XIII UT-1 Page # 1

PHYSICSXIII UNIT TEST-1 DATE : 17.06.2012

PART-A[SINGLE CORRECT CHOICE TYPE]

Q.1 to Q.7 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Q.1 Surface area is ________(A) Scalar (B*) Vector(C) Neither scalar nor vector (D) Both scalar and vector

Q.2 Hooke's law is valid upto(A) Elastic limit (B*) Proportionality limit (C) Yield point (D) No limit

Q.3 Two wires A and B of the same material have their lengths in the ratio of 1 : 2 and their diameters in theratio of 2 : 1. If they are stretched with the same force, the ratio of the increase in the length of A to thatof B will be(A) 1 : 4 (B*) 1 : 8 (C) 2 : 1 (D) 1 : 2

Q.4 A piece of metal weighing 5.10 g at a temperature of 48.6 °C was placed in a calorimeter containing

20 mL of water at 22.1 °C, and the final equilibrium temperature was found to be 28.2 C. What is the

specific heat of the metal ?(A*) 4.9 J/gC (B) 0.49 J/gC (C) 9.8 J/gC (D) 2.4 J/gC

Q.5 Between which two points does water gain the most energy ? (sample of ice at A is heated at constantrate without loss of energy).

�100

0

100

A

BC

DE

Temp.(in °C)

(time)

(A) A and B (B) B and C (C) C and D (D*) D and E

Q.6 By what process is heat transferred through the iron bar shown in the diagram ?

Warm Cool

Heat source

Transfer of heat

(A*) Conduction (B) Convection (C) Radiation (D) None of these

XIII UT-1 Page # 2

PHYSICSQ.7 A brass wire with Young's modulus of 10 × 1010 Pa is 2 m long and has cross-sectional area of 5 mm2.

If a weight of 5kN is hung from the wire, by how much does it stretch (in cm).(A) 1 cm (B*) 2 cm (C) 3 cm (D) 4 cm

[Sol.l

l Y

A

F

l = AY

Fl = 610

3

1051010

2105

= 2 cm ]

[PARAGRAPH TYPE]

Q.8 to Q.10 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Paragraph for question nos. 8 to 10

Feynman discusses an interesting situation while discussing general theory of relativity. If we want tomeasure the length of a hot body with a scale, the answer would be wrong. The scale would elongateand we can't find the true length.Now suppose we take a circular hot plate whose temperature increases as we go radially outwards. Ifwe take a scale to measure the radius, it would give wrong value of the radius. Not only this if we drawa chord on the plate when it was at uniform temperature and see it in this situation, the chord would becurved. If we want to measure the circumference, it would not read true value.Assume here that scale = hot body.

Q.8 Choose correct statement(A) Measured radius = actual radius of hot plate > radius of cold plate(B) Measured radius > actual radius of hot plate > radius of cold plate(C*) Actual radius of hot plate > measured radius = radius of cold plate(D) Actual radius of hot plate > measured radius > radius of cold plate

Q.9 The chord looks like

(A*) (B) (C) (D)

Q.10 The measured circumference is C and measured radius is r and temperature at circumference is T0(original temperataure is T)(A) C = 2r (1 + (T0 � T)) (B) C > 2r (1 + (T0 � T))

(C) C = 2r (1 � (T0 � T)) (D*) C < 2r (1 + (T0 � T))

[Sol. C = 2r ]

XIII UT-1 Page # 3

PHYSICS[MULTIPLE CORRECT CHOICE TYPE]

Q.11 to Q.13 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct.

Q.11 The temperature () of "cup of coffee" in a "BARISTA" Restaurant was plotted as a function of time.Which of the following curves may represent the plot.

(A*) curve A (B) curve B (C) curve C (D) curve D

Q.12 A metal cylinder of mass 0.5 kg is heated electrically by a 12 W heater in a room at 15°C. The cylinder

temperature rises uniformly to 25°C in 5 min and finally becomes constant at 45°C. Assuming that the

rate of heat loss is proportional to the excess temperature over the surroundings,(A*) the rate of loss of heat of the cylinder to surrounding at 20°C is 2W

(B*) the rate of loss of heat of the cylinder to surrounding at 45°C is 12W

(C) the rate of loss of heat of the cylinder to surrounding at 20°C is 5W.

(D) the rate of loss of heat of the cylinder to surrounding at 45°C is 30W.

