____

(A) positive.

(B) zero.

(C) negative.

(D) undefined.

Question 2. If a quadratic equation $ax^2 + bx + c = 0$ (with real coefficients) has complex roots, these roots always appear as a pair of complex____

(A) numbers.

(B) conjugates.

(C) integers.

(D) real numbers.

Question 3. When the discriminant is negative, $\sqrt{b^2 - 4ac}$ becomes $\sqrt{|\Delta|} i$, leading to complex roots in the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The imaginary unit $i$ arises because we are taking the square root of a____ number.

(A) positive.

(B) zero.

(C) negative.

(D) real.

Question 4. If one root of a quadratic equation with real coefficients is $p+qi$, where $q \neq 0$, then the other root must be $p-qi$, its complex____, ensuring the coefficients $b$ and $c$ in $ax^2+bx+c=0$ are real.

(A) number.

(B) part.

(C) conjugate.

(D) reciprocal.

Question 5. The sum of the complex conjugate roots $p+qi$ and $p-qi$ is $(p+qi) + (p-qi) = 2p$, which is a____ number.

(A) complex.

(B) imaginary.

(C) real.

(D) pure imaginary.

Question 6. The product of the complex conjugate roots $p+qi$ and $p-qi$ is $(p+qi)(p-qi) = p^2 - (qi)^2 = p^2 - q^2i^2 = p^2 + q^2$, which is a____ number.

(A) complex.

(B) imaginary.

(C) real.

(D) negative real.

Question 7. If a quadratic equation $ax^2 + bx + c = 0$ has complex roots, then its graph does not intersect the____ axis.

(A) real.

(B) imaginary.

(C) x.

(D) y.

Question 8. When solving a quadratic equation like $x^2+1=0$, the solutions are $x = \pm \sqrt{-1} = \pm i$, which are purely imaginary complex roots because the real part of the discriminant is negative and the linear term coefficient $b$ is____

(A) positive.

(B) negative.

(C) zero.

(D) non-zero.

Question 9. The appearance of complex roots in quadratic equations is fundamental in various fields like electrical engineering, physics, and control theory, where they represent oscillating behaviours or exponential growth/decay combined with____

(A) real values.

(B) stability.

(C) periodicity.

(D) equilibrium.

Question 10. A quadratic equation with complex coefficients may have complex roots that are not conjugates, but if the coefficients are restricted to be real numbers, then the complex roots must always form conjugate____

(A) pairs.

(B) sums.

(C) products.

(D) differences.

Question 1. A linear inequality is a mathematical statement that compares two expressions using an inequality symbol such as $<, >, \leq, \geq,$ or____

(A) $=$.

(B) $\neq$.

(C) $\approx$.

(D) $\infty$.

Question 2. A linear inequality in one variable, like $ax+b < 0$, has a solution set that can be represented as an interval on the number line, typically excluding or including the boundary point depending on the inequality____

(A) coefficient.

(B) variable.

(C) sign.

(D) term.

Question 3. When solving a linear inequality, multiplying or dividing both sides by a negative number requires reversing the direction of the inequality sign to maintain the correctness of the____

(A) solution.

(B) equation.

(C) coefficient.

(D) variable.

Question 4. The graphical solution of a linear inequality in one variable, like $x \geq 5$, involves representing the solution set as a ray on the number line, using a closed circle at 5 to indicate that 5 is included in the solution set because the inequality is "greater than or____ to".

(A) less.

(B) equal.

(C) not equal.

(D) approximately equal.

Question 5. The graphical solution of a linear inequality in two variables, like $Ax + By \leq C$, involves graphing the boundary line $Ax + By = C$ and shading one of the two half-planes formed by the line, where the shaded region represents all the points $(x, y)$ that____ the inequality.

(A) equal.

(B) do not satisfy.

(C) satisfy.

(D) are outside.

Question 6. A dashed boundary line in the graphical solution of a linear inequality in two variables indicates that the points on the line are____ included in the solution set, corresponding to strict inequalities like $<$ or $>$.

