MP Board Class 12 Sample Paper 2024 Maths

माध्यमिक शिक्षा मंडल,
मध्य प्रदेश

sample Paper
2024

dsoy vH;kl gsrq uewuk ç'u i=
Sample Question Paper for Practice only
gk;j lsds.Mjh ijh{kk −2024
Higher Secondary Examination −2024
fo"k; − mPp xf.kr
Subject Name −Higher Mathematics
(Hindi & English Versions)
Total Question Total Printed Pages Time Maximum Marks
23 12 3 Hour 80

funsZ'k :
lHkh iz'u vfuok;Z gSaA
iz'u Øekad 1 ls 5 rd oLrqfu"B izdkj ds iz'u gSaA
iz'u Øekad 6 ls 23 esa vkarfjd fodYi fn, x, gSaA
Instructions %
(i) All questions are compulsory.
(ii) Question numbers 1 to 5 are objective type questions.
(iii) Internal options have been given in question numbers 6 to 23.
(1) lgh fodYi pqudj fyf[k, : 1x6=6
𝑖) Qyu 𝑓: 𝑅 → 𝑅, 𝑓(𝑥) = 𝑥 }kjk ifjHkkf"kr gS rc &
𝑎) 𝑓,dSdh vkPNknd gS] 𝑏)𝑓cgq,dd vkPNknd gS
𝑐) 𝑓,dSdh gS fdUrq vkPNknd ugha gS 𝑑)𝑓u rks ,dSdh gS vkSj u rks
vkPNknd gSA
𝑖𝑖) 𝑡𝑎𝑛 dh eq[; 'kk[kk dk ifjlj gS &
𝑎) (0, 𝜋) 𝑏) (− , )

𝑐) 𝑅 𝑑) (0,2𝜋)

𝑖𝑖𝑖) ;fn 𝐴 = 𝑐𝑜𝑠𝛼 −𝑠𝑖𝑛𝛼 rFkk 𝐴 + 𝐴 = 𝐼 rks 𝛼 dk eku gS &
𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼

𝑎) 𝑏)

𝑐) 𝜋 𝑑)

1

𝑥 𝑥+1
𝑖𝑣) dk eku gksxk &
𝑥−1 𝑥
𝑎) 0 𝑏) 𝑥
𝑐) 𝑥 + 1 𝑑) 1

𝑣) ;fn nks lfn'kksa 𝑎⃗ rFkk 𝑏⃗ ds chp dk dks.k 𝜃 gS rks
𝑎⃗. 𝑏⃗ = 𝑎⃗ × 𝑏⃗ tc 𝜃 cjkcj gS &
𝑎) 1 𝑏) 𝑐) 𝑑) 𝜋

𝑣𝑖) ;fn ,d js[kk ds fnd dkslkbu 𝑙, 𝑚, 𝑛 gS rks √𝑙 + 𝑚 + 𝑛
dk eku gksxk &
𝑎) 0 𝑏) 1 𝑐) 𝑑) ±
√

Choose and write correct option -
(i) If function 𝑓 defined by 𝑓: 𝑅 → 𝑅, 𝑓(𝑥) = 𝑥 then
𝑎) 𝑓is one-one onto 𝑏)𝑓is many one onto
𝑐) 𝑓is one-one but not onto 𝑑) 𝑓 is neither one-one nor onto.
𝑖𝑖)Range of principal value of 𝑡𝑎𝑛 is
𝑎) (0, 𝜋) 𝑏) (− , ) 𝑐) 𝑅 𝑑) (0,2𝜋)
𝑐𝑜𝑠𝛼 −𝑠𝑖𝑛𝛼
𝑖𝑖𝑖) If 𝐴 = 𝑎𝑛𝑑 𝐴 + 𝐴 = 𝐼 then value of s 𝛼 is
𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼
𝑎) 𝑏) 𝑐) 𝜋 𝑑)
𝑥 𝑥+1
𝑖𝑣) Value of is
𝑥−1 𝑥
𝑎) 0 𝑏) 𝑥 𝑐) 𝑥 + 1 𝑑) 1
v) If angle between 𝑎⃗ and 𝑏⃗ is 𝜃, 𝑎⃗. 𝑏⃗ = 𝑎⃗ × 𝑏⃗ when 𝜃 is equal to &