Q.13 Energy density in a stretched wire is:

(A*) Y

)stress(

2

1 2

(B*) Half of stress × strain

(C*) 2

1Y (strain)2 (D)

2

1Y stress × strain

PART-B[MATRIX TYPE]

Q.1 has three statements (A,B,C) given in Column-I and four statements (P,Q,R,S) given in Column-II. Anygiven statement in Column-I can have correct matching with one or more statement(s) given in Column-II.

Q.1 Match the columnColumn-I (Initial State) Column-II (Final State)

(A) 1 gm ice 0°C + 1 gm water 70°C (P) Water + ice

(B) 1 gm ice 0°C + 1 gm steam 100°C (Q) Water + steam

(C) 1 gm ice 0°C + 1 gm water 50°C (R) Final temperatrue greater than 0°C and

+ 1 gm steam 100°C less than 100°C

(S) Final temperature either 0°C or 100°C

[Ans. (A) P, S (B) Q, S (C) Q, S ]

XIII UT-1 Page # 4

PHYSICS

PART-C[INTEGER TYPE]

Q.1 to Q.4 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits)

Q.1 The coefficient of thermal conductivity of copper is 9 times that of steel. In the composite cylindrical barshown in the figure, what will be the temperature at the junction of copper and steel (in °C)?

100°C 0°C

Copper Steel

18cm 6cm

[Ans. 0075 ]Q.2 If the work done in stretching a wire by 1 mm is 2 J. Find the work necessary (in joules) for stretching

another wire of the same material but with double the radius of cross-section and half the length by1 mm. Assume that elastic limit is not exceeded.

[Ans. 0016 ]

Q.3 The length of sections AB and EF is l1 in the figure, and their coefficient of thermal expansion is same

(1 = 2 × 10�5 /°C). The length of section CD is l

2 and its expansion coefficient is

2 = 1 × 10�5 /°C.

What must be the value of 2

1

×102 to ensure that distance AF remains the same at all temperatures?

(Points A, B, E and F reside on the same line.) [Ans. 0025]

Q.4 In a mercury thermometer at the temperature of 0°C there is 210 mm3 of mercury ( of mercury is180 × 10�6 /°C). Increase in volume when temperature raised by 1°C is n × 10�4 mm3. Find n.

[Ans. 0378 ]

XIII UT-1 Page # 1

CHEMISTRYXIII UNIT TEST-1 DATE : 17.06.2012

PART-A[SINGLE CORRECT CHOICE TYPE]

Q.1 to Q.7 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Q.1 27 g C and 48 g O2 are allowed to react completely to form CO and CO2. The weight ratio of CO andCO2 formed, is :(A) 7 : 11 (B) 3 : 4 (C*) 14 : 11 (D) 9 : 8

Sol. C = 27 gm , O2 = 48 gm2C + O2 2CO

x moles2

x moles x moles

C + O2 CO2y moles y moles y molesTotal amount of 'C' used up = 12 (x + y)Total amount of 'O' used up =16 x + 32 yNow 12 x + 12 y = 27 (i)

16x + 32y = 48 x + 2y = 3 (ii)12x + 12 y = 2712x + 24 y = 36���������������

�12 y = �9

y = 4

3

x + 2 × 4

3 = 3 x =

2

3

WCO = 28 x, WCO2 = 44 y

4/3

3/2

11

7

y 44

28

W

W

2CO

CO x

WCO : 2COW = 14 : 111

Q.2 Which of the following species do not exists?(A) NCl5 (B) XeF3¯ (C) OF4 (D*) All of these

Q.3MG A light beam of wavelength 310 nm strikes a metal of work function 2 eV. If ejected electron comes outafter making 2 collisions & in each collision it lost 50% of its K.E. with which it is moving, then de-Broglie wavelength when it comes out metal plate is(A*) 17.32 Å (B) 50 Å (C) 14.14 Å (D) infinite

Q.4 Which of the following pair has same geometry but different hybridisation?(A) BeCl2, C2H2 (B) SnCl2, XeF2 (C*) ICl2¯, CO2 (D) CCl4, NH4

+

Q.5MG Ratio of de-Broglie wavelength of electrons in 2 energy levels in a H-like atom is 2 : 1, the ratio offrequency of revolution is

(A) 1

4(B)

1

2(C*)

8

1(D)

4

1

XIII UT-1 Page # 2

CHEMISTRY

Q.6 In which of the following cases 22 yxd

orbital is involved in their hybridisation.