(A) always.

(B) sometimes.

(C) not.

(D) partially.

Question 7. The solution of a system of linear inequalities is the region where the solution regions of all the inequalities in the system____ each other.

(A) intersect.

(B) are parallel to.

(C) are perpendicular to.

(D) are disjoint from.

Question 8. Numerical inequalities involve comparing numerical values using inequality symbols, such as $7 > 4$, providing a statement about the relative size or order of two____

(A) variables.

(B) constants.

(C) expressions.

(D) numbers.

Question 9. The solution region of a system of linear inequalities in two variables is often a polygonal region (bounded or unbounded), whose vertices can be found by solving the pairs of linear equations corresponding to the intersecting____ lines.

(A) interior.

(B) boundary.

(C) parallel.

(D) perpendicular.

Question 10. Applications of linear inequalities include solving problems involving constraints on resources, budgeting, or nutritional requirements, where quantities must be greater than, less than, or equal to certain____ values.

(A) variable.

(B) unknown.

(C) maximum or minimum.

(D) proportional.

Question 1. A sequence is a list of numbers or elements in a specific order, often defined by a rule or formula for the $n$-th term, while a series is the____ of the terms of a sequence.

(A) product.

(B) difference.

(C) sum.

(D) average.

Question 2. An Arithmetic Progression (AP) is a sequence in which each term after the first is obtained by adding a fixed number, called the common____, to the preceding term.

(A) ratio.

(B) product.

(C) difference.

(D) term.

Question 3. The general term of an AP with first term $a_1$ and common difference $d$ is given by the formula $a_n = a_1 + (n-1)d$, where $n$ is the position of the____ in the sequence.

(A) difference.

(B) ratio.

(C) term.

(D) sum.

Question 4. The sum of the first $n$ terms of an AP can be calculated using the formula $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$, where $a_1$ is the first term and $a_n$ is the____ term.

(A) common.

(B) last.

(C) previous.

(D) next.

Question 5. A Geometric Progression (GP) is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed non-zero number, called the common____

(A) difference.

(B) ratio.

(C) factor.

(D) multiple.

Question 6. The general term of a GP with first term $a_1$ and common ratio $r$ is given by the formula $a_n = a_1 r^{n-1}$, where $n-1$ is the power to which the common ratio is raised for the $n$-th____

(A) sum.

(B) ratio.

(C) difference.

(D) term.

Question 7. The sum of the first $n$ terms of a GP with common ratio $r \neq 1$ is given by $S_n = \frac{a_1(r^n-1)}{r-1}$. If $r=1$, the sum of the first $n$ terms is simply $n \times a_1$, as all terms are____

(A) zero.

(B) increasing.

(C) decreasing.

(D) equal.

Question 8. The Arithmetic Mean (AM) of two positive numbers $a$ and $b$ is $\frac{a+b}{2}$, and the Geometric Mean (GM) is $\sqrt{ab}$. For any two positive numbers, the AM is always greater than or equal to the____, with equality holding only when $a=b$.

(A) sum.

(B) product.

(C) difference.

(D) GM.

Question 9. The sum of an infinite Geometric Progression converges to a finite value only if the absolute value of the common ratio $|r|$ is strictly less than____

(A) 0.

(B) 1.

(C) $-1$.

(D) the first term.

Question 10. Sequences and series are used to model various real-world phenomena involving patterns of growth, decay, or accumulation over discrete intervals, such as calculating compound interest, population growth, or the sum of distances traveled in a bouncing____

(A) wave.

(B) pendulum.

(C) object.

(D) trajectory.

Question 1. The Principle of Mathematical Induction (PMI) is a method of proof used to establish that a given statement $P(n)$ is true for all positive integers $n$, consisting of two main steps: the base case and the____ step.

(A) deductive.

(B) recursive.

(C) inductive.

(D) axiomatic.

