TOPIC = VECTOR, 3D AND MIX PROBLEMS (COLLECTION # 1)

Single Correct Type

Que. 1. The number N = 61og10 2+log1031, lies between two successive integers whose sum is equal to

(a) 5 (b) 7 (C) 9 (d) 10 (code-VlT2PAQl)

Que. 2. Number of integers satisfying the inequality log1/2 |x — 3| > —1 is

(a) 5 (b) 3 (c) 2 (d) infinite (code-VlT4PAQl)

Que. 3. 10logpK(log'x)) = 1 and logq (logr (logp x)j = 0 then ‘p’ equals

(a) rq/r (b) rq (c) 1 (d) rr/q (code-VlT4PAQ7)

Q ue.4. a = logl2, b = log21, c = logll, and d = log22 then can be expressed in this form

P (a -b ) + Q (c-d ) where P and Q are integers then the value of (7P-Q ) equals

(a) 5 (b) 9 (C) 13 (d) 15 (code-VlT5PAQ3)

Que. 5. If the equation, xlog*x = , a > 0, a * 0, has has exactly one solution for x, then the sum of the twocl

possible values Of ‘k’ is (code-VlT5PAQ10)

(a) 4 (b) 10 (c) 12 (d )8>/2

Que. 6 . There exist a positive number k such that log2 x + log4 x + log8 x = logk x, for all positive real num­

bers x. If k = >/a where a, b e N , the smallest possible value of (a + b) is equal to (code-VlT7PAQ3)

(a) 75 (b) 65 (c) 12 (d) 63

Que. 7. Suppose that a,b,c,d are real numbers satisfying a > b > c > d > 0 ,a 2+d2 =l;b2+c2 =1 and

ac+ bd= l/3 . Thevalueof (ab-cd) isequalto (code-VlT7PAQ9)

2 2V2 2V2 J 3<a ) i T i <b)^ “ <c)±^ “ (d )T

Que. 8 . Let x and y be numbers in the open interval (0, 1). Suppose there exists a positive number’a’,

different from 1 such that logx a + logy a = 41ogxy a. Which of the following statements are necessarly

true ? (code-VlT10PAQl)

I (loga x + loga y )2 = 41oga x loga y

II x = y

III log x 2 a = logx Va

(a) I only (b) II only (c) I and II only (d) I, II and HI

Que. 9. Let T, E, K and O be positive real numbers such that log (T.O) + log (T.K) = 2; log (K.O)+ log (K.E) = 3;

log (E.T) + log (E.O) = 4 The value of the product (TECO) equals (base of the log is 10)

(a) IQ2 (b) IQ3 (C) 104 (d) 109 (code-VlT12PAQ4)

Xk -2

Que. 10. If P is the number of natural numbers whose logratihms to the base 10 have the characteristic p and Q is the number of natural numbers logarithms of whose reciprocals to the base 10 have thecharacteristic -q then log10 P - log10 Q has the value equal to (code-VlTl3PAQ24)

(a) p - q +1 (b )p - q (c)p + q - l ( d ) p - q - l .

Que. 11. P be a point interior to the acute triangle ABC. If PA + PB + PC is a null vector then w.r.t. the trangle ABC, the point P is, its (code-V2T8PAQ3)

(a) centroid (b) orthocentre (c) incentre (d) corcumcentre

Que. 12. L, and L2 are two lines whose vector equations are (code-V2T9PAQi)

L j: r = A,^cos0 + V3 )i + (-v/2 sin0) j + (cos0 - V5)£)

L2: r = |a(ai + bj + ckj,

where X and M are scalars and a is the acute angle between Lx and L r If the angle 'a ' is independent of 0 then the value of ’a ’ is

(a)71

(b)71

(c)71

(d)71

6 v ' 4 w 3 v ' 2

Que. 13. Which of the following are equation for the plane passing through the points P (1,1, -1), Q (3,0,2)

and R (-2,1,0) ? (code-V2T9PAQ2)

(a) (2 i-3 j + 3k).((x + 2)i + (y -l) j + zk) = 0

(b) x = 3 - 1, y = -1 It, z = 2 - 3t

(c) (x + 2) + l l ( y - l ) = 3z

(d) (2 i- j-2 k )x (-3 i + k).((x + 2)i + (y -l} j + zk) = 0

Que. 14. If u and v are two vectors such that

(a) u a vg (0,90°)

= 3; = 2 and uxv = 6 then the correct statement is

(b) u Ave (90°, 180°) (code-V2T9PAQ3)

(c) u A v = 90° (d) (u Av)xu = 6v

Que. 15. P(p) and Q(q) are the positive vectors of two fixed points and R (r) is the position vector of

avariable point. If R moves such that ( r - p )x (r-q j = 0 then the locus of R is (code-V2Tl2PAQl)

(a) a plane containing the origin ‘O’ and parallel to two the non collinear vectors o p and OQ(b) the surface of a sphere described on PQ as its diametere.(c) a line passing through the points P and Q(d) a set of lines parallel to the line PQ.