(A) NO2+ (B) I3¯ (C*) XeF5

+ (D) PCl3F2

Q.7 A mixture of CH4 and C2H4 occupies 22.4 litre at 1 atm and 273K. The mixture reacts completely with76.8 gm O2 to produce CO2 and H2O. Assuming ideal gas behaviour calculate the mole fraction ofC2H4 in the mixture.(A) 0.6 (B) 0.5 (C*) 0.4 (D) 0.75

Sol. 22.4 L at S.T.P. = 1 mole gas.

424 HCCH nn = 1 mole

32

8.76n

2O = 2.4

CH4 + 2O2 CO2 + 2H2Ox mole 2x moleC2H4 + 3O2 2CO2 + 2H2Oy 3y

Total No. of moles of O2 (used up) = 2x + 3yTotal No. of moles of CH4and C2H4 (used up) = x + y

x + y = 12x + 3y = 2.42x + 2y = 2

������������ y = 0.4

42HCX = 1

4.0

y

y

x

42HCX = 0.4

[PARAGRAPH TYPE]

Q.8 to Q.10 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Paragraph for question nos. 8 to 10

Bond order is associated with strength of bond and bond length. Higher the bond order smaller will bebond length and stronger will be bond.

Q.8TA Arrange the O2,

2O ,

2O and 22O in order of increasing bond order :

(A) 22

2222 OOOO (B*) 222

22 OOOO

(C) 22

222 OOOO (D)

22222 OOOO

Q.9TA Which fo the following species has longest O�O bond ?

(A) O2 (B)

2O (C)

2O (D*) 22O

Q.10TAWhich of the following has highest O�O bond energy ?

(A) O2 (B*)

2O (C)

2O (D) 22O

XIII UT-1 Page # 3

CHEMISTRY[MULTIPLE CORRECT CHOICE TYPE]

Q.11 to Q.13 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct.

Q.11TA Which of the following molecle(s) has/have zero dipole moment ?(A*) CH4 (B*) CBr4 (C*) C2H2 (D) None of these

Q.12 The atomic weight of two elements A and B are 20 and 40 respectively. Select correct statement(s) :(A) x gm of A contains y atoms which is equal to atoms present in x gm of B.(B*) x gm of A contains y atoms which is equal to atoms present in 2x gm of B.(C) At STP, mass of x L of monoatomic gas A is equal to that of x L of monoatomic gas B.(D*) At STP, ratio of mass of x L monoatomic gas A to mass of x L monoatomic gas B is 1 : 2.

Q.13TAWhich of the following species is/are diamagnetic ?(A*) Diavalent cation of Hg (B) Divalent cation of Ni(C) Trivalent cation of Fe (D*) Tetravalent cation of Ti

PART-B[MATRIX TYPE]

Q.1 has three statements (A,B,C) given in Column-I and four statements (P,Q,R,S) given in Column-II. Anygiven statement in Column-I can have correct matching with one or more statement(s) given in Column-II.

Q.1 Column-I Column-II

(A) Angular wave function (, ) of this orbital (P)

changes with change in &

(B) For this orbital, as distance from nucleus increases, (Q) radial probability of finding electron may increaseor decrease

(C) Number of radial node = 1 (R)

(S) 4d

[Ans. (A) QS, (B) PQRS, (C) PQS]

XIII UT-1 Page # 4

CHEMISTRY

PART-C[INTEGER TYPE]

Q.1 to Q.4 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits)

Q.1MG Find the principal quantum number for an orbital having 22

31

4r/'k rkrkrek)r( .

[Ans: 0005]

Q.2 Find the maximum number of atoms lying in the same plane in CH2= C = C = CH(CH3) and find thenumber of such plane in this molecule.[If the answers are 3 and 5, represent as 0035] [4]

[Ans. 0091]Q.3MG Calculate volume of Cl2 gas (in ml) liberated at 1 atm and 273 K when 2.61 gm MnO2 reacts with

2.92 gm HCl according to the following reaction (Given : At wt of Mn = 55) [Ans. 448]MnO2 + HCl MnCl2 + Cl2 + 2 H2O

[Sol. MnO2 + 4 HCl MnCl2 + Cl2 + 2 H2O0.03 0.08

L. R.0.02 × 22.4 × 103 = 448 ml

Q.4 Find the number of acid(s) from the following in which X � H bond is/are present. Given X is central atom