𝑎) 1 𝑏) 𝑐) 𝑑) 𝜋

𝑣𝑖) If direction cosine of a line are 𝑙, 𝑚, 𝑛 then value of

√𝑙 + 𝑚 + 𝑛 is

𝑎) 0 𝑏) 1 𝑐) 𝑑) ±
√
2

(2) fjDr LFkkuksa dh iwfrZ dhft, & 1x6=6
𝑖) ;fn leqPp; 𝐴 ij ifjHkkf"kr laca/k 𝑅 = 𝐴 × 𝐴 rc 𝑅……… laca/k dgykrk gSA
𝑖𝑖) 𝑠𝑖𝑛 (− ) dk eku……… gksxkA

𝑖𝑖𝑖) ;fn 𝐴 = 𝐴 rks 𝐴……… vkO;wg dgykrk gSA
𝑖𝑣);fn 𝐴, 3 × 3 dk dksfV dk oxZ vkO;wg gS rFkk |𝐴| = 5 rks |𝑎𝑑𝑗𝐴| dk eku………gSA

𝑣) 𝑠𝑖𝑛√𝑥 dk vodyu xq.kkad ……… gSA

𝑣𝑖) fdlh o`Rr ds {ks=Qy ifjorZu dh nj f=T;k ds lkis{k………gksxh tcfd f=T;k 4 lseh
gSA
Fill in the blanks -
i) If relation R = A × A defined on set A then relation R is called ……

ii) value of sin (− ) is. . . . ..

iii) If A = A then matrix A is called. . . ..…
iv) If A is a square matrix of order 3 × 3 and |A| = 5 then value
of |adjA| is. . . . . . . ..
v) Differential coefficient of sin√x is. . . . ..
vi) Rate of change in area of any circle with respect to the radius,
will be … . when radius is 4cm

(3) lR;@vlR; fyf[k, & 1x6=6
i) leqPp; 𝐴 ij ifjHkkf"kr laca/k 𝑅 LorqY; dgykrk gS ;fn izR;sd
𝑎 ∈ 𝐴 ds fy, (𝑎, 𝑎) ∉ R

ii) fdlh oxZ vkO;wg dks ,d lefer vkSj ,d fo"ke lefer vkO;wg
ds ;ksxQy ds :i esa fu:fir fd;k tk ldrk gS A

iii) izR;sd ifjes; Qyu larr gksrk gS A
iv) ;fn 𝐸 ,oa 𝐹 Lora= ?kVuk, gS rks 𝑃(𝐸 ∩ 𝐹) = 𝑃(𝐸). 𝑃(𝐹)

3

v) 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = 𝑥 + 𝑦 ,d le?kkr vody lehdj.k gS A
vi) rhu dksfV okys fdlh vody lehdj.k ds fof'k"V gy esa mifLFkr
LosPN vpjksa dh la[;k rhu gksrh gSA
𝐖𝐫𝐢𝐭𝐞 𝐭𝐫𝐮𝐞 𝐚𝐧𝐝 𝐟𝐚𝐥𝐬𝐞 -

i) Relation R defined on set A is called reflexive when for every a ∈ A, (a, a) ∉ R
ii) Any square matrix can represent in the form of addition of a symmetric and
a skew symmetric matrix.
iii) Every rational function is a continuous function.
iv) If E and F are independent events then P(E ∩ F) = P(E). P(F)
v) xdy − ydx = x + y is a homogeneous differential equation
vi) Number of arbitrary constants in particular solution of a
differential equation of order three are 3.

