Q.B.- Maths I + II - FYJC - Ver

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1 Q.B.- Maths I + II - FYJC - Ver -709 Q Find the equation of lous of a point, whih moves suh that the ratio of its distanes from (,0) and (, ) is :. ( : 9x + 9y + x - 0y ) Q Q Find the equation of lous of a point P suh that the differene of the squares of its distanes from the points (, 0) and (. ) is 0. ( : x - y - 0 or x + y - 0) If A (, ) and B (, -) are given points. Find the equation of lous of points P, suh that AP + BP 0. ( : x + y - x + 8y + 0) Q If A(, ), B(-, ), find the equation of the lous of the point P, suh that AP BP. ( : x + y + 8x + 0) Q Q 6 Q 7 Q 8 Q 9 Find the lous of the point P in eah of the ases : (i) absissa of P exeeds twie its ordinate by. The sum of its o-ordinates equals 0. : (i) x y + x + y 0 If the origin is shifted to the point (, -), the axes remaining parallel, find the new o-ordinate of the points: (i) (, ); (, -7); (iii) ( -0, ) : (i) (, ); (0, -); (iii) ( -, 6) If the origin is shifted to the point (-, ), the axes remaining parallel, find the old o-ordinate of the points: (i) (, ); (, -); (iii) (0, ) : (i) (0, ); (, -); (iii) ( -, ) A(, ) and B(6, -) are two verties of triangle ABC and C is a point on the lous x +y9. Find the equation of lous of the entroid of trianle ABC. ( : x + y - 6 0) Given A (-, ) and B is the point on the lous x + y - x + y If the points P divides segment AB externally in the ratio :, find the equation of the lous of P. ( : x + y - x + y ) Q 0 Find the values of 'a' and 'b', if the points (, ), (-, -) lie on the lous ax + by. (a, b -) Q Find the value of 'a' if the points (-6, ) lies on lous x ay. (a ) Q Q Q A(, 0) and B(-,0) are two given points.a vriable point P is suh that PA - PB 6. Find the equation of lous of the point P. ( : 6x - 9y ) 7 Express the following angles in degrees () () () () 7 9 () - 0 () (60/) () (900/) () - 0 Express the following angles in radians (i) 0 9 (iii) -08 (iv) - ANS : () () () () Q In LMN, m L / & m N 0. Find m M both in degrees and radians : Q 6 If x 0 and y -, find x and y. [ x 9/, y - ] Q 7 The differene between two aute angles of a right angled triangle is /0.. Find the angles in in degrees and radians. [ : 7º & 8º i.e./ & /0 ] Q 8 Q 9 One angle of a triangle has measure /9 and the measure of other two angles are in the ratio :, find their measures in degrees and radians. [ : 80º & 60º i.e./9 & / ] The radius of a irle is 9 ms. Find the length of an ar of this irle whih uts off a hord of length equal to length of radius. [ s ms. ]

2 Q 0 Q Q Q Q Q Q 6 Q 7 The area of the irle is sq.ms.find the length of its ar subtending an angle of º at the entre. Also find the area of the orresponding setor. [l(ar) ms. A(setor) 0 sq. ms.] OPQ is a setor of a irle with entre O and radius is ms. If m POQ 0º, find the area enlosed by ar PQ and hord PQ. [ : 7( - )/ ] The perimeter of a setor of a irle, of area 6 sq.ms. is 6 ms. Find the area of setor. [ : 60 sq.ms. ] Two irles, eah of radius 7 ms, interset eah other. The distane between their entres is 7 ms. Find the area ommon to both the irles. [ : 9( - )/ ] PQR is an equilateral triangle with side 8 ms. A irle is drawn on segment QR as diameter. Find Find the length of the ar of this irle inteepted with in the triangle. [ : ms ] If the two ars of the same lengths in two irles subtend angles 6 and 0 at the entres. Find the ratio of their radii. [ : : ] If the angles of a triangle are in AP and the greatest angle is 8, ind all the three angles in radians. [ : /, /, 7/ ] Evaluate : sin + os + sin + os - se - 6 ose [ : 6 ] Q 8 Evaluate : sin 0 + os 0 + sin + os Q 9 + se ose [ : 6 ] Find the value of sin 0º + os 60º + tan º + se 60º - ose 0º : 9 Q 0 Find \ the value of ot 0 º + 9sin 60 º - 6ose 60 º - tan 60º : tan Q (/6) + sin (/6) + os (/) Verify that : se se (/ - os + os () 6 Q If os θ + os θ, then find os θ. [ : os θ - (Invalid), os θ / ] Q If tan θ - tan θ + 0, then find tan θ [ : tan θ or tan θ / ] Q Find the values of the trigonometri funtions if os θ / and / < θ < [ ANS : sin θ - /, tan θ - /, se θ /, ose θ - /, ot - / ] ose θ + ot θ Q If os θ -/, < θ < /, find the value of : se θ - tan θ 6 sin A sin B Q 6 If where A and B are in the II nd quadrant, find the value of os A + os B [ : - ] se x + Q 7 Prove that se x - ose x - ot x Q 8 Prove that : se x - se x tan x + tan x Q 9 Prove that ( - tan x) + ( - ot x) (se x - ose x) Q 0 Prove that (ose x - sin x) (se x - os x) (tan x + ot x). Q Prove that sin x + os x (sin x + os x)( - sin x os x)

