MCQOPTIONS
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MCQOPTIONS
This section includes 72 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
If \(A=\left[ \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} \right]\) then A-1 is- |
| A. | -A |
| B. | A |
| C. | I |
| D. | 0 |
| Answer» C. I | |
| 2. |
If the determinant \(\left| {\begin{array}{*{20}{c}} x&1&3\\ 0&0&1\\ 1&x&4 \end{array}} \right| = 0\) then what is x equal to? |
| A. | 02 or 2 |
| B. | -3 or 3 |
| C. | -1 or 1 |
| D. | 3 or 4 |
| Answer» D. 3 or 4 | |
| 3. |
If A and B are square matrices of order 3 such that |A| = -1, |B| = 3 then |3 AB| is equal to - |
| A. | -9 |
| B. | -81 |
| C. | -27 |
| D. | 81 |
| Answer» C. -27 | |
| 4. |
If B is a non-singular matrix and A is a square matrix, then the value of det (B-1 AB) is equal to |
| A. | det (B) |
| B. | det (A) |
| C. | det (B-1) |
| D. | det (A-1) |
| Answer» C. det (B-1) | |
| 5. |
If [x] denotes the greatest integer ≤ x, then the system of linear equations [sin θ]x + [-cos θ]y = 0 and [cot θ]x + y = 0 |
| A. | Have infinitely many solutions if \(\theta \in \left( {\frac{\pi }{2},{\rm{\;}}\frac{{2\pi }}{3}} \right)\) and has a unique solution if \(\theta \in \left( {\pi ,{\rm{\;}}\frac{{7\pi }}{6}} \right)\). |
| B. | Has unique solution if \(\theta \in \left( {\frac{\pi }{2},{\rm{\;}}\frac{{2\pi }}{3}} \right) \cup \left( {\pi ,{\rm{\;}}\frac{{7\pi }}{6}} \right)\). |
| C. | Has unique solution if \(\theta \in \left( {\frac{\pi }{2},{\rm{\;}}\frac{{2\pi }}{3}} \right)\) and have infinitely many solutions if \(\theta \in \left( {\pi ,{\rm{\;}}\frac{{7\pi }}{6}} \right)\). |
| D. | Have infinitely many solutions if \(\theta \in \left( {\frac{\pi }{2},\;\frac{{2\pi }}{3}} \right) \cup \left( {\pi ,\;\frac{{7\pi }}{6}} \right)\). |
| Answer» B. Has unique solution if \(\theta \in \left( {\frac{\pi }{2},{\rm{\;}}\frac{{2\pi }}{3}} \right) \cup \left( {\pi ,{\rm{\;}}\frac{{7\pi }}{6}} \right)\). | |
| 6. |
If a, b, c are real numbers, then the value of the determinant \(\left| {\begin{array}{*{20}{c}} {1 - {\rm{a}}}&{{\rm{a}} - {\rm{b}} - {\rm{c}}}&{{\rm{b}} + {\rm{c}}}\\ {1 - {\rm{b}}}&{{\rm{b}} - {\rm{c}} - {\rm{a}}}&{{\rm{c}} + {\rm{a}}}\\ {1 - {\rm{c}}}&{{\rm{c}} - {\rm{a}} - {\rm{b}}}&{{\rm{a}} + {\rm{b}}} \end{array}} \right|\) is |
| A. | 0 |
| B. | (a - b) (b - c) (c - a) |
| C. | (a + b + c) 2 |
| D. | (a + b + c) 3 |
| Answer» B. (a - b) (b - c) (c - a) | |
| 7. |
If the value of the determinant \(\left[ {\begin{array}{*{20}{c}} {\rm{a}}&1&1\\ 1&{\rm{b}}&1\\ 1&1&{\rm{c}} \end{array}} \right]\) is positive, where a ≠ b ≠ c, then the value of abc is |
