MATHEMATICS. r Statement I Statement II p q ~p ~q ~p q q p ~(p ~q) F F T T F F T F T T F T T F T F F T T T F T T F F F T T

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Preston Hopkins
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1 MATHEMATICS Directions : Questions number to 5 are Assertion-Reason type questions. Each of these questions contains two statements : Statement- (Assertion) and Statement- (Reason). Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice.. Let p be the statement is an irrational number, q be the statement y is a transcendental number, and r be the statement is a rational number iff y is a transcendental number. Statement- : r is equivalent to either q or p. Sol. (..) Statement- : r is equivalent to ~(p ~q). () Statement- is true, Statement- is false () Statement- is false, Statement- is true () Statement- is true, Statement- is true; Statement- is a correct eplanation for Statement- () Statement- is true, Statement- is true; Statement- is not a correct eplanation for Statement- r Statement I Statement II p q ~p ~q ~p q q p ~(p ~q) F F T T F F T F T T F T T F T F F T T T F T T F F F T T r is not equivalent to either of the statements. In a shop there are five types of ice-creams available. A child buys si ice-creams available. A child buys si ice-creams Statement- : The number of different ways the child can buy the si ice-creams is 0 C 5. Statement- : The number of different ways the child can buy the si ice-creams is equal to the number of different ways of arranging 6 A s and B s in a row. () Statement- is true, Statement- is false () Statement- is false, Statement- is true () Statement- is true, Statement- is true; Statement- is a correct eplanation for Statement- () Statement- is true, Statement- is true; Statement- is not a correct eplanation for Statement- Sol. () a + b + c + d + e = 6 a ice creams of type I, b off type II,... AIEEE 008 SOLUTIONS MATHS Page

2 . Statement- : n r= 0 total ways = 0 C 6A S and B S can be arranged in a row in n n r (r + ) C = (n + ). Statement- : n r= 0 n r n n r (r + ) C = (+ ) + n (+ ). 0! 0 C 6!! = ways () Statement- is true, Statement- is false () Statement- is false, Statement- is true () Statement- is true, Statement- is true; Statement- is a correct eplanation for Statement- () Statement- is true, Statement- is true; Statement- is not a correct eplanation for Statement- Sol. () n n n n n n (r + ) Cr = r Cr + Cr r= 0 r= 0 r= 0 n = n + n Statement- is true = (n+ ) n n n n r n r n r (r + ) Cr = r Cr + Cr r= 0 r= 0 r= 0 n n = n(+ ) + (+ ) Statement- is true & Statement- eplains Statement-. Statement- : For every natural number n, > n. n Statement- : n(n + ) < n +. n Page AIEEE 008 SOLUTIONS MATHS

3 () Statement- is true, Statement- is false () Statement- is false, Statement- is true () Statement- is true, Statement- is true; Statement- is a correct eplanation for Statement- () Statement- is true, Statement- is true; Statement- is not a correct eplanation for Statement- Sol. () n > > n > n > n > n = n n Adding them > n n Statement- is True n(n + ) < n+ n < n+ (True) > n n+ 5. Let A be a matri with real entries. Let I be the identity matri. Denote by tr(a), the sum of diagonal entries of A. Assume that A = I. Statement- : If A IandA I, then det A =. Statement- : If A IandA I, then tr(a) 0. AIEEE 008 SOLUTIONS MATHS Page

4 () Statement- is true, Statement- is false () Statement- is false, Statement- is true () Statement- is true, Statement- is true; Statement- is a correct eplanation for Statement- () Statement- is true, Statement- is true; Statement- is not a correct eplanation for Statement- Sol. () a b a b 0 = c d c d 0 a + bc b(a+ d) 0 = 0 (a + d)c bc + d a + bc =, b(a+ d) = 0 (a + d)c = 0, bc + d = if b = 0 a = d = now if a + d = 0 a =, d = or a =, d = 0 0 or c c Then A Ior I Then tr(a) = 0, A =, if a+ d 0,c = 0 then A = I or I 6. The statement p (q p) is equivalent to () p (q p) () p (p q) () p (p q) () p (p q) Sol () p q p q q q p (q q) p (p q) F F F T T T F T T F T T T F T T T T T T T T T T Page AIEEE 008 SOLUTIONS MATHS

5 5 7. The value of cot cos ec + tan is () 5 7 () 7 () 6 7 () 7 Sol () cot tan + tan cot tan cot tan = The differential equation of the family of circles with fied radius 5 units and centre on the line y = is () ( ) y = 5 (y ) () ( ) y = 5 (y ) () (y ) y = 5 (y ) () (y ) y = 5 (y ) Sol () Let centre be (h, ) equation of circle becomes ( h) + (y ) = 5 ( h) + (y )y = 0 h = ( y) y (y ) (y ) + (y ) = 5 AIEEE 008 SOLUTIONS MATHS Page 5