3 Q Prove that - - ose θ - ot θ sin θ sin θ ose θ + ot θ Q Prove that sin θ + os θ + + os θ sin θ ose θ Q Prove that tan θ se θ + + ose θ se θ + tan θ Q Prove that : tan A tan A + se A + se A - se A - + tan A Q 6 Prove that : + ose A + ot A ose A + ot A - + ose A - ot A ot A - ose A + Q 7 If x r os θ os φ, y r osθ sinφ and z r sin θ then prove that x + y + z r. Q 8 If tan x + ot x then show that tan x + ot x 7 + os Q 9 A Prove that + ot sin A A ot A + ose A - + os A Q 0 Prove that : ot A - ose A + sin A Q If os x + 7 sin x, find the permissible values of sin x. Q Without using trigonometri tables, find (i) os 7. tan 7. Q Q If sin α 7 : (i) - + -, / < α < and os β, < β < /, find (i) sin (α - β). os (α - β). [ : (i) sin (α - β) -8/8 os (α - β) -/8 ] Prove that.os - A os A + sin A Q Prove that sin ( + θ).os ( - φ) - os ( + θ).sin( - φ) sin (θ + φ). Q 6 Prove that : sin + A os - B + sin - B.os + A sin (x + y) 6 6 Q 7 Prove that : sin A.sin (B - C) + sin B.sin(C - A) + sin C.sin (A - B) 0 tan A - tan A sin A Q 8 Prove that : tan A + tan A sin 8A Q 9 Prove that : tan 8º + tan 7º + tan 8º.tan 7º os 7º + sin 7º Q 60 Prove that : tan 7º os 7º - sin 7º os º - sin º Q 6 Prove that : os º + sin º Q 6 If tan, tan B, prove that : A + B 6 Q 6 Find the values of the following () tan 80º () tan (- 690º) () ose (-90)º () se 0 Q 6 [ANS : (i) tan (80 ) - tan (-690 ) - / (iii) ose (- 90 ) (iv) se (0 ) - / ] Show that : sin (-0).os(-0) + sin(-0).os (90)

4 Q 6 Show that : ot (0º).tan(- 9º) - tain(8º).ot (- 9º) 0 sin (90º + A) sin (80º - A) tan (70º + A) Q 66 Show that : + os (-A) sin (-A) ot (-A) ose (90º - A).sin (80º - A).ot (60º - A) Q 67 Show that : se (80º + A).tan (90º + A).sin (-A) Q 68 Prove that : sin x + sin + x 9 Q 69 Prove that : os + os + os + os Q 70 Prove that : sin + sin + sin + sin Q 7 If tan θ, < θ <, then find (i) sin θ os θ (iii) tan θ. [ : (i) sin θ / os θ -7/ (iii) tan θ - / ] Q 7 Prove that : sin A (tan A + or A) sin A os A Q 7 Prove that : se A sin A os A sin A os A Q 7 Prove that : + ot A os A sin A Q 7 Prove that : os 0º.os 0º.os 60º.os 80º 6 Q 76 Q 77 + os θ Prove that : ot sin θ θ Prove that : tan + θ se θ - tan θ.os A + os A Q 78 Prove that : ot A..sin A - sin A - - Q 79 Expand the determinants : (a) (b) [ : (a) - (b) 0 ] - 0 x 0 - x Q 80 Find x, if : (a) x - (b) x + 0 (a) x - 6 or : (b) x - or - i -i - Q 8 Show that i i - i, where i i Q 8 Without expanding the determinant show that the value of the following determinents is zero. (i) Hint : xy xy(x + y) yz yz(y + z) zx zx(z + x) x + a x + b x + y + a y + b y + z + a z + b z + (i) C C + (xyz)c. Take (x + y + z) Common. R R - R, R R - R.. Take (x - y) and (y - z) Common from R & R 0

