Matching Items MCQs for Sub-Topics of Topic 2: Algebra
Fundamentals of Algebra: Variables, Expressions, and Basic Concepts
Question 1. Match the algebraic term with its coefficient.
(i) $5x^2$
(ii) $-3y$
(iii) $ab$
(iv) $-p^3$
(a) $1$
(b) $-1$
(c) $-3$
(d) $5$
(A) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
Answer:
Question 2. Match the algebraic phrase with the corresponding expression.
(i) 5 more than a number $n$
(ii) 3 less than twice a number $x$
(iii) The product of 7 and a number $y$
(iv) A number $p$ divided by 4
(a) $2x - 3$
(b) $7y$
(c) $n + 5$
(d) $p/4$
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
Answer:
Question 3. Match the algebraic expression with its number of terms.
(i) $4x^2$
(ii) $2a + 3b$
(iii) $p^3 - 5pq + q^2$
(iv) $y^4 - 10 + 2y + 7y^2$
(a) 4 terms
(b) 3 terms
(c) 2 terms
(d) 1 term
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the term with one of its factors.
(i) $10xyz$
(ii) $7p^2q$
(iii) $-4m^3$
(iv) $15ab^2c$
(a) $m^2$
(b) $xy$
(c) $pq$
(d) $ab^2$
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 5. Match the expression evaluated at given values with the result.
(i) $2x + 5$ at $x=3$
(ii) $a^2 - b$ at $a=2, b=1$
(iii) $p/q$ at $p=10, q=5$
(iv) $m - n$ at $m=-2, n=-5$
(a) 3
(b) 2
(c) 11
(d) 1
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Operations on Algebraic Expressions
Question 1. Match the addition problem with its sum.
(i) $(3x + 2y) + (x - y)$
(ii) $(a^2 + 5a) + (2a^2 - 3a)$
(iii) $(p + q) + (p - q)$
(iv) $(m^2 - m) + (-m^2 + 2m)$
(a) $2p$
(b) $3a^2 + 2a$
(c) $m$
(d) $4x + y$
(A) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 2. Match the subtraction problem with its difference.
(i) $(5a - 3b) - (2a + b)$
(ii) $(x^2 + x) - (x^2 - 2x)$
(iii) $(7p) - (p - 4)$
(iv) $(m^2 + 2) - (m^2 - 5)$
(a) 7
(b) $3x$
(c) $3a - 4b$
(d) $6p + 4$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 3. Match the multiplication problem with its product.
(i) $2x \times 3y$
(ii) $a^2 \times a^3$
(iii) $4m (-2m^2)$
(iv) $-p \times -q$
(a) $pq$
(b) $a^5$
(c) $-8m^3$
(d) $6xy$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 4. Match the division problem with its quotient.
(i) $10x^5 \div 5x^2$
(ii) $-12a^4b^3 \div 3a^2b$
(iii) $7m^2n \div 7mn$
(iv) $p^6q^2 \div p^3q^2$
(a) $p^3$
(b) $m$
(c) $-4a^2b^2$
(d) $2x^3$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the expanded expression with its simplified form after combining like terms.
(i) $5x - 2x + 3y$
(ii) $a^2 + 2a - a^2 + 5a$
(iii) $p - q + 2p - 3q$
(iv) $m^2 + n^2 - m^2 + n^2$
(a) $2n^2$
(b) $7a$
(c) $3p - 4q$
(d) $3x + 3y$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Polynomials: Definition, Types, and Properties
Question 1. Match the polynomial with its type (based on the number of terms).
(i) $5x^3$
(ii) $y^2 - 4$
(iii) $a^3 + 2a - 7$
(iv) $m^4 + 3m^2 - m + 1$
(a) Polynomial (more than 3 terms)
(b) Trinomial
(c) Binomial
(d) Monomial
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 2. Match the polynomial with its degree.
(i) $7x^5 - 2x + 1$
(ii) $-3y^2 + 10y$
(iii) $4z - 9$
(iv) $6$
(a) 0
(b) 1
(c) 2
(d) 5
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 3. Match the polynomial with its leading coefficient.
(i) $2x^4 - 5x^3 + 1$
(ii) $-y^2 + 3y - 8$
(iii) $15z - 1$
(iv) $-w^5 + 2w^2$
(a) $-1$
(b) $15$
(c) $-1$
(d) $2$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(C) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the polynomial with one of its zeroes.
(i) $x - 5$
(ii) $2y + 4$
(iii) $z^2 - 9$
(iv) $p^2 - p - 6$
(a) $-2$
(b) $-2$
(c) $5$
(d) $3$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Question 5. Match the description of the polynomial's graph with the number of real zeroes.
(i) Graph of a linear polynomial (not horizontal)
(ii) Graph of a quadratic polynomial touching the x-axis
(iii) Graph of a quadratic polynomial intersecting the x-axis at two points
(iv) Graph of a quadratic polynomial not intersecting the x-axis
(a) 0 real zeroes
(b) 1 real zero (repeated)
(c) 2 real zeroes
(d) 1 real zero
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Polynomial Theorems and Division
Question 1. Match the theorem with its statement or implication.
(i) Remainder Theorem
(ii) Factor Theorem
(iii) Division Algorithm for Polynomials
(iv) If $P(a) = 0$
(a) $P(x) = Q(x)D(x) + R(x)$ with deg $R(x) <$ deg $D(x)$
(b) $(x-a)$ is a factor of $P(x)$
(c) The remainder when $P(x)$ is divided by $(x-a)$ is $P(a)$
(d) $(x-a)$ is a factor of $P(x)$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Question 2. Match the polynomial and divisor with the remainder using the Remainder Theorem.
