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Steven Eaton
5 years ago
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1 . An aeroplane flying at a constant speed, parallel to the horizontal ground, km above it, is observed at an elevation of 6º from a point on the ground. If, after five seconds, its elevation from the same point, is º, then the speed (in km/hr) of the aeroplane, is : () 7 () () 7 () B B' kmk m kmk m 66 º º º º C A A' A Let from point C the angle of elevation of plane at B is 6º and after seconds it reach at B' In ABC AC cot 6º In CA'B' A'C cot º Hence distance AA' km 6 6 Speed Distance time km /hr. A box 'A' contains white, red and black balls. Another box 'B' contains white, red and black balls. If two balls are drawn at random, without replacement, from a randomly, selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box 'B' is : () 7 8 Probability that box A is selected P(A) Probability that box B is selected P(B) E be event that one ball is white while the other is red P(E) P(A). P(E/A) + P(B) P(E/B) C C P(B)P(E /B) P(B/E) P(E) () / 9 / () 7 6 () 9 Tel:+9--6 Page No.

2 . If a right circular cone, having maximum volume, is inscribed in a sphere of radius cm, then the curved surface area (in cm ) of this cone is : () 8 () 6 () 8 () 6 N C A m B V r h where r is radius and h is height of coin V ( sin) ( + cos) 7 sin cos dv d 7 [sin cos sin cos ] tan V max if tan Hence curved surface area S r r (sin) ( cos ) (sin ) 6sin 8 (sin cos ) If is one of the angles between the normals to the ellipse x + y 9 at the points (cos, sin) and cot ( sin, cos) ;, ; then is equal to : sin () x y 9 Normal at (cos, () sin) is sec. x cosec y 6..(i) normal at ( sin, cos) is cosec.x sec y 6..(ii) Angle between normal is tan tan cot () () sincos tan sin cot sin Tel:+9--6 Page No.

3 x. If f x x +, (x R {, }), then f(x) is equal to : (where C is a constant of integration) () log e x x + C () log e x x + C () log e x + x + C () log e x + x + C Ans. () x f x x + x f(x) x 6x 6 x 9 x x (x ) f(x) ( x) f(x) x x ( x) x x { n x x + C n x x + C 6. If R is such that the sum of the cubes of the roots of the equation, x + ( )x + ( ) is minimum, then the magnitude of the difference of the roots of this equation is : () () () () 7 x + ( ) x + ( ) Let roots are & + ( + ) ( + ) ( ) ( ) ( ) ( + ) + dz d 6 where + 8 ( ) ( + ), d z d 6 d z d ( ) < + max if d z d ( ) > + min. if Equation is x x i i Tel:+9--6 Page No.

4 7. Two parabolas with a common vertex and with axes along x-axis and y-axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is, then the equation of the common tangent to the two parabolas is : () (x + y) + () 8 (x + y) + () x + y + () (x + y) + Equation two parabola are y x and x y Let equation of tangent to y x is y mx + is also tangent to x y x mx + 9 m mx m x 9 have equal roots D m (m) ( 9) m + m m m Hence common tangent is y x (x + y) + 8. If f(x) cos x x sinx x x tan x x, then lim x Ans. () f '(x) x () does not exist () exists and is equal to () exists and is equal to () exists and is equal to. f(x) cos x x sinx x x tan x x x cosx + tanx. x x (tanx cosx) lim x lim x f '(x) x x(tan x cos x) x (sec x sin x) lim x x (tanx cosx) + x(sec x + sinx) Tel:+9--6 Page No.

5 9. The value of the integral sinx sin x log sin x is : () Ans. () () () () 8 sin x / sin x log / sin x...(i) Use proerties f(x) b b sinx f(a b x) a / sin x log / sin x...(ii) by (i) + (ii) / / / sin x sin x x n-digit number are formed using only three digit, and 7. The smallest value of n for which 9 such distinct numbers can be formed, is : () 9 () 7 () 8 () 6 Ans. n-digit number are formed using only three digits, and 7 with repetition is n 6 < 9 7 > 9 so n 7. If the tangents drawn to the hyperbola y x + intersect the co-ordinates axes at the distinct points A and B, then the locus of the mid point of AB is : () x y + x y () x y + x y () x y x y () x y x y Let tangent drawn at point (x, y) to the hyperbola y x + is : yy, xx + This tangent intersect co-ordinate axes at A and B respectively then A, and B, x y Let mid point is M (h,k) then of AB h x...(i) x h k y y 8k...(ii) Tel:+9--6 Page No.

6 Since point P(x, y ) lies on the hyperbola so y x + from (i) & (ii) 8k h + k h k ( + h ) x y + x y x y x y x y x y locus of M h +. If tan A and tanb are the roots of the quadratic equation, x x, then the value of sin (A + B) sin (A + B). cos (A + B) cos (A + B) is : () () () () Ans. () Since tana and tanb are roots of the equation x x so tana + tanb tanb.tanb tan(a + B) tan A tanb / tan A.tanB 8 so sin (A + B) and cos (A + B) 7 sin (A + B) sin (A + B) cos (A + B) cos (A + B) ( 8 96). Let y y(x) be the solution of the differential equation dy y f(x), where f(x), x [,], otherwise If y(), then y is : e () () e e dy + y f(x) is a linear differential equation If e e x solution of the above equation is y.e x x f(x).e C x x x x y(x) e f(x)e ce y() C x y(x) e x f(x)e x y(/) e x / e. e [e ] () e e e e () e e Tel:+9--6 Page No.6