5 Q 8 Q 8 Q 8 Without expanding the determinants, show that (i) (i) a a b a a b b a b b ab x + y y + z z + x z + x x + y y + z y + z z + x x + y x y z x y z x y z Hint : (i) Take a, b, from R, R, R. Then multiply C by ab. R R + R + R Take ommon from R, R R - R, R R - R, R R - R Without expanding the determinants, show that a b b a (a - b)(b - )( - a) x y z (x - y)(y - z)(z - x)(xy + yz + zx) ab x y z Hint : (i) R R - R, R R - R.Take (a - b)(b - ) from R & R. R R - R. Expand C C - C, C C - C. Take ( x - y ) ( y - z ) from C & C. Simplify, Fatorize then C C - C Take ( z - x ) from C then Expand Solve the following equations x - x + x + x + x + x + (i) x + x - x + 0 x + x + 0 (iii) x x + x + x - x + x + x + 9 Hint : (i) R R + R + R Take (x - ) from R. Expand : x -/ R R - R, R R - R. Take (x - ) & x from R & R. Expand : x 0,, - (iii) C C - C and C C - C. Expand : x 0, -8 0 Q 86 Solve the equations, x + y - 7 0, y - x, using Cramer's Rule [ : x - /, y /] Q 87 Find x and y using Cramer's Rule if 6, + 8 [ : x 7/, y - 7/0] x y x y Q 88 Q 89 Q 90 Q 9 Q 9 Solve the following equations using Cramer's Rule (i) x - y + z 9, x + y + z, x - y + z. [ : x, y, z. ] x + y + z 6, x + y + z 7, x + y + 9z. [ : x, y, z. ] Find x, y and z using Cramer's Rule if x + y + z, x y + z, and [ : x /, y -, z -/6] + x y z. The sum of three numbers is 6. Twie the third number when added to the first number gives 7. On adding the sum of seond and third number to three times the first number, we get. Find the three numbers using determinants. [ : The numbers are,, and ] Examine the onsisteny of the following equations. (i) x + 6y 7, x + y 8, x + y. x + y +, x + y + 7 0, x + y [ : Consistent ] [ : Not Consistent ] Show that the equations, x + y + 0, x + y + 0, x + y + 0 are onsistent

6 Q 9 Q 9 Q 9 Q 96 Q 97 Q 98 Find k if the following equations are onsistent. (i) 7x - ky, x + y 9 and x + y 8. ANS : x + y - k 0, kx - y + 0 and kx - y k k or - `Find the area of the following triangles whose verties are (i) (, ), (0, 7), (-, ) (, 6), (-, ), (, -) [ : (i) sq.units / sq.units] Find k if the area of the triangle with the verties at A(k, ), B(-, 7) and C(-, ) is Sq.Units [ : k or k - 7/ ] Find k if the area of the triangle with the verties at P(, - ), Q(-, k) and R(, ) is Sq.Units [ : k or k ] Without using tables, show that (i) < log 0 < < log 0 < log x log y log z If, then show that (i) xyz x b - - a a - b b + y + a z a + b Q 99 Evaluate : (i) i i 888 (iii) i 9 (iv) i 0 (v) i -8 (vi) i -888 (vii) i 0 + i 0 + i 0 + i 60 wers : (i) -i (iii) i (iv) -i (v) - (vi) (vii) 0 Q 00 Solve : (i) x - x + 0 x + ix (iii) x + ix + 0 (iv) ix - x - i 0 wers : (i) ± i i or -i (iii) i/ or -i (iv) -i Q 0 Simplify : wers : (i) ω 8 ω -0 (iii) ω -0 (iv) ( - ω)( - ω ) (v) ( + ω - ω ) 6 (vi) ( + ω) - ( + ω ) (vii) ( + ω)( + ω )( + ω )( + ω 8 ) (i) (iii) (iv) 7 (v) 6 (vi) 0 (vii) Q 0 If f(x) ax + bx + and f(), f(), then find a and b. ( : a, b -) Q 0 If f(x) f(x - ), where f(x) x - x +, then find the value of x. ( : a /, b /) Q 0 Find the domain and range : (i) f(x) x. f(x) x -, for - x (iii) f(x) 9 - x, for - x (iv) f(x) x - 6x + Q 0 wers : (i) R, [0, ) [-, ], [-7, ] (iii) [-, ], [-, 9] (iv) R, [, ) x - x + x + Find fog and gof : (i) f(x), g(x) f(x), g(x) x x + x - x - x + - x x + - x : fog gof x -, : fog, gof + x x - + x x + Q 06 If f(x), find f o f. ( : fof x ) x - Q 07 If f(x) x -, then find f - (9). ( : / ) Q 08 If f(x) x - 8, then find f - (). ( : ) Q 09 If f(x) x +, then find f - (). ( : 0 ) Q 0 Find whether the funtion f : R R given by f(x) x + for all x R is one-one,onto or not ( : one-one, onto) Q If the seventh term of an AP is 0 and the tenth term is, find the fourth term. ( t 9 ) Q If the third term of an AP is - and the nineth term is - find the n th term of the AP. (T n - n)