6 sin cos 9. Let I= d andj= d. Then which one of the following is true? 0 0 () I > and J < () () I< andj< () I > and J > I< andj> Sol () sin cos I= d,j= d 0 0 for 0 < < sin < sin < sin d < 0 0 d I < cos d < d 0 0 J < 0. The area of the plane region bounded by the curves + y = 0 and + y = is equal to () () 5 () () Sol () y = y = ( ) Page 6 AIEEE 008 SOLUTIONS MATHS

7 y y = (- ) y = On solving = ( ) = ( ) = = 6 = y = ( ) y = y =± Required Area = ( 0 ) dy = [( y ) ( y )]dy 0 = ( y )dy 0 y = y 0 AIEEE 008 SOLUTIONS MATHS Page 7

8 = =. The value of sin d is π sin π π () log cos( ) + c () + log cos( ) + c π π () log sin( ) + c () + log sin( ) + c Sol () sin d π sin = = π π sin + d π sin π π π π sin cos sin cos + d π sin π π π = cos d sin cot d + π = + log sin + c Page 8 AIEEE 008 SOLUTIONS MATHS

9 . AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60. He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 5. Then the height of the pole is () 7 m + () 7 m () 7 ( + )m () 7 ( )m Sol () Let height of pole is y m A y B 60 C 5 7m D y tan 60 = y = y & tan 5 = = y y 7 y + = y = 7 y = 7 7 y = ( + )m AIEEE 008 SOLUTIONS MATHS Page 9

10 . How many real solutions does the equation = 0 have? () 5 () 7 () () Sol () f() = f() > 0 f() is increasing f() = 0 has only one solution ( )sin if. Let f() = Then which one of the following is true? 0 if = () f is differentiable at = but not at = 0 () f is neither differentiable at = nor at = () f is differentiable at = 0 and at = () f is differentiable at = 0 but not at = Sol () f() = sin cos f() is differentiable at = 0 but not differentiable at = 5. The first two terms of a geometric progression add up to. The sum of the third and the fourth terms is 8. If the terms of the geometric progression are alternately positive and negative, then the first term is () () () () Sol () a + ar = a( + r) = ar + ar = 8 ar ( + r) = 8 r = r = ± r = a( ) = a = Page 0 AIEEE 008 SOLUTIONS MATHS

11 6. It is given that the events A and B are such that P(B) is P(A) =, P(A B) = and P(B A) =. Then () () 6 () () Sol. () P( A B) = P( A) P( B A) = P( B ).P( A B) = PB ( ) PB ( ) = 7. A Die is thrown. Let A be the event that the number obtained is greater than. Let B be the event that the number obtained is less than 5. Then P(A U B) is () 5 () 5 () 0 () s Sol. () A = {, 5, 6}, B = {,,, } A B = {,,,, 5, 6} P( A B) = 8. Suppose the cubic p + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds? () The cubic has maima at both p and p () The cubic has minima at p and maima at p () The cubic has minima at p and maima p () The cubic has minima at both p and p AIEEE 008 SOLUTIONS MATHS Page

12 Sol. () Let f() = p + q f'() = p= 0 = ± p f"() = 6 f"()at = p is positive Minima at = p and f"()at = p is negative So, maima at = p 9. How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent? () C. C () C. C () C () C Sol. () Total words which have no two s are adjacent 8 7! 8 6! = C = C 7!!!! 6 8 = 7 C C 0. The perpendicular bisector of the line segment joining P(, ) and Q(k, ) has y-intercept. Then a possible value of k is () () () () Page AIEEE 008 SOLUTIONS MATHS

13 Sol. () P R (, ) k+, 7 Q (k, ) Let the equation of perpendicular bisector is y = m k+ 7 Now, this line is passing through R, So, 7 k+ = m 7 = mk+ m 8... (i) Now, m = slope of PQ ( k) ( ) m = = k Putting value of m in equation (i) ( k ) k+ ( k ) 8 = 7 k = 5... (ii) k = 6 k = ±. A parabola has the origin as its focus and the line = as the directri. Then the verte of the parabola is at () (, 0) () (0, ) () (, 0) () (0, ) AIEEE 008 SOLUTIONS MATHS Page

14 Sol. () y (0,0) (,0) =. The point diametrically opposite to the point P(, 0) on the circle + y + + y = 0 is () (, ) () (, ) () (, ) () (, ) Sol. () Let the other point is (h, k). h+ = h = k = k =. A focus of an ellipse is at the origin. The directri is the line = and the eccentricity is. Then the length of the semi-major ais is () 5 () 8 () () Sol. () Distance between focus and directri = a ae e e a = e (given) Page AIEEE 008 SOLUTIONS MATHS

15 a = = 8 a. The solution of the differential equation dy + = y satisfying the condition y() = is d () y = ln + () y = ln + () y = ln + ( ) () y = e Sol. () dy y = + d dy y = d y. = d y = ln + c y = ln+ c Now, =.ln+ c c = y = ln+ 5. Let a, b, c be any real numbers. Suppose that there are real numbers, y, z not all zero such that = cy + bz, y = az + c, and z = b + ay. Then a + b + c + abc is equal to () () () () 0 Sol. () Q = 0 c b c a = 0 b a AIEEE 008 SOLUTIONS MATHS Page 5

