Algebra Study Resources for Competitive Exam Success

Algebra

Binomial theorem Complex numbers Determinants Inverse Trigonometric Functions Linear Inequalities Logarithms Matrices Permutations & combinations Quadratic equations Sequence & series Sets & Relations

Q. Find the value of the coefficient of x^2 in the expansion of (3x - 4)^4.

A. -144
B. -216
C. 216
D. 144

Solution

The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.

Correct Answer: A — -144

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Q. Find the value of the determinant \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) when \( a=1, b=2, c=3, d=4 \).

A. -2
B. 2
C. 0
D. 1

Solution

The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).

Correct Answer: A — -2

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Q. Find the value of the determinant \( |D| \) where \( D = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \).

A. -12
B. -10
C. -8
D. -6

Solution

The determinant is calculated as 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Correct Answer: A — -12

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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |

A. -1
B. 1
C. 2
D. 3

Solution

Using the determinant formula, we calculate it to be 1.

Correct Answer: B — 1

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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 2 |

A. -20
B. -10
C. 10
D. 20

Solution

Using the determinant formula, we calculate it to be -10.

Correct Answer: B — -10

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Q. Find the value of the determinant: | 2 3 1 | | 1 0 4 | | 5 6 7 |

A. -30
B. -20
C. 20
D. 30

Solution

Using the determinant formula, we calculate it to be -20.

Correct Answer: B — -20

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Q. Find the value of the determinant: | x 1 2 | | 3 x 4 | | 5 6 x | when x = 1.

A. -6
B. 0
C. 6
D. 12

Solution

Substituting x = 1 gives the determinant | 1  1  2 |  | 3  1  4 |  | 5  6  1 | = 6.

Correct Answer: C — 6

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Q. Find the value of x if 3x + 5 = 20.

A. 5
B. 10
C. 15
D. 20

Solution

Subtracting 5 from both sides gives 3x = 15, thus x = 15/3 = 5.

Correct Answer: A — 5

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Q. Find the value of z if z^2 + 4z + 8 = 0.

A. -2 + 2i
B. -2 - 2i
C. -4 + 0i
D. -4 - 0i

Solution

Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.

Correct Answer: A — -2 + 2i

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Q. Find the value of z if z^2 = -16.

A. 4i
B. -4i
C. 4
D. -4

Solution

Taking square root, z = ±√(-16) = ±4i.

Correct Answer: A — 4i

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Q. Find the value of z^2 if z = 1 + i.

A. 2i
B. 2
C. 0
D. 1

Solution

z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.

Correct Answer: B — 2

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Q. Find the value of \( k \) for which the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{vmatrix} = 0 \)

A. 5
B. 6
C. 7
D. 8

Solution

Setting the determinant to zero gives \( k = 6 \).

Correct Answer: B — 6

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Q. Find the value of \( \det \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).

A. 0
B. 1
C. 2
D. 3

Solution

The determinant of the identity matrix is always 1.

Correct Answer: B — 1

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Q. For the matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is the determinant \( |B| \)?

A. 0
B. 1
C. 2
D. 4

Solution

The determinant is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. For the quadratic equation 2x^2 - 4x + k = 0 to have real roots, what is the condition on k?

A. k >= 0
B. k <= 0
C. k >= 2
D. k <= 2

Solution

The discriminant must be non-negative: (-4)^2 - 4*2*k >= 0, which simplifies to k <= 2.

Correct Answer: C — k >= 2

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Q. For the quadratic equation ax^2 + bx + c = 0, if a = 1, b = -3, and c = 2, what are the roots?

A. 1 and 2
B. 2 and 1
C. 3 and 0
D. 0 and 3

Solution

The roots can be found using the quadratic formula: x = (3 ± √(9-8))/2 = 1 and 2.

Correct Answer: A — 1 and 2

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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the nature of the roots?

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant is 0, indicating that the roots are real and equal.