| A. | cannot be less than 1 |
| B. | is greater than -8 |
| C. | is less than -8 |
| D. | must be greater than 8 |
| Answer» C. is less than -8 | |
| 8. |
A matrix Mr is defined as \(M_r = \begin{bmatrix} r & r - 1 \\\ r - 1 & r \end{bmatrix} r \in N\), then the value of det (M1) + det(M2) + ... + det(M2015) is |
| A. | 20142 |
| B. | 20132 |
| C. | 2015 |
| D. | 20152 |
| Answer» E. | |
| 9. |
If \(A=\left[ \begin{matrix}{{e}^{t}} & {{e}^{-t}}\text{cos }\!\!~\!\!\text{ }t & {{e}^{-t}}\text{sin }\!\!~\!\!\text{ }t \\{{e}^{t}} & -{{e}^{-t}}\text{cos }\!\!~\!\!\text{ }t-{{e}^{-t}}\text{sin }\!\!~\!\!\text{ }t & -{{e}^{-t}}\text{sin }\!\!~\!\!\text{ }t+{{e}^{-t}}\text{cos }\!\!~\!\!\text{ }t \\{{e}^{t}} & 2{{e}^{-t}}\text{sin }\!\!~\!\!\text{ }t & -2{{e}^{-t}}\text{cos }\!\!~\!\!\text{ }t \\\end{matrix} \right]\) then A is: |
| A. | Invertible for all t ∈ R |
| B. | Invertible only if t = π |
| C. | Not invertible for any t ∈ R |
| D. | Invertible only if \(\text{t}=\frac{\pi }{2}\) |
| Answer» B. Invertible only if t = π | |
| 10. |
Factors of the determinant \(\left| {\begin{array}{*{20}{c}} a&{b + c}&{{a^2}}\\ b&{c + a}&{{b^2}}\\ c&{a + b}&{{c^2}} \end{array}} \right|\) |
| A. | (a - b), (b - c), (c - a), (a + b + c) |
| B. | (a + b), (b + c), (c + a), (a + b + c) |
| C. | (a + b), (b - c), (c + a), (a + b + c) |
| D. | (a2 + b2), (b2 + c2), (c2 + a2) |
| Answer» B. (a + b), (b + c), (c + a), (a + b + c) | |
| 11. |
Let \({\rm{a}}{{\rm{x}}^3} + {\rm{b}}{{\rm{x}}^2} + {\rm{cx}} + {\rm{d}} = \left| {\begin{array}{*{20}{c}} {{\rm{x}} + 1}&{2{\rm{x}}}&{3{\rm{x}}}\\ {2{\rm{x}} + 3}&{{\rm{x}} + 1}&{\rm{x}}\\ {2 - {\rm{x}}}&{3{\rm{x}} + 4}&{5{\rm{x}} - 1} \end{array}} \right|\)What is the value of a + b + c + d? |
| A. | 62 |
| B. | 63 |
| C. | 65 |
| D. | 68 |
| Answer» C. 65 | |
| 12. |
Let \({\rm{a}}{{\rm{x}}^3} + {\rm{b}}{{\rm{x}}^2} + {\rm{cx}} + {\rm{d}} = \left| {\begin{array}{*{20}{c}} {{\rm{x}} + 1}&{2{\rm{x}}}&{3{\rm{x}}}\\ {2{\rm{x}} + 3}&{{\rm{x}} + 1}&{\rm{x}}\\ {2 - {\rm{x}}}&{3{\rm{x}} + 4}&{5{\rm{x}} - 1} \end{array}} \right|\)What is the value of c? |
| A. | -1 |
| B. | 34 |
| C. | 35 |
| D. | 50 |
| Answer» D. 50 | |
| 13. |
If \(\left| {\begin{array}{*{20}{c}} 5&a\\ a&2 \end{array}} \right| = \left| {\begin{array}{*{20}{c}} 2&1\\ 3&2 \end{array}} \right|\), then the values of a are: |
| A. | ± 1 |
| B. | ± 2 |
| C. | ± 3 |
| D. | ± 4 |
| Answer» D. ± 4 | |
| 14. |