7 . If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval : () [, ) () (, ] () (,) () (, ) If b is the first term and r is the common ratio of an infinite G.P. then sum is b r ] r b r b r b < r < < b < b < < b < < b < < b < b (,) <. Consider the following two binary relations on the set A {a, b, c} : R {(c,a), (b,b), (a,c), (c,c), (b,c), (a,a)} and R {(a,b), (b,a), (c,c), (c,a), (a,a), (b,b), (a,c)}. Then : () R is symmetric but it is not transitive () both R and R are not symmetric () both R and R are transitive. () R is not symmetric but it is transitive Ans. () R (b, c) but R (c,b) Example R is not symmetric in R ; (b,c) R and (c,a)r but (b,a) R So R is not transitive R is symmetric is R ; (b,a) R and (a,c) R but (b,c) R So R is not transitive Tel:+9--6 Page No.7

8 . A circle passes through the points (,) and (,). If its centre lies on the line, y x +, then its radius is equal to : () () () () Let centre of circle is c(.) it lies is line y x + B c(, ) (, ) C A (, ) B (, ) CA CB ( ) + ( 6) ( ) + ( 8) ( ) 8 c(,) r 7. In a triangle ABC, coordinates of A are (,) and the equations of the medians through B and C are respectively, x + y and x. Then area of ABC (in sq. units) is : () () () 9 () x A(,) x+y, B (7 ) Area of 7 C (, 8 9 sq.unit. ( 8 )z 8. The set of all R, for which w z and Re z, is : () {} (),, is a purely imaginary number, for all z C satisfying z () equal to R () an empty set Tel:+9--6 Page No.8

9 Ans. () ( 8 )z ( 8 ) z z z z + ( 8)z ( 8) + z + ( 8) z ( 8) (z + z ) + ( 8) (z + z ) + 8 (z + z ) z + z or For all z C we have 9. If x, x,., x n and,,..., h h hn are two A.P. such that x h 8 and x 8 h 7, then x. h equals : () () () 6 () 6 x, x, x,.. x n in AP. x 8 & x 8 6 x h, h, h, h n, in HP h 8, h 7 h 6 8 x h 6. If a, b and c are unit vectors such that a + b + c, then a c () () a + b + c o a + c b a + c + a.c b () is equal to : () + + cos sin cos Now a c a c sin. A variable plane passes through a fixed point (,,) and meets x, y and z axes at A, B and C respectively. A plane is drawn parallel to yz - plane through A, a second plane is drawn parallel zx-plane through B and a third plane is drawn parallel to xy-plane through C. Then the locus of the point of intersection of these three planes, is : () () () x + y + z 6 () x y z Ans. () x y z Let plane is x y z a b c x y z 6 it passes through (,,) a b c Now A (a,,), B (, b, ), C (,,c) Locus of point of intersection of planes x a y b, z c is x y z Tel:+9--6 Page No.9

10 . Let A be a matrix such that A. is a scalar matrix and A 8. Then A equals : () 6 Ans. () A a b c d Now A. a b c d a a b c c d is scalar 6 () 6 () c, a + b, a c + d a d a 9d 6 A 8 A ad bc ad d Now A a b d a b a ab bd d d () 6. The area (in sq. units) of the region {x R : x, y, y x and y x }, is : (A) () 8 () () (, ) (,) (,) (,) x x x x x x x x 8( ) Tel:+9--6 Page No.

11 . If x + y d y + sin y, then the value of at the point (, ) is : () () () () Ans. () x + y + siny dy x + (y + cosy) dy x dy at (, ) y cos y dy and (y + cosy) x d dy (y+cosy) sin y y d y d y. An angle between the plane, x + y + z and the line of intersection of the planes, x + y + z and x + 8y + z +, is : () cos 7 i j k 8 i() j (6 ) + k ( ) ĵ kˆ Angle cos () cos 7 cos 7 7 sin () sin 7 7 () sin 7 6. Let S {, µ) R R : f(t) ( e t µ). sin ( t ), t R, is a differentiable function}. Then S is a subset of : () (, ) R () R [, ) () [, ) R () R (,) Ans. () Let s {, ) RR} + f(t) { e t ) sin t f( h) RHD lim h h h sin h h lim e e h h f( h) e LHD lim h sin h lim e h h h h e h R h Tel:+9--6 Page No.

12 7. Let S be the set of all real values of k for which the system of linear equations x + y + z x + y x + y + kz Has a unique solution. Then S is : () equal to R {} () an empty set () equal to R () equal to {} for k (k+) (k+) + ( ) k+ k + k k 8. If n is the degree of the polynomial, + x x x x coefficient of x n in it, then the ordered pair (n,m) is equal to : () (8, () ) () (, 8() ) () (,() ) () (, () 8 ) x x 8 x x and m is the C x C x x C x x C6 x x C8 x x 8x x 7x x 8 x x x h & m ( ) () 9. The mean of a set of observations is 7. If each observations is multiplied by a non-zero number and then each of them is decreased by, their mean remains the same. Then is equal to : () () () Ans. () x + x + + x 7 Now given (x + x + x +.+ x ) 7 ( 7 ) (). If (p ~ q) (p r) ~p q is false, then the truth values of p, q and r are, respectively : () T,T,T () F,T,F () T,F,T () F,F,F p ~ q p ~ r~ p q () T F T T F T F T T F T T F F F F () T T F T T T T T T () F F T F F F T F F () T F F F T F T T Tel:+9--6 Page No.

4.Let A be a matrix such that A. is a scalar matrix and Then equals :

4.Let A be a matrix such that A. is a scalar matrix and Then equals : 1.Consider the following two binary relations on the set A={a, b, c} : R1={(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R2={(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}. Then : both R1