H3PO2 , H4P2O7, H2S2O6, H3PO3 H3BO3, H2SO4 ,HNO3, H2S2O7

[Ans. 2]

[Sol.* H3PO2

P

O

H OH

H

H4P2O7

H2S2O6

OO

O

O

O

HO � S � S � OH * H3PO3P

O

H OH

OH

H3BO3 B

OH

HO OH

H2SO4S

O

HO O

OH

]

MATHEMATICS

XIII UT-1 Page # 1

PART-A[SINGLE CORRECT CHOICE TYPE]

Q.1 to Q.7 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Q.1 Let , be two distinct real roots of quadratic equation x2 � 4x + 2 = 0. The sum of the coefficients of

a quadratic trinomial with leading coefficient unity whose roots are 2 + 2 and 3 + 3 is(A) 329 (B*) 429 (C) 431 (D) 469

[Sol. + = 4 ; = 2 [ST-1]2 + 2 = 16 � 4 = 12

3 + 3 = 4(2 � + 2) = 4 (16 � 3 × 2) = 40

Quadratic equation is (x � 12) (x � 40) P(x) ; Sum is P(1) = (� 11) (� 39) = 429. Ans.]

Q.2 Let f (x) = � x2 + 2px � (3p + 4). If f (x) is positive for atleast one real x then the smallest positive integral

value of p is(A) 2 (B) 3 (C) 4 (D*) 5

[Sol. f (x) > 0 � x2 + 2px � (3p + 4) > 0 for atleast one real x

x2 � 2px + (3p + 4) < 0 for atleast one real x

Hence D > 0 p2 � 3p � 4 > 0 (p � 4)(p + 1) > 0

p (� , � 1) (4, ) Hence smallest positive integral value of p is 5 Ans.]

Q.3 The value of the expression

100cos222

20sin20cos is equal to

(A) 1 (B) 21

(C*) 2

1(D) 2

[Sol. We have

100cos222

20sin20cos =

50cos222

20sin70sin

2 =

50cos22

20sin70sin =

25sin22

20sin70sin2

=

25sin225sin45cos2

= cos 45° = 2

1. Ans.]

Q.4 If k1 and k2 are the two values of 'k' where k1 < k2 for which the expressionf(x, y) = x2 + 2xy + 4y2 + 2kx � 6y + 3 can be resolved as a product of two linear factors

then the value of (k2 � k1), is equal to

(A) 2

1(B*)

2

3(C) 2 (D)

2

5

[Sol. We have A = 1 ; B = 4 ; C = 3 ; F = �3 ; G = k ; H = 1 [11th, 16-08-2009, P-2]Now, ABC + 2FGH � AF2 � BG2 � CH2 = 0

k = 0, 23

; k1 = 23

, k2 = 0 ; (k2 � k1) = 23

Ans. ]

Q.5 Number of values of x [0, 2) satisfying the equation 2sin2x = cos x + 1 is(A) 1 (B) 2 (C*) 3 (D) 4

[Sol. 2cos2x + cos x � 1 = 0 (2cos x � 1) (1 + cos x) = 0 cos x = 21

or cos x = � 1

i.e. 3

5,,

3

. Ans.]

MATHEMATICS

XIII UT-1 Page # 2

Q.6 Let , , and be the roots (real or non-real) of equation x4 � 3x + 1 = 0. The value of

3 + 3 + 3 + 3 is equal to(A) 6 (B*) 9 (C) 12 (D) 15

[Sol. x4 = 3x � 1

x3 = 3 � x

1

3 = 3 �

1; 3 = 3 �

1; 3 = 3 �

1; 3 = 3 �

1

on adding

1111123

=

12 = 12 � 3 = 9 Ans.]

Q.7 A triangle ABC is inscribed in a circle with centre O. The vertices oftriangle divide the circle into 3 arcs of length 3, 4 and 5 as shown infigure. If the angles subtended by chords AB, BC and CA are 1, 2and 3 respectively then the value of (sin 1 + sec 2 + cosec 3), is equal to(A) 2 (B*) 1 C

B

A

3

4

5

12

3

O

(C) 23

(D) 25

[Sol. As, l1 = r1 ; l2 = r2 ; l3 = r3

r

5

r

4

r

3 = 2

r

12 = 2 r =

6

1 = r1l =

6

3 =

2

; 2 =

r2l =

6

4 =

3

2 and 3 =

r3l =

6

5 =

6

5

Hence, sin 1 + sec 2 + cosec 3 = 6

5cosec

3

2sec

2sin

= 1 � 2 + 2 = 1. Ans.]