Area of the triangle formed by the lines 7x - 2y + 10 = 0, 7x + 2y - 10 = 0 and y + 2 = 0 is |
| A. | 8 |
| B. | 14 |
| C. | 16 |
| D. | 18/7 |
| Answer» C. 16 | |
| 15. |
If \(A = \left[ {\begin{array}{*{20}{c}} a&b&c\\ b&c&a\\ c&a&b \end{array}} \right],\) where a, b, c are real positive numbers such that abc = 1 and ATA = I then the equation that holds true among the following is |
| A. | a + b + c = 1 |
| B. | a2 + b2 + c2 = 1 |
| C. | ab + bc + ca = 0 |
| D. | a3 + b3 + c3 = 4 |
| Answer» C. ab + bc + ca = 0 | |
| 16. |
If a, b, c are the roots of equation \(x^3-3x^2 + 3x + 7=0\), then the value of \(\begin{vmatrix} 2bc-a^2 & c^2 & b^2 \\\ c^2 & 2ac-b^2 & a^2 \\\ b^2 & a^2 & 2ab-c^2 \end{vmatrix}\) is |
| A. | 9 |
| B. | 27 |
| C. | 81 |
| D. | 0 |
| Answer» E. | |
| 17. |
Let \(A = \left| {\begin{array}{*{20}{c}} p&q\\ r&s \end{array}} \right|\)where p, q, r and s are any four different prime numbers less than 20. What is the maximum value of the determinant? |
| A. | 215 |
| B. | 311 |
| C. | 317 |
| D. | 323 |
| Answer» D. 323 | |
| 18. |
If p + q + r = a + b + c = 0, then the determinant \(\left| {\begin{array}{*{20}{c}} {{\rm{pa}}}&{{\rm{qb}}}&{{\rm{rc}}}\\ {{\rm{qc}}}&{{\rm{ra}}}&{{\rm{pb}}}\\ {{\rm{rb}}}&{{\rm{pc}}}&{{\rm{qa}}} \end{array}} \right|\) equals |
| A. | 0 |
| B. | 1 |
| C. | pa + qb + rc |
| D. | pa + qb + rc + a + b + c |
| Answer» B. 1 | |
| 19. |
If \(\left[ {\begin{array}{*{20}{c}} x&{ - 3i}&1\\ y&1&i\\ 0&{2i}&{ - i} \end{array}} \right] = 6 + 11i\), then what are the values of x and y respectively? |
| A. | -3, 4 |
| B. | 3, 4 |
| C. | 3, -4 |
| D. | -3, -4 |
| Answer» B. 3, 4 | |
| 20. |
An equilateral triangle has each side equal to a. If the co-ordinates of its vertices are (x1, y1); (x2, y2): (x3, y3) then the square of the determinant \(\begin{vmatrix} x_1 & y_1 & 1 \\\ x_2 & y_2& 1 \\\ x_2 & y_2 & 1 \end{vmatrix}\) equals: |
| A. | None of these |
| B. | 4a2 |
| C. | 3a4 |
| D. | \(\dfrac{3a^4}{4}\) |
| Answer» E. | |
| 21. |
If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {\rm{\alpha }}&2\\ 2&{\rm{\alpha }} \end{array}} \right]\) and det (A3) = 125, then α is equal to |
| A. | ± 1 |
| B. | ± 2 |
| C. | ± 3 |
| D. | ± 5 |
| Answer» D. ± 5 | |
| 22. |
If a square matrix A is such that AAT = I = ATA, then |A| is equal to - |
| A. | 0 |
| B. | ± 1 |
| C. | ± 2 |
| D. | None of these |
| Answer» C. ± 2 | |
| 23. |
Let \(\mathop \sum \limits_{k = 1}^{10} f\left( {a + k} \right) = 16\left( {{2^{10}} - 1} \right),{\rm{}}\) where the function \(f\) satisfies f(x + y) = f(x)f(y) for all natural numbers x, y and f(1) = 2. Then the natural number 'a' is: |
