(English+Hindi) MATHEMATICS ªÁáÊà 1. Let f (x)=2 10 x+1 and g(x)=3 10 x 1. If (fog)(x)=x, then x is equal to : (1) (2) (3) (4)

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1 MATHEMATICS ªÁáÊÃ. Let f (x)= 0 x and g(x)=3 0 x. If (fog)(x)=x, then x is equal to : Let p(x) be a quadratic polynomial such that p(0)=. If p(x) leaves remainder when divided by x and it leaves remainder 6 when divided by x; then : p= p=9 p( )=9 p( )=. ÊŸÊ f (x)= 0 x ÃÕÊ g(x)=3 0 x ÿᜠ(fog)(x)=x Ò, ÃÊapple x UÊ U Ò ÊŸÊ p(x) apple Ê ÁmÉÊÊÃË È Œ Ò Á apple Á ÿapple p(0)= Ò ÿᜠp(x) Êapple x apple ʪ ŒappleŸapple U Êapple U ÃÊ Ò ÃÕÊ x apple ʪ ŒappleŸapple U 6 Êapple øãê Ò, ÃÊapple p= p=9 p( )=9 p( )= VI - MATHEMATICS

2 3. Let z C, the set of complex numbers. Then the equation, z3i z i =0 represents : a circle with radius ÊŸÊ z C, Êapple Áê üê ÅÿÊ Êapple Ê ÈìÊÿ Ò, ÃÊapple Ë UáÊ z3i z i =0 ŒÁ Ê Ã UÃÊ Ò flîûê Á Ë ÁòÊíÿÊ 8 3 Ò a circle with diameter 0 3. an ellipse with length of major axis 6 3. an ellipse with length of minor axis The number of real values of λ for which the system of linear equations xy λz=0 xλyz=0 λxyz=0 has infinitely many solutions, is : Let A be any 3 3 invertible matrix. Then which one of the following is not always true? adj (A)= A A adj (adj(a))= A A adj (adj(a))= A (adj(a)) adj (adj(a))= A (adj(a)) flîûê Á Ê ÿê 0 3 Ò ŒËÉÊ flîûê Á apple ŒËÉÊ ˇÊ Ë Êß 6 3 Ò ŒËÉÊ flîûê Á apple ÉÊÈ ˇÊ Ë Êß 6 9 Ò. λ apple Ÿ flêsãáfl ÊŸÊapple Ë ÅÿÊ Á Ÿ apple Á ÒUÁπ Ë UáÊ ÁŸ Êÿ xy λz=0 xλyz=0 λxyz=0 apple Ÿ à Ò, Ò ÊŸÊ A Êappleß 3 3 Ê ÿèà áêëÿ Ê ÿí Ò ÃÊapple ÁŸêŸ apple apple ÊÒŸ- Ê ŒÊ àÿ Ÿ Ë Ò? adj (A)= A A adj (adj(a))= A A adj (adj(a))= A (adj(a)) adj (adj(a))= A (adj(a)) VI - MATHEMATICS

3 6. If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is : th 5 th 6 th th. If () 999 is divided by, then the remainder is : If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then a b is equal to : a b ÊéŒ QUEEN apple Ë ˇÊ UÊapple Ê ÿêappleª U apple ŸŸapple flê apple Ë ÊéŒ (Á Ÿ Ê Õ Ò ÕflÊ flapple Õ ËŸ Ò ) Êapple ªapple Ë ÊéŒ Êapple apple ŸÈ Ê U ªÊŸapple U, ÊéŒ QUEEN Ê SÕÊŸ Ò flê 5 flê 6 flê flê. ÿᜠ() 999 Êapple apple ʪ ÁŒÿÊ Ê, ÃÊapple Êapple» Ò ÿᜠŒÊapple ÅÿÊ Êapple a ÃÕÊ b, a > b > 0 Ê Ê Ã U Êäÿ (A.M.) Ÿ apple ªÈáÊÊappleûÊ U Êäÿ (G.M.) Ê 5 ªÈŸÊ Ò, ÃÊapple a b UÊ U Ò a b VI - MATHEMATICS

4 9. If the sum of the first n terms of the series is35 3, then n equals : ÿᜠüêappleáêë apple Õ n ŒÊapple Ê ÿêappleª 35 3 Ò, ÃÊapple n UÊ U Ò lim x x 3 is equal to : x 3. The tangent at the point (, ) to the curve, x y x=( y) does not pass through the point :, 3 (8, 5) (, 9) (, ) 0. lim x x 3 x 3 UÊ U Ò. fl x y x=( y) apple Á ŒÈ (, ) U πë øë ªß S Ê appleuπê ÁŸêŸ apple apple Á Á ŒÈ apple Ÿ Ë ªÈ UÃË Ò, 3 (8, 5) (, 9) (, ) VI - MATHEMATICS

