List of top Mathematics Questions asked in KEAM

\( \int \frac{(\sin x + \cos x)(2 - \sin 2x)}{\sin^2 2x} \, dx \)

\( \lim_{x\to0} \frac{1+x-e^x}{x^2} \)

The differential equation whose general solution is \( y = e^x(A\cos x + B\sin x) \) is

The real part of \( (i - \sqrt{3})^{13} \) is

\( \int_{\pi/4}^{3\pi/4} \frac{x}{1+\sin x} \, dx \) is equal to

\( \int_{0}^{\pi/2} \frac{2\sin x}{2\sin x + 2\cos x} dx \)

Area bounded by \( y=\sin^2 x \), \( x=\frac{\pi}{2} \), \( x=\pi \)

\( \lim_{x\to0} \frac{\int_0^{x^2} \sin(\sqrt{t}) \, dt}{x^2} \)

Maximum value of \( 2x^3 -15x^2 +36x +4 \)

\( \lim_{x\to\infty} \left(\sqrt{x^2+1} - \sqrt{x^2-1}\right) \)

If \( f \) is differentiable and \( \lim_{h\to0} \frac{f(1+h)-f(1)}{h}=5 \), find \( f'(1) \)

If \( \int f(x)\cos x \, dx = \frac{1}{2}\{f(x)\}^2 + c \), then \( f\left(\frac{\pi}{2}\right) \) is

Equation of circle with centre \( (2,2) \) passing through \( (4,5) \)

Point equidistant from \( (2,0,3), (0,3,2), (0,0,1) \)

Let \( f:(-1,1)\to(-1,1) \) be continuous, \( f(x)=f(x^2) \), and \( f(0)=\frac{1}{2} \). Find \( 4f\left(\frac{1}{4}\right) \)

If \( ^nC_{r-1}=36, ^nC_r=84, ^nC_{r+1}=126 \), then \( n= \)

Let \( f:\mathbb{N}\to\mathbb{N} \) be such that \( f(1)=2 \) and \( f(x+y)=f(x)f(y) \). If \( \sum_{k=1}^{n} f(a+k)=16(2^n-1) \), then \( a \) is

If \( \vec{a} \) and \( \vec{b}=3\hat{i}+6\hat{j}+6\hat{k} \) are collinear and \( \vec{a}\cdot\vec{b}=27 \), then \( \vec{a} \) is

If \( |\vec{a}|=13, |\vec{b}|=5 \) and \( \vec{a}\cdot\vec{b}=30 \), then \( |\vec{a}\times\vec{b}| \) is

Distance between two parallel lines \( y = 2x + 4 \) and \( y = 2x - 1 \) is

If \( ^{56}P_{r+6} : \, ^{54}P_{r+3} = 30800 : 1 \), then \( r \) is equal to

\( \binom{7}{0}+\binom{7}{1} + \binom{7}{2}+\binom{7}{3} + \cdots + \binom{7}{7} \)

Coefficient of \( x \) in \( (1 - 3x + 7x^2)(1 - x)^{16} \)

\( \sin 15^\circ = \)

If \( \theta \) is the angle between the pair of straight lines \( x^2 - 5xy + 4y^2 + 3x - 4 = 0 \), then \( \tan^2\theta \) is equal to