1. Tardigrade
  2. Question
  3. Mathematics

Mathematics Questions

Mathematics Questions

1. If $\alpha $ and $\beta $ are the solutions of $cot x=-\sqrt{3}$ in $\left[0,2 \pi \right]$ and $\alpha $ and $\gamma $ are the solutions of $cosec x=-2$ in $\left[0,2 \pi \right],$ then the value of $\frac{\left|\alpha - \beta \right|}{\beta + \gamma }$ is equal to

NTA Abhyas 2022

2. If the 4th term in the expansion of $ \pi $ ; $ \frac{\pi }{2} $ , is $ \frac{13}{56} $ and three normals to the parabola $ \frac{3}{56} $ are drawn through a point (q,0), then

Jamia 2015

3. Which of the following is the equation of a circle?

Conic Sections

4. If $\alpha, \beta(\beta>\alpha)$ are the roots of $f(x) \equiv a x^{2}+b x+c=0, a \neq 0$ and $f(x)$ is an even function and $I=\int_{\alpha}^{\beta} \frac{e_{f}^{f\left(\frac{f(x)}{x-\alpha}\right)}}{f\left(\frac{f(x)}{x-\alpha}\right)+e^{f}\left(\frac{f(x)}{x-\beta}\right)}$, then $|I|$ is equal to

Integrals

5. The ellipse $x^2 + 4y^2 = 4$ is inscribed in a rectangle aligned with the coordinate axes, which is in turn inscribed in another ellipse that passes through the point $(4, 0)$. Then the equation of the ellipse is

Conic Sections

6. If $a, b$ and $c$ form a geometric progression with common ratio $r$, then the sum of the ordinates of the points of intersection of the line $a x+b y+c=0$ and the curve $x+2 \,y^{2}=0$ is

EAMCET 2012

7. Match the following definite integrals in column I with their corresponding values in column II and choose the correct option from the codes given below.
Column I Column II
A $\int\limits_{-5}^5|x+2| d x$ 1 5
B $\int\limits_2^8|x-5| d x$ 2 $\frac{16 \sqrt{2}}{15}$
C $\int\limits_0^2 x \sqrt{2-x} d x$ 3 29
D $\int\limits_0^4|x-1| d x$ 4 9

Integrals

8. If $n \in N$, then $n^{3} + 2n$ is divisible by

Principle of Mathematical Induction

9. Match the following integrals in column I with their corresponding values in column II and choose the correct option from the codes given below.
Column I Column II
A $\int \frac{1}{x-x^3} d x$ 1 $\frac{-2}{a} \sqrt{\frac{a-x}{x}}+C$
B $\int \frac{1}{\sqrt{a+x}+\sqrt{x}+b} d x$ 2 $\frac{1}{2} \log \left|\frac{x^2}{1-x^2}\right|+C$
C $\int \frac{1}{x \sqrt{a x-x^2}} d x$ 3 $-\left(1+x^{-4}\right)^{1 / 4}+C$
D $\int \frac{1}{x^2\left(x^4+1\right)^{3 / 4}} d x$ 4 $\frac{2}{3(a-b)}\left[(x+a)^{3 / 2} -(x+b)^{3 / 2}\right]+C$

Integrals

10. If $f(x)=a+b x+c x^{2}$ and $\alpha, \beta, \gamma$ are roots of the equation $x^3=1$, then $ \begin{vmatrix}a&b&c\\ b&c&a\\ c&a&b\end{vmatrix}$ is equal to

Determinants

11. If $\vec{a}+2\vec{b}+3\vec{c} = \vec{0}$, then $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=$

AIRCH 2021

12. Statement 1: The variance of first n odd natural numbers is $\frac{n^2 - 1}{3}$.
Statement 2: The sum of first n odd natural number is $n^2$ and the sum of square of first n odd natural numbers is $\frac{ n (4n^2 + 1)}{3}$ .