5 5 5. If y= x x x x, d y dy then ( x ) x is equal to : dx d x 5 y y 5 y 5 y 5 5. ÿᜠy= x x x x Ò, ÃÊapple d y dy ( x ) x dx d x 5 y y 5 y 5 y UÊ U Ò 3. If a point P has co-ordinates (0, ) and Q is any point on the circle, x y 5x y5=0, then the maximum value of (PQ) is : ÿᜠÁ Ë Á ãœè P apple ÁŸŒapple ÊÊ (0, ) Ò ÃÕÊ Êappleß Á ãœè Q flîûê x y 5x y5=0 U ÁSÕà Ò, ÃÊapple (PQ) Ê Áœ à ʟ Ò VI - MATHEMATICS

6 . The integral cot x(cosec x cot x) dx 0 < x < π is equal to : (where C is a constant of integration) log sin C log sin C log cos C log cos C 5. The integral equals : π π 8 cos x dx 3 (tan x cot x). Ê cot x(cosec x cot x) d x, 0 < x < π UÊ U Ò ( Ê C Ê Ÿ ø U Ò) log sin C log sin C log cos C log cos C 5. Ê π π cos x dx 3 (tan x cot x) UÊ U Ò 6 VI - MATHEMATICS

7 6. The area (in sq. units) of the smaller portion enclosed between the curves, x y = and y =3x, is : 6. fl Êapple x y = ÃÕÊ y =3x apple Ëø ÁÉÊ appleu UÊapple appleu ʪ Ê ˇÊappleòÊ» (flª ß ÊßÿÊapple apple ) Ò π 3 3 π 3 3 π 3 3 π 3 3. The curve satisfying the differential equation, ydx (x3y )dy=0 and passing through the point (, ), also passes through the point :,, 3 3, 3 3, π 3 3 π 3 3 π 3 3 π 3 3. fl Ë UáÊ ydx (x3y )dy=0 Êapple ÃÈc U UŸapple flê Ë flêapple fl, Êapple Á ŒÈ (, ) apple Êapple U ÊÃË Ò, ÁŸêŸ apple apple Á Á ŒÈ apple Ë Êapple U ÊÃË Ò,, 3 3, 3 3, VI - MATHEMATICS

8 8. The locus of the point of intersection of the straight lines, tx y 3t=0 x ty3=0 (t R), is : 8. appleuπê Êapple tx y 3t=0 x ty3=0 (t R) apple ÁÃë appleuœÿ Á ŒÈ Ê Á ŒÈ Õ Ò an ellipse with eccentricity 5 an ellipse with the length of major axis 6 a hyperbola with eccentricity 5 a hyperbola with the length of conjugate axis 3 9. If two parallel chords of a circle, having diameter units, lie on the opposite sides of the centre and subtend angles cos and sec () at the centre respectively, then the distance between these chords, is : ŒËÉÊ flîûê Á Ë à apple 㜠ÃÊ 5 Ò ŒËÉÊ flîûê Á apple ŒËÉÊ ˇÊ Ë Êß 6 Ò Áà Ufl ÿ Á Ë à apple 㜠ÃÊ 5 Ò Áà Ufl ÿ Á apple ÿèç Ë ˇÊ (conjugate axis) Ë Êß 3 Ò 9. ÿᜠflîûê Á Ê ÿê ß Êß Ò Ë ŒÊapple Ê Ã U ËflÊ, Êapple flîûê apple apple Œ Ë Áfl UËà ÁŒ ÊÊ Êapple apple Ò ÃÕÊ apple 㜠U Ê cos ÃÕÊ sec () apple ÊappleáÊ ÃÁ Uà UÃË Ò, ÃÊapple ߟ ËflÊ Êapple apple Ëø Ë ŒÍ UË Ò VI - MATHEMATICS

9 0. If the common tangents to the parabola, x =y and the circle, x y = intersect at the point P, then the distance of P from the origin, is : ( 3 ) ( ) 3 0. ÿᜠUfl ÿ x =y ÃÕÊ flîûê x y = Ë ÿáÿc U S Ê appleuπê Á ŒÈ P U ÁÃë appleuœ UÃË Ò, ÃÊapple P Ë Í Á ŒÈ apple ŒÍ UË Ò ( 3 ) ( ) 3. Consider an ellipse, whose centre is at the origin and its major axis is along the x-axis. If its eccentricity is 3 and the 5 distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is : ŒËÉÊ flîûê, Á Ê apple Œ Í Á ŒÈ U Ò ÃÕÊ ŒËÉÊ ˇÊ x- ˇÊ Ë ÁŒ ÊÊ apple Ò, U ÁfløÊ U ËÁ ÿáœ Ë à apple 㜠ÃÊ 3 ÃÕÊ ŸÊÁ ÿêapple apple Ëø Ë ŒÍ UË 5 6 Ò, ÃÊapple øãè È, Êapple ŒËÉÊ flîûê apple ã㪠à ŸÊß ªß Ò ÃÕÊ Á apple ÊË, ŒËÉÊ flîûê apple ÊË ÊappleZ U Ò, Ê ˇÊappleòÊ» (flª ß ÊßÿÊapple apple ) Ò VI - MATHEMATICS