Que. 16. The range of values of m for which the line y = mx and the curve y = ——- enclose a region, isX + 1

(a) (-1,1) (b) (0,1) (c) [0,1] (d)(l,oo) (code-V2T17PAQ3)

Comprehesion Type # 1 Paragraph for Q. 1 to Q. 3

Vertices of a parallelogram taken in order are A (2, -1,4); B (1,0,-1); C (1,2,3) and D.

1. The distance bertween the parallel lines AB and CD is (code-V2T9PAQ4,5,6)

(a) V6 (b) 3 ^ | (c) 2V2 (d) 3

2. Distance of the point P (8,2,-12) form the plane of the parallelogram is

, , W 6 32>/6 , , 16V6(a) — (b) —— (c) - y - (d) None

3. The areas of the orthogonal projections of the parallelogram on the three coordinates planes xy, yz and zx respectively(a) 14, 4, 2 (b) 2, 4, 14 (c) 4, 2, 14 (d) 2, 14, 4

# 2 Paragraph for Q. 4 to Q. 6

The sides of a triangle ABC satsfy the relations a + b - c = 2 and 2ab - c2 = 4 and f (x) = ax2 + bx + c.

4. Area of the triangle ABC in square units, is (code-V2Tl5PAQl,2,3)

(a) V3 (b) ^ (c) (d) 4V3

5. If x g [0,1] then maximum value of f(x) is(a) 3/2 (b) 2 (c) 3 (d) 6

6. The radius of the circle opposite to the angle A is

(a) 1 (b) V3 (c) V3/2 (d) 1/V3

Assertion & Reason TypeIn this section each que. contains STATEMENT-1 (Assertion) & STATEMENT-2(Reason).Each

question has 4 choices (A), (B), (C) and (D), out of which only one is correct.Bubble (A) STATEMENT-1 is true, STATEMENT-2 is True; STATEMENT-2 is a correct

explanation for STATEMENT-1.Bubble (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct

explanation for STATEMENT-1.Bubble (C) STATEMENT-1 is True, STATEMENT-2 is False.Bubble (D) STATEMENT-1 is False, STATEMENT-2 is True.

Que. 1. Consider the following statements (code-VlT2PAQ7)

Statement - 1:

Number of cyphers after decimal before a significant figure comes in n = 2 10101 is 30.becauseStatement - 2:

Number of cyphers after decimal before a significant figure comes in n = 2 101 is 3.

Que. 2 Consider the following statements Statement - 1 :

There exists some value of 0 forwhich sec0 =

because Statement - 2

(code-VlT2PAQ8)

log i 7 + log71 -

y + -v y j

< - i

(code-VlT4PAQ10)

If y is negative then ^

Que. 3. Statement - 1 :

^logx cos(27tx) is a meaningful quantity only if xe (0 ,l/4 )u (3 /4 ,l).

because Statement - 2 :If the number N > 0 and the base of thelogarithm b (greater then zero not equal to 1) both lie on

the same side of unity then logbN < 0 and it they lie on differnt side of unity then logb N < 0

( i Y °Que. 4. Statement - 1 : If N = — then N contains 7 digitis before decimal. (code-VlTlOPAQ8)1.0.4 J

because

Statement - 2 : Characteristic of the logarithm of N to the base 10 is 7. [use log10 2 = 0.3010 ]

Que. 5. Consider the following statements (code-VlTl2PAQlO)

Statement - 1 : because

Statement - 2 :

The equation 5log5 x3+l = 1 has two distinct real solutions.