Question 2. The base case in a proof by PMI involves verifying that the statement $P(n)$ is true for the smallest positive integer for which the statement is claimed to hold, typically $n=1$, but it could be another integer depending on the statement's____

(A) form.

(B) variable.

(C) range.

(D) complexity.

Question 3. The inductive hypothesis in a proof by PMI is the assumption that the statement $P(n)$ is true for some arbitrary positive integer $k \geq \text{base case}$, serving as the premise for proving the next____

(A) statement.

(B) integer.

(C) condition.

(D) variable.

Question 4. The inductive step in a proof by PMI requires showing that if the inductive hypothesis $P(k)$ is true, then the statement $P(n)$ must also be true for the integer immediately following $k$, which is $k+1$. This step demonstrates the "hereditary" nature of the____

(A) base case.

(B) hypothesis.

(C) property.

(D) variable.

Question 5. If both the base case and the inductive step are successfully demonstrated in a proof by PMI, then the principle concludes that the statement $P(n)$ is true for all positive integers starting from the base case, by logically extending the truth from one integer to the____

(A) previous one.

(B) next one.

(C) base case.

(D) variables.

Question 6. PMI is often used to prove formulas for sums of series, divisibility properties, inequalities, and statements about sequences or sets defined recursively, as these types of statements often involve properties that hold for all____ integers.

(A) negative.

(B) rational.

(C) real.

(D) positive.

Question 7. The analogy of falling dominoes is commonly used to illustrate PMI: proving the base case is like pushing over the first domino, and proving the inductive step is like showing that if any domino falls, it will knock over the____ domino.

(A) previous.

(B) next.

(C) last.

(D) adjacent.

Question 8. There is a variant called Strong Induction, which assumes that $P(i)$ is true for all integers $i$ from the base case up to $k$ (instead of just $P(k)$) to prove $P(k+1)$, and this stronger assumption can be useful for statements where the truth for $k+1$ depends on the truth of previous cases other than just____

(A) the base case.

(B) $P(k)$.

(C) $P(k+1)$.

(D) all integers.

Question 9. When applying PMI, it is crucial that both the base case and the inductive step are proven correctly. Failing to prove either step means the induction is incomplete and the statement cannot be concluded as true for all____ integers.

(A) natural.

(B) positive.

(C) integer.

(D) real.

Question 10. The principle of Mathematical Induction is based on the well-ordering principle of natural numbers, which states that every non-empty set of positive integers has a least____

(A) element.

(B) subset.

(C) upper bound.

(D) set.

Question 1. The Fundamental Principle of Counting, also known as the multiplication principle, states that if there are $m$ ways to do one thing and $n$ ways to do another thing, there are $m \times n$ ways to do both. This principle applies when the events are____

(A) dependent.

(B) mutually exclusive.

(C) sequential or independent.

(D) identical.

Question 2. The factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$, with $0!$ defined as 1. Factorials are used to calculate the number of ways to arrange $n$ distinct objects, which is $n!$, representing the number of____

(A) combinations.

(B) selections.

(C) permutations.

(D) divisions.

Question 3. A permutation is an arrangement of objects in a specific order, and the number of permutations of $n$ distinct objects taken $r$ at a time is given by $P(n, r) = \frac{n!}{(n-r)!}$, where the order of selection or arrangement____

(A) does not matter.

(B) matters.

(C) is irrelevant.

(D) is the same as combination.

Question 4. A combination is a selection of objects where the order does not matter, and the number of combinations of $n$ distinct objects taken $r$ at a time is given by $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$. This formula is used when we are interested in the subsets of size $r$ from a set of size $n$, regardless of the____ in which the elements are chosen.

(A) order.

(B) type.

(C) quantity.

(D) repetition.

Question 5. The relationship between permutations and combinations is given by $P(n, r) = C(n, r) \times r!$, which reflects that the number of permutations is the number of combinations multiplied by the number of ways to____ the $r$ selected objects.

(A) select.

(B) choose.

(C) combine.

(D) arrange.

