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The equation to the directrix of a parabola if the two extremities of its latus rectum are (2, 4) and (6, 4) and the parabola passes through the point (8, 1) is
focus is (4, 4) & D can be y = 6 or y = 2
where ‘O’ is origin and S is the focus and D is directrix
Apply R_{3} → R_{3} → R_{1}, we get
= (3cos θ – sin θ)^{2} So, maximum value of Δ equals 10.
The number of solution(s) of the equation
(where z = x + iy, x, y ∈ R, i^{2} = –1 and x ≠ 2)
⇒ x = – √2
∴ z = – √2
Hence only one z will satisfy above equation.
Two circles of radii r_{1} and r_{2} are both touching the coordinate axes and intersecting each other orthogonally. The value of r_{1}/r_{2} (where r_{1} > r_{2}) equals
Circle is (x – r)^{2} + (y – r)^{2} = r^{2}
⇒ x^{2} + y^{2} – 2xr – 2yr + r^{2} = 0
Hence the circles are x^{2} + y^{2} – 2xr_{1} – 2yr_{1} + r_{1}^{2} = 0 ......(1)
x^{2} + y^{2} – 2xr_{2} – 2yr_{2} + r_{2}^{2} = 0 .....(2)
As (1) and (2) are orthogonal so
2r_{1}r^{2} + 2r_{1}r^{2} = r_{1}^{2} + r_{2}^{2}
Let X and Y be two matrices satisfying this relations
then Tr.(X) – Tr.(Y) equals
[Note : Tr.(P) denotes trace of matrix P.]
The differential equation dx/dy = 3y/2x represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity can be
2x dx – 3y dy = 0 gives, on integration,
The solution represents a family of hyperbolas given by
whose eccentricity
and eccentricity
it gives a pair of lines which are the asymptotes of the hyperbolas.
Line L, perpendicular to the line with equation y = 3x – 5, contains the point (1, 4). The xintercept of L, is
put y = 0, x = 13
Let A = [a_{ij}] (1 ≤ i , j ≤ 3) be a 3 × 3 matrix and B = [b_{ij}] (1 ≤ i , j ≤ 3) be a 3 × 3 matrix such that
If det. A = 4, then the value of det. B is
B = AA^{T}.
Hence, det.
B = AA^{T} = A A^{T} = A^{2} = 4^{2} = 16.
Number of words that can be formed using all the letters of the word GARGEE if no two alike letters are together, is
Total – n(A ∪ B)
Set A represents number of ways when G's are together
Set B represents number of ways when E's are together
Aliter: GG EE A R
Number of words when
Number of words when G's are separated but E's are together = 3! × ^{4}C_{2} = 36
∴ Number of ways when no two alike letters are together = 120 – 36 = 84
If acute angle between the line and xy plane is α and acute angle between the planes x + 2y = 0 and 2x + y = 0 is β then (cos^{2}α + sin^{2}β) equals
If area of pentagon PQRST be 7, where P(–1, –1), Q(2, 0), R(3, 1), S(2, 2) and T(–1, t), t > 0, then the value of t is
Area of pentagon PQRST = 7
⇒ ar.(trapezium PQST) + ar.(ΔQRS) = 7
⇒ t = 1
The sum of all value of λ for which the lines 2x + y + 1 = 0; 3x + 2λy + 4 = 0; x + y  3λ = 0 are concurrent, is
(3λ + 1)(2λ + 1) + 3λ(2λ  4) = 0
⇒ 6λ^{2} + 5λ + 1 + 6λ^{2}  12λ = 0
⇒ 12λ^{2}  7λ + 1 = 0
⇒ (3λ – 1)(4λ – 1) = 0
⇒ Sum = 7/12
If A and B are two independent events such that P(A' ∩ B') = 2/15, P(A ∩ B') = 1/6 then P(B) =
A hyperbola has centre at origin and one focus at (6, 8). If its two directrices are 3x + 4y + 10 = 0 and 3x + 4y –10 = 0 and eccentricity is e,then the value of 4e^{2}/5 is equal to
Distance between centre and focus = ae = 10
Distance between directrices = 2a/e = 4
Number of integral values of 'k' for which the chord of the circle x^{2} + y^{2} = 125 passing through P(8, k) gets bisected at P (8, k) and has integral slope is
The slope of the chord is
⇒ k = ± 1, ± 2, ± 4, ± 8 but (8, k) must also lie inside the circle x^{2} + y^{2} = 125
⇒ 64 + k^{2} – 125 < 0
⇒ k^{2} < 61
⇒ k can be equal to ± 1, ± 2, ± 4
⇒ 6 values
Locus of the feet of perpendiculars drawn from points (1, 2) and (3, 4) on a variable tangent to the conic  z  (1+ 2i)    z  (3+ 4i)  = 2 is
represents a hyperbola with foci (1, 2) and (3, 4) and length of transverse axis = 2.