, l og. N = N when a > 0, a ^ 1 and N > 0

Que. 6. Statement -1 : log10 x < log x < loge x < log2 x (x > 0 and x * 1)

because

Statement - 2 : If 0 < x < 1, then lo g x a < l o g x b=> 0 < a < b

x - 4 y + 5 z — 1 x - 2 y + 1 z

(code-V1T14PAQ4)

Que. 7. Given lines and12 4 - 3

Statement 1: The lines intersect, becauseStatement 2: They are not parallel.

Que. 8. Consider three vectors a, b and c

Statement 1: axb = ^ ix a j.b j^ jx a j.b j j + kxaj.b |k

because

Statement 2: c = (l.cji + ( j.c jj + (k.cjk

(code-V2T9PAQ7)

(code-V2T9PAQ8)

(code-VlT2PAQ12)

More than One May Correct Type Que. 1. In which of the following case(s) the real number ‘m’ greater than the real number ‘n’ ?

(a) m = (log2 5)2 and n = log2 20

(b) m = log10 2 and n = log10 l/lQ

(c) m = log105.1og10 20 + (log10 2)2 andn = l

(d) m = logi/2 j and n = log1/3 Q-j

Que. 2. If log2 (log3 (log4 2n)) = 2 then the value of n can be equal to (code-VlT10PAQ10)

(a)27

log27 tan 4n

4si<b)T (c)

162

log 2 sec 57t (d)81

log423 ) V 3

Que. 3. The value of x satisfying the equations 22x — 8.2X = —12, is (code-VlT12PAQll)

(a) 1 + log 3 log 2 (b) “ log 6 (c) 1 + log- (d) 1

Que. 4. The expression (tan4 x + 2tan2 x + l).cos2 x when x = n/12 can be equal to (code-VlTl2PAQl3)

(a) 4(2 — >/3) (b)4(V2 + l) (c) 16cos2 — v ; 12

(d) 16sin2^

Que. 5. Given a and b are positive numbers satisfying 4 (log10 a)2 + (log2 b)2 = 1 then which of the following statement(s) are correct ? (code-viTi9PAQ2i)

(a) Greatest and least possible values of ‘a’ are reciprocal of each other.(b) Greatest and least possible values of ‘b’ are reciprocal of each other.(c) Greatest value of ‘a’ is the square value of ‘b’(d) Least value of ‘b’ is the square of the least value of ‘a’.

Que. 6. Let a,b,c are non zero vectors and Vi = a x (bxc) and V2 = (axb)xc. The vectors y , and V2 are

equal (code-V2T9PAQ9)

(a) a and b are orthogonal (b) a and c are collinear

(c) b and c are orthogonal (d) b = A,(axc) when X is a scalar.

Que. 7. If A,B,C and D are four non zero vectors in the same plane no two of which are collinear then which of the following hold(s) good ? (code-V2T9PAQlO)

(a) (AxB).(CxD) = 0

(c) (A x b ) x (C x d ) = 0

(b) (A x C).(b x d ) * 0

(d) (a x c )x (b x d ) * o

Que. 8. Let Px denotes the equation of the plane to which the vector (i + j) is normal and which contains the

line L whose equation is r = i + j + k + A ,(i-j-k ),P 2 denotes the equation of the plane containing the

line L and a point with position vector j. Which of the following holds good ? (code-V2T9PAQll)

(a) equation of Px is x + y = 2

(b) equation of P2 is r (i - 2 j + k j = 2

(c) The acute angle between Px and P2 is cot 1 (>/3)

(d) The angle between the plane P2 and the line L is tan 1 73

Que. 9. If a,b,c be three non zero vectors satisfying the condition axb = c& bxc = a then which of the following alwyas hold(s) good ? (code-V2Ti2PAQi3)

(a) a,b,c are orthogonal in pairs

(c) a b c

(b)

(d)

a b c

Que. 10. If loga x = b for permissible values of a and x then identify the statement(s) which can be correct?

(a) If a and b are two irrational numbers then x can be rational.

(b) If a rational and b irrational then x can be rational

(c) If a irrational and b rational then x can be rational

(d) If a rational and b rational then x can be rational.

Match Matrix TypeQue. 1. Column - 1

sin(410°-A)cos(400° + A) + cos(4100-A)sin(4000 + A)

cos2l° -cos2 2°

(code-V2T15PAQ8)

A

B.

P.

Column - II

- 1 (code-VlT2PBQl)

2 sin 3°. sin 1° is equal to Q. 0

C.