Question 6. When dealing with permutations of objects where some are identical, the formula is adjusted by dividing the total permutations of $n$ objects ($n!$) by the factorials of the counts of each set of identical objects to avoid overcounting, as arranging identical objects among themselves does not result in distinct____

(A) combinations.

(B) selections.

(C) arrangements.

(D) sets.

Question 7. Counting principles are widely applied in probability theory to calculate the number of possible outcomes and the number of favourable outcomes for events, allowing us to determine the likelihood of specific events occurring based on combinations and____

(A) sums.

(B) differences.

(C) means.

(D) permutations.

Question 8. Problems involving the division of items into groups or their distribution among individuals often utilise concepts of combinations, where the order of items within a group or the order of assignment to individuals may or may not____

(A) matter.

(B) exist.

(C) apply.

(D) simplify.

Question 9. The symbol $\binom{n}{r}$ is read as "n choose r" and represents the number of combinations of $n$ items taken $r$ at a time, and these coefficients also appear in the expansion of binomials as coefficients, linking counting principles to the Binomial____

(A) series.

(B) formula.

(C) theorem.

(D) identity.

Question 10. Counting techniques are essential tools in fields like statistics, computer science, and operations research for analyzing possibilities, enumerating structures, and optimising processes by quantifying the number of ways events can occur under specific____

(A) variables.

(B) constraints.

(C) terms.

(D) operations.

Question 1. The Binomial Theorem provides a formula for expanding the power of a binomial $(a+b)^n$ for any positive integer $n$, stating that the expansion is a sum of terms involving powers of $a$ and $b$ multiplied by binomial____

(A) variables.

(B) constants.

(C) terms.

(D) coefficients.

Question 2. In the expansion of $(a+b)^n$, the general term, also known as the $(r+1)$-th term, is given by $T_{r+1} = \binom{n}{r} a^{n-r} b^r$, where $r$ is an integer ranging from 0 to____

(A) $n-1$.

(B) $n$.

(C) $n+1$.

(D) $r+1$.

Question 3. The binomial coefficients $\binom{n}{r}$ that appear in the expansion of $(a+b)^n$ can be calculated using the formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, which is the same formula used for combinations, $\binom{n}{r} = C(n, r)$, highlighting the link between expansion and____ principles.

(A) algebraic.

(B) geometrical.

(C) counting.

(D) probabilistic.

Question 4. The number of terms in the binomial expansion of $(a+b)^n$ is $n+1$, corresponding to the values of $r$ from 0 to $n$, where each term is defined by a unique combination of powers of $a$ and $b$ whose sum always equals the power of the____

(A) term.

(B) binomial.

(C) coefficient.

(D) variable.

Question 5. In the expansion of $(a+b)^n$, if $n$ is even, there is one middle term, which is the $(\frac{n}{2} + 1)$-th term, found by setting $r = n/2$ in the general____, $T_{r+1}$.

(A) coefficient.

(B) exponent.

(C) term.

(D) variable.

Question 6. In the expansion of $(a+b)^n$, if $n$ is odd, there are two middle terms, the $(\frac{n+1}{2})$-th term and the $(\frac{n+3}{2})$-th term, which correspond to $r$ values of $\frac{n-1}{2}$ and $\frac{n+1}{2}$ respectively in the formula for the general____

(A) coefficient.

(B) exponent.

(C) term.

(D) variable.

Question 7. The Binomial Theorem is used not only for expanding binomials but also for finding specific terms in an expansion, approximating values (like $(1.02)^4$), and in applications involving probability (Binomial Distribution) and other areas of____?

(A) geometry.

(B) statistics.

(C) chemistry.

(D) history.

Question 8. The sum of the binomial coefficients in the expansion of $(a+b)^n$ is obtained by setting $a=1$ and $b=1$, resulting in $(1+1)^n = 2^n$. This sum represents the total number of subsets of a set with $n$ elements, reinforcing the connection with____?

(A) permutations.

(B) combinations.

(C) factorials.

(D) identities.