[PARAGRAPH TYPE]

Q.8 to Q.10 has four choices (A), (B), (C), (D) out of which ONLY ONE is correct.

Paragraph for question nos. 8 to 10Let f(x) = px2 + px � 2, g(x) = x2 + (2p + 1) x + (p + 2) and h(x) = 2x2 + (p + q)x + 2q x R, where p, q are real constants.

Q.8 Number of integral values of p so that f(x) > g(x) x R, is(A*) 0 (B) 1 (C) 2 (D) infinite

Q.9 Number of integral values of p for which the equation g(x) = h(x) has non-real rootsfor all real values of q is(A*) 0 (B) 1 (C) 2 (D) infinite

Q.10 The set of all possible real values of p for which the equation f(x) = h(x) has real rootsfor all real values of q is

(A)

2

5,2 (B*)

2

5,2 (C)

2

5,2 (D)

2

5,2

[Sol.(i) Given

MATHEMATICS

XIII UT-1 Page # 3

f(x) > g(x) x R px2 + px � 2 > x2 + (2p + 1) x + p + 2 x R(p � 1) x2 � (p + 1) x � p � 4 > 0 x RSo, p � 1 > 0 p > 1and, D < 0 (p + 1)2 + 4 (p � 1) (p + 4) < 0

p2 + 2p + 1 + 4 (p2 + 3p � 4) < 0

5p2 + 14p � 15 < 0

1030019614

p10

30019614p < 0

)8.0p()6.3(p < 0

p (� 3.6, 0.8) but p > 1

Hence, there is no value of 'p' for which f(x) > g(x) x R. Ans.(ii) Given, g(x) = h(x)

x2 + (2p + 1) x + p + 2 = 2x2 + (p + q) x + 2q x2 + (p + q � 2p � 1) x + 2q � p � 2 = 0

x2 + (q � p � 1) x + 2q � p � 2 = 0, has non real roots q RSo, D < 0 x R (q � p � 1)2 � 4(2q � p � 2) < 0 q R q2 + (p + 1)2 � 2q (p + 1) � 8q + 4(p + 2) < 0

q2 � 2q (p + 5) + p2 + 6p + 9 < 0 q R, which is not possible q R.So, number of integral values of p is zero. Ans.

(iii) Given, f(x) = h(x) px2 + px � 2 = 2x2 + (p + q)x + 2q(p � 2)x2 � qx � 2 � 2q = 0, has real roots q RFor p � 2 0So, D 0 q2 + 4(p � 2) (2 + 2q) 0 q Rq2 + 8 (p + pq � 2 � 2q) 0 q R q2 + 8q (p � 2) + 8(p � 2) 0 q RSo, D' 0 64(p � 2)2 � 4 · 8 (p � 2) 0 (p � 2) (2p � 4 � 1) 0

(p � 2) (2p � 5) 0 2 p 2

5

but p 2, therefore p

2

5,2

now for p = 2 equation reduces � qx � 2 � 2q = 0 x = q

q22

for q = 0 equation does not have any real solution.

Hence p

25

,2 Ans. ]

[MULTIPLE CORRECT CHOICE TYPE]

Q.11 to Q.13 has four choices (A), (B), (C), (D) out of which ONE OR MORE may be correct.

Q.11 Find the number of principal solution of the equation |tanx| = 1(A) 2 (B*) 4 (C) 3 (D) none of these

[Sol: |tanx | = 1 ; tanx = +1 x = 4

7,

4

5,

4

3,

4

= 4 solutions]

MATHEMATICS

XIII UT-1 Page # 4

Q.12 Let P(x) = cot2x

xcotxcot1

xtanxtan12

2

+ 2

)x2cosx2(sin2

xsinx3sinx3cosxcos

. Then which of the

following is(are) correct?(A) Range of P(x) is [1, 2]. (B*) The value of P(18°) + P(72°) is 3.

(C) Range of P(x) is [0, 1]. (D) The value of P(18°) + P(72°) is 5.