(i) $x^2 - 3x + 5$ divided by $(x-1)$
(ii) $x^3 + 1$ divided by $(x+1)$
(iii) $2x^2 - x + 4$ divided by $(x-2)$
(iv) $x^4 - 16$ divided by $(x+2)$
(a) 0
(b) 10
(c) 3
(d) 0
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 3. Match the polynomial with the linear factor(s) according to the Factor Theorem.
(i) $x^2 - 4$
(ii) $x^2 - 5x + 6$
(iii) $x^3 - 1$
(iv) $x^2 + 2x + 1$
(a) $(x-1)$
(b) $(x-2)$
(c) $(x-2)$
(d) $(x+1)$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Question 4. Match the polynomial division result components.
(i) $x^2 - 5x + 6 \div (x-2)$
(ii) $x^3 + 1 \div (x+1)$
(iii) $2x^2 + 3x + 1 \div (x-1)$
(iv) $x^3 - 8 \div (x-2)$
(a) Quotient $x^2+2x+4$
(b) Remainder 6
(c) Quotient $x^2-x+1$
(d) Quotient $x-3$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the condition with the implication based on Polynomial Theorems.
(i) $P(a)=0$
(ii) $P(x)$ divided by $(x-a)$ leaves remainder $R$
(iii) Degree of $R(x)$ is less than degree of $D(x)$
(iv) $(x-a)$ is a factor of $P(x)$
(a) $a$ is a zero of $P(x)$
(b) Remainder Theorem
(c) Division Algorithm
(d) $P(a)=0$
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(C) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Algebraic Identities
Question 1. Match the identity with its correct expansion or factorization.
(i) $(a+b)^2$
(ii) $(a-b)^2$
(iii) $a^2 - b^2$
(iv) $(x+a)(x+b)$
(a) $x^2 + (a+b)x + ab$
(b) $a^2 - 2ab + b^2$
(c) $(a-b)(a+b)$
(d) $a^2 + 2ab + b^2$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 2. Match the expression with the identity primarily used for its expansion or factorization.
(i) $x^2 - 100$
(ii) $(y+8)^2$
(iii) $(p-3q)^2$
(iv) $(m+2)(m+5)$
(a) $(x+a)(x+b)$
(b) $(a-b)^2$
(c) $a^2 - b^2$
(d) $(a+b)^2$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 3. Match the numerical calculation with the identity that makes it easier.
(i) $101^2$
(ii) $97^2$
(iii) $52 \times 48$
(iv) $103 \times 105$
(a) $(x+a)(x+b)$
(b) $(a-b)^2$
(c) $a^2 - b^2$
(d) $(a+b)^2$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 4. Match the identity with its correct expansion or factorization (Cubic identities).
(i) $(a+b)^3$
(ii) $(a-b)^3$
(iii) $a^3 + b^3$
(iv) $a^3 - b^3$
(a) $(a-b)(a^2+ab+b^2)$
(b) $a^3 - 3a^2b + 3ab^2 - b^3$
(c) $(a+b)(a^2-ab+b^2)$
(d) $a^3 + 3a^2b + 3ab^2 + b^3$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 5. Match the expression with its factorization using cubic identities.
(i) $x^3 + 8$
(ii) $y^3 - 27$
(iii) $8a^3 + 125b^3$
(iv) $64m^3 - n^3$
(a) $(4m-n)(16m^2+4mn+n^2)$
(b) $(y-3)(y^2+3y+9)$
(c) $(x+2)(x^2-2x+4)$
(d) $(2a+5b)(4a^2-10ab+25b^2)$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
Answer:
Factorisation of Algebraic Expressions and Polynomials
Question 1. Match the expression with its factorization by taking out the common factor.
(i) $6x + 9y$
(ii) $a^2b - ab^2$
(iii) $10m - 15mn$
(iv) $p^3 - p^2 + p$
(a) $p(p^2 - p + 1)$
(b) $ab(a-b)$
(c) $5m(2 - 3n)$
(d) $3(2x + 3y)$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 2. Match the expression with its factorization by grouping.
(i) $ax + ay + bx + by$
(ii) $m^2 - mn + pm - pn$
(iii) $xy - yz + wx - wz$
(iv) $a^3 - a^2 - a + 1$
(a) $(a^2-1)(a-1)$
(b) $(y+w)(x-z)$
(c) $(m+p)(m-n)$
(d) $(a+b)(x+y)$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 3. Match the expression with its factorization using algebraic identities.
(i) $p^2 - 1$
(ii) $4x^2 + 4x + 1$
(iii) $a^3 + 125$
(iv) $y^3 - 8$
(a) $(y-2)(y^2+2y+4)$
(b) $(2x+1)^2$
(c) $(a+5)(a^2-5a+25)$
(d) $(p-1)(p+1)$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 4. Match the quadratic trinomial with its factorization by splitting the middle term.
(i) $x^2 + 5x + 6$
(ii) $a^2 - 7a + 12$
(iii) $y^2 + y - 2$
(iv) $m^2 - 2m - 8$
(a) $(m-4)(m+2)$
(b) $(y+2)(y-1)$
(c) $(a-3)(a-4)$
(d) $(x+2)(x+3)$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the polynomial with one of its linear factors.
(i) $x^3 - 2x^2 - x + 2$ (Hint: $x=1$ is a zero)
(ii) $y^3 + 2y^2 - y - 2$ (Hint: $y=-1$ is a zero)
(iii) $a^3 - 3a^2 + 3a - 1$
(iv) $m^4 - 16$
(a) $(m-2)$
(b) $(a-1)$
(c) $(y+1)$
(d) $(x-1)$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Linear Equations in One Variable
Question 1. Match the equation with its solution (root).