Question 9. Pascal's triangle provides a triangular arrangement of the binomial coefficients, where each number is the sum of the two numbers directly above it, offering a visual and systematic way to generate the coefficients for the expansion of $(a+b)^n$ for small values of____

(A) $a$.

(B) $b$.

(C) $n$.

(D) $r$.

Question 10. The coefficient of $x^k$ in the expansion of $(1+x)^n$ is given by $\binom{n}{k}$, which is a direct consequence of the Binomial Theorem applied to the specific binomial $1+x$ raised to the power____

(A) $k$.

(B) $n-k$.

(C) $n$.

(D) $1$.

Question 1. A matrix is a rectangular arrangement of numbers or functions arranged in rows and columns, often enclosed in brackets, used to represent and manipulate data or systems of____

(A) variables.

(B) expressions.

(C) equations.

(D) functions.

Question 2. The order or dimension of a matrix specifies the number of rows and the number of columns it contains, written as $m \times n$, where $m$ is the number of rows and $n$ is the number of____

(A) elements.

(B) entries.

(C) columns.

(D) variables.

Question 3. A square matrix is a matrix in which the number of rows is equal to the number of columns, such as a $2 \times 2$ matrix or a $3 \times 3$ matrix. Square matrices are particularly important as they have properties like determinants and possibly an____

(A) transpose.

(B) adjoint.

(C) inverse.

(D) element.

Question 4. Two matrices are considered equal if they have the same order and their corresponding elements in the same positions are identical. This strict definition of equality is crucial when performing operations like addition or checking if a matrix is the result of a specific____

(A) transformation.

(B) type.

(C) size.

(D) operation.

Question 5. Addition and subtraction of matrices are defined only for matrices of the same order, and these operations are performed by adding or subtracting the corresponding____ in the same positions.

(A) rows.

(B) columns.

(C) elements.

(D) matrices.

Question 6. Scalar multiplication of a matrix involves multiplying every element of the matrix by a single number (scalar), resulting in a new matrix of the same order where each element is the scalar times the corresponding element of the original____

(A) variable.

(B) coefficient.

(C) matrix.

(D) vector.

Question 7. Matrix multiplication $AB$ is defined only if the number of columns in matrix $A$ is equal to the number of rows in matrix $B$. The element in the $i$-th row and $j$-th column of the product matrix $AB$ is obtained by taking the dot product of the $i$-th row of $A$ and the $j$-th column of____

(A) $A$.

(B) $B$.

(C) $AB$.

(D) $BA$.

Question 8. Matrix multiplication is generally not commutative, meaning $AB$ is usually not equal to $BA$, even if both products are defined. This is a key difference compared to the multiplication of____ numbers.

(A) complex.

(B) rational.

(C) real.

(D) whole.

Question 9. The zero matrix is a matrix of any order whose all elements are zero. It acts as the additive identity in matrix addition, meaning $A + 0 = 0 + A = A$ for any matrix $A$ of the same____

(A) element.

(B) dimension.

(C) type.

(D) value.

Question 10. Matrices provide a powerful tool for representing linear transformations, solving systems of linear equations, and organizing data in various fields like computer graphics, engineering, economics, and____

(A) literature.

(B) biology.

(C) philosophy.

(D) linguistics.

Question 1. The transpose of a matrix $A$, denoted $A'$ or $A^T$, is obtained by interchanging its rows and columns, such that the element in the $i$-th row and $j$-th column of $A'$ is the element in the $j$-th row and $i$-th column of____

(A) $A'$.

(B) $A$.

(C) $I$.

(D) 0.

Question 2. A square matrix $A$ is called a symmetric matrix if it is equal to its transpose, i.e., $A' = A$, meaning that the elements $a_{ij}$ and $a_{ji}$ are always____?

(A) equal.

(B) opposite in sign.

(C) zero.

(D) non-zero.

Question 3. A square matrix $A$ is called a skew symmetric matrix if its transpose is equal to the negative of the matrix, i.e., $A' = -A$. A key property of skew symmetric matrices is that all their diagonal elements must be____

(A) one.