| A. | 2 |
| B. | 16 |
| C. | 4 |
| D. | 3 |
| Answer» E. | |
| 24. |
If A is a square matrix of order 3 and det A = 5, then what is det [(2A)-1] equal to ? |
| A. | 1/10 |
| B. | 2/5 |
| C. | 8/5 |
| D. | 1/40 |
| Answer» E. | |
| 25. |
If A is an invertible matrix of order n and k is any positive real number, then the value of [det(kA)]-1 det A is |
| A. | k-n |
| B. | k-1 |
| C. | kn |
| D. | nk |
| Answer» B. k-1 | |
| 26. |
If a + b + c = 4 and ab + bc + ca = 0, then what is the value of the following determinant?\(\left| {\begin{array}{*{20}{c}} {{a}}&{{b}}&{{c}}\\ {{b}}&{{c}}&{{a}}\\ {{c}}&{{a}}&{{b}} \end{array}} \right|\) |
| A. | 32 |
| B. | -64 |
| C. | -128 |
| D. | 64 |
| Answer» C. -128 | |
| 27. |
\(\mathop {\lim }\limits_{x \to 1} \frac{{1 - \sqrt x }}{{{{\cos }^{ - 1}}x}}\) is equal to |
| A. | 0 |
| B. | \(\frac{1}{2}\) |
| C. | \(\frac{1}{4}\) |
| D. | 1 |
| Answer» B. \(\frac{1}{2}\) | |
| 28. |
If \(A = \left( {\begin{array}{*{20}{c}} 9&6\\ 8&7 \end{array}} \right)\) then det (A99 – A98) is |
| A. | 1 |
| B. | 48 |
| C. | 0 |
| D. | 299 |
| Answer» D. 299 | |
| 29. |
Consider the following statements in respect of the determinant \(\left| {\begin{array}{*{20}{c}} {{{\cos }^2}\frac{\alpha }{2}}&{{{\sin }^2}\frac{\alpha }{2}}\\ {{{\sin }^2}\frac{\beta }{2}}&{{{\cos }^2}\frac{\beta }{2}} \end{array}} \right|\)Where α, β are complementary angles1. The value of the determinant is \(\frac{1}{{√ 2 }}\cos \left( {\frac{{\alpha - \beta }}{2}} \right)\;\)2. The maximum value of the determinant is \(\frac{1}{\sqrt2}\)Which of the above statements is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» D. Neither 1 nor 2 | |
| 30. |
If A = \(\left| {\begin{array}{*{20}{c}} 1& -2\\ 3&\rm -k \end{array}} \right|\) is a singular matrix, then value of k |
| A. | 0 |
| B. | -6 |
| C. | 6 |
| D. | 8 |
| Answer» D. 8 | |
| 31. |
If a1, a2, a3, _ _ _ _ _, a9 are in GP, then what is the value of the following determinant?\(\left| {\begin{array}{*{20}{c}} {{ln\:a_1}}&{{ln\:a_2}}&{{ln\:a_3}}\\ {{ln\:a_4}}&{{ln\:a_5}}&{{ln\:a_6}}\\ {{ln\:a_7}}&{{ln\:a_8}}&{{ln\:a_9}} \end{array}} \right|\) |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» B. 1 | |
| 32. |
If a complex number is z = \(\left\{ {\frac{{3\; + \;4i}}{{1\; - \;2i}}} \right\}\), then |z| will be: |
| A. | \(\sqrt{5}\) |
| B. | 1 |
| C. | 2\(\sqrt{5}\) |
| D. | \(\frac{1}{\sqrt{5}}\) |
| Answer» B. 1 | |
| 33. |