∴ 2a = 2 ⇒ a = 1
∵ Feet of perpendiculars from foci on any tangent lie on auxilliary circle of the hyperbola.
∴ Locus will be auxilliary circle.
∴ Centre = mid point of foci = (2, 3)
and radius = semi transverse axis = 1
∴ Equation of auxilliary circle is z  (2 + 3i) = 1
Number of numbers greater than a million and divisible by 5 which can be formed by using only the digits 1, 2,1, 2, 0, 5 and 2 is :
In dual statement v replace by ∧ and ∧ replace by v so answer is (p ∧ ~ q) v (~ p).
A normal is drawn to the parabola y^{2} = 9x at the point P(4, 6), S being the focus, a circle is described on the focal distance of the point P as diameter. The length of the intercept made by the circle on the normal at P is
Required intercept will be equal to the perpendicular distance from the focus on the tangent at P.
Tangent at P,
⇒ 12y = 9x + 36
⇒ 9x  12y + 36 = 0
Consider ellipse Let C is centre of the ellipse and P is a variable point lying on the ellipse. If the angle between CP and tangent at P is minimum, then P may be
and C (0, 0)
∴ angle is minimum, when θ = 45°
(Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)
If then the value of a^{5} + b^{4} must be
The value of x satisfying the equation
then k is
The sum of squares of all integral values of a for which the quadratic expression (x−a)(x−10)+1 can be factored as a product (x+α)(x+β) of two factors and α, β ∈ I must be equal to
If f(x) = tan^{1} (sin x + cos x)^{3} is an increasing function, then the value of x in (0, 2π) is x ∈ Then the value of a + 10b + 100c + 1000d must be
The sum of two digit even numbers which do not end with zero is
Required sum
= (12 + 14 + 16 + …. + 98) – (20 + 30 + 40 + …. + 90)
44/2 (12 + 98) – 8/2 (20 + 90)
= 22(110) – 4(110)
= (110) (18)
= 1980
A nonconducting rod AB of length l has a total charge q. The rod is rotated about an axis passing through its center of mass with a constant angular velocity ω as shown in the figure. The magnetic moment of the rod is
The distance between two parallel plates of a capacitor is a. A conductor of thickness b(b < a) is inserted between the plates as shown in the figure. The variation of effective capacitance between the plates of the capacitor as a function of the distance (x) is best represented by
A solid sphere of radius R, and dielectric constant ‘k’ has spherical cavity of radius R/4. A point charge q_{1} is placed in the cavity. Another charge q_{2} is placed outside the sphere at a distance of r from q_{1}. Then Coulombic force of interaction between them is found to be ‘F_{1}’. When the same charges are separated by same distance in vacuum then the force of interaction between them is found to be F_{2} then
Coulombic force between them remains same.
Charge flown from battery = CV
Work done = CV^{2}
Heat produced ΔH = ΔU + ΔW
Energy stored in the capacitor in it’s steady state is
Potential across capacitor is zero, hence energy stored is zero.
A point charge of 0.1C is placed on the circumference of a nonconducting ring of radius 1m which is rotating about an axis passing from centre and perpendicular to the plane of ring with a constant angular acceleration of 1 rad/sec^{2}. If ring starts from rest at t = 0, the magnetic field at the centre of the ring at t = 10 sec, is
ω = 0 + 1 × 10 = 10 rad/sec^{2}
∴ v = rω = 1 × 10 = 10 m/s
In an L – C circuit shown in the figure, C = 1F, L = 4H. At time t = 0, charge in the capacitor is 4C and it is decreasing at a rate of √5 C/s. Choose the correct statements.
A solid conducting sphere of radius r is having a charge Q and point charges +q and –q are kept at distances d from the center of sphere as shown in the figure. The electric potential at the centre of solid sphere
Consider the circuit in the adjacent figure. What will be potential difference between A and B in the steady state
There will be no current any where in the circuit.