(4) lgh tksMh+ cukb;s & 1x7=7
LrEHk v LrEHk c
i) ∫ 𝑑𝑥 a) −𝑐𝑜𝑡𝑥 + 𝑐

ii) ∫ 𝑑𝑥 b) 𝑡𝑎𝑛𝑥 + 𝑐

iii) ∫ 𝑡𝑎𝑛 𝑥𝑑𝑥 c) −𝑐𝑜𝑡𝑥 − 𝑥 + 𝑐
iv) ∫ 𝑐𝑜𝑡 𝑥𝑑𝑥 d) 𝑡𝑎𝑛𝑥 − 𝑥 + 𝑐

v) ∫ 𝑑𝑥 e) −𝑐𝑜𝑠𝑥 + 𝑐

vi) ∫ 𝑑𝑥 f) 𝑠𝑖𝑛𝑥 + 𝑐

vii) ∫ 𝑑𝑥 g) −𝑙𝑜𝑔|𝑐𝑜𝑠 𝑥| + 𝑐

h) 𝑙𝑜𝑔|𝑠𝑒𝑐𝑥| + 𝑐

i) 𝑙𝑜𝑔|𝑠𝑖𝑛 𝑥| + 𝑐

4

Match the correct column -
𝐶𝑜𝑙𝑢𝑚𝑛 𝐴 𝐶𝑜𝑙𝑢𝑚𝑛 𝐵
1 a) −𝑐𝑜𝑡𝑥 + 𝑐
𝑖) 𝑑𝑥
𝑐𝑜𝑠 𝑥
1 b) 𝑡𝑎𝑛𝑥 + 𝑐
𝑖i) 𝑑𝑥
𝑠𝑖𝑛 𝑥
c) −𝑐𝑜𝑡𝑥 − 𝑥 + 𝑐
𝑖𝑖𝑖) 𝑡𝑎𝑛 𝑥𝑑𝑥

d) 𝑡𝑎𝑛𝑥 − 𝑥 + 𝑐
𝑖𝑣) 𝑐𝑜𝑡 𝑥𝑑𝑥

1 e) −𝑐𝑜𝑠𝑥 + 𝑐
𝑣) 𝑑𝑥
𝑠𝑒𝑐𝑥
1 f) 𝑠𝑖𝑛𝑥 + 𝑐
𝑣𝑖) 𝑑𝑥
𝑐𝑜𝑠𝑒𝑐𝑥
𝑠𝑖𝑛𝑥 g) −𝑙𝑜𝑔|𝑐𝑜𝑠 𝑥| + 𝑐
𝑣𝑖𝑖) 𝑑𝑥
𝑐𝑜𝑠𝑥
h) 𝑙𝑜𝑔|𝑠𝑒𝑐𝑥| + 𝑐
i) 𝑙𝑜𝑔|𝑠𝑖𝑛 𝑥| + 𝑐

(5) ,d okD;@'kCn esa mRrj fyf[k, & 1x7=7
i)∫ 𝑑𝑥 dk eku fyf[k,A

ii) 𝑠𝑖𝑛𝑥 fdl varjky esa o?kZeku Qyu gS fyf[k,A

iii) + 𝑦 = 𝑥 dk lekdyu xq.kd fyf[k,A

iv) vody lehdj.k = 1 dk O;kid gy fyf[k,A

v) nks lfn'kksa 𝑎⃗ rFkk 𝑏⃗ ds yEcor~ gksus dh 'krZ fyf[k,A
vi) ;fn 𝑎⃗. 𝑏⃗ = 0vkSj 𝑎⃗ × 𝑏⃗ = 0 rks lfn'k 𝑎⃗ vkSj 𝑏⃗ ckjs esa vki D;k fu"d"kZ

fudky ldrs gS fyf[k, A
𝐹
vii) ;fn P(E)=0.6 rFkk 𝑃(𝐸 ∩ 𝐹) = 0.2 rks 𝑃( ) Kkr dhft, A
𝐸
5

𝐖𝐫𝐢𝐭𝐞 𝐚𝐧𝐬𝐰𝐞𝐫 𝐢𝐧 𝐨𝐧𝐞 𝐰𝐨𝐫𝐝/𝐬𝐞𝐧𝐭𝐞𝐧𝐜𝐞 -

i) Write the value of ∫ dx

ii) Write in which interval 𝑠𝑖𝑛𝑥 is an increasing function.

iii) Write integrating factor of + y = x.

iv) Write general solution of differential equation =1

v) write the condition of perpendicularity of two vectors a⃗ and b⃗
vi) If a⃗. b⃗ = 0 and 𝑎⃗ × 𝑏⃗ = 0 then what result you can find about a⃗ and b⃗
vii) If P(E)=0.6 and𝑃(𝐸 ∩ 𝐹) = 0.2 then write the value of 𝑃( )