D.

sin (-870°) + cos ec (-660°) + tan (-85 5°)

+2 cos (840°) + cos (480°) + sec (900°)

R. 2

nIf cos0 = -j where anc cos(t) = where (t)e [ 0,— | S.

then cos (0 - §) has the value equal to

Que. 2.

A

B.

C.

D.

Que. 3.A.

B.

C.

D.

Que. 4.A.

B.

C.

D.

Column - 1

Let f (x) = (a2 + a + 2) x2 - (a + 4) x - 7. if unity lies between the

roots of f(x) = 0 then possible integral value(s) of ‘a’ is/are The possible integral value(s) of x satisfying the inequality

Column - II

P. 1

(code-VlT4PBQl)

Q. 2

1< 3x -2 x + 8 x2 + l

< 2, is/are

If sinxcos4x + 2sin22x = l ,^ s in 2 — - —U 2

then the value(s) of sin x, is/areThe possible integral value(s) of x satistying theequation

(log5x)2 + log5;c ^ j = 1, is/are

Column - 1Let ABCDEFGHUKL be a regular dodecagon.

AB AF The value of —- + —7- is equal to AF AB HAssume that 0 is a rational multiple of n such 0 is a distinct rational

Number of values of cos0 is

log2 3.1og45.1og6 7 . log43.1og651og87 lsec*ual

Number of values o f ‘t’ satistying the equation

cos (sin (cos t)) = 1 for te [0,2 :]

Column - 1If x and y are positive real numbers and p,q are any positive integers then the possible value which the xpression

(l + x2p)(l + y2q)

R. 4

S. 5.

Column - II P. 2

(code-VlT6PBQl)

Q. 3

R. 4

S. 5

Column - II P. 3(code-V1T6PB Q2)

xpyqcan take is

Given a and b are positive numbers not equal to 1 such that

logb (alog2b) = loga (blog4a) and loga (p - (b - 9)2 j = 2 then the minimum

integral value for p isPossible values of x simultaneouly satisfying the system of

(x -6 )(x -3 ) x -5inequalties---------------- ^ 0 and ------ < 3 is

x + 2 x +1

If the numbers, (31+x +31 X),(a/2),(9X + 9 x) form an A.P , then

possible value(s) of ‘a’ is/are (xe R)

Q. 5

R. 9

10

Que. 5.

A.

B.

C.

D.

Que. 6.A.

B.

C.

D.

8 sin 40°. sin 50°. tan 10

Column - 1

cos 80°

The value of -

equals

log 2 loga log4

is equal to

P.

Q.

1

Column - II

41ogi8 (>/2 ) + —log18 (729) is equal to R.

Number of expression(s) which simplifies to (sec2 0cosec20), is are S.

3

4

(i)

(iii)

sec2 0 + cos ec20

tan 0 + 1

(ii) (tan 0 + cot 0)2

(iv) cosec2 201 - cos2 0Column - 1

If a, b, c and d are four non zero real numbers P.

such that (d+ a-b )2+ (d + b -c )2 =0 and the

roots of the equation a(b - c)x2 + b(c - a)x + c(a - b) = 0

are real and equal thenIf a, b, c are real non zero positive numbers such Q.that log a, log b, log c are in A.P. then

If the equation ax2 + bx + c = 0 and x2 _ 3x2 + 3x _ \ = 0 R. have a common real root thenLet a, b, c be positive real numbers such that the S.

(code-VlT10PBQl)

Column - II

a + b + c = 0

(code-VlT14PBQ2)

a, b, c are in A.P.

a, b, c are in G.P

a, b, care in H.P.

expression bx2 + |^ (a + c)2 +4b2 jx + (a + c) is

non negative V x e R then Que. 7. Column - I (code-VlT16PBQl) Column - II

A. Sum of the positive integers which satisty the inequality P.

-2x-15<0

B.

C.

D.

( x 2 - 3 x + 4 ) ( x - 2 ) 1S

The difference between the sum of the first k terms of theI3 + 23 + 33 + .......+ n3 and the sum of the first k terms of1 + 2 + 3 +.........+ n is 1980. ThevlaueofkisSides of a triangle ABC are 5, 12,13. The radius of the escribed circle touching the side of length 12, is If x2 + x +1 = o then the value of

Q.

R.

S.

10

11

12

Number ‘a’ and ‘b’, then ^ +(^2 is equal to S. 6 .

Que. 8. Column - 1 (code-VlTl8PBQl) Column - IIA. The value o f ‘a’ for which the equation P. 2.