A charge q is placed at some distance along the axis of a uniformly charged disc of surface charge density σ C/m^{2}. The flux due to the charge q through the disc is ϕ. The electric force on charge q exerted by the disc is
In the given circuit diagram, find the heat generated on closing the switch S. (Initially the capacitor of capacitance C is uncharged)
Only charge is that capacitor 'C' will get charged.
A metallic ring of radius R moves in a vertical plane in the presence of a uniform magnetic field B perpendicular to the plane of the ring. At any given instant of time its centre of mass moves with a velocity v while ring rotates in its COM frame with angular velocity ω as shown in the figure. The magnitude of induced e.m.f. between points O and P is
An object is placed at 30 cm from a convex lens of focal length 15cm. On the other side of the lens a convex mirror of focal length 12cm is placed so that the principal axis of both coincide. It is observed that the object and image coincide (autocollimation). What is the separation between the lens and mirror?
For image to be coincident, either the rays should retrace or the image due to the lens should formed just at the pole of the mirror in thin case. The image formed due to lens is at 30 cm (2f) be from the lens. Thus either this image should be at centre of curvature of the convex mirroror at the pole of the mirror. Hence 6cm or 30cm should be the separation between the lens and the mirror.
The relation between R and r (internal resistance of the battery) for which the power consumed in the external part of the circuit is maximum
Its a wheat stone bridge with equivalent 2R.
A capacitor and resistor are connected with an A.C. source as shown in figure. Reactance of capacitor is X_{C} = 3W and resistance of resistor is 4Ω. Phase difference between current I and I_{1} is
Let I_{2} be current in capacitor
Find the stress at distance R/2 from centre in a uniformly charged non conducting sphere having radius R and charge density ρ.
The capacitor is initially uncharged. Find ratio of current through the 10 Ω resistance and through the 20 Ω resistance initially.
In the diagram below, light is incident on the interface between media 1 and 2 as shown and is totally reflected. The light is then also totally reflected at the interface between media 1 and 3, after which it travels in a direction opposite to its initial direction. The two interfaces are perpendicular. The refractive indices are related as
∴ from (1) and (2), n_{1}^{2} – n_{2}^{2} > n_{3}^{2}
Which of the following transitions of He^{+} ion will give rise to spectral line which has same wavelength as some spectral line in hydrogen atom ?
If n_{2} → n_{1} in H (z = 1) gives λ then z n_{2} → z n_{1} gives λ in Hlike ion for He^{+} ion, z = 2
Two imaginary spherical surfaces of radius R and 2R respectively surround a positive point charge Q located at the center of the concentric spheres. When compared to the number of field lines N_{1} going through the sphere of radius R, the number of electric field lines N_{2} going through the sphere of radius 2R is
A radioactive sample contains two radioactive nucleus A and B having decay constant λ hr^{–1} and 2λ hr^{–1}. Initially 20% of decay comes from A. How long (in hr) will it take before 50% of decay comes from A. [Take λ = ln 2]
∴ from (1) and (2) , e^{λt} = 9
⇒ λt = 2ln3 ⇒ t = 2.
(Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)
A block is placed on an inclined plane moving towards right horizontally with an acceleration a_{0}=g. The length of the plane AC = 1m. Friction is absent everywhere. Find the time taken (in seconds) by the block to reach from C to A.
In the given figure, find the horizontal velocity u (in ms^{1}) of a projectile so that it hits the inclined plane perpendicularly. Given H = 6.25 m:
A uniform cylinder rests on a cart as shown. The coefficient of static friction between the cylinder and the cart is 0.5. If the cylinder is 4 cm in diameter and 10 cm in height, then what is the minimum acceleration (in m/s^{2}) of the cart needed to cause the cylinder to tip over?
In the circuit shown in figure, find the ratio of currents i_{1}/i_{2}
The centers of two identical small conducting spheres are 1m apart. They carry charges of opposite kind and attract each other with a force F. When they are connected by a conducting thin wire, they repel each other with a force F/3. The ratio of the magnitude of charges carried by the spheres initially is n : 1. Find the value of n.
A 0.1 M solution of which salt is most acidic?
Which is the strongest acid amongst the following compounds?
Select the correct order of metallic character :
As we move left to right metallic character decreases and as we move top to bottom metallic character increases, so correct is
In hydrogen atom which transition produces a photon with highest energy?