10 . The coordinates of the foot of the perpendicular from the point (,, ) on the plane containing the lines, x y z 3 = = and 6 8 x y z 3 = =, is : 3 5 (,, ) (,, ) (0, 0, 0) (,, ) 3. The line of intersection of the planes ( i j k) ( i j k) r. 3 = and r. =, is : x 5 y z = = 3 x 5 y z = = 3 x y = 3 = z 3 x y = 3 = z 3. Ã, Á apple ŒÊappleŸÊapple appleuπê x y z = = ÃÕÊ x y z = = ÁSÕà Ò, U Á ãœè (,, ) apple «UÊ apple ª ê apple ÊŒ apple ÁŸŒapple ÊÊ Ò (,, ) (,, ) (0, 0, 0) (,, ) 3. à Êapple ( i j k) r. ( i j k) r. 3 = ÃÕÊ = Ë ÁÃë appleuœë appleuπê Ò x 5 y z = = 3 x 5 y z = = 3 x y = 3 = z 3 x y = 3 = z 3 0 VI - MATHEMATICS

11 . The area (in sq. units) of the parallelogram whose diagonals are along the vectors 8 i 6 j and 3 i j k, is : The mean age of 5 teachers in a school is 0 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is : Ê Ã U øãè È, Á apple Áfl áê, ÁŒ ÊÊapple 8 i 6 ÃÕÊ 3 i j k j Ë ÁŒ ÊÊ Êapple apple Ò, Ê ˇÊappleòÊ» (flª ß ÊßÿÊapple apple ) Ò ÁfllÊ ÿ apple 5 äÿê Êapple Ë Êäÿ- ÊÿÈ 0 fl Ò äÿê 60 fl Ë ÊÿÈ apple appleflê ÁŸflÎûÊ ÊappleÃÊ Ò ÊÒ U apple SÕÊŸ U Ÿÿapple äÿê Ë ÁŸÿÈÁÄà ÊappleÃË Ò ÿáœ ß ÁfllÊ ÿ apple äÿê Êapple Ë Êäÿ- ÊÿÈ 39 fl Ò ÃÊapple Ÿÿapple äÿê Ë ÊÿÈ (fl ÊappleZ apple ) Ò VI - MATHEMATICS

12 6. Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are 3, and 5 8 respectively, then the probability that the target is hit by P or Q but not by R is : An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is : ÃËŸ ÿáäã P, Q ÃÕÊ R Sflà òê M apple ÁŸ ÊÊŸapple Êapple appleœÿapple Ê ÿê UÃapple Ò ÿᜠŸ apple ÁŸ ÊÊŸapple Êapple appleœ ÊŸapple Ë ÊÁÿ ÃÊ Ê 3, ÃÕÊ 5 Ò, ÃÊapple 8 P ÕflÊ Q apple ÁŸ ÊÊŸÊ appleœ ÊŸapple UãÃÈ R apple ÁŸ ÊÊŸÊ Ÿ appleœ ÊŸapple Ë ÊÁÿ ÃÊ Ò ŸÁ ŸÃ (unbiased) Á Ä apple Êapple Ê U Ê U UÊ Ê ÊÃÊ Ò, ÃÊapple apple ÁøûÊ ÃÕÊ apple U Êåà UŸapple Ë ÊÁÿ ÃÊ Ò VI - MATHEMATICS

13 8. If 0 cos x sin x S = x [ 0, π] : sin x 0 cos x = 0, cos x sin x 0 8. ÿᜠ0 cos x sin x S = x [ 0, π] : sin x 0 cos x = 0 cos x sin x 0 then tan π x is equal to : x S 3 Ò, ÃÊapple tan x S 3 π UÊ U Ò x x x 9. The value of tan, x x x <, x 0, is equal to : π cos x π cos x π cos x π cos x x x tan x x x <, x 0, Ê ÊŸ Ò π cos x π cos x π cos x π cos x 3 VI - MATHEMATICS

14 30. The proposition (~p) (p ~q) is equivalent to : p ~q p ~q p ~q q p - o 0 o ÕŸ (~p) (p ~q) ÃÈÀÿ Ò p ~q p ~q p ~q q p - o 0 o - VI - MATHEMATICS