15 extra multiple-choice questions for Algebra (10th Standard Mathematics, Samacheer Kalvi), beyond the ones printed in the textbook — each with the correct option highlighted and a clear, worked explanation. Free to read in English and Tamil.
Q1
In a system of three linear equations in three variables, if the elimination process results in a false statement like \(0 = 5\), what does this indicate about the nature of the solutions?
- A. Unique solution
- B. Infinitely many solutions
- C. No solutionCorrect
- D. Two distinct solutions
Explanation. A false statement like \(0 = 5\) indicates that the system is inconsistent and has no solution because the planes represented by the equations do not intersect at any common point.
Q2
What is the Greatest Common Divisor (GCD) of the polynomials \(12x^2 y^3 z\) and \(18x^3 y^2\)?
- A. \(6x^2 y^2\)Correct
- B. \(6x^3 y^3 z\)
- C. \(36x^3 y^3 z\)
- D. \(3x^2 y^2\)
Explanation. The GCD of numerical coefficients 12 and 18 is 6. For the variables, we take the lowest power of common variables, which are \(x^2\) and \(y^2\). Thus, the GCD is \(6x^2 y^2\).
Q3
What is the Least Common Multiple (LCM) of the algebraic expressions \((x - 1)^2\) and \(x^2 - 1\)?
- A. \((x - 1)(x + 1)\)
- B. \((x - 1)^2(x + 1)\)Correct
- C. \((x - 1)(x + 1)^2\)
- D. \((x - 1)^2(x + 1)^2\)
Explanation. Factoring the expressions gives \((x - 1)^2\) and \((x - 1)(x + 1)\). The LCM is the product of the highest powers of all distinct factors, which is \((x - 1)^2(x + 1)\).
Q4
What are the excluded values of the rational expression \(\frac{x + 3}{x^2 - 16}\)?
- A. \(-3\) and \(3\)
- B. \(-4\) and \(4\)Correct
- C. \(-3\) and \(4\)
- D. \(-4\) and \(3\)
Explanation. Excluded values are the values of \(x\) that make the denominator of the rational expression zero. Setting \(x^2 - 16 = 0\) gives \(x^2 = 16\), which yields \(x = \pm 4\).
Q5
If the rational expression \(\frac{2x + 1}{x^2 + kx + 9}\) has no real excluded values, what is the range of values that the constant \(k\) can take?
- A. \(-6 < k < 6\)Correct
- B. \(k > 6\) or \(k < -6\)
- C. \(-3 < k < 3\)
- D. \(k = \pm 6\)
Explanation. No real excluded values means the denominator has no real roots. For \(x^2 + kx + 9 = 0\), the discriminant \(\Delta = k^2 - 36\) must be less than zero, which gives \(-6 < k < 6\).
Q6
What is the square root of the algebraic expression \(16x^4 - 24x^2 + 9\)?
- A. \(|4x^2 - 3|\)Correct
- B. \(|4x^2 + 3|\)
- C. \(|2x^2 - 3|\)
- D. \(|4x - 3|\)
Explanation. The polynomial can be written as \((4x^2)^2 - 2(4x^2)(3) + 3^2 = (4x^2 - 3)^2\). Taking the square root of this perfect square expression yields \(|4x^2 - 3|\).
Q7
If the discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is equal to zero, what is the nature of its roots?
- A. Real and unequal
- B. Real and equalCorrect
- C. No real roots
- D. Imaginary and unequal
Explanation. When the discriminant \(\Delta = b^2 - 4ac\) is equal to zero, the quadratic equation always has real and equal roots.
Q8
Find the quadratic equation whose sum of the roots is \(-5\) and the product of the roots is \(6\).
- A. \(x^2 - 5x + 6 = 0\)
- B. \(x^2 + 5x + 6 = 0\)Correct
- C. \(x^2 - 5x - 6 = 0\)
- D. \(x^2 + 5x - 6 = 0\)
Explanation. The general form of a quadratic equation with sum \(S\) and product \(P\) of roots is \(x^2 - Sx + P = 0\). Substituting \(S = -5\) and \(P = 6\) yields \(x^2 + 5x + 6 = 0\).