(i) $3x - 6 = 0$
(ii) $2(y+1) = 8$
(iii) $\frac{p}{5} - 2 = 0$
(iv) $4m + 7 = m + 13$
(a) $m=2$
(b) $p=10$
(c) $y=3$
(d) $x=2$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 2. Match the word problem description with its corresponding linear equation in one variable. Let the number be $x$.
(i) A number increased by 8 is 20.
(ii) Twice a number is 14.
(iii) 5 less than a number is 12.
(iv) A number divided by 3 is 4.
(a) $x/3 = 4$
(b) $x - 5 = 12$
(c) $2x = 14$
(d) $x + 8 = 20$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 3. Match the equation transformation with the operation applied.
(i) From $x + 3 = 7$ to $x = 7 - 3$
(ii) From $2x = 10$ to $x = 5$
(iii) From $x/4 = 2$ to $x = 8$
(iv) From $x - 5 = 1$ to $x = 1 + 5$
(a) Multiplying both sides by 4
(b) Transposing -5
(c) Transposing 3
(d) Dividing both sides by 2
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the equation with the result of checking a given potential solution.
(i) $2x - 5 = 3$, check $x=4$
(ii) $y + 10 = 7$, check $y=-3$
(iii) $3p = 12$, check $p=3$
(iv) $\frac{m}{2} = 5$, check $m=10$
(a) $10/2 = 5$, True
(b) $3(3) = 9 \neq 12$, False
(c) $-3+10 = 7$, True
(d) $2(4) - 5 = 8 - 5 = 3$, True
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the property of linear equations in one variable with its description.
(i) Solution/Root
(ii) Transposition
(iii) Equivalent equations
(iv) Degree of variable
(a) Equations with the same solution set
(b) Moving a term to the other side by changing its sign
(c) The value that satisfies the equation
(d) Always 1 in a linear equation
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Linear Equations in Two Variables
Question 1. Match the equation with its type.
(i) $2x - 3y = 7$
(ii) $y = 5$
(iii) $x = -2$
(iv) $x + y = 0$
(a) Linear equation in two variables, passes through origin
(b) Linear equation in two variables, parallel to y-axis
(c) Linear equation in two variables, parallel to x-axis
(d) General linear equation in two variables
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 2. Match the linear equation in two variables with a point that is a solution.
(i) $x + y = 5$
(ii) $2x - y = 1$
(iii) $y = 3x$
(iv) $x - 2y = 0$
(a) $(2, 1)$
(b) $(3, 9)$
(c) $(4, 1)$
(d) $(4, 2)$
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 3. Match the description of the graph with the equation.
(i) A line parallel to the y-axis
(ii) The x-axis
(iii) A line passing through the origin
(iv) The y-axis
(a) $x = 0$
(b) $y = mx$
(c) $y = 0$
(d) $x = c$ (where $c$ is a constant)
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the point with the equation of a line parallel to an axis passing through it.
(i) $(3, 5)$ (line parallel to x-axis)
(ii) $(3, 5)$ (line parallel to y-axis)
(iii) $(-1, 2)$ (line parallel to x-axis)
(iv) $(-1, 2)$ (line parallel to y-axis)
(a) $x = -1$
(b) $y = 2$
(c) $x = 3$
(d) $y = 5$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the word problem description with its corresponding linear equation in two variables.
(i) The sum of two numbers is 10.
(ii) The difference of two numbers is 3.
(iii) The cost of an apple ($a$) and a banana ($b$) is $\textsf{₹}25$.
(iv) The length ($l$) of a rectangle is twice its width ($w$).
(a) $l = 2w$
(b) $x - y = 3$
(c) $x + y = 10$
(d) $a + b = 25$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Pair of Linear Equations in Two Variables: Systems and Solutions
Question 1. Match the ratio condition with the type of lines represented by a pair of linear equations $\begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases}$.
(i) $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
(ii) $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
(iii) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
(iv) Lines intersect at one point
(a) Coincident lines
(b) Intersecting lines
(c) Parallel and distinct lines
(d) Unique solution
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(b)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
Answer:
Question 2. Match the type of system with the number/type of solutions.
(i) Consistent system with unique solution
(ii) Inconsistent system
(iii) Consistent system with infinitely many solutions
(iv) Lines are coincident
(a) Infinitely many solutions
(b) No solution
(c) Unique solution
(d) Infinitely many solutions
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(a)
(D) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
Answer:
Question 3. Match the system of equations with the number of solutions.
(i) $\begin{cases} x + y = 4 \\ x - y = 2 \end{cases}$
(ii) $\begin{cases} 2x + 3y = 5 \\ 4x + 6y = 8 \end{cases}$
(iii) $\begin{cases} x - y = 1 \\ 3x - 3y = 3 \end{cases}$
(iv) $\begin{cases} x = 5 \\ y = 2 \end{cases}$
(a) Infinitely many solutions
(b) No solution
(c) Unique solution
(d) Unique solution
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
Answer:
Question 4. Match the system of equations with its solution $(x, y)$.
(i) $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$
(ii) $\begin{cases} 2x + y = 4 \\ x + y = 3 \end{cases}$
(iii) $\begin{cases} 3x - y = 5 \\ x + 2y = 4 \end{cases}$
(iv) $\begin{cases} y = 2x \\ x + y = 6 \end{cases}$
(a) $(2, 4)$
(b) $(2, 1)$
(c) $(1, 2)$
(d) $(3, 2)$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the algebraic method for solving linear systems with its description.
