FIRST TERMINAL EXAMINATION 2079

Subject: Opt. Mathematics F.M.: 100

Class: 10(Ten) Time: 3 hrs.
All questions are compulsory. ;a}
k|Zgx? clgjfo{ 5g\ .
;d"x ‘s’ Group 'A' [5x(1+1)=10]
1. a. 𝑦 = 𝑚𝑥 + 𝑐 df 𝑚 sf] dfg slt x'Fbf of] cr/ kmng x'G5 <
For what value of m in 𝑦 = 𝑚𝑥 + 𝑐 makes it a constant function.
𝑥
b. olb 𝑓(𝑥) = 2 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 eP 𝑓𝑜 𝑔(𝑥) sf] dfg lgsfNg'xf];\ .
𝑥
If 𝑓(𝑥) = 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 2 , find the value of 𝑓𝑜 𝑔(𝑥).
2
2. a. z]if ;fWosf] syg n]Vg'xf];\ .
State the remainder theorem.
b. b'O{ ;+Vofx? 𝑎 / 𝑏 sf] cª\sul0ftLo dWos slt x'G5 <
What is the arithmetic mean between two numbers 𝑎 and 𝑏.
3. a. s'g cj:yfdf kmng 𝑓(𝑥), laGb' 𝑥 = 𝑎 df lg/Gt/ ePsf] dflgG5 <
Under what condition, the function 𝑓(𝑥) is said to be
continuous at the point 𝑥 = 𝑎 ?
b. (2, 5) sf] HofldtLo :j?k n]Vg'xf];\ .
Write down the geometrical form of (2, 5) .
4. a. Psn d]l6«S; eg]sf] s] xf] < What is singular matrix ?
b. olb 𝐴 = [2 0] / 𝐴 Psn d]l6«S; eP 𝑥 sf] dfg kQf nufpg'xf];\ .
𝑏 𝑥
2 0
If 𝐴 = [ ] and A is a singular matrix then find the value of x.
𝑏 𝑥
5. a. olb b'O{ /]vfx? 𝑦 = 𝑚1 𝑥 + 𝑐1 / 𝑦 = 𝑚2 𝑥 + 𝑐2 cfk;df ;dfgfGt/ eP,
logLx?sf] em'sfjsf] ;DaGw n]Vg'xf];\ .
If the lines 𝑦 = 𝑚1 𝑥 + 𝑐1 and 𝑦 = 𝑚2 𝑥 + 𝑐2 are parallel to each
other, write the relation between their slopes.
b. /]vf 𝑦 = 4𝑥 − 1 ;Fu nDa x'g] /]vfsf] em'sfj slt x'G5 <
What is the slope of the line perpendicular to the line = 4𝑥 − 1 .
;d"x ‘v’ Group B [3x(2+2+2)+2x(2+2)=26]
6. a. olb kmng 𝑓(𝑥) = 2𝑥 − 7 sf] lj:tf/ If]q {−2, 3} eP, To;sf] If]q
lgsfNg'xf];\ .
If the range of the function 𝑓(𝑥) = 2𝑥 − 7 is {−2,3}, find its
domain.
b. olb 𝑓(𝑥) = 2𝑥 − 3 / 𝑔(𝑥) = 𝑥 2 + 1 eP 𝑓𝑜 𝑔(𝑥) sf] dfg lgsfNg'xf];\.
If 𝑓(𝑥) = 2𝑥 − 3 and 𝑔(𝑥) = 𝑥 2 + 1, find the value of 𝑓𝑜 𝑔(𝑥).
 c. ax'kbLo 𝑓(𝑥) = 𝑥 3 − (𝑝 − 2)𝑥 2 − 𝑝𝑥 + 28 nfO{ (𝑥 + 3) n] efu ubf{
z]if 10 cfpF5 eg] z]if ;fWosf] k|of]u u/L 𝑝 sf] dfg kQf nufpg'xf];\ .
The polynomial 𝑓(𝑥) = 𝑥 3 − (𝑝 − 2)𝑥 2 − 𝑝𝑥 + 28 leaves a
remainder 10J when divided by (𝑥 + 3). Find the value of 𝑝,
using remainder theorem.
7. a. olb 𝑥 3 − 21𝑥 − 20 = (𝑥 + 1)𝑄(𝑥) eP ;ª\lIfKt ljlw k|of]u u/]/ 𝑄(𝑥)
kQf nufpg'xf];\ .
If 𝑥 3 − 21𝑥 − 20 = (𝑥 + 1)𝑄(𝑥), find 𝑄(𝑥) by using synthetic
division method.
b. olb >]0fL 80 + 79 + 60 + ⋯ … . …. sf] 𝑛 cf}+ kb z'Go eP 𝑛 sf] dfg
lgsfNg'xf];\ .
