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QUESTION: 1

A uniform metallic wire has a resistance of 18 Ω and is bent into an equilateral triangle. Then, the resistance between any two vertices of the triangle is :

Solution:

R_{eq} between any two vertex will be

QUESTION: 2

A satellite is moving with a constant speed v in circular orbit around the earth. An object of mass 'm' is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time of ejection, the kinetic energy of the object is :

Solution:

At height r from center of earth. orbital velocity

∴ By energy conservation

(At infinity, PE = KE = 0)

QUESTION: 3

A solid metal cube of edge length 2 cm is moving in the positive y-direction at a constant speed of 6 m/s. There is a uniform magnetic field of 0.1 T in the positive z-direction. The potential difference between the two faces of the cube perpendicular to the x-axis is :

Solution:

Potential difference between two faces perpendicular to x-axis will be

QUESTION: 4

A parallel plate capacitor is of area 6 cm^{2} and a separation 3 mm. The gap is filled with three dielectric materials of equal thickness (see figure) with dielectric constants K_{1}, = 10, K_{2} = 12 and K_{3} = 14. The dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance would be :

Solution:

Let dielectric constant of material used be K.

⇒ K = 12

QUESTION: 5

A 2 W carbon resistor is color coded with green, black, red and brown respectively. The maximum current which can be passed through this resistor is :

Solution:

P = i^{2}R.

∴ for i_{max}, R must be minimum

from color coding R = 50×10^{2}Ω

∴ i_{max} = 20mA

QUESTION: 6

In a Young's double slit experiment with slit separation 0.1 mm, one observes a bright fringe at angle 1/40 rad by using light of wavelength λ_{1}. When the light of wavelength λ_{2} is used a bright fringe is seen at the same angle in the same set up. Given that λ_{1 }and λ_{2 }are in visible range (380 nm to 740 nm), their values are :

Solution:

Path difference = d sinθ ≈ dθ

= 0.1 x 1/40 mm = 2500nm

or bright fringe, path difference must be integral multiple of λ.

∴ 2500 = nλ_{1} = mλ_{2}

∴ λ_{1} = 625, λ_{2} = 500 (from m=5) (for n = 4)

QUESTION: 7

A magnet of total magnetic moment 10^{-2} î A-m^{2 }is placed in a time varying magnetic field, B î (costωt) where B = l Tesla and ω = 0.125 rad/ s. The work done for reversing the direction of the magnetic moment at t = 1 second, is :

Solution:

Work done,

= 2 × 10^{–2} × 1 cos(0.125)

= 0.02 J

∴ correct answer is (2)

QUESTION: 8

To mop-clean a floor, a cleaning machine presses a circular mop of radius R vertically down with a total force F and rotates it with a constant angular speed about its axis. If the force F is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is μ the torque, applied by the machine on the mop is :

Solution:

Consider a strip of radius x & thickness dx, Torque due to friction on this strip.

QUESTION: 9

Using a nuclear counter the count rate of emitted particles from a radioactive source is measured. At t = 0 it was 1600 counts per second and t = 8 seconds it was 100 counts per second. The count rate observed, as counts per second, at t = 6 seconds is close to:

Solution:

at t = 0, A_{0} = dN/dt = 1600 C/s

at t = 8s, A = 100 C/s

Therefor half life is t_{1/2} = 2 sec

∴ Activity at t = 6 will be 1600 (1/2)^{3 }= 200 C/s

QUESTION: 10

If the magnetic field of a plane electromagnetic wave is given by (The speed of light = 3 × 10^{8}/m/s)

then the maximum electric field associated with it is :

Solution:

E_{0} = B_{0} × C

= 100 × 10^{–6} × 3 × 10^{8}

= 3 × 10^{4} N/C

QUESTION: 11

A charge Q is distributed over three concentric spherical shells of radii a, b, c (a < b < c ) such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre, where r < a, would be :

Solution:

QUESTION: 12

Water flows into a large tank with flat bottom at the rate of 10^{–4} m3s^{–1}. Water is also leaking out of a hole of area 1 cm^{2} at its bottom. If the height of the water in the tank remains steady, then this height is:

Solution:

Since height of water column is constant therefore, water inflow rate (Q_{in})

= water outflow rate

Q_{in} = 10^{–4} m^{3}s^{–1}

QUESTION: 13

A piece of wood of mass 0.03 kg is dropped from the top of a 100 m height building. At the same time, a bullet of mass 0.02 kg is fired vertically upward, with a velocity 100 ms^{–1}, from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is : (g =10ms^{–2})