(i) Substitution Method
(ii) Elimination Method
(iii) Graphing Method
(iv) Finding point(s) of intersection
(a) Graphing Method
(b) Make coefficients of one variable equal to cancel
(c) Solve one equation for one variable and plug into the other
(d) Find where the lines cross
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Quadratic Equations: Introduction and Solving Methods
Question 1. Match the equation with its type.
(i) $3x - 5 = 0$
(ii) $x^2 - 4 = 0$
(iii) $y^3 + 2y^2 - 1 = 0$
(iv) $z^2 + z + 1 = 0$
(a) Quadratic Equation
(b) Cubic Equation
(c) Linear Equation
(d) Quadratic Equation
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 2. Match the quadratic equation $ax^2 + bx + c = 0$ with its coefficients.
(i) $x^2 - 5x + 6 = 0$
(ii) $2y^2 + y - 3 = 0$
(iii) $-p^2 + 4p = 0$
(iv) $m^2 + 7 = 0$
(a) $a=1, b=0, c=7$
(b) $a=-1, b=4, c=0$
(c) $a=2, b=1, c=-3$
(d) $a=1, b=-5, c=6$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 3. Match the quadratic equation with the sum of its roots.
(i) $x^2 - 7x + 10 = 0$
(ii) $2y^2 + 4y + 1 = 0$
(iii) $3p^2 - 6p = 0$
(iv) $-m^2 + 5m - 2 = 0$
(a) 2
(b) $-2$
(c) 7
(d) 5
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the quadratic equation with the product of its roots.
(i) $x^2 + 2x - 8 = 0$
(ii) $3y^2 - 9y + 6 = 0$
(iii) $p^2 + 16 = 0$
(iv) $5m^2 - 10m = 0$
(a) 0
(b) 16
(c) 2
(d) $-8$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the quadratic equation with its roots.
(i) $x^2 - 4 = 0$
(ii) $x^2 - 3x + 2 = 0$
(iii) $x^2 - 6x + 9 = 0$
(iv) $x^2 + x - 12 = 0$
(a) $x=3, x=-4$
(b) $x=3$ (repeated)
(c) $x=1, x=2$
(d) $x=2, x=-2$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Complex Numbers: Introduction and Algebra
Question 1. Match the power of $i$ with its value.
(i) $i^1$
(ii) $i^2$
(iii) $i^3$
(iv) $i^4$
(a) $1$
(b) $-i$
(c) $-1$
(d) $i$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 2. Match the complex number operation with its result.
(i) $(2 + 3i) + (1 - i)$
(ii) $(5 - 2i) - (3 + i)$
(iii) $(1 + i)(1 - i)$
(iv) $\frac{4}{i}$
(a) $2 - 3i$
(b) $2$
(c) $3 + 2i$
(d) $-4i$
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
Answer:
Question 3. Match the complex number with its real part.
(i) $7 + 5i$
(ii) $-3i$
(iii) $10$
(iv) $-1 - 2i$
(a) $-1$
(b) 10
(c) 0
(d) 7
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
Answer:
Question 4. Match the complex number with its imaginary part.
(i) $4 - 3i$
(ii) $-i$
(iii) $2$
(iv) $-6 + 7i$
(a) 7
(b) 0
(c) $-1$
(d) $-3$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the complex number with its additive inverse or multiplicative inverse.
(i) $2 + 5i$ (Additive Inverse)
(ii) $i$ (Multiplicative Inverse)
(iii) $1$ (Additive Inverse)
(iv) $-1$ (Multiplicative Inverse)
(a) $-1$
(b) $-i$
(c) $-2 - 5i$
(d) $-1$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Complex Numbers: Representation and Properties
Question 1. Match the complex number with the point representing it in the Argand plane.
(i) $3 + 4i$
(ii) $-2 + i$
(iii) $5$
(iv) $-2i$
(a) $(5, 0)$
(b) $(3, 4)$
(c) $(0, -2)$
(d) $(-2, 1)$
(A) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(B) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 2. Match the complex number with its modulus.
(i) $1 + i$
(ii) $3 - 4i$
(iii) $-5$
(iv) $2i$
(a) 2
(b) 5
(c) $\sqrt{2}$
(d) 5
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 3. Match the complex number with its conjugate.
(i) $2 + 3i$
(ii) $5 - i$
(iii) $-4i$
(iv) $7$
(a) 7
(b) $4i$
(c) $5 + i$
(d) $2 - 3i$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
Answer:
Question 4. Match the complex number with its principal argument (in $(-\pi, \pi]$).
(i) $1$
(ii) $i$
(iii) $-1$
(iv) $-i$
(a) $\pi$
(b) $\pi/2$
(c) $0$
(d) $-\pi/2$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the equation/condition with the locus of the complex number $z=x+iy$ in the Argand plane.
(i) $|z| = 5$
(ii) $z = \bar{z}$
(iii) $z + \bar{z} = 0$
(iv) $z = 0$
(a) Imaginary axis
(b) Real axis
(c) Origin
(d) Circle centered at origin with radius 5
(A) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Quadratic Equations with Complex Roots
Question 1. Match the quadratic equation with its discriminant $D = b^2 - 4ac$.
(i) $x^2 + 1 = 0$
(ii) $x^2 - 2x + 2 = 0$
(iii) $x^2 + x + 1 = 0$
(iv) $x^2 - 4x + 5 = 0$
(a) $-3$
(b) $-4$
(c) $-4$
(d) $-8$
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Question 2. Match the discriminant value with the nature of roots for a quadratic equation with real coefficients.