(6) fl) dhft, fd leqPp; {12,3} es 𝑅 = {(1,2), (2,1)} }kjk iznRr laca/k
lefer gS fdUrq u rks LorqY; gS vkSj u ladzked gSA 2
Prove that relation given by R = (1,2), (2,1) in set {1,2,3} is
symmetric but neither reflexive nor transitive
fl) dhft, fd ,d ,dSdh Qyu 𝑓: {1,2,3} → {1,2,3} vfuok;Z :i ls vkPNknd
Hkh gSA
Prove that a one − one function f: {1,2,3} → {1,2,3} is necessarily onto
(7) fn, x;s Qyu dks ljyre :i esa fyf[k,A 2

𝑡𝑎𝑛 , 0<𝑥<𝜋

Write given function in the simplest form

𝑡𝑎𝑛 0<𝑥<𝜋

𝑡𝑎𝑛 [2cos (2𝑠𝑖𝑛 )] dk eku Kkr dhft, A

Find the value of tan 2 cos 2sin

6

(8) fuEufyf[kr dk eku Kkr dhft, & 2
𝑡𝑎𝑛 (1) + 𝑐𝑜𝑠 − + 𝑠𝑖𝑛 − +

𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 −+

𝑡𝑎𝑛 (1) + 𝑐𝑜𝑠 − + 𝑠𝑖𝑛 − +

fn, x;s Qyu dks ljyre :i esa fyf[k, &
𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 𝜋 3𝜋
𝑡𝑎𝑛 ,− <𝑥<
𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 4 4
Write given function in the simplest form −

𝑡𝑎𝑛 ,− < 𝑥 <

] (9) ;fn 𝑥 = 2𝑎𝑡 , 𝑦 = 𝑎𝑡 rks Kkr dhft,A 2
dy
If 𝑥 = 2at , y = at then find
dx
dk 𝑥 ds lkis{k vodyu dhft,A
e
Find differentiation of with respect to x
sinx
(10) fl) dhft, fd 𝑅 ij 𝑓(𝑥) = 3𝑥 + 17 ls iznRr Qyu o?kZeku gSA 2
Prove that function on R given by f(𝑥) = 3𝑥 + 17 is increasing
varjky Kkr dhft, ftlesa𝑓(𝑥) = 2𝑥 + 3𝑥 ls iznRr Qyu 𝑓 o?kZeku gSA
Find interval in which function f given by f(𝑥) = 2𝑥 + 3𝑥 is increasing.
(11) ,d o`Rr dh f=T;k leku :i ls 3𝑐𝑚/𝑠 dh nj ls c<+ jgh gS Kkr dhft, o`Rr dk

{ks=Qy fdl nj ls c<+ jgk gS tc f=T;k 10lseh gSA 2
Radius of a circle is increasing at the rate of 3𝑐𝑚/𝑠𝑒𝑐 Find rate of
increasing area of circle when radius of circle is 10cm
7

fdlh mRikn dh 𝑥 bdkbZ;ksa ds fodz; ls izkIr dqy vk; 𝑅(𝑥):i;ksa esa 𝑅(𝑥) = 3𝑥 +
26𝑥 + 15 ls izkIr gS lhekar vk; Kkr dhft, tc 𝑥 = 5 gSA
The total revenue in Rupees received from the sale of x units of a product is given
by R(𝑥) = 3𝑥 + 26𝑥 + 15. Find the marginal revenue when 𝑥 = 5

(12) vody lehdj.k 𝑥 = −𝑦 dk O;kid gy Kkr dhft,A 2
dy
Find general solution of the differential equation 𝑥 5 = −y5
dx
vFkok @OR
vody lehdj.k = 4−𝑦 , (−2 < 𝑦 < 2) dk O;kid gy Kkr dhft,A

Find general solution of the differential equation

= 4−𝑦 , (−2 < 𝑦 < 2)

