Mathematics Question and Answers

1. If $(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d}) \cdot(\vec{a} \times \vec{d})=0$, then which of the following is always true?

VITEEE 2017

2. Let $f(x)=\begin{cases}\tan x & \text { if } 0 \leq x \leq \frac{\pi}{4} \\ a x+b & \text { if } \frac{\pi}{4}< x< \frac{\pi}{2}\end{cases}$. If $f(x)$ is differentiable at $x=\frac{\pi}{4}$, then the values of $a$ and $b$ are respectively

KEAM 2022

3. A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O the ground is '45$\circ$' It 'flies' off horizontally straight a Way from 1 the point O. After one second, the elevation of,the bird from O is reduced to 30$^\circ$. Then the 'speed' (m m/s) of the bird is

4. A line $L_1$ passing through a point with position vector $\vec{p}=\hat{i}+2 \hat{j}+3 \hat{k}$ and parallel to $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ . Another line $L _2$ passing through a point with position vector $\vec{q}=2 \hat{i}+3 \hat{j}+\hat{k}$ and parallel to $\vec{b}=3 \hat{i}+\hat{j}+2 \hat{k}$.
Equation of plane equidistant from lines $L_1$ and $L_2$ is

Vector Algebra

5. If $x^{2}-x+1=0$, then value of $x^{3 n}$ is

Complex Numbers and Quadratic Equations

6. Using differentials, find the approximate value of $\sqrt{25.2}$.

Application of Derivatives

7. A & B play a game where each is asked to select a number from 1 to 25. If the two numbers match both of them win a prize. The probability that they will not win a prize in a single trial is :

Probability - Part 2

8. $\int \frac{(x^2 -1)}{(x^2 + 1) \sqrt{x^4 + 1}} dx $ is equal to

BITSAT 2013

9. For the differential equation $\frac{d^2 y}{d x^2}+y=0$, if there is a function $y=\phi(x)$ that will satisfy it, then the function $y=\phi(x)$ is called

Differential Equations

10. If $\cos h^{-1} x = 2 \log_e ( \sqrt{2} + 1 )$, then x =

TS EAMCET 2017

11. Let $f$ be a differentiable function satisfying $\int\limits_0^{f(x)} f^{-1}(t) d t-\int\limits_0^x(\cos t-f(t)) d t=0$ and $f(0)=1$
The value of $\underset {x \rightarrow 0}{\text{Lim}}\left(\left[\frac{\cos x}{f(x)}\right]+\left[\frac{\cos 2 x}{f(2 x)}\right]+\left[\frac{\cos 3 x}{f(3 x)}\right]+\ldots \ldots+\left[\frac{\cos (100 x)}{f(100 x)}\right]\right)$ is equal to [Note : where $[k]$ denotes greatest integer less than or equal to $k$.]

Differential Equations

12. If $[x]^2-5[x]+6=0$, where $[x]$ denotes the greatest integer function, then

KCET 2024 Relations and Functions

13. If the foci of an ellipse are $( \pm \sqrt{5}, 0)$ and its eccentricity is $\frac{\sqrt{5}}{3}$, then the equation of the ellipse is

Conic Sections

14. The greatest value of $(2 \sin \theta+3 \cos \theta+4)^{3} \cdot(6-2 \sin \theta-3 \cos \theta)^{2}$, as $\theta \in R$, is

Trigonometric Functions

15. The simultaneous equations
$ k x+2 y-z=1 $
$(k-1) y-2 z=2 $
$(k+2) z=3$
have only one solution when

Determinants

16. The vertices of $\triangle ABC$ are $A (2,0,0), B (0,1,0), C (0,0,2)$. Its orthocentre is $H$ and circumcentre is $S$. $P$ is a point equidistant from $A , B , C$ and the origin $O$.
$PA =$

Vector Algebra

17. Let $A$ be a matrix of order $3$ such that $A^{2}=3A-2I$ where, $I$ is an identity matrix of order $3.$ If $A^{5}=αA+\beta I,$ then $\alpha \beta $ is equal to