The correct basicity order of indicated atoms P, Q and R is 
Basicity order of indicated atoms P, Q, R is P > Q > R
Select the incorrect statement regarding N^{3–}, O^{2–}, F^{}, Na^{+} and Mg^{2+}.
Dioxygen difluoride (O_{2}F_{2}) is a highly oxidising and unstable liquid. At 300 K it decomposes back to oxygen and fluorine, which are both gases at this temperature. The equation for the reaction is given below. 0.1 g of O_{2}F_{2} was left for 24 hours and the 24.9 ml of gas mixture evolved was collected at 300 K and 100 kPa. What % by mass of dioxygen difluride has decomposed by this time?
O_{2}F_{2}(l) → O_{2}(g) + F_{2}(g)
Which of the following cyclic dienes does not show geometrical isomerism?
Cyclohexene does not show
GeometricalIsomerism
Which of the following property is different for two different isoelectronic homonuclear diatomic species?
In an analysis of solutions containing Barium ions (Ba^{2+}), 50 ml of solution gave 0.233 g of BaSO_{4} upon addition of sufficient sulphuric acid to precipitate all the Ba^{2+} ions present. What is concentration (in M) of Ba^{2+} ions in the solution.
Ethers are more volatile than same no. of carbon containing alcohol due to 
Ethers are more volatile than same number of carbon containing alcohol due to absence of Hbonding.
Select the species which becomes Bent due to lone pair – bond pair repulsion.
(Bent, due to lone pairbond pair repulsion)
Quinaldine red is a useful acidbase indicator which is red in solution of pH greater than 3.5 but colorless below pH = 1.5. Which of the following solution would turn red if a few drops of quinaldine red were added?
(i) 0.1 mol L^{–1} HCl
(ii) 0.05 mol L^{–1} NH_{3}
(iii) 0.0001 mol L^{–1} CH_{3}COOH
Oxidising nature of H_{2} is observed on reaction with
(3) Cu_{2}O + H_{2} → 2Cu + H_{2}O
→ act as reducing agent
(4) RCHO + H_{2} → R – CH_{2} – OH
→ act as reducing agent
A brownblack compound of thallium was found to contain 89.5% Tl and 10.5% oxygen. What is oxidation number of thallium in this compound? [Atomic weight of Tl = 204]
Empirical formula of the compound = Tl
Identify R in the following series of reaction.
Select the incorrect match regarding action of H_{2}O.
In the mono chlorination of 3Ethyl pentane, how many total products will be formed?
Total products = 4
(Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)
From the following sets of quantum numbers, state which are not possible.
a. n = 0, l = 0, m = 0, s = +1/2
b. n = 1, l = 0, m = 0, s = –1/2
c. n = 1, l = 1, m = 0, s = +1/2
d. n = 1, l = 0, m = +1, s = +1/2
e. n = 3, l = 0, m = –1, s = –1/2
f. n = 2, l = 2, m = 0, s = –1/2
g. n = 2, l = 1, m = 0, s = –1/2
a. is not possible because n = 0 is wrong.
b. is possible.
c. is not possible because l = n is wrong.
d. is not possible because m = + 1 is wrong.
e. is not possible
f. is not possible because l = 2 is wrong.
g. is possible
200g impure CaCO_{3} on heating gives 5.6L. CO_{2} gas at STP. Find the percentage of Calcium in the limestone sample.
CaCO_{3} → CaO + CO_{2}
5.6/ 22.4 = 0.25 mole
Mole of CaO = mole of Ca = 0.25
Mass of Ca= 0.25 x 40 = 10
% of Calcium = 10 x 100/ 200 = 5%
NO → N_{2}O + NO_{2}
n this reaction equivalent weight of NO is x/y M then x + y will be
NO → N_{2}O (2 to + 1 for one atom ) = 3
NO → NO_{2 }(+2 to + 4) = 2
10 g impure NaOH is completely neutralised by 1000 ml of 1/10 N HCI. Calculate the percentage purity of the impure NaOH.
Then weight of MgSO_{4} will be formed in this series is
MgCO_{3} Molar mass: 84.3139
Moles = 0.05 moles produced 0.05 moles MgO and
MgSO_{4} molar mass: 120 x 0.05 = 6
2 videos327 docs203 tests

2 videos327 docs203 tests