Let A = [aij] and B = [bij] be two square matrices of order n and det(A) denote the determinant of A. Then, which of the following is not correct: |
| A. | If A is a diagonal matrix, then det(A) = a11 a22 ... ann. |
| B. | det(AB) = det(A) det(B). |
| C. | det(cA) = c [det(A)]. |
| D. | det(A) = det(AT), where AT denotes the transpose of the matrix A. |
| Answer» D. det(A) = det(AT), where AT denotes the transpose of the matrix A. | |
| 34. |
If the system of equation x + 2y - 3z = 2, (k + 3) z = 3, (2k + 1) y + z = 2 is consistent, then K is |
| A. | -3 and \( - \frac{1}{2}\) |
| B. | \( - \frac{1}{2}\) |
| C. | 1 |
| D. | 2 |
| Answer» B. \( - \frac{1}{2}\) | |
| 35. |
(cos 5θ - i sin 5θ)2 is same as |
| A. | cos 10θ + i sin 10θ |
| B. | cos 25θ - i sin 25θ |
| C. | (cos θ + i sin θ)-10 |
| D. | (c0s θ - i sin θ)-10 |
| Answer» D. (c0s θ - i sin θ)-10 | |
| 36. |
If \(\left| {\begin{array}{*{20}{c}} {a - b - c}&{2a}&{2a}\\ {2b}&{b - c - a}&{2b}\\ {2c}&{2c}&{c - a - b} \end{array}} \right| = \left( {a + b + c} \right){(x + a + b + c)^2}\), x ≠ 0 and a + b + c ≠ 0, then ‘x’ is equal to: |
| A. | abc |
| B. | -(a + b + c) |
| C. | 2(a + b + c) |
| D. | -2(a + b + c) |
| Answer» E. | |
| 37. |
If a + b + c = 0, then one of the solutions of\(\left| {\begin{array}{*{20}{c}} {a - x}&c&b\\ c&{b - x}&a\\ b&a&{c - x} \end{array}} \right| = 0\) is |
| A. | x = a |
| B. | \(x = \sqrt {\frac{{3\left( {{a^2} + {b^2} + {c^2}} \right)}}{2}} \) |
| C. | \(x = \sqrt {\frac{{2\left( {{a^2} + {b^2} + {c^2}} \right)}}{3}} \) |
| D. | x = 0 |
| Answer» E. | |
| 38. |
Find the condition on k, so that the system of equations: x + 3y = 5 and 2x + ky = 8 has a unique solution. |
| A. | k = 6 |
| B. | k ≠ 6 |
| C. | k ≠ 4 |
| D. | k = 4 |
| Answer» C. k ≠ 4 | |
| 39. |
If x + a + b + c = 0, then what is the value of \(\left| {\begin{array}{*{20}{c}} {x + a}&b&c\\ a&{x + b}&c\\ a&b&{x + c} \end{array}} \right|?\) |
| A. | 0 |
| B. | (a + b + c)2 |
| C. | a2 + b2 + c2 |
| D. | a + b + c - 2 |
| Answer» B. (a + b + c)2 | |
| 40. |
Let A and B be two invertible matrices of order 3 × 3. If det(ABAT) = 8 and det(AB(-1)) = 8, then det(BA(-1) BT) is equal to: |
| A. | \(\frac{1}{4}\) |
| B. | 1 |
| C. | \(\frac{1}{16}\) |
| D. | 16 |
| Answer» D. 16 | |
| 41. |
Let p, q and r be three distinct positive real numbers. If \(\rm D = \left| {\begin{array}{*{20}{c}} \rm p&\rm q&\rm r\\ \rm q&\rm r&\rm p\\ \rm r&\rm p&\rm q \end{array}} \right|,\) then which one of the following is correct? |
| A. | D < 0 |
| B. | D ≤ 0 |
| C. | D > 0 |
| D. | D ≥ 0 |
| Answer» C. D > 0 | |
| 42. |