Solution:

Time taken for the particles to collide,

Speed of wood just before collision = gt = 10 m/s & speed of bullet just before collision v-gt = 100 – 10 = 90 m/s

Now, conservation of linear momentum just before and after the collision -

–(0.02) (1v) + (0.02) (9v) = (0.05)v

⇒ 150 = 5v

⇒ v = 30 m/s

Max. height reached by body h = v^{2}/2g

∴ Height above tower = 40 m

QUESTION: 14

The density of a material in SI units is 128 kg m^{-3}. In certain units in which the unit of length is 25 cm and the unit of mass is 50 g, the numerical value of density of the material is :

Solution:

= 40 units

QUESTION: 15

To get output '1' at R, for the given logic gate circuit the input values must be :

Solution:

To make O/P P + Q must be 'O' SO, y = 0 x = 1

QUESTION: 16

A block of mass m is kept on a platform which starts from rest with constant acceleration g/2 upward, as shown in fig. Work done by normal reaction on block in time t is :

Solution:

QUESTION: 17

A heat source at T= 10^{3} K is connected to another heat reservoir at T=10^{2 }K by a copper slab which is 1 m thick. Given that the thermal conductivity of copper is 0.1 WK^{-1} m^{-1}, the energy flux through it in the steady state is :

Solution:

QUESTION: 18

A TV transmission tower has a height of 140 m and the height of the receiving antenna is 40 m. What is the maximum distance upto which signals can be broadcasted from this tower in LOS(Line of Sight) mode ? (Given : radius of earth = 6.4 x 10^{6}m).

Solution:

Maximum distance upto which signal can be broadcasted is

where h_{T} and h_{R} are heights of transmiter tower and height of receiver respectively.

Putting all values -

on solving, d_{max} = 65 km

QUESTION: 19

A potentiometer wire AB having length L and resistance 12 r is joined to a cell D of emf ε and internal resistance r. A cell C having emf ε/2 and internal resistance 3r is connected. The length AJ at which the galvanometer as shown in fig. shows no deflection is:

Solution:

QUESTION: 20

An insulating thin rod of length ℓ has a x linear charge density λ (x) = on it. The rod is rotated about an axis passing through the origin (x = 0) and perpendicular to the rod. If the rod makes n rotations per second, then the time averaged magnetic moment of the rod is :

Solution:

QUESTION: 21

Two guns A and B can fire bullets at speeds 1 km/s and 2 km/s respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is :

Solution:

QUESTION: 22

A string of length 1 m and mass 5 g is fixed at both ends. The tension in the string is 8.0 N. The string is set into vibration using an external vibrator of frequency 100 Hz. The separation between successive nodes on the string is close to :

Solution:

Velocity of wave on string

Now, wavelength of wave

Separation b/w successive nodes,

= 20 cm

QUESTION: 23

A train moves towards a stationary observer with speed 34 m/s. The train sounds a whistle and its frequency registered by the observer is f_{1}. If the speed of the train is reduced to 17 m/s, the frequency registered is f_{2}. If speed of sound is 340 m/s, then the ratio f_{1}/f_{2} is :

Solution:

QUESTION: 24

In an electron microscope, the resolution that can be achieved is of the order of the wavelength of electrons used. To resolve a width of 7.5 × 10^{–12}m, the minimum electron energy required is close to :

Solution:

{λ = 7.5 × 10^{–12}}

KE = 25 Kev

QUESTION: 25

A homogeneous solid cylindrical roller of radius R and mass M is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is:

Solution:

QUESTION: 26

A plano convex lens of refractive index µ_{1} and focal length f_{1} is kept in contact with another plano concave lens of refractive index µ_{2} and focal length f_{2}. If the radius of curvature of their spherical faces is R each and f_{1} = 2f_{2}, then µ1 and µ_{2} are related as

Solution:

QUESTION: 27

Two electric dipoles, A, B with respective dipole moments and placed on the x-axis with a separation R, as shown in the figure

The distance from A at which both of them produce the same potential is :

Solution:

QUESTION: 28

In the given circuit the cells have zero internal resistance. The currents (in Amperes) passing through resistance R_{1}, and R_{2} respectively, are:

Solution:

i_{1} = 10/20 = 0.5A

i_{2} = 0

QUESTION: 29

In the cube of side 'a' shown in the figure, the vector from the central point of the face ABOD to the central point of the face BEFO will be:

Solution:

QUESTION: 30

Three Carnot engines operate in series between a heat source at a temperature T_{1} and a heat sink at temperature T_{4} (see figure). There are two other reservoirs at temperature T_{2}, and T_{3}, as shown, with T_{2} > T_{2} > T_{3} > T_{4} . The three engines are equally efficient if:

Solution:

QUESTION: 31

Two pi and half sigma bonds are present in:

Solution:

⇒ BO = 2.5 ⇒ [π - Bond = 2 & σ - Bond = 1/2]

N_{2} ⇒ B.O. = 3.0 ⇒ [π - Bond = 2 & σ - Bond = 1]

= B.O. ⇒ 2.5 ⇒ [π - Bond = 1.5 & σ - Bond = 1]

O_{2} ⇒ B.O. ⇒ 2 ⇒ [π - Bond ⇒ 1 & σ - Bond = 1]

QUESTION: 32

The chemical nature of hydrogen preoxide is :-

Solution:

H_{2}O_{2} act as oxidising agent and reducing agent in acidic medium as well as basic medium.

H_{2}O_{2} Act as oxidant :-

(In acidic medium)

(In basic medium)

H_{2}O_{2} Act as reductant :-

(In acidic medium)

(In basic medium)

QUESTION: 33

Which dicarboxylic acid in presence of a dehydrating agent is least reactive to give an anhydride :

Solution:

Adipic acid CO_{2}H–(CH_{2})_{4}–CO_{2}H

7 membered cyclic anhydride (Very unstable)

QUESTION: 34

Which premitive unit cell has unequal edge lenghs (a ≠ b ≠ c) and all axial angles different from 90° ?

Solution:

In Triclinic unit cell

a ≠ b ≠ c & α ≠ β ≠ γ ≠ 90°

QUESTION: 35

Wilkinson catalyst is :

Solution:

Wilkinsion catalyst is [(ph_{3}P)_{3}RhCl]

QUESTION: 36

The total number of isotopes of hydrogen and number of radioactive isotopes among them, respectively, are :

Solution:

Total number of isotopes of hydrogen is 3

and only ^{3}_{1}H or ^{3}_{1 }T is an Radioactive element.

QUESTION: 37

The major product of the following reaction is

Solution:

Example of E_{2} elimination and conjugated diene is formed with phenyl ring in conjugation which makes it very stable.

QUESTION: 38

The total number of isomers for a square planar complex [M(F)(Cl)(SCN)(NO_{2})] is :

Solution:

The total number of isomers for a square planar complex [M(F)(Cl)(SCN)(NO_{2})] is 12.

QUESTION: 39

Hall-Heroult's process is given by "

Solution:

In Hall-Heroult's process is given by

2Al_{2}O_{3} + 3C → 4Al + 3CO_{2}

QUESTION: 40

The value of K_{p}/K_{C} for the following reactions at 300K are, respectively :

(At 300K, RT = 24.62 dm^{3}atm mol^{–1})

N_{2}(g) + O_{2}(g) ⇔ 2NO(g)

N_{2}O_{4}(g) ⇔ 2NO_{2}(g)

N_{2}(g) + 3H_{2}(g) ⇔ 2NH_{3}(g)

Solution:

QUESTION: 41

If dichloromethane (DCM) and water (H_{2}O) are used for differential extraction, which one of the following statements is correct?

Solution:

QUESTION: 42

The type of hybridisation and number of lone pair(s) of electrons of Xe in XeOF_{4}, respectively, are :

Solution:

QUESTION: 43

The metal used for making X-ray tube window is :

Solution:

"Be" Metal is used in x-ray window is due to transparent to x-rays.

QUESTION: 44

Consider the given plots for a reaction obeying Arrhenius equation (0°C < T < 300°C) : (k and E_{a} are rate constant and activation energy, respectively)

Solution:

On increasing E_{a}, K dec reases

QUESTION: 45

Water filled in two glasses A and B have BOD values of 10 and 20, respectively. The correct statement regarding them, is :

Solution:

Two glasses "A" and "B" have BOD values 10 and "20", respectively.

Hence glasses "B" is more polluted than glasses "A".

QUESTION: 46

The increasing order of the pKa values of the following compounds is :

Solution:

Acidic strength is inversely proportional to pka.