(B) negative.

(C) positive.

(D) zero.

Question 4. Elementary operations on a matrix, such as interchanging rows, multiplying a row by a non-zero scalar, or adding a multiple of one row to another, are operations that do not change the solution set of the corresponding system of linear equations and are used in methods like Gaussian elimination to transform the matrix into a simpler____

(A) variable.

(B) equation.

(C) form.

(D) determinant.

Question 5. An invertible matrix is a square matrix $A$ for which there exists a matrix $A^{-1}$, called the inverse, such that $AA^{-1} = A^{-1}A = I$, the identity matrix. A square matrix is invertible if and only if its determinant is non-zero, meaning the matrix is____

(A) singular.

(B) non-singular.

(C) symmetric.

(D) skew symmetric.

Question 6. Any square matrix $A$ can be uniquely expressed as the sum of a symmetric matrix and a skew symmetric matrix, given by $A = \frac{1}{2}(A+A') + \frac{1}{2}(A-A')$, where $\frac{1}{2}(A+A')$ is symmetric and $\frac{1}{2}(A-A')$ is skew____

(A) symmetric.

(B) adjoint.

(C) transpose.

(D) invertible.

Question 7. Elementary row operations can be used to find the inverse of a matrix by augmenting the matrix with the identity matrix $[A | I]$ and applying row operations to transform $[A | I]$ into $[I | A^{-1}]$. If the left side cannot be transformed into the identity matrix, then the matrix $A$ is not____

(A) square.

(B) symmetric.

(C) invertible.

(D) singular.

Question 8. The inverse of the transpose of a matrix is equal to the transpose of the inverse, i.e., $(A')^{-1} = (A^{-1})'$. This property is useful in matrix algebra and in dealing with transposed____

(A) rows.

(B) columns.

(C) matrices.

(D) elements.

Question 9. If $A$ and $B$ are invertible matrices of the same order, then their product $AB$ is also invertible, and the inverse of the product is the product of the inverses in reverse order, i.e., $(AB)^{-1} = B^{-1}A^{-1}$. This rule is important in matrix multiplication and the properties of matrix____

(A) addition.

(B) transpose.

(C) inversion.

(D) determinant.

Question 10. The identity matrix $I$ is a square matrix with ones on the main diagonal and zeros elsewhere. It is the multiplicative identity for matrices, meaning multiplying any matrix $A$ by the appropriately sized identity matrix results in the matrix____

(A) 0.

(B) $A$.

(C) $A'$.

(D) $A^{-1}$.

Question 1. A determinant is a scalar value that can be computed from the elements of a square matrix, providing useful information about the matrix, such as whether it is invertible or if the corresponding system of linear equations has a unique____

(A) variable.

(B) coefficient.

(C) solution.

(D) determinant.

Question 2. For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated as the difference of the products of the elements on the main diagonal and the anti-diagonal, i.e., $ad - bc$. This value is $\det(A)$, denoted by $|A|$ or____

(A) $A'$.

(B) $\text{adj}(A)$.

(C) $\det(A)$.

(D) $A^{-1}$.

Question 3. One application of determinants is to calculate the area of a triangle given the coordinates of its vertices, using a specific formula involving a determinant derived from the coordinate____

(A) plane.

(B) points.

(C) system.

(D) vertices.

Question 4. The adjoint of a square matrix $A$ is the transpose of the cofactor matrix, where the cofactor $C_{ij}$ of an element $a_{ij}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th____

(A) element.

(B) column.

(C) row.

(D) matrix.

Question 5. A key property relating a matrix, its adjoint, and its determinant is $A(\text{adj}(A)) = (\text{adj}(A))A = \det(A)I$, where $I$ is the identity matrix. This property is fundamental in understanding matrix inverses, as it shows that if $\det(A) \neq 0$, the inverse $A^{-1}$ can be expressed in terms of the adjoint and the____

(A) matrix.

