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Mathematics Questions

Mathematics Questions

1. Assertion (A): Domain of $f ( x )=\sin ^{-1} x+\cos ^{-1} x$ is $[-1,1]$
Reason (R): Domain of a function is the set of all possible values for which function will be defined.

Inverse Trigonometric Functions

2. The area enclosed between the curves $y = x^3$ and $y = \sqrt {x}$ is, in square units

KCET 2004 Application of Integrals

3. $3[\sin x -\cos x]^4 + 6[\sin x + \cos x]^2 + 4[\sin^6 x + \cos^6 x] =$

Trigonometric Functions

4. If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4\left(x-a^2\right)=0$ and the other two vertices are the points of intersection of the parabola and y -axis, is $250$ sq. units, then a value of ' $a$ ' is

JEE Main 2025

5. The average value of a function $f ( x )$ over the interval, $[ a , b ]$ is the number
$\mu=\frac{1}{b-a} \int\limits_a^b f(x) d x$
The square root $\left\{\frac{1}{b-a} \int\limits_a^b[f(x)]^2 d x\right\}^{1 / 2}$ is called the root mean square of $f$ on $[ a, b].$ The average value of $\mu$ is attained if $f$ is continuous on $[a, b]$.
The average value of $f(x)=\frac{\cos ^2 x}{\sin ^2 x+4 \cos ^2 x}$ on $[0, \pi / 2]$ is -

Integrals

6. Three lines $L_1: \vec{r}=\lambda \hat{i}, \lambda \in R, L_2: \vec{r}=\hat{k}+\mu \hat{j}, \mu \in R$ and $L_3: \bar{r}=\hat{i}+\hat{j}+v \hat{k}, v \in R$ are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P, Q$ and $R$ are collinear?

Vector Algebra

7. The extremities of latus rectum of a parabola are $(1,1)$ and $(1,-1)$. Then the equation of the parabola can be

Conic Sections

8. Let $S = \{(a, b) : a , b, \in Z, 0 \le a, a , b \le 18\}$. The number of elements $(x, y)$ in $S$ such that $3x + 4y + 5$ is divisible by $19$, is

KVPY 2014

9. General solution of the equation $\tan \theta \tan 2 \theta=1$ is given by

Trigonometric Functions

10. Consider a quadratic polynomial ' $C^{\prime}$ and a line ' $l$ ' as $C : y = x ^2-2 x \cos \theta+\cos 2 \theta+\cos \theta+\frac{1}{2}$ where $\theta \in\left[0,360^{\circ}\right]$ and $l: y = x$.
If $C$ touches the line $l$ then sum of all possible values of $\theta$, is

Straight Lines

11. If $\log _{1 / 3} \frac{3 x-1}{x+2}$ is less than unity then $x$ must lie in the interval -

Complex Numbers and Quadratic Equations

12. The range of the function $f\left(x\right)=\frac{x^{2}+8}{x^{2}+4}$, $x \in R$ is

Relations and Functions

13. Water is flow at rate of 5m3<\sup> /min into a conical vassal whose semi vertical angle is tan$^{-1} (\frac{1}{2})$Then there rate of change in height of water level. When height of water level is 10 m is

14. Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R=\{(A, B): A \cap B \neq \phi ; A, B \in M\}$ is :

Relations and Functions - Part 2

15. Let $f: R \rightarrow R$ be defined as $f(x)=x^3+3 x+2$ and $g(x)$ be a function such that $g(x)=x f^{-1}(x)-\int\limits_0^{f^{-1}(x)} f(t) d t$, then

Integrals

16. Let $A$ and $B$ be two events such that $P(A)=0.3$ and $P(A \cup B)=0.8$. If $A$ and $B$ are independent events. Then, $P(B)$ is equal to

Bihar CECE 2011

17. If $\sin x+\cos x=\sqrt{y+\frac{1}{y}}, x \in(0, \pi)$, then

Trigonometric Functions

18. The last two digits of the number $(23)^{14}$ are

Binomial Theorem

19. The number of common terms in the progressions $4,9,14$, $19, \ldots \ldots$, up to $25^{\text {th }}$ term and $3,6,9,12, \ldots \ldots$., up to $37^{\text {th }}$ term is:

Sequences and Series

20. Which of the following statements is/are true?
I. The values of median and mode can be determined graphically.
II. Mean, median and mode have the same unit.
III. Range is the best measure of dispersion.

Statistics

21. If $\tan \theta=\frac{\sin \alpha-\cos \alpha}{\sin \alpha+\cos \alpha}$, then which among the following is true?
I. $\sin \alpha+\cos \alpha=\sqrt{2}$
II. $\sin \alpha+\cos \alpha=\sqrt{2} \sec \theta$
III. $\sin \alpha+\cos \alpha=\sqrt{2} \cos \theta$
IV. $\sin \alpha+\cos \alpha=\sqrt{2} \sin \theta$

Trigonometric Functions

22. Number of points of intersection of $n$ straight lines if $n$ satisfies ${ }^{n+5} P_{n+1}=\frac{11(n-1)}{2} \times{ }^{n+3} P_n$ is

Permutations and Combinations

23. In a certain town $25 \%$ families own a cell phone, $15 \%$ families own a scooter and $65 \%$ families own neither a cell phone nor a scooter. If $1500$ families own both a cell phone and a scooter, then the total number of families in the town is

Sets

24. Area of the triangle formed by the vertex, focus and one end of latusrectum of the parabola $(x+2)^2=-12(y-1)$ is

Conic Sections

25. If the mean of $2, x$ and $y$ is 8 , then the mean of $x, y$ and 8 is

Statistics

26. The interval in which the function $f(x) = 2x^2 - \log \, x $, for $x > 0$ decreases, is

AP EAMCET 2018

27. If $A =\{ x , y \}$ then the power set of $A$ is:

Sets

28. If $x$ is real, then the range of $\frac{x^2 + 2x + 1}{x^2 + 2x + 7}$ is

AP EAMCET 2018

29. In the following figure, $O$ is the centre of the circle, $A C$ is the diameter and if $\angle A P B=120^{\circ}$, then find $\angle B Q C$.

Geometry

30. If $\cos 3 x+\sin \left(2 x-\frac{7 \pi}{6}\right)=-2$, then $x=$

Trigonometric Functions