QUESTION: 47

Liquids A and B form an ideal solution in the entire composition range. At 350 K, the vapor pressures of pure A and pure B are 7 × 10^{3} Pa and 12 × 10^{3} Pa, respectively. The composition of the vapor in equilibrium with a solution containing 40 mole percent of A at this temperature is :

Solution:

y_{B} = 0.72

QUESTION: 48

Consider the following reduction processes :

Zn^{2+} + 2e^{–} → Zn(s); E° = – 0.76 V

Ca^{2+} + 2e^{–} → Ca(s); E° = – 2.87 V

Mg^{2+} + 2e^{–} → Mg(s); E° = – 2.36 V

Ni^{2+} + 2e^{–} → Ni(s); E° = – 0.25 V

The reducing power of the metals increases in the order :

Solution:

Higher the o xidation potentia l bet ter will be reducing power.

QUESTION: 49

The major product of the following reaction is:

Solution:

QUESTION: 50

The electronegativity of aluminium is similar to :

Solution:

E.N. of Al = (1.5) ≌ Be (1.5)

QUESTION: 51

The decreasing order of ease of alkaline hydrolysis for the following esters is :

Solution:

More is the electrophilic character of carbonyl group of ester faster is the alkaline hydrolysis.

QUESTION: 52

A process has ΔH = 200 Jmol^{–1} and ΔS = 40 JK^{–1}mol^{–1}. Out of the values given below, choose the minimum temperature above which the process will be spontaneous :

Solution:

QUESTION: 53

Which of the graphs shown below does not represent the relationship between incident light and the electron ejected form metal surface ?

Solution:

QUESTION: 54

Which of the following is not an example of heterogeneous catalytic reaction?

Solution:

Then is no catalyst is required for combustion of coal.

QUESTION: 55

The effect of lanthano id contraction in the lanthanoid series of elements by and large means :

Solution:

Due to Lanthanoid contraction both atomic radii and ionic radii decreases gradually in the lanthanoid series.

QUESTION: 56

The major product formed in the reaction given below will be :

Solution:

QUESTION: 57

The correct structure of product 'P' in the following reaction is :

Solution:

Asn–Ser is dipeptide having following structure

QUESTION: 58

Which hydrogen in compound (E) is easily replaceable during bromination reaction in presence of light :

Solution:

QUESTION: 59

The major product 'X' formed in the following reaction is :

Solution:

QUESTION: 60

A mixture of 100 m mol of Ca(OH)_{2} and 2g of sodium sulphate was dissolved in water and the volume was made up to 100 mL. The mass of calcium sulphate formed and the concentration of OH^{–} in resulting solution, respectively, are: (Molar mass of Ca(OH)_{2}, Na_{2}SO_{4} and CaSO_{4} are 74, 143 and 136 g mol^{–1}, respectively; K_{sp }of Ca(OH)_{2} is 5.5 × 10^{–6})

Solution:

Ca(OH)_{2} + Na_{2}SO_{4} → CaSO_{4} + 2NaOH 100 m mol 14 m mol — — — — 14 m mol 28 m mol

QUESTION: 61

Consider a triangular plot ABC with sides AB=7m, BC=5m and CA=6m. A vertical lamp-post at the mid point D of AC subtends an angle 30° at B. The height (in m) of the lamp-post is:

Solution:

QUESTION: 62

Let f : R → R be a function such that f(x) = x^{3}+x^{2}f'(1) + xf''(2)+f'''(3), x∈R. Then f(2) equal :

Solution:

f(x) = x^{3} + x^{2} f'(1) + x f''(2) + f'''(3)

⇒f'(x) = 3x^{2} + 2xf'(1) + f''(x) .. ...(1)

⇒ f''(x) = 6x + 2f'(1) .....(2)

⇒ f'''(x) = 6 .....(3)

put x = 1 in equation (1) :

f'(1) = 3 + 2f'(1) + f''(2) .....(4)

put x = 2 in equation (2) :

f''(2) = 12 + 2f'(1) .....(5)

from equation (4) & (5) :

–3 – f'(1) = 12 + 2f'(1)

⇒ 3f'(1) = –15

⇒ f'(1) = –5 ⇒ f''(2) = 2 .. ..(2)

put x = 3 in equation (3) :

f''(3) = 6

∴ f(x) = x^{3 }– 5x^{2} + 2x + 6

f(2) = 8 – 20 + 4 + 6 = –2

QUESTION: 63

If a circle C passing th rough the point (4, 0) touches the circle x^{2} + y^{2} + 4x – 6y = 12 externally at the point (1, –1), then the radius of C is :