(B) inverse.

(C) determinant.

(D) transpose.

Question 6. Properties of determinants help in simplifying calculations and understanding their behaviour under matrix operations. For instance, $\det(A') = \det(A)$, meaning the determinant of a matrix is equal to the determinant of its____

(A) adjoint.

(B) inverse.

(C) transpose.

(D) product.

Question 7. If the determinant of a square matrix is zero, the matrix is called singular, which implies that the columns (and rows) of the matrix are linearly dependent, and the matrix does not have an____

(A) adjoint.

(B) determinant.

(C) inverse.

(D) transpose.

Question 8. The determinant of a diagonal matrix or a triangular matrix (upper or lower) is simply the product of its elements on the main diagonal. This property significantly simplifies the calculation of determinants for these specific types of____

(A) vectors.

(B) matrices.

(C) scalars.

(D) elements.

Question 9. The adjoint matrix is used in Cramer's Rule and in the formula for the inverse. Calculating the adjoint for larger matrices involves finding the determinants of numerous submatrices and their respective cofactors, making it computationally more intensive than some other methods for finding the inverse, especially for large-dimensional____

(A) vectors.

(B) scalars.

(C) matrices.

(D) determinants.

Question 10. Determinants have a geometric interpretation; for a $2 \times 2$ matrix representing a linear transformation, the absolute value of the determinant represents the scaling factor of the area. For a $3 \times 3$ matrix, the absolute value represents the scaling factor of the____

(A) length.

(B) area.

(C) volume.

(D) angle.

Question 1. The inverse of a square matrix $A$, denoted $A^{-1}$, is the matrix such that when multiplied by $A$, it yields the identity matrix $I$, i.e., $AA^{-1} = A^{-1}A = I$. The inverse exists if and only if the matrix is non-singular, meaning its determinant is not equal to____

(A) 1.

(B) -1.

(C) 0.

(D) undefined.

Question 2. One method to calculate the inverse of a square matrix $A$ is using the formula $A^{-1} = \frac{1}{\det(A)}\text{adj}(A)$, which requires first computing the determinant and the____ of $A$.

(A) transpose.

(B) cofactor.

(C) adjoint.

(D) trace.

Question 3. Another method to find the inverse of a matrix $A$ is by using elementary row operations on the augmented matrix $[A | I]$, transforming it into the form $[I | A^{-1}]$. This method relies on the fact that applying row operations to $A$ is equivalent to multiplying $A$ by a sequence of invertible elementary____, which are also applied to $I$.

(A) vectors.

(B) scalars.

(C) matrices.

(D) elements.

Question 4. The solution of a system of linear equations $AX=B$ can be found using the matrix inverse method, where $X = A^{-1}B$, provided the coefficient matrix $A$ is invertible. This method is applicable when the system has a unique____, which is guaranteed if $\det(A) \neq 0$.

(A) variable.

(B) coefficient.

(C) solution.

(D) equation.

Question 5. Cramer's Rule is another method for solving a system of $n$ linear equations in $n$ variables using determinants. If $\det(A) \neq 0$, the solution for each variable is given by the ratio of two determinants: the determinant of the matrix $A_i$ (where the $i$-th column of $A$ is replaced by $B$) divided by the determinant of the coefficient____

(A) vector.

(B) matrix.

(C) variable.

(D) solution.

Question 6. If the determinant of the coefficient matrix $A$ is zero, the system of linear equations $AX=B$ does not have a unique solution. It either has no solution (inconsistent system) or infinitely many solutions (dependent consistent system), and neither the matrix inverse method nor Cramer's Rule can be used to find a single unique____

(A) matrix.

(B) determinant.

(C) inverse.

(D) solution.

Question 7. For a $2 \times 2$ system $a_{11}x + a_{12}y = b_1$, $a_{21}x + a_{22}y = b_2$, the solution for $x$ using Cramer's Rule is $x = \frac{\begin{vmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{vmatrix}}{\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}}$, where the denominator is the determinant of the coefficient matrix and must be non-zero for a unique____

(A) variable.

