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Mathematics Questions

Mathematics Questions

1. The distance of the point $(1,0,2)$ from the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane

JEE Main 2015 Three Dimensional Geometry

2. Find the area enclosed by the parabola $4y = 3x^{2}$ and the line $2y = 3x + 12$.

Application of Integrals

3. In a group of 400 people, 160 are smokers and non- vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35 \%, 20 \%$ and $10 \%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

JEE Main 2021 Probability - Part 2

4. If $\underset{x \rightarrow 0}{lim}\left(\frac{a sin 2 x - x^{x + 1}}{ln \left(2 x + 1\right)}\right)^{\frac{2 x}{x^{2} + x}}=\frac{9}{4},a\in R,$ then sum of all possible values of $\_{}^{'}a_{}^{'}$ is

NTA Abhyas 2022

5. The locus of the centre of a circle which cuts orthogonally the parabola $y^{2}=4 x$ at $(1,2)$ will pass through

Conic Sections

6. $\int e^{\tan \theta}(\sec \theta-\sin \theta) d \theta$ is equal to

Integrals

7. If $x^{2}+x+1=0$, then the value of $\left(x+\frac{1}{x}\right)+\left(x^{2}+\frac{1}{x^{2}}\right)+\ldots+\left(x^{52}+\frac{1}{x^{52}}\right)$ equals

Complex Numbers and Quadratic Equations

8. The function $f\left(x\right)=tan x+\frac{1}{x},\forall x\in \left(0 , \frac{\pi }{2}\right)$ has

NTA Abhyas 2020 Application of Derivatives

9. Let $A=\begin{pmatrix} 0&0 &-1 \\[0.3em] 0 &-1 &0 \\[0.3em] -1 &0&0 \end{pmatrix}$ . The only correct statement about the matrix A is

AIEEE 2004 Matrices

10. In a classroom, one-fifth of the boys leave the class and the ratio of the remaining boys to girls is $2 : 3$. If further $44$ girls leave the class, then class the ratio of boys to girls is $5 : 2$. How many more boys should leave the class so that the number of boys equals that of girls?

KVPY 2017

11. Consider the function $f(x)=\begin{cases} 1, \text { if } x \leq 0 \\ 2, \text { if } x>0\end{cases}$ some statements are given below based on the above function. Identify the correct combination of true (T) and false (F) of the given statements.
(i) Left and right hand limit of the function is equal at $x=0$.
(ii) Limit of the function does not exist at $x=0$.
(iii) The left and right hand limit of $f$ at 0 is 1 and 2 , respectively.

Continuity and Differentiability

12. Inverse of the matrix $\begin{bmatrix}\cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta\end{bmatrix}$ is

Determinants

13. If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x \in R$, then :

JEE Main 2019 Application of Derivatives

14. Among $2$ children, a child may equally be a boy or a girl if the probability that exactly one of them is a boy is $p$, then $6 p=$

Probability - Part 2

15. Statement-1 : The equation $\sin x=e^x$ has infinite solutions.
Statement-2 : $y = e ^{ x }$ is unbounded functions.

Relations and Functions - Part 2

16. The lines $\vec{r}=\hat{i}+\hat{j}-\hat{k}+\lambda\left(3\hat{i}-\hat{j}\right)$ and $\vec{r}=4\hat{i}-\hat{k}+\mu\left(2\hat{i}+3\hat{k}\right)$ intersect at the point

KEAM 2011

17. Matrix A is such that $ T\propto \frac{1}{V} $ , where is the identity matrix. Then, for $ \text{6}\times \text{1}{{0}^{-\text{7}}}\text{A}-{{\text{m}}^{\text{2}}} $ , $ {{A}^{n}} $ is equal to

Jamia 2014

18. The value of $p$, so that the three lines $3 x+y-2=0$, $p x+2 y-3=0$ and $2 x-y-3=0$ may intersect at one point, is

Straight Lines

19. If $\sec \theta=m$ and $\tan \theta=n$, then $\frac{1}{m}\left[(m+n)+\frac{1}{(m+n)}\right]=$

AIRCH 2021

20. If $A=\int\limits_1^{\sin \theta} \frac{t}{1+t^2} d t, B=\int\limits_1^{\operatorname{cosec} \theta} \frac{1}{t\left(1+t^2\right)} d t$, then $\begin{vmatrix}A & A^2 & B \\ e^{A+B} & B^2 & -1 \\ 1 & A^2+B^2 & -1\end{vmatrix}$ equals

Integrals

21. The value(s) of $x$ satisfying the equation $\left(sin\right)^{- 1} \left(1 - x\right)-2\left(sin\right)^{- 1} ⁡ x=\frac{\pi }{2}$ is/are

NTA Abhyas 2020 Inverse Trigonometric Functions

22. $\left(\sin ^{-1} x\right)^2+\left(\sin ^{-1} y\right)^2+2\left(\sin ^{-1} x\right)\left(\sin ^{-1} y\right)=\pi^2$, then $x^2+y^2$ is equal to -

Inverse Trigonometric Functions

23. A non-zero vector $\vec{a}$ is such that its projections along vectors $\frac{\hat{i}+\hat{j}}{\sqrt{2}}, \frac{-\hat{i}+\hat{j}}{\sqrt{2}}$ and $\hat{k}$ are equal, then unit vector along $\vec{a}$ is

Vector Algebra

24. Let $S = \{(a, b) a, b \in Z , 0 \le a, b \le 18\}$. The number of lines in $R^2$ passing through $(0, 0)$ and exactly one other point in $S$ is

KVPY 2014

25. If $ f(x)={{\cot }^{-1}}\left[ \sqrt{\frac{1+\sin x}{1-\sin x}} \right],\,0\le x\le \frac{\pi }{4}, $ then $ f'\left( \frac{\pi }{4} \right) $ is equal to

J & K CET 2010 Continuity and Differentiability

26. $\begin{aligned} & -4,3 \\ & 6,3 \\ & -6,3 \\ & -4,9 \\ & 9,-4\end{aligned}$

KEAM 2021

27. If $f\left(x\right)=\prod_{k = 1}^{999} \left(x^{2} - 47 x + k\right)$ , then product of all real roots of $f\left(x\right)=0$ is

NTA Abhyas 2020 Complex Numbers and Quadratic Equations

28. If the lines $2 x+y-3=0,5 x+K y-3=0$ and $3 x-y-2=0$ are concurrent, then the value of $K$ is

Straight Lines

29. The distance between circumcentre and the orthcentre of the triangle formed by the points $(2,1,5),(3,2,3)$ and $(4,0,4)$ is

AIRCH 2021

30. The two curves $x^3-3xy^2+2=0$ and $3x^2y-y^3-2=0$

Application of Derivatives