Vardaan Learning Institute

Evaluate: $\begin{vmatrix} \cos 15^\circ & \sin 15^\circ \\ \sin 75^\circ & \cos 75^\circ \end{vmatrix}$.

If $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}$, then find the value of $x$.

Evaluate $\begin{vmatrix} \log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9 \end{vmatrix}$.

Let $\omega$ be a complex cube root of unity. Evaluate $\begin{vmatrix} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{vmatrix}$.

Find the maximum value of a $2 \times 2$ determinant whose elements belong to the set $\{0, 1\}$.

Evaluate the determinant $\Delta = \begin{vmatrix} 1 & a & a^2 \\ 0 & 1 & a \\ 0 & 0 & 1 \end{vmatrix}$.

Evaluate $\Delta = \begin{vmatrix} a+ib & c+id \\ -c+id & a-ib \end{vmatrix}$, where $i^2 = -1$.

If $A$ is a square matrix of order 3 and $|A| = -4$, find the value of $|-3A|$.

Determine the roots of the equation $\begin{vmatrix} x & 0 & 8 \\ 4 & 1 & 3 \\ 2 & 0 & x \end{vmatrix} = 0$.

Find the value of $x$ for which $\begin{vmatrix} 3 & x \\ x & 1 \end{vmatrix} = \begin{vmatrix} 3 & 2 \\ 4 & 1 \end{vmatrix}$.

Let $A = \begin{bmatrix} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{bmatrix}$. If $|A| = 4$, find the value of $\lambda$.

Evaluate: $\begin{vmatrix} \sin^2 \theta & \cos^2 \theta & 1 \\ \cos^2 \theta & \sin^2 \theta & 1 \\ -10 & 12 & 2 \end{vmatrix}$.

Evaluate: $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}$.

Evaluate $\Delta = \begin{vmatrix} 0 & x & y \\ -x & 0 & z \\ -y & -z & 0 \end{vmatrix}$.

Solve for $x$: $\begin{vmatrix} 1 & 2 & x \\ 1 & 1 & 1 \\ 2 & 1 & -1 \end{vmatrix} = 0$.

Without expanding, prove that $\begin{vmatrix} x & a & x+a \\ y & b & y+b \\ z & c & z+c \end{vmatrix} = 0$.

Using properties of determinants, prove that $\begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix} = (a-b)(b-c)(c-a)$.

Prove that $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix} = 0$.

Show that $\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix} = 0$.

Prove that $\begin{vmatrix} a & a^2 & a^3 \\ b & b^2 & b^3 \\ c & c^2 & c^3 \end{vmatrix} = abc(a-b)(b-c)(c-a)$.

Let $A$ and $B$ be $3 \times 3$ matrices such that $|A| = 5$ and $|B| = -2$. Find the value of $|2AB|$.

Without expanding, evaluate $\begin{vmatrix} 41 & 1 & 5 \\ 79 & 7 & 9 \\ 29 & 5 & 3 \end{vmatrix}$. (Hint: Express $C_1$ in terms of $C_2, C_3$).

If $A$ is a skew-symmetric matrix of order 3, prove using properties that $|A| = 0$.

Show that $\begin{vmatrix} x+4 & 2x & 2x \\ 2x & x+4 & 2x \\ 2x & 2x & x+4 \end{vmatrix} = (5x+4)(4-x)^2$.

Evaluate $\begin{vmatrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{vmatrix}$ by taking common factors from rows/columns.

Prove that $\begin{vmatrix} b+c & a & a \\ b & c+a & b \\ c & c & a+b \end{vmatrix} = 4abc$.

Without expanding, find the value of $\begin{vmatrix} \sin \alpha & \cos \alpha & \sin(\alpha+\delta) \\ \sin \beta & \cos \beta & \sin(\beta+\delta) \\ \sin \gamma & \cos \gamma & \sin(\gamma+\delta) \end{vmatrix}$.

If $a, b, c$ are in A.P., evaluate $\begin{vmatrix} 2y+4 & 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{vmatrix}$.

Evaluate $\begin{vmatrix} 1 & \omega^n & \omega^{2n} \\ \omega^n & \omega^{2n} & 1 \\ \omega^{2n} & 1 & \omega^n \end{vmatrix}$ where $\omega$ is a complex cube root of unity and $n$ is not a multiple of 3.

Prove that $\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix} = (1-x^3)^2$.

Find the area of the triangle whose vertices are $A(3, 8)$, $B(-4, 2)$, and $C(5, 1)$ using determinants.

The area of a triangle is $9$ sq units. Two of its vertices are $(3, 0)$ and $(0, 2)$. If the third vertex lies on the y-axis, find its coordinates.

Find the value of $k$ if the points $(k, 2-2k)$, $(-k+1, 2k)$, and $(-4-k, 6-2k)$ are collinear.

Find the equation of the line joining $A(1, 3)$ and $B(0, 0)$ using determinants and find $k$ if $D(k, 0)$ is a point such that area of $\triangle ABD$ is 3 sq units.