16 ( ) ( ) ( ) a + c c ab b ac+ b = 0 a c abc abc b = 0 a + b + c + abc = 6. Let A be a square matri all of whose entries are integers. Then which one of the following is true? () If det A =±, then A need not eist () If det A =±, then A eists but all its entries are not necessarily integers () If det A ±, then A eists and all its entries are non-integers () If det A =±, then A eists and all its entries are integers Sol. () Q det A =± inverse of matri A eists and since all entries of matri A are integers and Adj A is matri of transpose of co-factors of matri A. Entries of Adj A is also integers. 7. The quadratic equations 6 + a = 0 and c + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio :. Then the common roots is () () () () Sol. () Let α, β are roots of 6+ a = 0... (i) and α, γ are roots of c+ 6 = 0... (ii) α+β= and α+γ= 6, αβ= a c, αγ= 6 αβ a β a a Q = = = a = 8 αγ 6 γ 6 6 From (i) 6+ 8 = 0 =, ; Q αγ = 6 α cannot be ( Q γ is integer) α= 8. The mean of the numbers a, b, 8, 5, 0 is 6 and the variance is Then which one the following gives possible values of a and b? () a =, b = () a = 0, b = 7 () a = 5, b = () a =, b = 6 Page 6 AIEEE 008 SOLUTIONS MATHS

17 Sol. () a+ b = 6 5 a+ b = 7... (i) ( ) ( ) a 6 + b = ( ) a + b + 9 a + b = a + b = 5... (ii) By equation (i) and (ii), we get a =, b = 9. The vector a =α ˆi + j ˆ +β kˆ lies in the plane of the vectors b = ˆi + ˆj and c = ˆj + kˆ and bisects the angle between bandc. Then which one of the following gives possible values of α and β? () α=, β= () α=, β= () α=, β= () α=, β= Sol. () r r r Q a, b, c are co-planar α β 0 = 0 0 α+β=... (i) and angle between a r and b r is same as a r and c r α+ +β = α =β α + +β α + +β From (i) and (ii), we get α =, β =... (ii) AIEEE 008 SOLUTIONS MATHS Page 7

18 0. The non-zero vectors a, b and c are related by a = 8b and c = 7 b. Then the angle between a and c () π () 0 π () π () Sol. () r r Q aandb are parallel with same direction and b r and c r are parallel with opposite direction. Angle between a r and c r is π.. The line passing through the points (5,, a) and (, b, ) crosses the yz-plane at the point 7 0,,. Then () a = 8, b = () a =, b = 8 () a =, b = 6 () a = 6, b = Sol () Equation of line is 5 y z a = = b a 7 It passes through 0,, 7 5 = = b a From Ist and nd ratio 5+ 5b = 7 b = and from Ist and IIrd ratio 5a + 5 = a a = 8 a = 6.. If the straight lines = y = z and = y = z intersect at a point, then the k k integer k is equal to () () 5 () 5 () Page 8 AIEEE 008 SOLUTIONS MATHS

19 Sol () Lines are intersecting k = 0 k = 0 k k ( k) (k 9) (k 6) = 0 k k + 9 k + = 0 k + 5k 5 = 0 k = 5,5/. The conjugate of a comple number is () () i i+. i Then that comple number is () i () i+ Sol () zz = z z = + i + i i z = i + = = ( i) + i. Let R be the real line. Consider the following subsets of the plane R R : S = {(, y) : y = + and 0 < < } T = {(, y) : y is an integer}. Which one of the following is true? () T is an equivalence relation on R out S is not () Neither S nor T is an equivalence relation on R () Both S and T are equivalence relations on R () S is an equivalence relation on R but T is not AIEEE 008 SOLUTIONS MATHS Page 9

20 Sol () Q S is not refleive S in not equivalance relation. T is refleive, Symmetric, and transitive T is equivalance relation. 5. Let f:n Y be a function defined as f() = + where Y = {y N:y = + for some N}. Show that f is invertible and its inverse is y () g(y) = () y + g(y) = y+ y+ () g(y) = + () g(y) = Sol () y = + y = y g(y) g(y) = Page 0 AIEEE 008 SOLUTIONS MATHS

02. If (x, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) x + y = 0 (B) bx ay = 0 (C) ax by = 0 (D) bx + ay = 0 (E) ax + by =

02. If (x, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) x + y = 0 (B) bx ay = 0 (C) ax by = 0 (D) bx + ay = 0 (E) ax + by = 0. π/ sin d 0 sin + cos (A) 0 (B) π (C) 3 π / (D) π / (E) π /4 0. If (, y) is equidistant from (a + b, b a) and (a b, a + b), then (A) + y = 0 (B) b ay = 0 (C) a by = 0 (D) b + ay = 0 (E) a + by = 0 03.