(13) lfn'k 𝚤̂ + 𝚥̂ + 2𝑘 ds vuqfn'k ,d ek=d lfn'k Kkr dhft,A 2
Find the unit vector in the direction of vector ı̂ + ȷ̂ + 2k
lfn'k 𝚤̂ − 𝚥̂ ij lfn'k 𝚤̂ + 𝚥̂ dk iz{ksi Kkr dhft, A
Find the projection of the vector ı̂ + ȷ̂ on the vector ı̂ − ȷ̂
(14) ml js[kk ds fnd dkslkbu Kkr dhft, tks funsZ'kk{kksa ds lkFk leku

dks.k cukrh gSA 2
Find the direction cosines of a line which makes equal angles with the
coordinate axes
vFkok @OR
fcUnq (1,2,3) ls xqtjus okyh js[kk dk lfn'k lehdj.k Kkr dhft, tks
3𝚤̂ + 2𝚥̂ − 2𝑘 ds lekUrj gSA

Find the equation of the line which passes through the point (1,2,3)
and is parallel to the 3𝚤̂ + 2𝚥̂ − 2𝑘

8

(15) ;fn 𝑎⃗ + 𝑏⃗ . (𝑎⃗ − 𝑏⃗) = 8 vkSj |𝑎⃗| = 8 𝑏⃗ gks rks |𝑎⃗|,oa 𝑏⃗ Kkr dhft,A 2
Find |𝑎⃗| and 𝑏⃗ , if 𝑎⃗ + 𝑏⃗ . 𝑎⃗ − 𝑏⃗ = 8 and |𝑎⃗| = 8 𝑏⃗

;fn ,d ek=d lfn'k 𝑎⃗ ds fy, (𝑥⃗ − 𝑎⃗). (𝑥⃗ + 𝑎⃗) = 12 rks |𝑥⃗| dk eku Kkr dhft,A
If 𝑎⃗ is a unit vector and (𝑥⃗ − 𝑎⃗). (𝑥⃗ + 𝑎⃗) = 12 , then find |𝑥⃗|
2 1
(16) ;fn 𝐴= 3 2 rFkk 𝐵 = 1 0 1
rks 𝐴 rFkk 𝐵 dk xq.kuQy Kkr
−1 2 1
−1 1
dhft,A 3
2 1
1 0 1
If 𝐴 = 3 2 and 𝐵 = then find the product of 𝐴 and 𝐵
−1 2 1
−1 1
;fn 2𝑋 + 3𝑌 = 2 3 rFkk 3𝑋 + 2𝑌 = 2−2
rks 𝑋 rFkk 𝑌 Kkr dhft,A
4 0 −1 5
2 3 2 −2
If 2𝑋 + 3𝑌 = and 3𝑋 + 2𝑌 = then find 𝑋 and 𝑌
4 0 −1 5
(17) o`Rr 𝑥 + 𝑦 = 9 ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 3
Find the area enclosed by the circle 𝑥 + 𝑦 = 9
nh?kZo`Rr + = 1ls f?kjs {ks= dk {ks=Qy Kkr dhft,A
𝑥 𝑦
Find the area enclosed by the ellipse + =1
𝑎 𝑏
(18) vkys[kh; fof/k }kjk fuEu jSf[kd çksxzkeu leL;k dks gy dhft,: 3
fuEu O;ojks/kksa ds varxZr
𝑥 + 3𝑦 ≥ 10; 3𝑥 + 4𝑦 ≤ 24; 𝑥 ≥ 0: 𝑦 ≥ 0
𝑧 = 200𝑥 + 500𝑦 dk U;wure eku Kkr dhft,A

Solve the following linear programming problem graphically:
Minimise 𝑧 = 200𝑥 + 500𝑦 subject to constraints:
𝑥 + 3𝑦 ≥ 10; 3𝑥 + 4𝑦 ≤ 24; 𝑥 ≥ 0: 𝑦 ≥ 0