Correct Answer: B — Real and equal

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Q. For the quadratic equation x^2 + 2x + 1 = 0, what is the vertex of the parabola?

A. (-1, 0)
B. (-1, 1)
C. (0, 1)
D. (1, 1)

Solution

The vertex can be found using the formula (-b/2a, f(-b/2a)). Here, vertex is (-1, 0).

Correct Answer: A — (-1, 0)

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Q. For the quadratic equation x^2 + 2x + k = 0 to have no real roots, k must be:

A. < 0
B. ≥ 0
C. ≤ 0
D. > 0

Solution

The discriminant must be negative: 2^2 - 4*1*k < 0 => 4 < 4k => k > 1.

Correct Answer: A — < 0

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Q. For the quadratic equation x^2 + 4x + 4 = 0, what is the nature of the roots?

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant is 0 (b^2 - 4ac = 16 - 16 = 0), indicating real and equal roots.

Correct Answer: B — Real and equal

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Q. For the quadratic equation x^2 + 4x + k = 0 to have no real roots, k must be:

A. 0
B. 1
C. 2
D. 3

Solution

The discriminant must be negative: 4^2 - 4*1*k < 0 => 16 < 4k => k > 4.

Correct Answer: A — 0

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Q. For the quadratic equation x^2 + 4x + k = 0 to have real roots, what is the condition on k?

A. k >= 4
B. k <= 4
C. k > 0
D. k < 0

Solution

The discriminant must be non-negative: 4^2 - 4*1*k >= 0, which simplifies to k <= 4.

Correct Answer: A — k >= 4

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Q. For the quadratic equation x^2 + 6x + 8 = 0, what are the roots?

A. -2 and -4
B. -4 and -2
C. 2 and 4
D. 0 and 8

Solution

Factoring gives (x+2)(x+4) = 0, hence the roots are -2 and -4.

Correct Answer: B — -4 and -2

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Q. For the quadratic equation x^2 + 6x + 9 = 0, what is the nature of the roots?

A. Two distinct real roots
B. One real root
C. No real roots
D. Complex roots

Solution

The discriminant is 0, indicating one real root (a repeated root).

Correct Answer: B — One real root

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Q. For the quadratic equation x^2 + mx + n = 0, if the roots are 2 and 3, what is the value of n?

A. 5
B. 6
C. 7
D. 8

Solution

The product of the roots is n = 2 * 3 = 6.

Correct Answer: B — 6

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Q. For the quadratic equation x^2 + px + q = 0, if the roots are 1 and -3, what is the value of p?

A. 2
B. -2
C. 3
D. -3

Solution

The sum of the roots is 1 + (-3) = -2, hence p = -2.

Correct Answer: A — 2

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Q. For the quadratic equation x^2 - 10x + 25 = 0, what is the double root?

A. 5
B. 10
C. 0
D. 25

Solution

The equation can be factored as (x-5)^2 = 0, hence the double root is x = 5.

Correct Answer: A — 5

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Q. For the quadratic equation x^2 - 6x + k = 0 to have equal roots, what must be the value of k?

A. 6
B. 9
C. 12
D. 0

Solution

Setting the discriminant to zero: (-6)^2 - 4*1*k = 0 gives k = 9.

Correct Answer: B — 9

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Q. For which value of k does the equation x^2 + kx + 16 = 0 have real and distinct roots?

A. -8
B. -4
C. 0
D. 4

Solution

The discriminant must be positive: k^2 - 4*1*16 > 0 => k^2 > 64 => k > 8 or k < -8. Thus, k = -4 is valid.

Correct Answer: B — -4

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Q. For which value of k does the equation x^2 + kx + 4 = 0 have one root equal to 2?

A. -4
B. -2
C. 0
D. 2

Solution

Substituting x = 2 into the equation gives 2^2 + 2k + 4 = 0, leading to k = -4.

Correct Answer: B — -2

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Algebra

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