( . \ V 3 x -l _ / r. \ V -9 x 2- 3 x + 2 3(sinxj + 2 (cos2xj +log1/3x =a

has atleast one solution is

B. Let function f (x) = ax3 + bx2 + cx + d has 3 positive roots. Q. 4.If the sum of the roots of f(x) is 4, the largest possible integral value of c/a is

C. If Aj be the A.M. and G1;G2 be two G.M.’s between two positive R. 5.

G ^ AQue. 9. Column - 1 (code-VlTl8PBQ2) Column - II

A. Let x2 + y2 + xy +1 > a(x + y) V x, y e R then the possible integer(s) P. -1. in the range of a can be

B. Let a, (3, y be measures of angle such that sin a + sin (3 + sin y > 2 then Q. 0. the possible integral values which cos a + cos (3 + cos y can attain

C. The terms a1, a2, a3 form an arithmetic sequence whose sum is 18. R. 1.

The terms a +1, a2, a3 + 2, in that order, form ageometric sequence.The sum of all possible common difference of the A.P., is

D. The elements in the solution set of Vx2 -4 x + 4<3 and \ < —— < , is S. 2.4 3 - x 2

Que. 10. Match the Statement / Expression in Column - 1 with the Statements /Expressions in Column - II.Column - 1 (code-VlT20PBQ2) Column - II

A. If a2 - 4a +1 = 4, then the value of - — \ +a—-( a2 ^ l) is equal toa -1 v '

P. 1.

Q. 2 .

R. 4.

S. 11

B. The value(s) of x satisfying the equation ^ |x -3 |x+1 = ^ |x -3 |x 2 is

C. The value(s) of x satisfying the equation 3X +1 - |3X - 1| = 2 log516 — x| is

D. If the sum of the first 2n terms of the A.P.2,5,8............is equal tothe sum of the first n terms of the A.P.57,59,61,....... , then n equals

Que. 11. Column - 1 (code-VlT20PBQ3) Column - II

A. If the roots of i 0x3 - nx2 - 54x - 27 = 0 are in harmonic progression, then ‘n’ equal

B. A preson has ‘n’ friends. The minimum value of ‘n’ so that a preson can invite a different pair of friends every day for four weeks in a row is

C. If 2n+1Pn_! :2n 1 Pn =3:5, then ‘n’ equalsD. There are two sets of parallel lines, their equation being

xcosa + ysina = p and xsina + ycosa = p; p = l,2,3,....,n and

a e (0,71/2). If the number of rectangles formed by these two sets of lines is 225 then the vlaue of n is equal to

P. 4.

Q. 6 .

R. 8 .S. 9.

Que. 12. Column - I (code-V2T3PBQ2)

A. The set of values of x satisfying the equation

x

j t 2.sin(x-t)dt = x2, is/are0

B. The set of values of ‘x’ satisfying the equation

cos4x + 6 = 7cos2x, is/are

C. The set of values of ‘x’ for which the expression

Column - II

P. n7t(nel)

71Q. (4n + l) —(ne I) 4

R. y ( n e I )

■ X X .sin —+ cos----itanx2 2_______

1 — 2i sin —is real, is/are

D.

Que. 13.

A.

B.

C.

D.Que. 14.

A.

B.

D.

The set of values of ‘x’ satisfying the equation,

sinx + sin5x = sin2x + sin4x is/are

S.

Column I (code-V2T6PBQl)

(3sinx-sin3x )4li™- t-------------- is equal tox ° (secx-cosxj n

Given that x, y, z are positive reals such that xyz = 32.

The mininum value of x2 + 4xy + 4y2 + 2z2 is equal The number of ways in which 6 men can be seated so that 3 particular men are consecutive is

The number N = 6loglo40.5logl°36 isColumn I (code-V2T6PBQ2)

1x - 3 x + 2Let f (x) = —---------- . The value of x for which f(x) is equal to — is

x + x + 6 5

Given f(x) is a function such that f (x) =x sin—

x0

if x > 0

if x = 0

where a is a constant. If f(x) is a derivable V x > 0, then a is Let A and B be 3 x 3 matrices with integer entries, such that AB = A+B. The value of det (A - 1) can be (where I denotes a 3x 3 unit matrix)

If x = 2008 (a - b), y = 2008(b - c) and z = 2008(c - a) then the2 . 2 , 2 x + y +z

numerical value o f -------------- is equal to 'p '(xy + yz + zx * 0).xy + yz + zx H v \ >

The value of ‘p’ so obtained is less than

2mi; (ne I)

Column II

P.