(i) $D > 0$
(ii) $D = 0$
(iii) $D < 0$
(iv) Discriminant is negative
(a) Real and distinct roots
(b) Real and equal roots
(c) Complex conjugate roots
(d) Non-real roots
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 3. Match the quadratic equation with its complex roots.
(i) $x^2 + 4 = 0$
(ii) $x^2 - 2x + 5 = 0$
(iii) $x^2 + 6x + 13 = 0$
(iv) $x^2 + 2x + 2 = 0$
(a) $-1 \pm i$
(b) $3 \pm 2i$
(c) $1 \pm 2i$
(d) $\pm 2i$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 4. Match the complex root (for an equation with real coefficients) with its conjugate pair.
(i) $1 + i$
(ii) $-2 + 3i$
(iii) $5i$
(iv) $p - qi$
(a) $p + qi$
(b) $-5i$
(c) $-2 - 3i$
(d) $1 - i$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the pair of complex conjugate roots with the quadratic equation having these roots (with leading coefficient 1).
(i) $\pm i$
(ii) $\pm 3i$
(iii) $1 \pm i$
(iv) $-1 \pm 2i$
(a) $x^2 + 2x + 5 = 0$
(b) $x^2 - 2x + 2 = 0$
(c) $x^2 + 9 = 0$
(d) $x^2 + 1 = 0$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Linear Inequalities
Question 1. Match the inequality with the number line representation of its solution set for $x \in \mathbb{R}$.
(i) $x > 3$
(ii) $x \le -1$
(iii) $2 \le x < 5$
(iv) $-2 < x \le 0$
(a) Segment with open circle at -2, closed at 0
(b) Ray to the right of 3, open circle at 3
(c) Ray to the left of -1, closed circle at -1
(d) Segment with closed circle at 2, open at 5
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Question 2. Match the inequality in two variables with the description of its solution region in the coordinate plane.
(i) $y > 2$
(ii) $y \le 0$
(iii) $x < -3$
(iv) $x \ge 0$
(a) Region to the right of the y-axis (including boundary)
(b) Region below the x-axis (including boundary)
(c) Region above the line $y=2$ (excluding boundary)
(d) Region to the left of the line $x=-3$ (excluding boundary)
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Question 3. Match the operation performed on the inequality $x > y$ with the resulting inequality.
(i) Add 5 to both sides
(ii) Multiply both sides by 2
(iii) Multiply both sides by -1
(iv) Subtract 10 from both sides
(a) $-x < -y$
(b) $x - 10 > y - 10$
(c) $2x > 2y$
(d) $x + 5 > y + 5$
(A) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 4. Match the system of inequalities with the description of its feasible region.
(i) $\begin{cases} x \ge 0 \\ y \ge 0 \end{cases}$
(ii) $\begin{cases} x \le 0 \\ y \ge 0 \end{cases}$
(iii) $\begin{cases} x \ge 0 \\ y \le 0 \end{cases}$
(iv) $\begin{cases} x \le 0 \\ y \le 0 \end{cases}$
(a) Third quadrant (including boundaries)
(b) Fourth quadrant (including boundaries)
(c) Second quadrant (including boundaries)
(d) First quadrant (including boundaries)
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the word problem constraint with the inequality that represents it. Let the number of items be $x$.
(i) At least 10 items
(ii) No more than 20 items
(iii) More than 5 items
(iv) Less than 15 items
(a) $x < 15$
(b) $x > 5$
(c) $x \le 20$
(d) $x \ge 10$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Sequences and Series
Question 1. Match the sequence with its type.
(i) 3, 7, 11, 15, ...
(ii) 2, 6, 18, 54, ...
(iii) 1, 4, 9, 16, ...
(iv) 10, 7, 4, 1, ...
(a) Neither AP nor GP
(b) AP
(c) AP
(d) GP
(A) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(B) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 2. Match the AP with its common difference.
(i) 5, 10, 15, ...
(ii) 8, 6, 4, ...
(iii) 1/2, 1, 3/2, ...
(iv) $-2, -5, -8, ...$
(a) $-3$
(b) $-2$
(c) 5
(d) $1/2$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Question 3. Match the GP with its common ratio.
(i) 4, 12, 36, ...
(ii) 100, 50, 25, ...
(iii) $-1, 2, -4, ...$
(iv) $1/3, 1, 3, ...$
(a) 3
(b) $-2$
(c) $1/2$
(d) 3
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the sequence description with its $n$-th term formula (assuming $n \ge 1$).
(i) AP with $a=3, d=2$
(ii) GP with $a=2, r=3$
(iii) Sequence of squares
(iv) Sequence of odd numbers (starting with 1)
(a) $2n-1$
(b) $n^2$
(c) $2 \cdot 3^{n-1}$
(d) $3 + (n-1)2$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the sum of the series description with the formula for the sum of the first $n$ terms.
(i) Sum of first $n$ natural numbers
(ii) Sum of first $n$ odd numbers
(iii) Sum of first $n$ even numbers
(iv) Sum of first $n$ terms of AP ($a, d$)
(a) $n^2$
(b) $\frac{n(n+1)}{2}$
(c) $n(n+1)$
(d) $\frac{n}{2}(2a + (n-1)d)$
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Principle of Mathematical Induction
Question 1. Match the step of Mathematical Induction with its purpose.