[Sol. Given, P(x) = cot2x

xcotxcot1

xtanxtan12

2

+ 2

)x2cosx2(sin2

xsinx3sinx3cosxcos

= xcotxcot1

1xcotxcot2

2

+

2

)x2cosx2(sin2

)x2cosx2(sinxsin2

= 1 + sin2 x

P(18°) + P(72°) = (1 + sin218°) + (1 + sin2 72°)

= 1 + 1 + (sin2 18° + cos2 18°)

= 3. Ans.]Q.13 The graph of a quadratic polynomial y = ax2 + bx + c is as shown in

the adjacent figure. Which of the following quantities is(are) negative?(A*) b � c (B*) bc

x

y

O(C) c � a (D*) ab2

[Sol.7/qe a < 0; � a

b < 0 b < 0;

a

c < 0 c > 0 [11th, 24-06-2007]

now b � c = (� ve) � (+ ve) must be negative

bc = (�) (+ve) must be negative

ab2 = (�) (+ve) must be negative

c � a = (+) � (� ) must be positive]

PART-B[MATRIX TYPE]

Q.1 has Three statements (A, B, C) given in Column-I and four statements (P, Q, R, S) given in Column-II.Any given statement in Column-I can have correct matching with one or more statement(s) given in Column-II.

Q.1 Let P(x) = 2x2 � 12x + c x R where c is a real constant, thenColumn-I Column-II

(A) If greatest value of p(x) for x [1, 2] is 1, then c equals (P) 8(B) If smallest value of P(x) for x [1, 5] is � 1, then c equals (Q) 11(C) If the greatest value of P(x) for x [1, 4] is 2, then c equals (R) 12

(S) 17[Ans. (A) Q, (B) S, (C) R]

[Sol.(A) Maximum value of P(x) in [1, 2] occurs at x = 1.

Hence, P(1) = 1 c � 10 = 1 c = 11. Ans.(B) Minimum value of P(x) in [1, 5] occurs at x = 3.

Hence, P(3) = � 1 c � 18 = � 1 c = 17. Ans.(C) Maximum value of P(x) in [1, 4] occurs at x = 1.

Hence, P(1) = 2 c � 10 = 2 c = 12. Ans. ]

MATHEMATICS

XIII UT-1 Page # 5

PART-C[INTEGER TYPE]

Q.1 to Q.4 are "Integer Type" questions. (The answer to each of the questions are upto 4 digits)

Q.1 If sin 2 + sin 4 + sin 8 + sin 16 = 2

p(p N) where =

15

, then find the value of p.[Ans. 0015]

[Sol. = 15

= 12°, so

L.H.S. = sin 24° + sin 48° + sin 96° + sin 192° = (sin 24° + sin 96°) + (sin 48° + sin 192°)

= 2sin 60° cos 36° + 2sin 120° cos 72° = 3 (cos 36° + cos 72°)

=

4

15

4

153 =

4

523 =

2

15 =

2

p (given) ;Hence, p = 15. Ans.]

Q.2 Find the value of expression

º89

º1

2424 sin4coscos4sin . [Ans. 0]

[Sol.

º89

º1

2424 sin4coscos4sin =

89

1

2222 cos2sin2

=

89

1

2424 cos44cossin44sin =

89

22

1

)cos2()sin2(

=

89

1

2cos [ST-9] ; =

seriescosine

178cos......6cos4cos2cos ;

=

1sin

89sin·21782

cos

= 0. Ans.]

Q.3 Find the number of integral values of a for which (x � 3a) (x � a � 3) 0 for all x [1, 3].[Ans. 1]

[Sol. f (x) = (x � 3a) (x � a � 3) = x2 � x (3 + 4a) + 3a2 + 9a

1 3x-axis

Given f(x) 0 x [1, 3] f(1) 0 and f(3) 0Now, f(1) = 1 � (3 + 4a) + 3a2 + 9a 0 3a2 + 5a � 2 0

(a + 2) (3a � 1) 0 a

3

1,2 ...........(1)

�2 1/3

Now, f(3) = 9 � 3( 3 + 4a) + 3a2 + 9a 0

MATHEMATICS

XIII UT-1 Page # 6

3a2 � 3a 03a (a � 1) 0 ...........(2)

0 1(1) (2)

a

3

1,0

Number of integral values of a is one i.e., a = 0. Ans.]

Q.4 Find the number of solutions of the equation 2222

x

1xxsin

2

xcos2 in [0, 2] . [Ans: 0]

[Sol. 2xsin2

xcos2 22

& 2x

1x

22

equation hold only when LHS & RHS = 2but RHS = 2 when x = +1& for x = +1 LHS 2 No solution.]