Write down the first four terms of the expansion of \((1-x)^{-\frac{1}{2}}\) in ascending powers of \(x\) and also give the coefficient of \(x^n\). Prove that, when \(x\) is very small, \[ \frac{(1+2x)^{\frac{1}{2}}(1-5x)^{\frac{1}{3}}}{(1-11x)^{\frac{1}{4}}} = 1-\frac{1}{2}x. \]
Calculate to four places of decimals \[ (\cdot 0035)^{-\frac{1}{2}} \times (32\cdot 17)^{\frac{1}{5}} \quad \text{and} \quad \operatorname{cosec} 2^\circ 17'. \]
Find the conditions that the equation \(ax^2+2bx+c=0\) should have (i) both its roots positive and (ii) two equal roots.
By drawing the graph of \(y=\sin x\), prove that the equation \(x=10\sin x\) has seven real roots.
Prove that in a triangle \(\tan\frac{A-B}{2} = \frac{a-b}{a+b}\cot\frac{C}{2}\). In a triangle \(a=7\cdot 5, b=5, C=36^\circ 12'\), prove that \(A=103^\circ 22'\).
Find the condition that \(lx+my+n=0\) should touch the circle \(x^2+y^2+2ax=0\).
Differentiate \(x^{x^2}\), \((ax^2+b)^n\), \(x^2 \sin x\) and \(\frac{x+2}{(x+1)(x+3)}\).
Trace the curves (i) \(y^2(a-x)=x^3\), (ii) \(r=a+b\cos\theta\) (\(b>a\)).
Evaluate \[ \int x^2 e^x dx, \quad \int \frac{dx}{1+2x^2}, \quad \int_0^\infty xe^{-x^2}dx, \quad \int_0^{\frac{\pi}{2}} \sin^2 x dx. \]
Find the volume of the portion of the paraboloid formed by rotating the parabola \(y^2=4ax\) about the axis of \(x\), contained between the planes \(x=h\) and \(x=k\).