(i) Base Case ($P(n_0)$)
(ii) Inductive Hypothesis ($P(k)$)
(iii) Inductive Step ($P(k) \implies P(k+1)$)
(iv) Conclusion by PMI
(a) To show the property holds for all integers $\ge n_0$
(b) To show the property propagates from one integer to the next
(c) To show the property holds for the starting value
(d) To assume the property holds for an arbitrary integer $k \ge n_0$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 2. Match the statement $P(n): 1+3+\dots+(2n-1) = n^2$ with the expression representing a step in its proof by induction (starting at $n=1$).
(i) $P(1)$
(ii) $P(k)$
(iii) LHS of $P(k+1)$
(iv) RHS of $P(k+1)$
(a) $(k+1)^2$
(b) $1 + 3 + \dots + (2k-1)$
(c) $1+3+\dots+(2k-1) + (2(k+1)-1)$
(d) $1 = 1^2$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 3. Match the type of statement with whether PMI can be used to prove it.
(i) For all positive integers $n$, $n! > 2^n$.
(ii) For all real numbers $x$, $x^2 \ge 0$.
(iii) For all positive integers $n$, $n^3+2n$ is divisible by 3.
(iv) Every even number greater than 2 is the sum of two primes (Goldbach Conjecture).
(a) Can be proven by PMI
(b) Cannot be proven by PMI (Statement is false for small $n$)
(c) Cannot be proven by PMI (Not a statement about properties of positive integers in the required form)
(d) Can be proven by PMI
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(c)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
Answer:
Question 4. Match the condition for a statement $P(n)$ ($n \ge n_0$) proven by induction with its implication.
(i) $P(n_0)$ is false
(ii) $P(n_0)$ is true and $P(k) \implies P(k+1)$ for all $k \ge n_0$
(iii) $P(n_0)$ is true and $P(k) \implies P(k+1)$ for all $k \ge m$ where $m > n_0$
(iv) $P(n)$ is true for all integers $n \ge n_0$
(a) Need to check $P(n_0), \dots, P(m-1)$ separately; P(n) true for $n \ge m$
(b) PMI proves the statement
(c) PMI cannot be used to prove the statement for $n \ge n_0$
(d) This is the conclusion reached by a successful induction proof
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the concept related to PMI with its brief explanation.
(i) Principle of Mathematical Induction
(ii) Strong Induction
(iii) Inductive Hypothesis
(iv) Base Case
(a) The starting point of the induction proof
(b) Assuming $P(k)$ is true for some $k \ge n_0$
(c) Assuming $P(j)$ is true for all $n_0 \le j \le k$
(d) A method to prove statements about positive integers
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Counting Principles: Factorial, Permutations, and Combinations
Question 1. Match the term with its definition or formula.
(i) Factorial ($n!$)
(ii) Permutation ($\text{P}(n, r)$)
(iii) Combination ($\text{C}(n, r)$)
(iv) Fundamental Principle of Counting
(a) Number of ways to select $r$ items from $n$ (order does not matter)
(b) If one task can be done in $m$ ways and another in $n$ ways, both can be done in $mn$ ways
(c) Product of first $n$ positive integers
(d) Number of ways to arrange $r$ items from $n$ (order matters)
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
Answer:
Question 2. Match the scenario with the counting method used.
(i) Arranging books on a shelf
(ii) Selecting a committee from a group
(iii) Choosing lottery numbers (order doesn't matter)
(iv) Forming passwords with distinct characters
(a) Combination
(b) Permutation
(c) Combination
(d) Permutation
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 3. Match the calculation with the result.
(i) $4!$
(ii) $\text{P}(5, 3)$
(iii) $\text{C}(6, 2)$
(iv) $\text{C}(7, 7)$
(a) 1
(b) 15
(c) 60
(d) 24
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the scenario with the correct counting expression.
(i) Number of ways to choose 2 flavours from 5
(ii) Number of ways to award 1st, 2nd, 3rd prizes in a race of 10
(iii) Number of ways to arrange the letters in 'CAT'
(iv) Number of handshakes among 7 people
(a) $\text{P}(10, 3)$
(b) $3!$
(c) $\text{C}(7, 2)$
(d) $\text{C}(5, 2)$
(A) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(B) (i)-(a), (ii)-(d), (iii)-(b), (iv)-(c)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the counting expression with an equivalent expression or value.
(i) $\text{C}(n, r)$
(ii) $\text{P}(n, r)$
(iii) $\text{C}(n, n-r)$
(iv) $\text{C}(5, 3)$
(a) $\text{C}(5, 2)$
(b) $\frac{n!}{(n-r)!}$
(c) $\text{C}(n, r)$
(d) $\frac{n!}{r!(n-r)!}$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Binomial Theorem
Question 1. Match the binomial expansion property with its description.
(i) Index $n$ (positive integer)
(ii) Number of terms
(iii) General term in $(a+b)^n$
(iv) Sum of coefficients in $(x+y)^n$
(a) $2^n$
(b) $n+1$
(c) The power to which the binomial is raised
(d) $T_{r+1} = \text{C}(n, r) a^{n-r} b^r$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 2. Match the expansion with the specified term or coefficient.
(i) $(x+y)^4$, 3rd term
(ii) $(a-b)^5$, 2nd term
(iii) $(1+x)^6$, coefficient of $x^3$
(iv) $(p+q)^7$, coefficient of $p^3q^4$
(a) $\text{C}(7, 4) = 35$
(b) $\text{C}(6, 3) = 20$
(c) $\text{C}(5, 1)a^4(-b)^1 = -5a^4b$
(d) $\text{C}(4, 2)x^2y^2 = 6x^2y^2$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 3. Match the index $n$ with the position(s) of the middle term(s) in the expansion of $(a+b)^n$.