7 Q Find three no.s in AP suh that their sum is and the sum of their squares is 00. (6,8,0 or 0, 8,6) Q Find four numbers in AP suh that the sum of the first and the last is 8 and the produt of the seond and the third is. ( : -,, 6, 0 or 0, 6,, - ) Q Find the sum to terms ( : 7) Q 6 Find the value of n if to n terms 90 ( : 0) Q 7 Find the value of n if to n terms 6 ( : n or n ) Q 8 For a GP if a 7 and r /, find t 6. 7 Q 9 For a GP if a and r, find t Q 0 For a GP if a / and t , find r. ( r ) Q For a GP if t 8 60 and r, find a. ( : a ) Q Find three numbers in GP suh that their sum is and their produt is 000. :, 0, 0 0, 0, Q Find three numbers in GP suh that their sum is and the sum of their squares is 9 9 ( :,, / or /,, ) Q Find four numbers in GP suh that their produt is and the sum of the seond and the third is ( /7, /,, 7 or 7,, /, /7 ) Q Find five numbers in GP suh that their produt is and the produt of the last two is 08. ( : /9, /,, 6, 8 ) Q 6 For a GP if a and r -/, find S. : 6 Q 7 For a GP if S 0 and r, find a. ( : a ) Q 8 For a GP if a and r and S n, find n. ( : n ) Q 9 For a GP if t 8 and t 6 86, find S. ( : S ) 0(0 n - ) - 8n Q 0 Find the sum to n terms. : 8 Q Find the sum to n terms. : 0(0 n - ) - 9n 7 Q Find the sum to n terms. : 0(0 n - ) - 8n 8 Q Find the sum to n terms. : [ 9n - + (0.) n ] 9 Q Find the sum to n terms. : [ 9n - + (0.) n ] n n 8 n Q Evaluate : (i) (r - 8) (r - ) (r + ) (iii) r (6r - r + 6) r n : n - 7n : [ n + 6n - ] : n(n + n + ) r 0

8 Q 6 Find the sum using sigma notation. ( : 600 ) Q 7 Find the sum to terms using sigma notation. ( : 0 ) Q 8 Find the sum to n terms using sigma notation. : n (n + ) Q 9 Find the sum to n terms using sigma notation. : n [ n + 6n - ] Q 0 Find the sum to n terms using sigma notation. : n(n + )(n + ) Q Find the sum using sigma notation. ( : 70) Q lim x - x - 6 lim x - lim x - Evaluate : (i) x x - x + x - x x (iii) + x - x / x - Q Q Q Q 6 Q 7 Q 8 Q 9 Q 0 : / : / : lim x - 6 lim x /7 - a /7 x x - 6 x a x /7 - a /7 (iii) lim x - Evaluate : (i) x x - a /7 wers : (i) 6 (iii) -/ lim x - a If 0, find all possible values of a ( : a ± ) x a x - a lim x 9 - a 9 If 9, find all possible values of a ( : a ± ) x a x - a lim x + - lim x lim + x + x - Evaluate : (i) (iii) x x x x - x 0 x 0 lim - os x lim sin x - sin x lim - tan x Evaluate : (i) x 0 x x 0 x (iii) x - sin x lim sin x - sin a lim - tan x lim ose x - Evaluate : (i) (iii) x a x - a x x - x - x os a Evaluate : (i) lim x - x lim x - x - x + x 0 x x 0 x (iii) lim a b x - a x b x x - : log : (log )(log ) lim x lim - x Evaluate : (i) - (iii) x 0 x 0 + x -- End -- x : : e / e x : ab b log a lim log ( - x) - log ( + x) x 0 x : -

STD. XI Sci. Perfect Mathematics - I

STD. XI Sci. Perfect Mathematics - I Written as per the revised syllabus presribed by the Maharashtra State Board of Seondary and Higher Seondary Eduation, Pune. STD. XI Si. Perfet Mathematis - I Fifth Edition: May 0 Salient Features Exhaustive