Statistics

13. If $a>0, b>0$ then the maximum area of the parallelogram whose three vertices are $O(0,0) \cdot A(a \cos \theta, b \sin \theta)$ and $B(a \cos \theta,-b \sin \theta)$ is

WBJEE 2021

14. "Paris is in England" is a ...X... . Here, X refers to

Mathematical Reasoning

15. If a line segment $A M=a$ moves in the plane $X O Y$ remaining parallel to $O X$ so that the left end point $A$ slides along the circle $x^{2}+y^{2}=a^{2}$, the locus of $M$ is

Conic Sections

16. Find the derivative of the $sin\,x$ at $x = 0$

Limits and Derivatives

17. A coin is tossed once, then the sample space is

Probability

18. A father with 8 childrens takes them 3 at a time to the zoological gardens, as often as he can without taking the same 3 childrens together more than once. The number of times he will go the garden is

Permutations and Combinations

19. If $\frac{sin\,A}{sin\,B}=m$ and $\frac{cos\,A}{cos\,B}=n$ then find the value of $tan \,B$;
$n^2 < 1 < m^2$.

Trigonometric Functions

20. In a $\Delta A B C$, if $\angle A=90^{\circ}$, then $\cos ^{-1}\left(\frac{R}{r_{2}+r_{3}}\right)$ is equal to

AP EAMCET 2016

21. If $A$ is a skew symmetric matrix of order $3$ , $B$ is a $3\times 1$ column matrix and $C=B^{T}AB$ , then which of the following is false?

NTA Abhyas 2020 Matrices

22. If $ {{\log }_{2}}({{5.2}^{x}}+1),{{\log }_{4}}({{2}^{1-x}}+1) $ and 1 are in AP, then $ x $ equals

Jamia 2009

23. If $0 \le x < \frac{\pi}{2}$ , then the number of values of $ x$ for which $sin \,x-sin\,2x+sin\,3x = 0$, is

JEE Main 2019 Trigonometric Functions

24. If $ y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right), $ then $ \frac{dy}{dx} $ is equal to

KEAM 2011

25. State $T$ for true and $F$ for false.
(i) The equation of the circle having centre at ($3, -4)$ and touching the line $5x + 12y - 12 = 0$ is $\left(x-3\right)^{2}+\left(y+4\right)^{2}=\left(\frac{45}{13}\right)^{2}\cdot$
(ii) The equation of the circle circumscribing the triangle whose sides are the lines $y = x + 2$, $3y = 4x$, $2y = 3x$ is $x^{2} - y^{2} - 46x + 2 2y = 0$.
(iii) The equation of the parabola having focus at $(-1, -2)$ and directrix is $x - 2 y + 3 = 0$ is $x^{2 }+y^{2} + 3x +32y + 4 xy + 1 6 = 0$.
(i) (ii) (iii)
(a) $F$ $\,$ $F$ $\,$ $F$ $\,$
(b) $F$ $\,$ $T$ $\,$ $F$ $\,$
(c) $F$ $\,$ $F$ $\,$ $T$ $\,$
(d) $T$ $\,$ $F$ $\,$ $F$ $\,$

Conic Sections

26. Let $S=\left\{1, 2,3,....... ,n \right\}$ and $A=\left\{\left(a,b|\right)1 \le\,a,b\,\le\,n\right\}=S \times S $ A subset $B$ of $A$ is said to be a good subset if $\left(x, x\right)\in\,B$ for every $x \,\in S$ Then, the number of good subsets of $A$ is

KVPY 2012

27. If relation $R$ defined on the set $A=\{1,2,3,4,5\}$ by $R=\left\{(x, y):\left|x^{2}-y^{2}\right|<16\right\}$ then $R$ is

NTA Abhyas 2022

28. If the roots of the equation $\frac{a}{x-a}+\frac{b}{x-b}=1$ are equal in magnitude and opposite in sign, then

Complex Numbers and Quadratic Equations

29. The distance between $ (2, 1, 0)$ and $2x + y + 2z + 5 = 0$ is

KEAM 2017 Three Dimensional Geometry

30. The sum of the infinite series $\sin^{-1}\frac{1}{\sqrt2}+\sin^{-1}\left(\frac{\sqrt2-1}{\sqrt6}\right)+\sin^{-1}\left(\frac{\sqrt3-\sqrt2}{\sqrt{12}}\right)+...+\sin^{-1}\left(\frac{\sqrt n-\sqrt{n-1}}{\sqrt{n(n+1)}}\right)$ is

Inverse Trigonometric Functions