If the points $(a, 0)$, $(0, b)$, and $(1, 1)$ are collinear, prove that $\frac{1}{a} + \frac{1}{b} = 1$.

Show that points $A(a, b+c)$, $B(b, c+a)$, $C(c, a+b)$ are collinear.

A triangle has vertices at $(a, c)$, $(b, c)$, and $(c, a)$. Calculate its area using determinants.

Check whether the points $(2, 5)$, $(4, 6)$, and $(8, 8)$ form a triangle.

Find the area of the quadrilateral whose vertices are $(-3, 2)$, $(5, 4)$, $(7, -6)$ and $(-5, -4)$ by splitting it into two triangles.

If $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ are vertices of an equilateral triangle whose sides are of length $a$, then prove that $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}^2 = \frac{3a^4}{4}$.

If the points $(x, -2)$, $(5, 2)$, and $(8, 8)$ are collinear, find $x$.

Find the equation of the line passing through $(-1, 3)$ and $(2, -4)$ using the determinant method.

If the area of a triangle with vertices $(2, 4)$, $(2, 6)$ and $(2, k)$ is $0$, what can you deduce about $k$?

If the points $(2, -3)$, $(\lambda, -1)$ and $(0, 4)$ are collinear, find $\lambda$.

Let the vertices of a triangle be $(1, -3)$, $(4, p)$, and $(-9, 7)$. If its area is $15$ sq units, find the values of $p$.

Find the minor and cofactor of the element $6$ in the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$.

For the matrix $A = \begin{bmatrix} 2 & -1 & 4 \\ 0 & 3 & -2 \\ -1 & 5 & 1 \end{bmatrix}$, evaluate $|A|$ using the cofactors of the second row.

If $A$ is a square matrix of order 3 and $|A| = 8$, find the value of $|adj A|$.

Find $adj(A)$ for $A = \begin{bmatrix} 3 & -4 \\ 2 & 1 \end{bmatrix}$.

If $A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$, find $A(adj A)$ without calculating the entire adjoint matrix.

If $A$ is an invertible matrix of order 2, then find $|A^{-1}|$ in terms of $|A|$.

Let $A$ be a non-singular matrix of order $3 \times 3$. If $|adj A| = 64$, find $|A|$.

Find the inverse of the diagonal matrix $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{bmatrix}$.

For the determinant $\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$, verify that $a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33} = 0$.

Show that if $A$ is a symmetric matrix, then $adj A$ is also a symmetric matrix.

If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$, find $A^{-1}$.

If $A$ is a square matrix of order 3 such that $A(adj A) = 10I$, find $|adj A|$.

Find the adjoint of $A = \begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix}$.

Verify $(AB)^{-1} = B^{-1}A^{-1}$ for $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}$.

Evaluate $|adj(adj A)|$ if $A$ is a $3 \times 3$ matrix and $|A| = 3$.

Use matrix method to solve: $5x + 2y = 4$, $7x + 3y = 5$.

Examine the consistency of the system: $x + 3y = 5$, $2x + 6y = 8$.

Find the inverse of $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$.

Solve the system of equations by Cramer's Rule: $x + y + z = 1$, $2x + 2y + 3z = 6$, $x + 4y + 9z = 3$.

Determine the values of $\lambda$ for which the system of equations $3x - y + z = 0$, $15x - 6y + 5z = 0$, $\lambda x - 2y + 2z = 0$ has non-trivial solutions.

Find the matrix $X$ such that $X \begin{bmatrix} 1 & 2 \\ 5 & 3 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ -1 & 0 \end{bmatrix}$.

If $A = \begin{bmatrix} 2 & -3 \\ 3 & 4 \end{bmatrix}$, show that $A^2 - 6A + 17I = O$. Hence find $A^{-1}$.

Solve the system of equations using matrix method: $\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4$, $\frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1$, $\frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2$.

Given $A = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$, find $AB$. Use this to solve $x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7$.

A sum of Rs 10,000 is invested in three different schemes at 2%, 3% and 5% per annum. The total annual interest is Rs 340. The interest from the third scheme is Rs 100 more than the first. Formulate equations and solve using matrices.

Check the consistency: $x + y + z = 1$, $2x + 3y + 2z = 2$, $ax + ay + 2az = 4$. For what value of $a$ is the system inconsistent?

Find the derivative w.r.t $x$ of $\Delta(x) = \begin{vmatrix} x^2 & x & 1 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{vmatrix}$.

Evaluate $\int_0^1 \begin{vmatrix} 2x & 1 \\ 1 & x^2 \end{vmatrix} dx$.

Solve the homogeneous system: $x + y - z = 0$, $2x - 3y + z = 0$, $x - 4y + 2z = 0$.

Solve using matrices: $3x - 2y + 3z = 8$, $2x + y - z = 1$, $4x - 3y + 2z = 4$.