9

vkys[kh; fof/k }kjk fuEu jSf[kd çksxzkeu leL;k dks gy dhft,:
fuEu vojks/kksa ds varxZr 𝑧 = 3𝑥 + 4𝑦 dk vf/kdrehdj.k dhft,:
𝑥 + 𝑦 ≤ 8, 𝑥 ≥ 0, 𝑦 ≥ 0
Solve the following linear programming problem graphically:
Maximise 𝑧 = 3𝑥 + 4𝑦 subject to the constraints: 𝑥 + 𝑦 ≤ 8, 𝑥 ≥ 0, 𝑦 ≥ 0
(19) ,d ifjokj esa nks cPps gSa ;fn ;g Kkr gks fd cPpksa esa ls de ls de ,d cPpk yM+dk
gS rks nksuksa cPpksa ds yM+dk gksus dh D;k izkf;drk gSA 3
A family has two children , what is the probability that both the children
are boys given that at least one of them is a boy.
vFkok @OR
,d U;k¸; ikals dks mNkyk x;k gS ?kVukvksa 𝐸 = {1,3,5}, 𝐹 = {2,3} vkSj 𝐺 = {2,3,4,5}
ds fy, fuEu Kkr dhft,&
𝑃( ) vkSj 𝑃( )
A fair die is rolled . Consider events 𝐸 = {1,3,5}, 𝐹 = {2,3} and
𝐸 𝐺
𝐺 = {2,3,4,5} then find 𝑃 and 𝑃(𝐸)
𝐺

(20) 𝐾 ds ml eku dks Kkr dhft, ftlls iznRr Qyu 4
य द𝑥 ≠
𝑓(𝑥) = 𝑥= ij larr gksA
3 यद 𝑥=

Find the value of k for which given function.

if 𝑥 ≠
𝑓(𝑥) = is continuous at 𝑥 =
3 if 𝑥 =

10

D;k 𝑓(𝑥) = 𝑥 + 5 य द 𝑥 ≤ 2 }kjk ifjHkkf"kr Qyu ,d larr Qyu gSA
3 यद 𝑥>2
𝑥+5 य द𝑥 ≤2
Is function defined by 𝑓(𝑥) = is a continuous function.
3 यद 𝑥>2
(21) fn, x;s lehdj.k fudk; dks vkO;wg fof/k ls gy dhft, & 4
5𝑥 − 2𝑦 = 4
7𝑥 + 3𝑦 = −5
Solve given system of equation by matrix method .
5𝑥 − 2𝑦 = 4
7𝑥 + 3𝑦 = −5
vFkok @OR
;fn 𝐴 = 2 3 rks iznf'kZr dhft, 𝐴 − 4𝐴 + 𝐼 = 0 tgk¡ 𝐼 dksfV 2 × 2 rRled
4 0
vkO;wg gS] rFkk 0 dksfV 2 × 2 dk 'kwU; vkO;wg gSA bldh lgk;rk ls 𝐴 Kkr dhft,A
2 3
If 𝐴 = then show that 𝐴 − 4𝐴 + 𝐼 = 0 where 𝐼 𝑖𝑠 𝑎𝑛 identity matrix
4 0
of order 2 × 2 and 0 is a zero matrix of order 2 × 2 and hence find 𝐴

√
(22) ∫ 𝑑𝑥 dk eku Kkr dhft,A 4
√ √
𝜋 3
√𝑠𝑖𝑛𝑥
Find the value of ∫0 3 2
𝑑𝑥
√𝑠𝑖𝑛𝑥+ 3√𝑐𝑜𝑠𝑥

∫ 𝑥𝑠𝑖𝑛 𝑥 𝑑𝑥 dk eku Kkr dhft,A
Find the value of ∫ 𝑥𝑠𝑖𝑛 𝑥 𝑑𝑥
(23) fuEufyf[kr js[kkvksa ds chp dh U;wure~ nwjh Kkr dhft,& 4
𝑟⃗ = 𝚤̂ + 2𝚥̂ + 3𝑘 +++λ 𝚤̂ − 3𝚥̂ + 2𝑘 ;
𝑟⃗ = 4𝚤̂ + 5𝚥̂ + 6𝑘 +++μ(𝚤̂ − 3𝚥̂ + 2𝑘)

Find the shortest distance between following lines.
𝑟⃗ = 𝚤̂ + 2𝚥̂ + 3𝑘 +++λ 𝚤̂ − 3𝚥̂ + 2𝑘 ;
𝑟⃗ = 4𝚤̂ + 5𝚥̂ + 6𝑘 +++μ(𝚤̂ − 3𝚥̂ + 2𝑘)
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;fn js[kk,¡ = = vkSj = = & ijLij yac gks rks 𝑘 dk
eku Kkr dhft, A
If lines = = and = =

are mutually perpendicular then find the value of k .

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