Determinants - Maharashtra Board STD 12 Science Maths Questions

Q 1MCQ1 Mark

The value of $\begin{vmatrix}5^2&5^3&5^4\\5^3&5^4&5^5\\5^4&5^5&5^6\end{vmatrix}$ is:

A
$5^2$
✓
$0$
C
$5^{13}$
D
$5^9$

Answer: B.

Q 2MCQ1 Mark

$\begin{vmatrix}\log_3512&\log_43\\\log_38&\log_49\end{vmatrix}\times\begin{vmatrix}\log_23&\log_83\\\log_34&\log_34\end{vmatrix}$

A
$7$
✓
$10$
C
$1$
D
$17$

Answer: B.

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Q 3MCQ1 Mark

If $\text{D}_\text{k}=\begin{vmatrix}1&\text{n}&\text{n}\\2\text{k}&\text{n}^2+\text{n}+2&\text{n}^2+\text{n}\\2\text{k}-1&\text{n}^2&\text{n}^2+\text{n}+2\end{vmatrix} $ and $\sum\limits_{\text{k}=1}^\text{n}\text{D}_\text{k}=48,$ then $n$ equals:

✓
$4$
B
$6$
C
$8$
D
None of these.

Answer: A.

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Q 4MCQ1 Mark

Let $\begin{vmatrix}\text{x}^2+3\text{x}&\text{x}-1&\text{x}+ 3\\\text{x}+1&-2\text{x}&\text{x}-4\\\text{x}-3&\text{x}+4&3\text{x}\end{vmatrix}=\text{ax}^4+\text{bx}^3+\text{cx}^2+\text{dx}+\text{e}$ be an identity in $x,$ where $a, b, c, d, e$ are independent of $x.$ Then the value of $e$ is:

A
$4$
✓
$0$
C
$1$
D
None of these.

Answer: B.

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Q 5MCQ1 Mark

If $\begin{vmatrix}\text{a}&\text{p}&\text{x}\\\text{b}&\text{q}&\text{y}\\\text{c}&\text{r}&\text{z}\end{vmatrix}=16,$ then the value of $\begin{vmatrix}\text{p}+\text{x}&\text{a}+\text{x}&\text{a}+\text{p}\\\text{q}+\text{y}&\text{b}+\text{y}&\text{b}+\text{q}\\\text{r}+\text{z}&\text{c}+\text{z}&\text{c}+\text{r}\end{vmatrix}$ is:

A
$4$
B
$8$
C
$16$
✓
$32$

Answer: D.

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Q 6Solve the Following Question.(2 Marks)2 Marks

If $\begin{vmatrix}2\text{x}+5&3\\5\text{x}+2&9\end{vmatrix}=0,$ find x.

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Q 7Solve the Following Question.(2 Marks)2 Marks

Find the value of the determinant $\begin{vmatrix}4200&1201\\4205&4203\end{vmatrix}$

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Q 8Solve the Following Question.(2 Marks)2 Marks

If $\text{A}=\begin{bmatrix}1&2\\3&-1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\-1&0\end{bmatrix},$ find |AB|.

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Q 9Solve the Following Question.(2 Marks)2 Marks

If A is a square matrix of order $3$ with determinant $4$, then write the value of $|-A|$.

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Q 10Solve the Following Question.(2 Marks)2 Marks

If $\begin{vmatrix}2\text{x}&\text{x}+3\\2(\text{x}+1)&\text{x}+1\end{vmatrix}=\begin{vmatrix}1&5\\3&3\end{vmatrix},$ then write the value of x.

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Q 11Solve the Following Question.(3 Marks)3 Marks

Evaluate $\triangle=\begin{vmatrix}0&\sin\alpha&-\cos\alpha\\-\sin\alpha&0&\sin\beta\\\cos\alpha&-\sin\beta&0 \end{vmatrix}$

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Q 12Solve the Following Question.(3 Marks)3 Marks

Show that $\begin{vmatrix}\sin10^\circ&-\cos10^\circ\\\sin80^\circ&\cos80^\circ \end{vmatrix}=1$

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Q 13Solve the Following Question.(3 Marks)3 Marks

Evaluate $\begin{vmatrix}2&3&7\\13&17&5\\15&20&12\end{vmatrix}^2$

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Q 14Solve the Following Question.(3 Marks)3 Marks

Using determinants, find the equation of the line joining the points:
(1, 2) and (3, 6)

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Q 15Solve the Following Question.(3 Marks)3 Marks

Using determinants, find the value of k so that the points $(k, 2 - 2k), (-k + 1, 2k)$ and $(-4 - k, 6 - 2k)$ may be collinear.

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Q 16Solve the Following Question.(5 Marks)5 Marks

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission.

Month	Sale of units			Total commission drawn (in Rs.)
	A	B	C
Jan	90	100	20	800
Feb	130	50	40	900
March	60	100	30	850

Find out the rates of commission on items A, B and C by using determinant method.

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Q 17Solve the Following Question.(5 Marks)5 Marks

Solve the following systems of linear equations by cramer's rule:

$x + y + z + 1 = 0,$

$ax + by + cz + d = 0,$

$a^2x + b^2y + x^2z + d^2 = 0$

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Q 18Solve the Following Question.(5 Marks)5 Marks

Solve the following systems of linear equations by cramer's rule:

$6x + y - 3z = 5,$

$x + 3y - 2z = 5,$

$2x + y + 4z = 8$

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Q 19Solve the Following Question.(5 Marks)5 Marks

Solve the following systems of linear equations by cramer's rule:

5x - 7y + z = 11,

6x - 8y - z = 15,

3x + 2y - 6z = 7

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Q 20Solve the Following Question.(5 Marks)5 Marks

If $\sin(\text{xy})+\frac{\text{y}}{\text{x}}=\text{x}^2-\text{y}^2,$ find $\frac{\text{dy}}{\text{dx}}$

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Determinants Maths - Determinants

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