A is a square matrix of order 3 such that its determinate is 4. What is the determinant of its transpose? |
| A. | 64 |
| B. | 36 |
| C. | 32 |
| D. | 4 |
| Answer» E. | |
| 43. |
If \(\left| {\begin{array}{*{20}{c}} {\rm{x}}&{\rm{y}}&0\\ 0&{\rm{x}}&{\rm{y}}\\ {\rm{y}}&0&{\rm{x}} \end{array}} \right| = 0\), then which one of the following is correct? |
| A. | \(\frac{{\rm{x}}}{{\rm{y}}}\) is one of the cube roots of unity |
| B. | x is one of the cube roots of unity |
| C. | y is one of the cube roots of unity |
| D. | \(\frac{{\rm{x}}}{{\rm{y}}}\) is one of the cube roots of -1 |
| Answer» E. | |
| 44. |
If A + B + C = \(\pi \), then, the value of \(\left| {\begin{array}{*{20}{c}} {\sin \left( {A + B + C} \right)}&{\sin B}&{\cos C}\\ { - \sin B}&0&{\tan A}\\ {\cos \left( {A + B} \right)}&{ - \tan A}&0 \end{array}} \right|\) is |
| A. | 0 |
| B. | 1 |
| C. | 2 sin A sin B |
| D. | 2 |
| Answer» B. 1 | |
| 45. |
If x, y, z are distinct real numbers and \(\left| {\begin{array}{*{20}{c}} x&{{x^2}}&{2 + {x^3}}\\ y&{{y^2}}&{2 + {y^3}}\\ z&{{z^2}}&{2 + {z^3}} \end{array}} \right| = 0\), then xyz = |
| A. | 1 |
| B. | -1 |
| C. | 2 |
| D. | -2 |
| Answer» E. | |
| 46. |
Let \(f(x) = \left| {\begin{array}{*{20}{c}} {{x^3}}&{\sin x}&{\cos x}\\ 6&{ - 1}&0\\ p&{{p^2}}&{{p^3}} \end{array}} \right|\), where p is a constant, then \(\frac{{{d^3}}}{{d{x^3}}}\left( {f(x)} \right)\) at x = 0 is |
| A. | p |
| B. | p + p2 |
| C. | p + p3 |
| D. | independent of p |
| Answer» E. | |
| 47. |
Let matrix B be the adjoint of a square matrix A, l be the identify matrix of same order as A. If k (≠ 0) is the determinate of the matrix A, then what is AB equal to? |
| A. | l |
| B. | kl |
| C. | k2l |
| D. | (1/k)l |
| Answer» C. k2l | |
| 48. |
Let \(Δ = \left| {\begin{array}{*{20}{c}} 1&{\sin \theta }&1\\ { - \sin \theta }&1&{\sin \theta }\\ { - 1}&{ - \sin \theta }&1 \end{array}} \right|\) The Δ lies in the interval |
| A. | [3, 4] |
| B. | [2, 4] |
| C. | [1, 4] |
| D. | None of these |
| Answer» C. [1, 4] | |
| 49. |
If A is a square matrix of order n > 1, then which one of the following is correct? |
| A. | det (-A) = det A |
| B. | det (-A) = (-1)n det A |
| C. | det (-A) = -det A |
| D. | det (-A) = n det A |
| Answer» C. det (-A) = -det A | |
| 50. |
If u, v and w (all positive) are the pth, qth and rth terms of a GP, then the determinant of the Matrix \(\left( {\begin{array}{*{20}{c}} {lnu}&p&1\\ {lnv}&q&1\\ {lnw}&r&1 \end{array}} \right)is\) |
| A. | 0 |
| B. | 1 |
| C. | (p - q) (q - r)(r - p) |
| D. | ln u × ln v × ln w |
| Answer» B. 1 | |