(i) $n=5$
(ii) $n=6$
(iii) $n=7$
(iv) $n=8$
(a) 5th term
(b) 4th and 5th terms
(c) 4th term
(d) 4th and 5th terms
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the sum of coefficients property with the example.
(i) Sum of coefficients in $(x+y)^3$
(ii) Sum of coefficients in $(a-b)^4$
(iii) Sum of coefficients in $(2p+3q)^2$
(iv) Sum of coefficients in $(m-n)^5$
(a) $(1-1)^5 = 0$
(b) $(2+3)^2 = 25$
(c) $(1-1)^4 = 0$
(d) $(1+1)^3 = 8$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the binomial coefficient property with its formula or implication.
(i) Symmetry property
(ii) Pascal's Identity
(iii) Sum of all coefficients
(iv) $\text{C}(n, 0)$
(a) 1
(b) $2^n$
(c) $\text{C}(n, r) = \text{C}(n, n-r)$
(d) $\text{C}(n, r) + \text{C}(n, r-1) = \text{C}(n+1, r)$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Matrices: Introduction, Types, and Basic Operations
Question 1. Match the matrix with its order.
(i) $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$
(ii) $\begin{pmatrix} 5 & 6 & 7 \end{pmatrix}$
(iii) $\begin{pmatrix} 8 \\ 9 \end{pmatrix}$
(iv) $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$
(a) $2 \times 3$
(b) $2 \times 1$
(c) $1 \times 3$
(d) $2 \times 2$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 2. Match the description with the type of matrix.
(i) A matrix with only one row
(ii) A matrix with only one column
(iii) A matrix where number of rows equals number of columns
(iv) A square matrix with 1s on the diagonal, 0s elsewhere
(a) Identity matrix
(b) Square matrix
(c) Column matrix
(d) Row matrix
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 3. Match the matrix operation with the condition required for it to be defined.
(i) $A+B$
(ii) $A-B$
(iii) $AB$
(iv) $BA$
(a) Number of columns in B = Number of rows in A
(b) Number of columns in A = Number of rows in B
(c) A and B have the same order
(d) A and B have the same order
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the matrices with the result of the specified operation.
(i) $\begin{pmatrix} 1 & 2 \end{pmatrix} + \begin{pmatrix} 3 & 4 \end{pmatrix}$
(ii) $3 \begin{pmatrix} 1 \\ 2 \end{pmatrix}$
(iii) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$
(iv) $\begin{pmatrix} 1 & 2 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix}$
(a) $\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$
(b) $\begin{pmatrix} 4 & 6 \end{pmatrix}$
(c) $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$
(d) $\begin{pmatrix} 11 \end{pmatrix}$
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
Answer:
Question 5. Match the description with the corresponding matrix type.
(i) All elements are zero
(ii) Square matrix with non-zero elements only on the main diagonal
(iii) A matrix $A$ such that $A=A'$
(iv) A matrix $A$ such that $A=-A'$
(a) Skew-symmetric matrix
(b) Symmetric matrix
(c) Diagonal matrix
(d) Zero matrix
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Matrices: Properties and Advanced Operations
Question 1. Match the matrix with its transpose.
(i) $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$
(ii) $\begin{pmatrix} 5 & 6 \end{pmatrix}$
(iii) $\begin{pmatrix} 7 \\ 8 \end{pmatrix}$
(iv) $\begin{pmatrix} a & b & c \end{pmatrix}$
(a) $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$
(b) $\begin{pmatrix} 7 & 8 \end{pmatrix}$
(c) $\begin{pmatrix} 5 \\ 6 \end{pmatrix}$
(d) $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 2. Match the property of matrix operations with its statement.
(i) $(A+B)'$
(ii) $(kA)'$
(iii) $(AB)'$
(iv) $(A')'$
(a) $A$
(b) $B'A'$
(c) $kA'$
(d) $A' + B'$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 3. Match the square matrix with its classification (Symmetric/Skew-Symmetric).
(i) $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$
(ii) $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
(iii) $\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$
(iv) $\begin{pmatrix} 0 & 3 & -4 \\ -3 & 0 & 6 \\ 4 & -6 & 0 \end{pmatrix}$
(a) Skew-symmetric
(b) Symmetric
(c) Symmetric
(d) Skew-symmetric
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 4. Match the matrix expression with its property (for a square matrix A).
(i) $A + A'$
(ii) $A - A'$
(iii) $\frac{1}{2}(A+A')$
(iv) $\frac{1}{2}(A-A')$
(a) Skew-symmetric part of A
(b) Symmetric part of A
(c) Skew-symmetric matrix
(d) Symmetric matrix
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the property of invertible matrices (A and B are invertible of same order) with its statement.
(i) $(A^{-1})^{-1}$
(ii) $(A')^{-1}$
(iii) $(AB)^{-1}$
(iv) $|A^{-1}|$
(a) $B^{-1}A^{-1}$
(b) $A$
(c) $(A^{-1})'$
(d) $1/|A|$
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Determinants and Adjoint
Question 1. Match the matrix with its determinant.
(i) $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$
(ii) $\begin{pmatrix} -1 & 5 \\ 2 & -3 \end{pmatrix}$
(iii) $\begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix}$
(iv) $\begin{pmatrix} 5 \end{pmatrix}$
(a) 5
(b) 0
(c) 10
(d) $-7$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 2. Match the property of a square matrix's determinant with its description.
(i) Two rows are identical
(ii) A row is multiplied by $k$
(iii) A multiple of a row is added to another row
(iv) The matrix is singular
(a) Determinant does not change
(b) Determinant is multiplied by $k$
(c) Determinant is 0
(d) Determinant is 0
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
Answer:
Question 3. Match the matrix property with its determinant property.