If the 𝑛𝑡ℎ term of the series 80 + 79 + 60 + ⋯ … . …. is zero, find the
value of n.
8. a. olb 2,4, 𝑚 Pp6f u'0ff]Q/ cg'qmd eP 𝑚 sf] dfg lgsfNg'xf];\.
If 2,4, 𝑚 is a geometric sequence then find the value of m.
b. pb\ud laGb'df zLif{laGb' ePsf] Pp6f kf/faf]nf laGb' (4,32) eP/ hfG5 eg]
;f] kf/faf]nfsf] ;dLs/0f kQf nufpg'xf];\ .
Find the equation of parabola which pass through the point
(4,32) and vertex lies at the origin.
𝑥 2 −9
9. a. 𝑓(𝑥) = 𝑥−3
𝑎𝑡 𝑥 = 3 df lg/Gt/tf jf ljlR5Ggtf kl/If0f ug'{xf];\ .
𝑥 2 −9
Test the continuity and discontinuity of 𝑓(𝑥) = 𝑥−3 𝑎𝑡 𝑥 = 3.

b. olb 𝐴 = [2 3] / B=[1 0 ] eP |𝐴𝐵| kQf nufpg'xf];\ .

1 5 2 −3
2 3 1 0
If 𝐴 = [ ] and B=[ ], find |𝐴𝐵|
1 5 2 −3
c. olb | 𝑥 −2𝑥| = 100 eP 𝑥 sf] dfg lgsfNg'xf];\.
3𝑥 4𝑥
𝑥 −2𝑥
If | | = 100, then find the value of x.
3𝑥 4𝑥
10. a. olb 𝐴 = [ 1 2] / 𝐵 = [2 0 ] eP |𝐴𝐵| = |𝐴||𝐵|k|dfl0ft ug'{xf];\ .
−3 5 4 −3
1 2 2 0
If 𝐴 = [ ] 𝑎𝑛𝑑 𝐵 = [ ] then verify that |𝐴𝐵| = |𝐴||𝐵|.
−3 5 4 −3
b. /]vfx? 2𝑥 − 𝑦 + 4 = 0, / 3𝑥 + 𝑦 + 3 = 0 aLrsf] clwssf]0f
lgsfNg'xf];\ .
Find the obtuse angle between the lines 2𝑥 − 𝑦 + 4 = 0,
and 3𝑥 + 𝑦 + 3 = 0.
 c. olb /]vfx? 𝑙1 𝑥 + 𝑚1 𝑦 + 𝑛1 = 0 / 𝑙2 𝑥 + 𝑚2 𝑦 + 𝑛2 = 0 Ps cfk;df
nDa eP 𝑙1 𝑙2 + 𝑚1 𝑚2 = 0 x'G5 egL k|dfl0ft ug'{xf];\ .
If the lines 𝑙1 𝑥 + 𝑚1 𝑦 + 𝑛1 = 0 and 𝑙2 𝑥 + 𝑚2 𝑦 + 𝑛2 = 0 are
perpendicular to each other prove that 𝑙1 𝑙2 + 𝑚1 𝑚2 = 0.
;d"x u Group C [11×4=44]
11. olb 𝑓(𝑥) = 2𝑥 − 5, 𝑔(𝑥) = 3𝑥+5
2
/ 𝑓(𝑥) = 𝑔−1 (𝑥) eP 𝑥 sf] dfg
lgsfNg'xf];\.
3𝑥+5
If 𝑓(𝑥) = 2𝑥 − 5, 𝑔(𝑥) = 𝑎𝑛𝑑 𝑓(𝑥) = 𝑔−1 (𝑥), then find the
2
value of x.
12. xn ug'{xf];\ . Solve: 2𝑥 3 − 9𝑥 2 + 7𝑥 + 6 = 0
13. olb s'g} ;dfgfGt/ >]0fLsf] rf}yf] kb 1 / klxnf] cf7f}F kb ;Ddsf] of]ukmn 18
eP ;f] >]0fLsf] bzf}+ kb kQf nufpg'xf];\ .
If the fourth term of an AP is 1 and the sum of its eight terms is
18, find the tenth term of the series.
2
14. 3
/ 162 sf] lardf /x]sf 4 cf]6f u'0ff]Q/ dWodfx? lgsfNg'xf];\ .
2
Write 4 𝐺𝑀𝑠 between and 162 .
3
15. ju{ ;dLs/0f 𝑥 2 − 3𝑥 = 10 nfO{ n]vflrqaf6 xn ug'{xf];\ .
Solve graphically the quadratic equation 𝑥 2 − 3𝑥 = 10.