(B) determinant.

(C) system.

(D) solution.

Question 8. The matrix inverse method and Cramer's Rule are powerful techniques for solving systems of linear equations, particularly for systems with a moderate number of variables, offering systematic algebraic approaches compared to graphical methods which are limited to____ variables.

(A) one.

(B) two.

(C) three.

(D) any number of.

Question 9. If a system of linear equations $AX=B$ is consistent and has infinitely many solutions, the matrix $A$ is singular ($\det(A)=0$), and the equations are linearly dependent, meaning one equation can be obtained as a linear combination of the____

(A) variables.

(B) coefficients.

(C) other equations.

(D) constants.

Question 10. The concept of the inverse of a matrix is also used in linear transformations; applying a matrix $A$ transforms a vector, and applying $A^{-1}$ performs the reverse transformation, taking the transformed vector back to its original____, provided $A$ is invertible.

(A) position.

(B) dimension.

(C) value.

(D) coefficient.

Question 1. Solving word problems based on linear equations often involves representing unknown quantities with variables, translating the relationships described in the problem into one or more linear equations, and then solving the resulting____ or system of equations.

(A) variables.

(B) expressions.

(C) inequalities.

(D) equation.

Question 2. Word problems involving age, cost, speed-distance-time, mixtures, or work done by pipes or people can often be modelled and solved using linear equations or systems of linear equations by setting up relationships between the given and unknown____

(A) variables.

(B) quantities.

(C) coefficients.

(D) terms.

Question 3. Word problems leading to quadratic equations often involve scenarios where the product of two quantities is given, or areas of shapes are related, leading to an equation with the highest power of the variable being____, which can be solved by factorisation, completing the square, or the quadratic formula.

(A) 1.

(B) 2.

(C) 3.

(D) any even number.

Question 4. In problems involving boats in rivers or planes in wind, the speed upstream is the speed in still water minus the speed of the stream/wind, and the speed downstream/with the wind is the speed in still water plus the speed of the stream/wind. These relationships can be used to set up equations involving time, distance, and____

(A) acceleration.

(B) force.

(C) speed.

(D) velocity.

Question 5. Problems involving mixtures of different concentrations or substances, or distribution of amounts based on certain conditions, can often be solved by setting up linear equations based on the total quantity and the quantity of a specific component or the total value of the distributed____

(A) variables.

(B) proportions.

(C) amounts.

(D) ingredients.

Question 6. When solving word problems, it is important to define the variables clearly, including the units of measurement (e.g., metres, kilograms, $\textsf{₹}$, hours), and to make sure the solution obtained from the equation(s) makes sense in the context of the original____

(A) numbers.

(B) variables.

(C) equations.

(D) problem.

Question 7. Work problems often involve the concept of rate of work (amount of work done per unit of time). If a person or pipe can complete a job in $t$ units of time, their rate of work is $1/t$ job per unit of time. If multiple entities work together, their rates of work are typically added to find the combined____

(A) time.

(B) amount of work.

(C) rate.

(D) efficiency.

Question 8. Problems involving percentage increases or decreases, such as discounts, profits, or interest calculations, can be translated into algebraic equations where percentages are represented as decimals or fractions, allowing calculation of original amounts, final amounts, or percentage____

(A) values.

(B) points.

(C) rates.

(D) variables.

Question 9. Numerical problems involving relationships between numbers, such as sums, differences, products, quotients, or relationships between digits in a multi-digit number, are classic applications of algebraic equations, particularly linear and____ equations.

(A) cubic.

(B) polynomial.

(C) exponential.

(D) quadratic.

Question 10. Applications of algebraic equations extend across various disciplines, providing essential tools for modelling relationships, solving problems, and making predictions in science, engineering, finance, and other areas by transforming complex real-world scenarios into solvable mathematical____

(A) variables.

(B) formulas.

(C) models.

(D) operations.