Solution:

x^{2 }+ y^{2} + 4x – 6y – 12 = 0

Equation of tangent at (1, –1)

x – y + 2(x + 1) – 3(y – 1) – 12 = 0

3x – 4y – 7 = 0

∴ Equation of circle is

(x^{2} + y^{2} + 4x – 6y – 12) + λ(3x – 4y – 7) = 0

It passes through (4, 0) :

(16 + 16 – 12) + λ(12 – 7) = 0

⇒ 20 + λ(5) = 0

⇒ λ = –4

∴ (x^{2} + y^{2} + 4x – 6y – 12) – 4(3x – 4y – 7) = 0

or x^{2} + y^{2} – 8x + 10y + 16 = 0

Radius =

QUESTION: 64

In a class of 140 students numbered 1 to 140, all even numbered students opted mathematics course, those whose number is divisible by 3 opted Physics course and theose whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is :

Solution:

Let n(A ) = number of students opted Mathematics = 70,

n(B) = number of students opted Physics = 46,

n(C) = number of students opted Chemistry = 28,

n(A ∩ B) = 23,

n(B ∩ C) = 9,

n(A ∩ C) = 14,

n(A ∩ B ∩ C) = 4,

Now n(A ∪ B ∪ C)

= n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C)

– n(A ∩ C) + n(A ∩ B ∩ C)

= 70 + 46 + 28 – 23 – 9 – 14 + 4 = 102

So number of students not opted for any course

= Total – n(A ∪ B ∪ C)

= 140 – 102 = 38

QUESTION: 65

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is :

Solution:

= 7 × 90 + 24 = 654

Total = 654 + 702 = 1356

QUESTION: 66

Let and be three vectors such thatis perpendicular to Then a possible value of (λ_{1},λ_{2},λ_{3}) is :-

Solution:

Now check the options, option (2) is correct

QUESTION: 67

The equation of a tangent to the hyperbola 4x^{2}–5y^{2} = 20 parallel to the line x–y = 2 is :

Solution:

slope of tangent = 1

equation of tangent

QUESTION: 68

If the area enclosed between the curves y=kx^{2} and x=ky^{2}, (k>0), is 1 square unit. Then k is:

Solution:

Area bounded by y^{2} = 4ax & x^{2} = 4by, a, b ≠ 0 is

by using formula :

QUESTION: 69

Let

Let S be the set of points in the interval (–4,4) at which f is not differentiable. Then S:

Solution:

f(x) is not differentiable at x = {–2,–1,0,1,2}

⇒ S = {–2, –1, 0, 1, 2}

QUESTION: 70

If the parabolas y^{2}=4b(x–c) and y^{2}=8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a,b,c)

Solution:

Normal to these two curves are y = m(x - c) - 2bm - bm^{3}, y = mx - 4am - 2am^{3}

If they have a common normal (c + 2b) m + bm^{3} = 4am + 2am^{3} Now (4a - c - 2b) m = (b - 2a)m^{3}

We get all options are correct for m = 0 (common normal x-axis)

Ans. (1), (2), (3), (4)

**Remark :**

If we consider question as If the parabolas y^{2} = 4b(x - c) and y^{2} = 8ax

have a common normal other than x-axis, then

which one of the following is a valid choice for the ordered triad (a, b, c) ?

When m≠ 0 : (4a - c - 2b) = (b - 2a)m^{2}

QUESTION: 71

The sum of all values of satisfying sin^{2} 2θ + cos^{4} 2θ = 3/4 is:

Solution:

QUESTION: 72

Let z_{1} and z_{2} be any two non-zero complex numbers such that 3|z_{1}| = 4 |z_{2}|. If then :

Solution:

Now all options are incorrect

**Remark :**

There is a misprint in the problem actual problem should be :

" Let z_{1 }and z_{2} be any non-zero complex number such that 3|z_{1}| = 2|z_{2}|.

∴ Im(z) = 0

Now option (4) is correct.