Q9
If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(3x^2 - 5x - 2 = 0\), what is the value of \(\alpha^2 + \beta^2\)?
- A. \(\frac{13}{9}\)
- B. \(\frac{37}{9}\)Correct
- C. \(\frac{19}{9}\)
- D. \(\frac{25}{9}\)
Explanation. The sum \(\alpha + \beta = \frac{5}{3}\) and product \(\alpha\beta = -\frac{2}{3}\). Thus, \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \frac{25}{9} + \frac{4}{3} = \frac{37}{9}\).
Q10
If the variables \(x\) and \(y\) vary inversely such that \(xy = k\), where \(k\) is a positive constant, how does \(y\) change when \(x\) is doubled?
- A. \(y\) is doubled
- B. \(y\) is halvedCorrect
- C. \(y\) remains unchanged
- D. \(y\) is quadrupled
Explanation. Since \(xy = k \implies y = \frac{k}{x}\). If \(x\) is doubled to \(2x\), the new value of \(y\) is \(\frac{k}{2x} = \frac{y}{2}\), so \(y\) is halved.
Q11
A square matrix in which all the entries above and below the leading main diagonal are zero is called a:
- A. Scalar matrix
- B. Unit matrix
- C. Diagonal matrixCorrect
- D. Row matrix
Explanation. A diagonal matrix is defined as a square matrix in which all non-diagonal elements (those above and below the main diagonal) are zero.
Q12
If matrix \(A\) is of order \(3 \times 4\) and matrix \(B\) is of order \(4 \times 2\), what is the order of the product matrix \(AB\)?
- A. \(3 \times 2\)Correct
- B. \(4 \times 4\)
- C. \(2 \times 3\)
- D. \(4 \times 2\)
Explanation. For matrix multiplication \(A_{m \times n} B_{n \times p}\), the product matrix \(AB\) has the order \(m \times p\). Here, \(m=3\), \(n=4\), and \(p=2\), so the order of \(AB\) is \(3 \times 2\).
Q13
If \(A\) is a matrix of order \(m \times n\), what is the order of its transpose matrix \(A^T\)?
- A. \(m \times n\)
- B. \(n \times m\)Correct
- C. \(m \times m\)
- D. \(n \times n\)
Explanation. The transpose of a matrix is formed by swapping its rows with columns. Therefore, if \(A\) is of order \(m \times n\), then \(A^T\) is of order \(n \times m\).
Q14
If \(A\) and \(B\) are two matrices of the same order, which of the following matrix properties is always true?
- A. \(AB = BA\)
- B. \(A + B = B + A\)Correct
- C. \(A - B = B - A\)
- D. \((A + B)^T = A^T - B^T\)
Explanation. Matrix addition is commutative, meaning \(A + B = B + A\) for any matrices \(A\) and \(B\) of the same order. Matrix multiplication is generally not commutative.
Q15
If one root of the quadratic equation \(x^2 + px + q = 0\) is the square of the other, which of the following relations is correct?
- A. \(p^3 - q(3p - 1) + q^2 = 0\)Correct
- B. \(p^3 + q(3p - 1) + q^2 = 0\)
- C. \(p^3 - q(3p + 1) + q^2 = 0\)
- D. \(p^3 + q(3p + 1) + q^2 = 0\)
Explanation. Let roots be \(\alpha, \alpha^2\). Sum \(\alpha + \alpha^2 = -p\) and product \(\alpha^3 = q\). Cubing the sum: \((\alpha + \alpha^2)^3 = -p^3 \implies \alpha^3 + \alpha^6 + 3\alpha^3(\alpha + \alpha^2) = -p^3 \implies q + q^2 - 3pq = -p^3 \implies p^3 - q(3p - 1) + q^2 = 0\).