NTA Abhyas 2022

18. The relation, $R=\{(1,3),(3,5)\}$ is defined on the set with minimum number of elements of natural numbers. The minimum number of elements to be included in $R$ so that $R$ is equivalence, is:

Sets, Relations, and Functions

19. If lines $x-y+2=0$ and $2 x-y-2=0$ meet at a point $P$ then equation of tangent drawn to the parabola $y ^2=8 x$ from the point ' $P$ ' is

Conic Sections

20. If 1 lies between the roots of $3 x^{2}-3 \sin \theta-2 \cos ^{2} \theta=0$ then

Complex Numbers and Quadratic Equations

21. Consider the 10 digits $0, 1, 2, 3, ......., 9$
Statement-1: Number of four digit even numbers that can be formed if each digit is to used only once in the number is 2268.
because
Statement-2: Total 4 digit numbers that can be formed if each digit is used only once is 4536.

Permutations and Combinations

22. There are 18 hunters which are divided into four groups as,
$G _1$ : Consisting of 5 hunters each of whom can hit a target with the probability 0.8 .
$G _2$ : Consisting of 7 each hitting a target with probability 0.7 .
$G _3$ : Consisting of 4 each hitting a target with probability 0.6 , and
$G _4$ : Consisting of 2 each hitting a target with probability 0.5 .
A randomly selected hunter fires a shot and failed to hit the target. Most probable group of the hunter is

Probability - Part 2

23. The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What is the correct standard deviation?

Statistics

24. If $A=\{1,3,5,7\}, B=\{a, b, c\}$, then $n(A \times B)$ is equal to

NTA Abhyas 2022

25. The equation of the circle with origin as centre passing the vertices of an equilateral triangle whose median is of length $3a$ is

UPSEE 2012

26. The number of real negative terms in the binomial expansion of $(1+i x)^{4 n-2}, n \in N, x >0$ is

Binomial Theorem

27. The radius of the circle passing through the foci of the ellipse
$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and having its centre at $(0,3)$ is

AP EAMCET 2015

28. The length of major axis of the ellipse $(5 x-10)^2+(5 y+15)^2 =\frac{(3 x-4 y+7)^2}{4}$ is:

Conic Sections

29. If for a distribution $\Sigma(x - 5) = 3, \Sigma(x - 5)^2 = 43$ and the total number of items is $18$. Find the standard deviation.

Statistics

30. If any tangent to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ intercepts equal lengths $l$ on the axes, then $l=$

Conic Sections

Mathematics Questions

Mathematics Questions

1. If $(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d}) \cdot(\vec{a} \times \vec{d})=0$, then which of the following is always true?

2. Let $f(x)=\begin{cases}\tan x & \text { if } 0 \leq x \leq \frac{\pi}{4} \\ a x+b & \text { if } \frac{\pi}{4}< x< \frac{\pi}{2}\end{cases}$. If $f(x)$ is differentiable at $x=\frac{\pi}{4}$, then the values of $a$ and $b$ are respectively

3. A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O the ground is '45$\circ$' It 'flies' off horizontally straight a Way from 1 the point O. After one second, the elevation of,the bird from O is reduced to 30$^\circ$. Then the 'speed' (m m/s) of the bird is

5. If $x^{2}-x+1=0$, then value of $x^{3 n}$ is

6. Using differentials, find the approximate value of $\sqrt{25.2}$.

7. A & B play a game where each is asked to select a number from 1 to 25. If the two numbers match both of them win a prize. The probability that they will not win a prize in a single trial is :

8. $\int \frac{(x^2 -1)}{(x^2 + 1) \sqrt{x^4 + 1}} dx $ is equal to

9. For the differential equation $\frac{d^2 y}{d x^2}+y=0$, if there is a function $y=\phi(x)$ that will satisfy it, then the function $y=\phi(x)$ is called

10. If $\cos h^{-1} x = 2 \log_e ( \sqrt{2} + 1 )$, then x =

12. If $[x]^2-5[x]+6=0$, where $[x]$ denotes the greatest integer function, then

13. If the foci of an ellipse are $( \pm \sqrt{5}, 0)$ and its eccentricity is $\frac{\sqrt{5}}{3}$, then the equation of the ellipse is

14. The greatest value of $(2 \sin \theta+3 \cos \theta+4)^{3} \cdot(6-2 \sin \theta-3 \cos \theta)^{2}$, as $\theta \in R$, is

15. The simultaneous equations
$ k x+2 y-z=1 $
$(k-1) y-2 z=2 $
$(k+2) z=3$
have only one solution when

16. The vertices of $\triangle ABC$ are $A (2,0,0), B (0,1,0), C (0,0,2)$. Its orthocentre is $H$ and circumcentre is $S$. $P$ is a point equidistant from $A , B , C$ and the origin $O$.
$PA =$

17. Let $A$ be a matrix of order $3$ such that $A^{2}=3A-2I$ where, $I$ is an identity matrix of order $3.$ If $A^{5}=αA+\beta I,$ then $\alpha \beta $ is equal to

18. The relation, $R=\{(1,3),(3,5)\}$ is defined on the set with minimum number of elements of natural numbers. The minimum number of elements to be included in $R$ so that $R$ is equivalence, is:

19. If lines $x-y+2=0$ and $2 x-y-2=0$ meet at a point $P$ then equation of tangent drawn to the parabola $y ^2=8 x$ from the point ' $P$ ' is

20. If 1 lies between the roots of $3 x^{2}-3 \sin \theta-2 \cos ^{2} \theta=0$ then

21. Consider the 10 digits $0, 1, 2, 3, ......., 9$
Statement-1: Number of four digit even numbers that can be formed if each digit is to used only once in the number is 2268.
because
Statement-2: Total 4 digit numbers that can be formed if each digit is used only once is 4536.

23. The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What is the correct standard deviation?

24. If $A=\{1,3,5,7\}, B=\{a, b, c\}$, then $n(A \times B)$ is equal to

25. The equation of the circle with origin as centre passing the vertices of an equilateral triangle whose median is of length $3a$ is

26. The number of real negative terms in the binomial expansion of $(1+i x)^{4 n-2}, n \in N, x >0$ is

27. The radius of the circle passing through the foci of the ellipse
$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and having its centre at $(0,3)$ is

28. The length of major axis of the ellipse $(5 x-10)^2+(5 y+15)^2 =\frac{(3 x-4 y+7)^2}{4}$ is:

29. If for a distribution $\Sigma(x - 5) = 3, \Sigma(x - 5)^2 = 43$ and the total number of items is $18$. Find the standard deviation.

30. If any tangent to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ intercepts equal lengths $l$ on the axes, then $l=$

Mathematics Questions

Mathematics Questions

1. If $(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d}) \cdot(\vec{a} \times \vec{d})=0$, then which of the following is always true?

2. Let $f(x)=\begin{cases}\tan x & \text { if } 0 \leq x \leq \frac{\pi}{4} \\ a x+b & \text { if } \frac{\pi}{4}< x< \frac{\pi}{2}\end{cases}$. If $f(x)$ is differentiable at $x=\frac{\pi}{4}$, then the values of $a$ and $b$ are respectively

3. A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O the ground is '45$\circ$' It 'flies' off horizontally straight a Way from 1 the point O. After one second, the elevation of,the bird from O is reduced to 30$^\circ$. Then the 'speed' (m m/s) of the bird is

5. If $x^{2}-x+1=0$, then value of $x^{3 n}$ is

6. Using differentials, find the approximate value of $\sqrt{25.2}$.

7. A & B play a game where each is asked to select a number from 1 to 25. If the two numbers match both of them win a prize. The probability that they will not win a prize in a single trial is :

