Find the- domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1} \cdot \)

1

(a)

\(\text { Given function, } f(x)=\sin ^{-1} \sqrt{x-1}\)
\(\text { For domain of } f(x), 0 \leq \sqrt{x-1} \leq 1\)
\(\Rightarrow 0 \leq x-1 \leq 1 \Rightarrow 1 \leq x \leq\)
\(\therefore x \in[1,2]\)

Find the domain of the function \(\cos ^{-1}(2 x-1)\)

1

(a)

\(\text{ we have}, f(x)=cos \)
\(−1 (2x−1)\)
\(∵−1≤2x−1≤1\)
\(⇒ 0 ≤ 2 x ≤ 2\)
\(⇒0≤x≤1\)
\(∴x∈[0,1]\)

If A is a skew-symmetric matrix, then show that A² is a symmetric matrix.

1

(a)

Now,(A²)'=(AA)'
= A' A'= (-A)(-A) [.: A'= -A] = A²

If possible, then find the sum of the matrices A and B where \(\text {A models} \left[\begin{array}{cc}\sqrt{3} & 1 \\ 2 & 3\end{array}\right]\)and \(B=\left[\begin{array}{ccc}x & y & z \\ a & b & c\end{array}\right]\)

1

(a)

Not possible

If A is a matrix of order 3x 3, then find the number of minors in determinant A.

1

(a)

Since, the matrix have 9 elements, therefore its corresponding determinant have 9 elements.
Hence, there are 9 minors in determinant of A.

If A and B are matrix of order 3 and IAI = 5, IBI = 3, then find the value of |3AB|.

1

(a)

Clearly,\(|3 A B|=3^{3}|A||B|=27 \times 5 \times 3=405\) 
[if matrix A is of order n \times n, then \( |k A|=k^{n}|A|\)\(\text { and }|A B|=|A||B|\) ]

If A is a matrix of order 3 x 3,then show that (A²)^-1=(A^-1)²

1

(a)

\(\left(A^{2}\right)^{-1}=(A A)^{-1}=\left(A^{-1} A^{-1}\right) \quad\left[\because(A B)^{-1}=B^{-1} A^{-1}\right]\) 
\(=\left(A^{-1}\right)^{2}\)

If \(y=\tan (x+y), \text { then find } \frac{a y}{d x}\) 

1

(a)

\(-\operatorname{cosec}^{2}(x+y)\)

Find the derivative of sec(x + y) =xy w.r.t. x.

1

(a)

\(\frac{y-\sec (x+y) \tan (x+y)}{\sec (x+y) \tan (x+y)-x}\)

Integrate \(\left(\frac{2 a}{\sqrt{x}}-\frac{b}{x^{2}}+3 c \sqrt[3]{x^{2}}\right) \text { w.r.t. } x\)

1

(a)

Let \(I=\int\left(\frac{2 a}{\sqrt{x}}-\frac{b}{x^{2}}+3 c \sqrt[3]{x^{2}}\right) d x\) 
\( =\int \frac{2 a}{\sqrt{x}} d x-\int \frac{b}{x^{2}} d x+\int 3 c \sqrt[3]{x^{2}} d x \)
\(=2 a \int x^{-1 / 2} d x-b \int x^{-2} d x+3 c \int x^{2 / 3} d x \)
\(=2 a\left[\frac{x^{(-1 / 2)+1}}{(-1 / 2)+1}\right]-b\left[\frac{x^{-2+1}}{-2+1}\right]+3 c\left[\frac{x^{(2 / 3)+1}}{(2 / 3)+1}\right]+C \)
\(=2 a\left[\frac{x^{1 / 2}}{1 / 2}\right]-b\left[\frac{x^{-1}}{(-1)}\right]+3 c\left[\frac{x^{5 / 3}}{5 / 3}\right]+C\)
\( {\left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}\right]} \)
\(=4 a \sqrt{x}+\frac{b}{2}+\frac{9 c}{x} x^{5 / 3}+C \)

Evaluate the following integral.
\(\int_{-\pi / 3}^{\pi / 3} \sin ^{3} x d x\)

1

(a)

Use the property \(\int_{a}^{b} f(x) d x=\int_{a}^{b} f(a+b-x) d x\) 
= 3

Draw a rough sketch of the curve  \(y=\sqrt{x-1}\) the interval [1,5]. Find the area under the curve and the lines x = 1 and x = 5.

1

(a)

= \(\frac{16}{3} \text { sq units }\)