16. lgDg lnlvt ;dLs/0fsf] n]vflrq lvRg'xf];\ M 𝑦 = 4𝑥 2 + 8𝑥 + 5 /
𝑥+𝑦 =3
Draw the graph of the given equation: y = 4x 2 + 8x + 5 and x +
y = 3.
2
17. kmng 𝑓(𝑥) = { 𝑥 + 2 𝑥 ≤ 5 𝑎𝑡 𝑥 = 5 df lg/Gt/tf kl/If0f ug'{xf];\ .
3𝑥 + 12 𝑥 > 5
Examine the continuity of the function:
2
𝑓(𝑥) = { 𝑥 + 2 𝑥 ≤ 5 𝑎𝑡 𝑥 = 5
3𝑥 + 12 𝑥 > 5
18. olb lbOPsf] kmng 𝑥 = 2 df lg/Gt/ eP 𝑘 sf] dfg kQf nufpg'xf];\ .
If the given function is continuous at x=2 then find the value of
𝑘𝑥 − 1 𝑥 < 2
k. 𝑓(𝑥) = { .
2𝑥 − 3 𝑥 ≥ 2
19. olb [2𝑚 7] sf] ljk/Lt d]l6«S; [ 9 𝑛
] eP 𝑚 / 𝑛 sf dfgx? kQf
5 9 −5 4
nufpg'xf];\ .
2𝑚 7 9 𝑛
If the inverse of the matrix [ ] is the matrix [ ],
5 9 −5 4
Calculate the value of m and n.
20. laGb' (−2, −3) af6 hfg] / /]vf 5𝑥 + 7𝑦 = 14 ;Fu ;dfgfGt/ x'g] ;/n
/]vfsf] ;dLs/0f kQf nufpg'xf];\ .
Find the equation of a straight line parallel to the line with
equation 5𝑥 + 7𝑦 = 14 and passes through the point (−2, −3).
21. 𝐴(4,7) / 𝐵(5, −2) b'O{cf]6f laGb'x? x'g\ . 𝐴𝐵 df nDa x'g] / laGb' (1,2) eP/
hfg] /]vfsf] ;dLs/0f kQf nufpg'xf];\ .
𝐴(4,7) and 𝐵(5, −2) are two points. Find the equation of a
straight lines perpendicular to AB and passing through the point
(1,2).
;d"x 3 Group D [4×5=20]
𝑥−2 1 1
22. kmng 𝑓(𝑥) = 2𝑥+1 , 𝑥 ≠ − / 𝑔(𝑥) = , 𝑥 ≠ 0 lbOPsf 5g\ . olb
2 𝑥
𝑓 −1 (𝑥) = 𝑔𝑜 𝑓(𝑥) eP 𝑥 sf] dfgx? kQf nufpg'xf];\ .
𝑥−2 1 1
Function 𝑓(𝑥) = ,𝑥 ≠ − and 𝑔(𝑥) = , 𝑥 ≠ 0 are given. If
2𝑥+1 2 𝑥
𝑓 −1 (𝑥) = 𝑔𝑜 𝑓(𝑥), find the value of x.
23. Pp6f wgfTds ;dfg cg'kft ePsf] u'0ff]Q/ >]0fLsf klxnf rf/ kbx?sf] of]ukn
40 / klxnf b'O{ kbx?sf] of]ukmn 4 5 eg] ;f] >]0fLsf] klxnf] cf7 kbx?sf]
of]ukmn lgsfNg'xf];\ .
In a geometric series having positive value of common ratio, the
sum of first four terms is 40 and the sum of first two term is 4.
Find the sum of first eight term of the series.
24. p2]Zo kmng 𝑃 = 5𝑥 + 4𝑦 sf] lgDglnlvt cj:yfdf clwstd dfg lgsfNg'xf];\ .
2𝑥 + 5𝑦 ≤ 16, 2𝑥 + 𝑦 ≤ 8, 𝑥 ≥ 0, 𝑦 ≥ 0
Maximize 𝑃 = 5𝑥 + 4𝑦 under the following constraints:
2𝑥 + 5𝑦 ≤ 16, 2𝑥 + 𝑦 ≤ 8, 𝑥 ≥ 0, 𝑦 ≥ 0
25. olb ;/n /]vf 3𝑥 + 5𝑦 = 17 /] ;/n /]vf 3𝑥 − 𝑘𝑦 = 8 ;Fu 450 sf] sf]0f
agfpF5 eg] 𝑘 sf] dfg kQf nufpg'xf];\ .
Find the value of k if the straight line 3𝑥 + 5𝑦 = 17 makes an
angle of 450 with the straight line 3𝑥 − 𝑘𝑦 = 8.
;kmntfsf] z'esfdgf