JEE MAIN Mathematics Previous Year Questions (PYQs) – Page 9 of 275
A Place for Latest Exam wise Questions, Videos, Previous Year Papers,
Study Stuff for MCA Examinations
Study Stuff for MCA Examinations
Showing 2742 questions
Let $P$ be the point of intersection of the lines
$\dfrac{x-2}{1}=\dfrac{y-4}{5}=\dfrac{z-2}{1}$ and $\dfrac{x-3}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{2}$.
Then, the shortest distance of $P$ from the line $4x=2y=z$ is:
1
2
3
4
Solution
If $I=\displaystyle\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$, then $\displaystyle\int_{0}^{2I}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$ equals:
1
2
3
4
Solution
$I=\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$
$I+\int_{0}^{\pi/2}\frac{\cos^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx=\frac{\pi}{2}$
⇒ $2I=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}$
Required integral:
$J=\int_{0}^{2I}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx=\int_{0}^{\pi/2}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$
Using symmetry:
$2J=\frac{\pi}{2}\int_{0}^{\pi/2}\frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$
Now,
$\int_{0}^{\pi/2}\frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx=\frac{\pi}{4}$
So,
$2J=\frac{\pi}{2}\cdot\frac{\pi}{4}=\frac{\pi^2}{8}$
⇒ $J=\frac{\pi^2}{16}$
$\boxed{\frac{\pi^2}{16}}$
$I+\int_{0}^{\pi/2}\frac{\cos^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx=\frac{\pi}{2}$
⇒ $2I=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}$
Required integral:
$J=\int_{0}^{2I}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx=\int_{0}^{\pi/2}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$
Using symmetry:
$2J=\frac{\pi}{2}\int_{0}^{\pi/2}\frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$
Now,
$\int_{0}^{\pi/2}\frac{\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx=\frac{\pi}{4}$
So,
$2J=\frac{\pi}{2}\cdot\frac{\pi}{4}=\frac{\pi^2}{8}$
⇒ $J=\frac{\pi^2}{16}$
$\boxed{\frac{\pi^2}{16}}$
Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $n \geqslant 3$. Then the sum of all the elements of $\mathrm{A}^{50}$ is :
1
2
3
4
Solution
Given $A^n=A^{n-2}+A^2-I$ for $n\ge 3$
Let $S_n$ = sum of all elements of $A^n$
Taking sum on both sides:
$S_n=S_{n-2}+S_2-3$ (since sum of $I=3$)
Compute base values:
$A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$
⇒ $S_1=3$
$A^2=\begin{bmatrix}1&0&0\\1&1&0\\1&0&1\end{bmatrix}$
⇒ $S_2=5$
So recurrence:
$S_n=S_{n-2}+2$
Even terms:
$S_2=5,\ S_4=7,\ S_6=9,\dots$
⇒ $S_{2k}=5+(k-1)\cdot2=2k+3$
For $n=50$, $k=25$:
$S_{50}=2(25)+3=53$
Let $S_n$ = sum of all elements of $A^n$
Taking sum on both sides:
$S_n=S_{n-2}+S_2-3$ (since sum of $I=3$)
Compute base values:
$A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$
⇒ $S_1=3$
$A^2=\begin{bmatrix}1&0&0\\1&1&0\\1&0&1\end{bmatrix}$
⇒ $S_2=5$
So recurrence:
$S_n=S_{n-2}+2$
Even terms:
$S_2=5,\ S_4=7,\ S_6=9,\dots$
⇒ $S_{2k}=5+(k-1)\cdot2=2k+3$
For $n=50$, $k=25$:
$S_{50}=2(25)+3=53$
If $\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}$ and $\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}$ be two given vectors $\vec{a}$ and $\vec{b}$ which are non-collinear, then the value of $\lambda$ for which vectors $\vec{\alpha}$ and $\vec{\beta}$ are collinear, is –
1
2
3
4
Solution
The term independent of $x$ in the expansion of
$\left(\dfrac{1}{60} - \dfrac{x^{8}}{81}\right)\left(2x^{2} - \dfrac{3}{x^{2}}\right)^{6}$
is equal to:
1
2
3
4
Solution
If $g$ is the inverse of a function $f$ and $f'(x)=\dfrac{1}{1+x^{5}}$, then $g'(x)$ is equal to:
1
2
3
4
Solution
The system of equations
$3x + y + 4z = 3$
$2x + ay - z = -3$
$x + 2y + z = 4$
has no solution, then the value of $\alpha$ is equal to:
1
2
3
4
Solution
for no solution $\Delta = 0$
$\begin{vmatrix} 3 & 1 & 4 \ 2 & \alpha & -1 \ 1 & 2 & 1 \end{vmatrix} = 0$
$\Rightarrow 3(\alpha + 2) + 1(-1 - 2) + 4(4 - \alpha) = 0$
$\Rightarrow 3\alpha + 6 - 3 + 16 - 4\alpha = 0$
$\Rightarrow 19 - \alpha = 0 \Rightarrow \alpha = 19$
$\Delta_x = \begin{vmatrix} 3 & 1 & 4 \ -3 & \alpha & -1 \ 4 & 2 & 1 \end{vmatrix}$
$\neq 0$
$\therefore$ no solution for $\alpha = 19$
If f(x + y) = f(x)f(y) and $\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$ , x, y $ \in $ N, where N is the set of all natural number, then thevalue of${{f\left( 4 \right)} \over {f\left( 2 \right)}}$ is :
1
2
3
4
Solution
The value of the definite integral$ \int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $ is equal to :
1
2
3
4
Solution
Let $\beta=\lim_{x\to 0}\dfrac{\alpha x-(e^{3x}-1)}{\alpha x(e^{3x}-1)}$ for some $\alpha\in\mathbb{R}$.
Then the value of $\alpha+\beta$ is:
1
2
3
4
Solution
JEE MAIN
Online Test Series, Information About Examination,
Syllabus, Notification
and More.
View More
JEE MAIN
Online Test Series, Information About Examination,
Syllabus, Notification
and More.
View More
Ask Your Question or Put Your Review.