Q.

R.

96

144

216

S. 256 Column II

P. - 1

Q.

R.

S.

1.

2 .

non existent.

A.

B.

C.

D.

Que. 16.

A.

B.

C.

D.

Que. 17.

A

B.

C.

D.

Que. 15. Column - 1 (code-V2T7PBQl) Column - II

If the vlaue of l' rnx O(3/x) + l

can be expressed in the form of ep/ q(3 /x ) - l

where p and q are relative prime then (p + q) is equal to

The area of a triangle ABC is equal to (a2 + b2 - c2), where a,b and c Q.

are the sides of the triangle. The value of tan C equals

7 1If the value of y (greater than 1) satisfying the equation j x f n x dx = — R.

1 4

can be expressed in the form of em/n, where m and n are relative

prime then (m+n) is equal to

Number of integeral value of x satisfying log2 (1 + x ) = log3 (l + 2X) S.

Colum - 1 (code-V2TlOPBQl) Column - II

In a A ABC maximum value of cos2 A + cos2 B + cos2 C, is P.

If a , b are c are positive and 9a + 3b + c = 90 then the maximum Q.

value of (loga + logb + logc) is (base of the logarithm is 10)

2 .

3.

4.

tan x>/tan x -s in xVsinx lim --------------- :—f=-------------equals

\ \ -£n(sec x)If f (x) = cos | x cos and g(x) = ------------

x ) x sin xare

both continuous at x = 0 then f (0) + g(0) equals Colum - I (code-V2T10PBQ2)

5.

R.

S.

3/4

2

3.

Non existent

Column - II

Let f :r -^R and fn(x) = f (fn_1(x))Vn > 2,ne N, the roots of equaltion

f3 (x) f2 (x) f (x) - 25f2 (x).f (x) +175 f (x) = 375. Which also satisfy equation

f (x) = x will be

Let f : [5,10] onto [4,17], the integers in the range of y = f (f (f (x))is/are

Let f(x) = 8co r1 (cotx) + 5sin_1 (sinx) + 4tan_1 (tan x)-sin (s in 1 x) then

possible integral values which f (x) can take Let ‘a’ denote the roots of equation

cos (cos 1 x) + sin : sin ^1 + x2 v 2 ,

= 2 sec ^secx) then posible values

of [|l0a|] where [.] denotes the greatest integer function will be

Q.

R.

1.

5.

10.

15.

A.

B.

C.

Que. 19.

A.

B.

C.

D.

Que. 20.

A.

B.

C.

D.

Que. 18.

Let O be an interior point of AABC such that OA + 20B + 30C = 0,

then the ratio of the area of AABC to the area of AAOC, is

Let ABC be a triangle whose centroid is G orthocentre is H and circumcentre is the origin ‘O’. If D is any point in the plane of the triangle such that no three of 0,A, B, C and D are

If a, b, c and cj are non zero vectors such that no three of them are

in the same plane and no two are orthogonal then the value of the

(bxc).(axd) + (cxa).(bxd)

Column - I (code-V2T9PBQl)

scalar is(axb).(dxc)

Colum - I (code-V2T10PBQl)

In a AABC maximum value of cos2 A + cos2 B + cos2 C, is

If a , b are c are positive and 9a + 3b + c = 90 then the maximum

value of (loga + logb + logc) is (base of the logarithm is 10)

tan xVtanx - sin x-y/sinxlimXO equals

f i A -£n(sec x)If f(x) = cos xcos— andg(x)=------------ are

x j xsinx

both continuous at x = 0 then f (0) + g(0) equals

Colum - I (code-V2T10PBQ2)

Q. 1.

R. 2.

Column - II

P. 0.

Column - II

P. 3/4

Q. 2

R. 3.

S. Non existent

Column - II

Let f :R —>R and fn(x) = f (fn_1(x))Vn > 2,ne N, the roots of equaltion P. 1.

f3(x) f2 (x) f (x) - 25f2 (x).f (x) +175f (x) = 375. Which also satisfy equation

f (x) = x will be

Let f : [5,10] onto [4,17], the integers in the range of y = f (f (f (x))is/are Q. 5.