(i) Identity matrix $I_n$
(ii) Zero matrix
(iii) Transpose of matrix A ($A'$)
(iv) Product of matrices AB
(a) $\det(A)\det(B)$
(b) $\det(A)$
(c) 0
(d) 1
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 4. Match the matrix with its adjoint.
(i) $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$
(ii) $\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$
(iii) $\begin{pmatrix} -1 & 0 \\ 0 & 5 \end{pmatrix}$
(iv) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
(a) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
(b) $\begin{pmatrix} 5 & 0 \\ 0 & -1 \end{pmatrix}$
(c) $\begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix}$
(d) $\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the determinant/adjoint property with its formula (for square matrix A of order n, $|A| \neq 0$).
(i) $A \cdot \text{adj}(A)$
(ii) $|\text{adj}(A)|$
(iii) $\text{adj}(A')$
(iv) $\text{adj}(I_n)$
(a) $I_n$
(b) $|A| I$
(c) $|A|^{n-1}$
(d) $(\text{adj}(A))'$
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Inverse of a Matrix and Systems of Equations
Question 1. Match the condition with the property of a square matrix A.
(i) $|A| \neq 0$
(ii) $|A| = 0$
(iii) A is invertible
(iv) A is non-singular
(a) A is invertible
(b) A is singular
(c) A is non-singular
(d) $|A| \neq 0$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 2. Match the matrix with its inverse (if it exists).
(i) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
(ii) $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$
(iii) $\begin{pmatrix} 3 & -2 \\ 1 & 1 \end{pmatrix}$
(iv) $\begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix}$
(a) Does not exist
(b) $\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$
(c) $\begin{pmatrix} 1/5 & 2/5 \\ -1/5 & 3/5 \end{pmatrix}$
(d) $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 3. Match the system of linear equations $AX=B$ with the condition on A and B and the type of solution.
(i) $|A| \neq 0$
(ii) $|A| = 0$ and $(\text{adj}A)B \neq O$
(iii) $|A| = 0$ and $(\text{adj}A)B = O$
(iv) $B=O$ and $|A| \neq 0$
(a) Consistent, infinitely many solutions
(b) Consistent, unique solution (trivial)
(c) Consistent, unique solution
(d) Inconsistent
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 4. Match the system of equations with its solution $(x, y)$.
(i) $\begin{cases} x+y=3 \\ x-y=1 \end{cases}$
(ii) $\begin{cases} 2x+y=5 \\ x+y=3 \end{cases}$
(iii) $\begin{cases} 3x-y=0 \\ x+y=8 \end{cases}$
(iv) $\begin{cases} x=4 \\ y=5 \end{cases}$
(a) $x=2, y=6$
(b) $x=2, y=1$
(c) $x=4, y=5$
(d) $x=2, y=2$
(A) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 5. Match the property with its statement (for invertible matrices A and B of the same order, scalar $k$).
(i) $(A^{-1})'$
(ii) $(kA)^{-1}$
(iii) $|A^{-1}|$
(iv) $(\text{adj}(A))^{-1}$
(a) $1/|A|$
(b) $\frac{1}{k} A^{-1}$
(c) $(A')^{-1}$
(d) $\text{adj}(A^{-1})$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Applications of Algebraic Equations in Word Problems
Question 1. Match the word problem with the type of algebraic equation(s) generally used to solve it.
(i) Finding two numbers given their sum and difference
(ii) Finding the side of a square given its perimeter
(iii) Finding the dimensions of a rectangle given its area and a relationship between length and width
(iv) Finding the ages of two people given a relationship and how that changes over time
(a) System of linear equations
(b) Linear equation in one variable
(c) Quadratic equation
(d) System of linear equations
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(a)
(C) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(a), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 2. Match the word problem description with a simplified equation derived from the setup.
(i) Sum of two consecutive integers is 55.
(ii) The cost of 3 pens and 2 pencils is $\textsf{₹}60$. Let pen cost be $p$, pencil cost be $c$.
(iii) The product of a number and 5 more than the number is 24. Let the number be $x$.
(iv) A number increased by 10 is equal to twice the number. Let the number be $n$.
(a) $n + 10 = 2n$
(b) $x(x+5) = 24$
(c) $2x + 1 = 55$
(d) $3p + 2c = 60$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 3. Match the word problem with its solution.
(i) Find two numbers whose sum is 18 and difference is 4.
(ii) The product of two consecutive positive integers is 72.
(iii) If 5 is added to a number, the result is 17.
(iv) Find the side of a square whose area is 49 cm$^2$.
(a) 7 cm
(b) 12
(c) 8 and 9
(d) 11 and 7
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 4. Match the scenario with the concept of speed involved (Boat and Stream problems).
(i) Speed of boat in still water = $u$, speed of stream = $v$
(ii) Boat travelling against the stream
(iii) Boat travelling with the stream
(iv) Time taken to travel a distance D upstream
(a) $u+v$
(b) $u-v$
(c) $D/(u-v)$
(d) $u$ and $v$ are speeds
(A) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the work problem setup with the equation component representing rate.
(i) Pipe A fills a tank in 10 hours
(ii) Pipe B empties a tank in 15 hours
(iii) Person P completes a job in $x$ days
(iv) Person Q works at twice the rate of P
(a) Rate of Q is $2/x$ job/day
(b) Rate of P is $1/x$ job/day
(c) Rate of B is $-1/15$ tank/hour
(d) Rate of A is $1/10$ tank/hour
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer: