TS EAMCET 2022 (Online) 19th July Evening Shift | TS EAMCET Year Wise Previous Years Questions

Chemistry

1. If the radius and energy of the second Bohr orbit of hydrogen atom is $r_2$ and $E_2$. respectively. The radius and ener 2. When a radiation of 300 nm is shined on five metals, namely $\mathrm{Li}, \mathrm{Mg}, \mathrm{Ag}, \mathrm{Cu}$ and K , 3. The correct order of ionic radii for the given species is 4. Which of the following set of properties generally decreases along a period? 5. The correct order of the bond angles of the compounds $\mathrm{SiCl}_4, \mathrm{BF}_3, \mathrm{BeCl}_2$ and $\mathrm{SF} 6. Identify all the species that do not exist $\mathrm{H}_2^{+}, \mathrm{He}_2^{2+}, \mathrm{Li}_2^{2-}, \mathrm{Ne}_2, \ma 7. The compressibility factor of a real gas at high pressure is 8. The compressibility factor $(\mathrm{Z})$ is lower for $\mathrm{NH}_3$ and $\mathrm{CO}_2$ gases than that of $\mathrm{N 9. Combustion of 10 mL of a gaseous hydrocarbon gives 40 mL of $\mathrm{CO}_2$ and 50 mL of water vapour under the same con 10. The number of moles of ferrous oxalate oxidised by one mole of $\mathrm{KMnO}_4$ in acidic medium is 11. An air bag on adiabatic expansion undergoes $5 \%$ increase in its volume. The percentage change in pressure is $\left[\ 12. The $K_p$ value at equilibrium of $\mathrm{SO}_3$ formation reaction from $\mathrm{SO}_2(g)$ and $\mathrm{O}_2(g)$ is $5 13. $$ \text { Match the following } $$ .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;borde 14. The correct statements among the following are (A) $\mathrm{BeH}_2$ and $\mathrm{MgH}_2$ are polymeric in nature. (B) Li 15. The correct order of melting points of the following salts is $$ \begin{array}{lll} \text { LiCl } & \text { LiF } & \te 16. $\mathrm{Al}+$ aq. NaOH (excess) $\longrightarrow P+Q \cdot P$ and $Q$ are 17. Among the following elements, $X$ exhibits maximum catenation and $Y$ is the least abundant on earth. $X$ and $Y$ elemen 18. For the following radicals, the correct order of their stability is 19. The order of decreasing reactivity towards an electrophilic reagent, for the following compounds, is (i) benzene (iii) c 20. \text { The major product of the following reactions is } 21. If the length of the body diagonal of a FCC unit cell is $x \mathop {\rm{A}}\limits^{\rm{o}}$, the distance between two 22. Calculate the quantity of $\mathrm{CO}_2$ required to prepare 1 L of soda water when the soda water was packed under 2 a 23. Which of the following substances show the highest colligative properties? 24. In two separate experiments, the same quantity of electricity was passed through silver and gold solutions. [Assume ' $l 25. Which of the following is a zero order reaction? 26. The coagulation of 200 mL of a positive colloid took place when 0.73 g of HCl was added to it without changing the volum 27. The oxoacid of phosphorus which contains $4 \mathrm{P}-\mathrm{O}-\mathrm{H}, 2 \mathrm{P}=\mathrm{O}$ and one $\mathrm{ 28. Group 16 elements are also called 29. For the reaction, $\mathrm{Br}_2+\mathrm{F}_2$ (excess) ⟶ P , the molecular formula and structure of P , respectively, a 30. In the following reaction, $a, b, p, q, r$ and $s$ are $a \mathrm{XeF}_4+b \mathrm{H}_2 \mathrm{O} \longrightarrow p \ma 31. The increase in the atomic radii of the third (5d) series of transition elements is very small, which may be accounted f 32. A metal complex absorbed orange light. The colour in which it appears is 33. The major product of the following reaction is $$ \text { Glucose } \frac{\text { (i) } \mathrm{HI}, \Delta}{\text { (ii 34. $$ \text { The major product in the following reaction is } $$ 35. Arrange the following phenols in decreasing order of their $\mathrm{p} K_a$. (A) Phenol (B) ortho - nitrophenol (C) meta 36. Decreasing order of reactivity of the following compounds in the Williamson's ether synthesis is (II) $\mathrm{Cl}-\mat 37. $$ \text { Identify } X \text { and } Y \text { in the reactions, given below } $$ 38. A primary alcohol was reacted with pyridinium chlorochromate (PCC), which resulted in a product $P$. The product $P$ on 39. The major product of the following reactions 40. In the following reaction, the suitable starting reagent $P$ is

Mathematics

1. Let $f: A \rightarrow B$ be defined as $f(x)=\frac{1}{2}-\tan \left(\frac{\pi x}{2}\right)$ and $g: B \rightarrow C$ be 2. If $D$ is the domain and $G$ is the range of the real valued function $f(x)=\sqrt{\frac{1-x^2}{1+x^2}}$, then $D \cap G= 3. Let $A=\left[\begin{array}{ll}0 & 1 \\ 1 & k\end{array}\right], k \in R$ and $A^3=\left[\begin{array}{ll}a & b \\ c & d\ 4. Let $A$ and $B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ unit matrix such that $A B-C$ is a non-singular m 5. Let $A=\left[\begin{array}{ccc}0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0\end{array}\right], B=\left[\begin{array}{lll}0 & 1 6. Let $\alpha, \beta$ and $\gamma$ be real numbers. If $\left[\begin{array}{ccc}7 & 5 & \alpha \\ \beta & 2 & 11 \\ 3 & \g 7. If $z=\alpha+i \beta$ satisfies the equation $|z|-z=1+2 i$ and $|z|=\sqrt{\alpha^2+\beta^2}$, then $z \bar{z}=$ 8. If $-i$ and $\alpha$ are the roots of the equation $i z^2-2(i+1) z+(2-i)=0, \tan \theta=\frac{-1}{2}$ and $\theta \in 4$ 9. If $1, \alpha_1, \alpha_2, \alpha_3, \ldots \alpha_{n-1}$ are $n$th roots of unity then $\sum\limits_{1 \le i 10. Let $f(x)=A x^2+B x, g(x)=L x^2+M x+N$. Given that $f(2)-g(2)=1, f(3)-g(3)=4, f(4)-g(4)=9$. Then, a root of $f(x)-g(x)=0 11. If $f(x)=\frac{2 x-3}{(x-2)(x-3)}$ is a real valued function, then the value that $f(x)$ does not take is 12. If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-3 x^2+2 x-4=0$, then $\Sigma \alpha^2 \beta^2=$ 13. If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ such that $\alpha+\beta=0$, then $\al 14. If $(2-i)$ is one of the roots of the equation $x^4-9 x^3+31 x^2-49 x+30=0$ and $\alpha, \beta(\alpha 15. If $m$ and $M$ are respectively, the smallest and greatest rational roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 16. The number of ways of arranging the letters of the word LINEAR so that the letters N and R do not come together and E an 17. 15 lines are concurrent at a point $P$. A line $L$ is not passing through $P$ intersects all the 15 lines and forms tria 18. The expansion of $(a+x)^n$ contains 15 terms. When $x=1$ the ratio of the neighbouring terms to the middle term in this 19. If $\frac{d}{d x}\left(\frac{2 x+1}{(x+1)^2(x-2)}\right)=\frac{A}{(x-2)^2}+\frac{B}{(x+1)^3}+\frac{C}{(x+1)^2}$, then $A 20. If $\frac{x^2-2}{\left(x^2+1\right)\left(x^2+3\right)}=\frac{A x+B}{x^2+1}+\frac{C x+D}{x^2+3}$, then $D=$ 21. Let $\alpha$ be the period of $3 \sin \frac{\pi x}{3}-\cos \frac{\pi x}{2}+\tan \frac{\pi x}{4}, \beta$ be the period of 22. If $\theta$ does not lie in the second quadrant and $\tan \theta=\frac{-3}{4}$, then $\tan \frac{\theta}{2}+\sin 2 \thet 23. $$ \cos ^2 76^{\circ}+\sin ^2 46^{\circ}+\sin 76^{\circ} \cos 46^{\circ}= $$ 24. If $x=\log _e\left(\cot \left(\frac{\pi}{4}+\theta\right)\right)$, then $\lim _{\theta \rightarrow 0} \frac{\theta}{(\si 25. If $e^{i t}=\cos t+i \sin t$ and $e^{-i t}=\cos t-i \sin t$, then $\cosh (x+i y)-\cosh (x-i y)=$ 26. In a $\triangle A B C, A D$ and $B E$ are medians. If $A D=4, \angle D A B=\frac{\pi}{6}$ and $\angle A B E=\frac{\pi}{3 27. If $S$ is the circumentre of a $\triangle A B C, a=5, b=6, c=9$ and $S B=\frac{27}{4 \sqrt{2}}$, then $\sin 2 C=$ 28. In a $\triangle A B C$, if $\frac{r}{r_1}=\frac{1}{2}$, then $4 \tan \frac{A}{2}\left(\tan \frac{B}{2}+\tan \frac{C}{2}\ 29. If $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ are th 30. If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are the non-coplanar vectors and $\mathbf{a}-2 \mathbf{b}+3 \mathbf{c},-4 31. The point which lies on the plane passing through the point $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 3 \ 32. If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-3 \hat{\mathbf 33. If the angle between the planes $\mathbf{r} \cdot(11 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\alpha \hat{\mathbf{k}})=7$ and 34. If $2 \mathbf{i}+3 \mathbf{j}-4 \mathbf{k}$ and $-\mathbf{i}+2 \mathbf{j}+\mathbf{k}$ are the two diagonals of a paralle 35. There are $n$ observations and all of them are negative numbers. The ascending order of these observations is $x_1, x_2, 36. A bag contains 3 white and 6 red balls. Four balls are drawn at a time randomly. Then, the probability of getting at lea 37. $A$ and $B$ are two independent events $P(A)=\frac{2}{5}, P(B)=\frac{1}{3}$. Match the following : .tg {border-collap 38. Two players $A$ and $B$ are alternately throwing a coin and a die together. $A$ player who first throws head and 6 wins 39. If two dice are thrown and if $X$ denotes the sum of the numbers that show up on the faces of the dice, then the mean of 40. In a university campus, the probability that a person chosen at random is an engineering student is $\frac{1}{5}$. The p 41. If $A(1,1), B(-1,1)$ and $C(-1,-1)$ are three points and a point $P$ moves such that $(P A)^2=(P B)^2+(P C)^2$, then the 42. If $x^2=8 a y$ is the transformed equation of $x^2-4 y+6 x+15=0$ when the origin is shifted to the point $(\alpha, \beta 43. If a straight line $L$ passing through the point $(5,-3)$ is inclined at an angle of $60^{\circ}$ to the line $\sqrt{3} 44. Let $\alpha, \beta$ and $\gamma$ be three non-zero real constants and $a, b$ and $c$ be three arbitrary real numbers whi 45. If the algebraic sum of the perpendicular distances from the points $(2,0),(0,2)$ and $(1,1)$ to a variable line is zero 46. For $a, b, c \in R$, if $6 a^2-3 b^2-c^2+7 a b-a c+4 b c=0$ and $|a|+|b| \neq 0$, then all the lines given by $a x+b y+c 47. If $\theta$ is the acute angle between the pair of lines $H \equiv a x^2-x y+b y^2=0, \tan \theta=5$ and $(1,-1)$ is a p 48. The equation of the pair of straight lines passing through the point $(2,3)$ and perpendicular to the pair of lines $3 x 49. If $f(x, y)=0$ is the combined equation of the lines joining the origin to the points where the line $4 x-6 y-2=0$ meets 50. The radius of the circle passing through the points $(-1,1),(2,-1)$ and $(1,0)$ is 51. If $A=(0,-2)$ and $B$ is any point on the circle $x^2+y^2-2 x-2 y+1=0$, then the maximum value of $(\mathbf{A B})^2$ is 52. If $(\alpha, \beta)$ is the pole of the line $3 x-5 y+6=0$ with respect to the circle $x^2+y^2-10 x+14 y+46=0$, then $\a 53. $O(0,0)$ and $A(1,0)$ are centres of two units circles $C_1$ and $C_2$, respectively. $C_3$ is also a unit circle having 54. If the circles $x^2+y^2-16 x-20 y+164=r^2(r>0)$ and $x^2+y^2-8 x-14 y+29=0$ intersect in two distinct points, then the m 55. If the circle $x^2+y^2-6 x-12 y+1=0$ cuts another circle $C$ orthogonally and the centre of the circle $C$ is $(-4,2)$, 56. Let $L L^{\prime}$ be the latusrectum and $P Q$ be the focal chord of the parabola $y^2=16 x$. If $P=(1,4)$ and $P, L$ l 57. If $P\left(\frac{1}{2}, 4\right)$ and $Q$ are the ends of a focal chord of the parabola $y^2=32 x$ and $S$ is the focus 58. Statement I The equation of the directrix of the ellipse $4 x^2+y^2-8 x-4 y+4=0$ is $3 y=6-4 \sqrt{3}$ Statement II The 59. If $S$ is the focus of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ lying on the positive $X$ - axis and $P(\theta)$ is a 60. A hyperbola having its centre at the origin is passing through the point $(5,2)$ and has transverse axis of length 8 alo 61. If $e_1$ is the eccentricity of the hyperbola $x=\sec \theta$, $y=\sqrt{2} \tan \theta$ and $e_2$ is the eccentricity of 62. $A(27,-243,81)$ is a point in space, $B, C$ and $D$ are images of $A$ with respect to $X Y, Y Z$ and $Z X$ planes respec 63. Let $A(2,5,7)$ be the image of the point $B(1,-2,3)$ with respect to a plane $\pi$. Let $C$ be the point where $A B$ mee 64. If a plane $x+y+z-5=0$ intersects the line joining $A(1,1,1)$ and $B(2,2,2)$ at $P$, then $A P: P B=$ 65. $$ \mathop {\lim }\limits_{x \to 2}\left[\left(x^2-4 x+4\right) \cos \left(\frac{2}{x-2}\right)+\frac{x^2-4}{x^3-2 x-4}\ 66. $$ \lim _{x \rightarrow 0} \frac{\tan 2 x-2 \tan x}{(1-\cos x)\left(2^x-1\right)}= $$ 67. $$ \frac{d}{d x}\left[\left(x^{\frac{5}{2}}-x^{\frac{3}{2}}+1\right)\left(x^2-3 x+5\right)\right]= $$ 68. The value of $\frac{d}{d x}\left[\log \left(\sin \sqrt{\frac{x^2+1}{x^2+2}}\right)\right]$ when $x=\sqrt{2}$, is 69. If $f(x)=\frac{1+\sec x}{2(\sec x-1)}$ for $0 70. The equation of the normal to the curve $\sin y=\sqrt{3} x \sin \left(\frac{\pi}{6}+y\right)$ at $x=0$, is 71. Assertion (A) The curves $y^2=4 x$ and $x^2=-2 y$ intersect at $(1,2)$ orthogonally. Reason (R) If the product of the sl 72. Let $f(x)=\left\{\begin{array}{cc}1+6 x-3 x^2 & x \leq 1 \\ x+\log _2\left(b^2+7\right) & x>1\end{array}\right.$. Then, 73. Let $g(x)$ be the anti-derivative of $f(x)$. Then, the function for which $\log _e\left(1+(g(x))^2\right)+c$ is an anti- 74. If $f(x)=\int\left[\tan ^2 x+\cot ^2 x+\frac{4\left(\sin ^3 x+\cos ^3 x\right)}{\sin ^2 2 x}\right] d x$ and $f\left(\fr 75. $\int \sqrt{4 \cos ^2 x-5 \sin ^2 x} \cos x d x=$ 76. $\int_0^3\left(\sin \left(\frac{\pi}{3} x\right)-\cos \left(\frac{\pi}{3} x\right)\right) d x=$ 77. $$ \int_0^{\pi / 2} \sin ^4 \theta \cos ^3 \theta d \theta= $$ 78. Statement I The differential equation corresponding to the family of circles having their centres on $Y$-axis and fixed 79. If $m$ and $n$ are respectively the order and the degree of the differential equation representing the family of curves 80. The general solution of $\left(\left(1+x^2\right) y \sin x-2 x y\right) d x-\log y^{1+x^2} d y=0$ is

Physics

1. The range of the nuclear force is 2. Select the physical quantities in Column I and Column II having same dimensions .tg {border-collapse:collapse;border-sp 3. A rocket moves straight upward with zero initial velocity and with an acceleration $20 \mathrm{~m} / \mathrm{s}^2$. It r 4. Two cars, at a certain instant, are 50 km apart on a line running from south to north. The one farther north is moving w 5. A particle initially at origin starts moving in $X Y$ - plane has velocity component $\mathbf{v}=(6+2 t) \hat{\mathbf{i 6. A bullet is fired at time $t=0$ with velocity $20 \mathrm{~m} / \mathrm{s}$ and at an initial angle of $30^{\circ}$ with 7. A constant horizontal force $\mathbf{F}$ of magnitude 10 N is applied to a block $A$ and this produces an acceleration o 8. A body of mass $m$ slides down along a frictionless inclined plane from height $h$ and just completes motion in a vertic 9. Particle $A$ moving with a velocity $v=10 \mathrm{~m} / \mathrm{s}$ experienced a head on collision with a stationary pa 10. A metre stick is balanced on the knife edge at its centre. When four coins, each of mass 2 g are put one on top of the o 11. The amplitude of a damped oscillator varies with time as $A(t)=A_0 \exp (-b t / 2 \mathrm{~m})$, where $b=70 \mathrm{~g} 12. Four particles each of mass $m$ are placed at four vertices of a rectangle having side length as $3 l_0$ and $4 l_0$. Th 13. Two wires of same length having radius of 2 mm and 1.5 mm respectively, are loaded with same weights. Extension of the s 14. Consider an increase of $1 \%$ in each of radius of artery, viscosity of blood and density of blood, respectively. The p 15. A metal cube of side 10 cm rests on a film of a liquid of thickness 0.2 mm . If upon applying a horizontal force $\mathb 16. 176 g of $\mathrm{CO}_2$ can change its temperature from $0^{\circ} \mathrm{C}$ to $30^{\circ} \mathrm{C}$ by absorbing 17. A solution consists of ether and 5.0 g of water at $0^{\circ} \mathrm{C}$. If the ether evaporates completely to freeze 18. Assertion (A) Heat and work are modes of energy transfer to a system resulting in change in its internal energy. Reason 19. An ideal gas at pressure $p_0$ undergoes an isothermal expansion until its volume is 8.0 times its initial volume. The g 20. At what temperature is the root mean square rms speed of neon gas atoms is equal to the rms speed of helium gas atoms at 21. A wire of length 0.4 m stretched at both ends vibrates 250 times per second. If the length of the wire is increased by 0 22. A spherical glass is attached to a rigid wall as shown in the figure. An observer located at point $O$ is looking at a p 23. In a double slit experiment performed in air, the angular width of a fringe is found to be $0.15^{\circ}$ on a screen pl 24. $6 \mu \mathrm{C}$ charge is placed at the centre of a cube. What will be the electric flux at each face of the cube? $$ 25. There are two thin wire rings, each of radius $R$, whose axes coincide. The charges of the rings are $q$ and $-q$. The m 26. Statement I The temperature coefficient of resistance for most of metals in pure form is positive. Statement II A metal 27. A cylindrical resistor of radius 7.0 mm and length 4.0 cm is made of material that has a resistivity of $10^{-6} \Omega- 28. Statement I A uniform electric field and a uniform magnetic field are pointed in the same direction. If an electron is p 29. Two parallel conductor each 50 m long, separated by 0.2 m experience a force of 1 N . If the current in first conductor 30. The magnitude of axial field due to a bar magnet at a distance of 1 m , is found to be $5 \times 10^{-8} \mathrm{~T}$. T 31. The magnetic flux through the triangular loop shown in the figure below Where a uniform magnetic field of strength 2 T p 32. The $Q$ value of a series $L-C-R$ circuit with $L=2 \mathrm{H}$, $C=32 \mu \mathrm{~F}, R=20 \Omega$ is 33. A laser beam has intensity $2.1 \times 10^{15} \mathrm{~W} / \mathrm{m}^2$. The amplitude of magnetic field in the beam 34. In a photoelectric experiment, the wavelength of the light incident on the metal is changed from 200 nm to 400 nm . The 35. The de-Broglie wavelength of an electron with kinetic energy of 320 eV is (take, $h=6.0 \times 10^{-34}$ SI unit, mass o 36. Considering the Bohr's model of hydrogen atom, the ratio of velocities of electrons orbiting in the 4th orbit to that in 37. What is the mass number of the nucleus having radius equal to $\frac{1}{3}$ of that of ${ }^{189} \mathrm{Os}$ ? 38. The number of silicon atoms per $\mathrm{m}^3$ is $5 \times 10^{28}$. This is doped with $4.5 \times 10^{21}$ atoms $/ \ 39. The output of the following circuit is equivalent to $.......$ gate 40. Which of the following statements is not true?

TS EAMCET 2022 (Online) 19th July Evening Shift

MCQ (Single Correct Answer)

-0

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are the non-coplanar vectors and $\mathbf{a}-2 \mathbf{b}+3 \mathbf{c},-4 \mathbf{a}+5 \mathbf{b}-6 \mathbf{c}, x \mathbf{a}-9 \mathbf{b}+z \mathbf{c}$ are collinear points, then $2 x-z=$

-10

-9

TS EAMCET 2022 (Online) 19th July Evening Shift

MCQ (Single Correct Answer)

-0

The point which lies on the plane passing through the point $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, 3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ is

$-\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$

$7 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$

$-\hat{\mathbf{i}}+9 \hat{\mathbf{j}}+14 \hat{\mathbf{k}}$

$3 \hat{i}-7 \hat{j}+8 \hat{k}$

TS EAMCET 2022 (Online) 19th July Evening Shift

MCQ (Single Correct Answer)

-0

If $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$, then the component of $\mathbf{b}$ perpendicular to $\mathbf{a}$ is

$\frac{1}{3}(4 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})$

$\frac{1}{3}(8 \hat{\mathbf{i}}-13 \hat{\mathbf{j}}-\hat{\mathbf{k}})$

$\frac{2}{3}(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$

$\frac{1}{7}(\hat{\mathbf{i}}-5 \hat{\mathbf{j}}-17 \hat{\mathbf{k}})$

TS EAMCET 2022 (Online) 19th July Evening Shift

MCQ (Single Correct Answer)

-0

If the angle between the planes $\mathbf{r} \cdot(11 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\alpha \hat{\mathbf{k}})=7$ and $\mathbf{r} \cdot(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})=5$ is $\frac{\pi}{2}$, then $\alpha=$

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