QUESTION: 73

If the system of equations

x+y+z = 5

x+2y+3z = 9

x+3y+αz = β

has infinitely many solutions, then β–α equals:

Solution:

for infinite solutions D = 0 ⇒ α = 5

⇒ 2 +β -15 = 0 ⇒ β-13= 0

on β = 13 we get D_{y} = D_{z} = 0

α = 5, α = 13

QUESTION: 74

The shortest distance between the point and the curve y = √x, (x> 0) is :

Solution:

Let points

So minimum distance is

QUESTION: 75

Consider the quad ra tic equa tion (c–5)x^{2}–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :

Solution:

Let f(x) = (c – 5)x^{2} – 2cx + c – 4

∴ f(0)f(2) < 0 .....(1)

& f(2)f(3) < 0 .....(2)

from (1) & (2)

(c – 4)(c – 24) < 0

& (c – 24)(4c – 49) < 0

∴ s = {13, 14, 15, ..... 23}

Number of elements in set S = 11

QUESTION: 76

then k equals :

Solution:

QUESTION: 77

Let d∈R, and θ∈[0,2π]. If the minimum value of det(A) is 8, then a value of d is :

Solution:

= (2 + sinθ)(2 + 2d - sinθ) - d(2 sin θ - d)

=4 + 4d – 2sinθ + 2sinθ+2dsinθ – sin^{2}θ–2dsinθ+d^{2}

=d^{2} + 4d + 4 – sin^{2}θ

=(d + 2)^{2} – sin^{2}θ

For a given d, minimum value of det(A) = (d + 2)^{2} – 1 = 8

⇒ d = 1 or –5

QUESTION: 78

If the third term in the binomial expansion of equals 2560, then a possible value of x is :

Solution:

QUESTION: 79

If the line 3x + 4y – 24 = 0 intersects the x-axis at the point A and the y-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is

Solution:

7r – 24 = ±5r

2r = 24 or 12r + 24

r = 14, r = 2

then incentre is (2, 2)

QUESTION: 80

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is :

Solution:

Let two observations are x_{1} & x_{2}

mean =

⇒ 1 + 3 + 8 + x_{1} + x_{2} = 25

⇒ x_{1} + x_{2} = 13 ....(1)

variance

by (1) & (2)

(x_{1} + x_{2})2 – 2x_{1}x_{2} = 97

or x_{1}x_{2} = 36

∴ x_{1} : x_{2} = 4 : 9

QUESTION: 81

A point P moves on the line 2x – 3y + 4 = 0. If Q (1,4) and R (3,–2) are fixed points, then the locus of the centroid of ΔPQR is a line :

Solution:

Let the centroid of ΔPQR is (h, k) & P is (α, β), then

and

α = (3h – 4) β = (3k – 4)

Point P(α, β) lies on line 2x – 3y + 4 = 0

∴ 2(3h – 4) – 3(3k – 2) + 4 = 0

⇒ locus is 6x – 9y + 2 = 0

QUESTION: 82

If and then equals :

Solution:

Given

QUESTION: 83

The plane passing through the point (4, –1, 2) and parallel to the lines and also passes through the point :

Solution:

Let be the normal vector to the plane passing through (4, –1, 2) and parallel to the lines L_{1} & L_{2
}

∴ Equation of plane is

–1(x – 4) – 1(y + 1) + 1(z – 2) = 0

∴ x + y – z – 1 = 0

Now check options

QUESTION: 84

Let If I is minimum then the ordered pair (a, b) is :

Solution:

Let f(x) = x^{2}(x^{2} – 2)

As long as f(x) lie below the x-axis, definite integral will remain negative,

so correct value of (a, b) is (-√2,√2) for minimum of I

QUESTION: 85

If 5, 5r, 5r^{2} are the l engths of the sides of a triangle, then r cannot be equal to :

Solution:

r = 1 is obviously true.

Let 0 < r < 1

⇒ r + r^{2} > 1

⇒ r^{2} + r - 1 > 0

When r > 1

QUESTION: 86

Consider the statement : "P(n): n^{2} – n + 41 is prime." Then which one of the following is true?

Solution:

P(n) : n^{2} – n + 41 is prime

P(5) = 61 which is prime

P(3) = 47 which is also prime

QUESTION: 87

Let A be a point on the line and B(3, 2, 6) be a point in the space. Then the value of µ for which the vector is parallel to the plane x -4y +3z=1 is :

Solution:

Let point A is

and point B is (3, 2, 6)

then

which is parallel to the plane x – 4y + 3z = 1

∴ 2 + 3µ – 12 + 4µ + 12 – 15µ = 0

8µ = 2

µ = 1/4

QUESTION: 88

For each t∈R, let [t] be the greatest integer less than or equal to t. Then,

Solution:

QUESTION: 89

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1,2,3,...,9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is:

Solution:

QUESTION: 90

Let n≥2 be a natural number and 0<θ<π/2.

Then is equal to :

(Where C is a constant of integration)

Solution:

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