8. $\int \frac{(x^2 -1)}{(x^2 + 1) \sqrt{x^4 + 1}} dx $ is equal to

9. For the differential equation $\frac{d^2 y}{d x^2}+y=0$, if there is a function $y=\phi(x)$ that will satisfy it, then the function $y=\phi(x)$ is called

10. If $\cos h^{-1} x = 2 \log_e ( \sqrt{2} + 1 )$, then x =

12. If $[x]^2-5[x]+6=0$, where $[x]$ denotes the greatest integer function, then

13. If the foci of an ellipse are $( \pm \sqrt{5}, 0)$ and its eccentricity is $\frac{\sqrt{5}}{3}$, then the equation of the ellipse is

14. The greatest value of $(2 \sin \theta+3 \cos \theta+4)^{3} \cdot(6-2 \sin \theta-3 \cos \theta)^{2}$, as $\theta \in R$, is

15. The simultaneous equations $ k x+2 y-z=1 $ $(k-1) y-2 z=2 $ $(k+2) z=3$ have only one solution when

16. The vertices of $\triangle ABC$ are $A (2,0,0), B (0,1,0), C (0,0,2)$. Its orthocentre is $H$ and circumcentre is $S$. $P$ is a point equidistant from $A , B , C$ and the origin $O$. $PA =$

17. Let $A$ be a matrix of order $3$ such that $A^{2}=3A-2I$ where, $I$ is an identity matrix of order $3.$ If $A^{5}=αA+\beta I,$ then $\alpha \beta $ is equal to

18. The relation, $R=\{(1,3),(3,5)\}$ is defined on the set with minimum number of elements of natural numbers. The minimum number of elements to be included in $R$ so that $R$ is equivalence, is:

19. If lines $x-y+2=0$ and $2 x-y-2=0$ meet at a point $P$ then equation of tangent drawn to the parabola $y ^2=8 x$ from the point ' $P$ ' is

20. If 1 lies between the roots of $3 x^{2}-3 \sin \theta-2 \cos ^{2} \theta=0$ then

21. Consider the 10 digits $0, 1, 2, 3, ......., 9$ Statement-1: Number of four digit even numbers that can be formed if each digit is to used only once in the number is 2268. because Statement-2: Total 4 digit numbers that can be formed if each digit is used only once is 4536.

23. The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What is the correct standard deviation?

24. If $A=\{1,3,5,7\}, B=\{a, b, c\}$, then $n(A \times B)$ is equal to

25. The equation of the circle with origin as centre passing the vertices of an equilateral triangle whose median is of length $3a$ is

26. The number of real negative terms in the binomial expansion of $(1+i x)^{4 n-2}, n \in N, x >0$ is

27. The radius of the circle passing through the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and having its centre at $(0,3)$ is

28. The length of major axis of the ellipse $(5 x-10)^2+(5 y+15)^2 =\frac{(3 x-4 y+7)^2}{4}$ is:

29. If for a distribution $\Sigma(x - 5) = 3, \Sigma(x - 5)^2 = 43$ and the total number of items is $18$. Find the standard deviation.

30. If any tangent to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ intercepts equal lengths $l$ on the axes, then $l=$

15. The simultaneous equations
$ k x+2 y-z=1 $
$(k-1) y-2 z=2 $
$(k+2) z=3$
have only one solution when

16. The vertices of $\triangle ABC$ are $A (2,0,0), B (0,1,0), C (0,0,2)$. Its orthocentre is $H$ and circumcentre is $S$. $P$ is a point equidistant from $A , B , C$ and the origin $O$.
$PA =$

21. Consider the 10 digits $0, 1, 2, 3, ......., 9$
Statement-1: Number of four digit even numbers that can be formed if each digit is to used only once in the number is 2268.
because
Statement-2: Total 4 digit numbers that can be formed if each digit is used only once is 4536.

27. The radius of the circle passing through the foci of the ellipse
$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and having its centre at $(0,3)$ is