Let f(x) = 8cor' (cotx) + 5skr' (sinx) + 4tan~' (tan x)-sin (s in 1 x) then R. 10.

possible integral values which f (x) can take

Let ‘a’ denote the roots of equation S. 15.

cos (cos 1 x) + sin : sin2 A

= 2 sec ^secx) then posible values

of [|l0a|] where [.] denotes the greatest integer function will be

A.

B.

C.

D.

Que. 22.A.

B.

C.D.

Que. 23.

A

B.

C.

D.

Que. 21.

Let a,b are real number such that a + b = 1 then the minimum value

7tof the integral j (a sin x + b sin 2x )2 dx is equal to

0

, 1

Column - I (code-V2T12PBQl)

%R 2

Column - II

Jt/2

sin x — 2

dx is equal

RA rectangle is inscribed in a semicircle with one of its sides along the

diameter. If k times the area of the largest rectangle equals the area of the semicircle then the value of k equals

If a,b,c arenon-coplanar unit vectors such that ax (bxc) = - ^ ( b + c) S.

71

4

71

8"

3tz

~4

then the angle between the vectors a, b is Column - 1

Let f (x) is a derivable function satisfyingColumn - II P. -1.

f (x) = j e ‘ sin(x - t)dt and g(x) = f "(x) - f (x) then thepossible integers (code-V2Tl2PBQ2)

in the range of g(x) isIf the substitution x = tan_1(t) transforms the differential equation

d2 y dy 2— ^ + xy— + sec x = 0 in to adifferential equation dx dx

0 .

d2 a(l + 12) + (2t + y tan-1 (t)) = k then k is equal to

If a2 + b2 = 1 then (a3b - ab3) can be equal to If the system of equation

x - A,y - z = 0

A,x - y - z = 0

X + y - z = 0

has a unique solution, then the value of X can be Column - I (code-V2T15PBQl)

The sum ^ arc tan j \ J equalsn=l U )

lim nsin^2%^\ + n2 j (ne N) equals

Period of the function f (x) = sin2 2x + cos4 2x + 2, is

j (l + x)1/2( l - x ) 3/2dx equals

dy

R.S.

1.2 .

Column - II

P.%4 '

%Q. 2 ‘

3nR

4

S. 71.

A.

B.

C.

D.

Que. 25.

A.

B.

C.

D.

Que. 26.

A.

B.

C.

D.

Que. 24. Column - 1 (code-V2T16PBQl) Column - II

If the constant term in the binomial expansion of | x — J , n e N is 15

then the vlalue of n is equal toThe positive value of ‘c’ that makes the area bounded by the graph of

y = c (l - x2) and the x-axis equal to 1, can be expressed in the form

p/q where p,qe N and in their lowest form, then (p + q) equals

Suppose a, b, c are such that the curve y = ax2 + bx + c is tangent to

y = 3x -3 at (1,0) and is also tangent to y = x + 1 at (3,4) then the value

of (2 a -b -4 c ) equals

2 2 x ySuppose F , F are the foci of the ellipse — + — = 1 • P is a point on9 4

ellipse such that PFX: PF2 = 2: l.The area of the triangle Pl;l:2 is

Column - I (code-V2T16PBQ2)

Let f(x) = j xsmx (l + xcosxin x + sinx)dx and f j — j = — then the

value of f(7t) is

l + 2cosxLet g(x) = f-J i dx and g(0) = 0 then the value of g(n/2) is(cos x + 2)

If real numbers x and y satisfy (x + 5)2+ ( y - 12)2 =(14)2 then the

minimum value of ^ (x 2 + y2) is

f (x2+l)dx \Let k(x) = , = and k(-l) = —j= then the value of d(-2) is

Vx +3x + 6 7 2Column - 1

fxMIf log — =1 and log(x2y2) = 7 then log(|xy|) is equal to

f (x) = min.{|x|,x2,2},xe [-5,5] number of points where f(x)

is not derivable, is

j2x - 6x + 9 x -5

x -2 x + 5dx is equal to

If the range of the function f (x) = log2 (4X + 3(x '

then the value of ‘a’equals

Q.

R.

S.

P.

Q.

4.

6 .

R. 7.

S. 9.

Column - II

rational

Irrational

Integral

Prime

Column - II

P. 0.

Q. 3/2

(code-V2T19PBQl)

R. 3.

4.

Vector, 2D, 3D AND Mix SolvedQuestion For 11 & 12

Publisher : Faculty Notes Author : Panel Of Experts

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