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Given that and A^{2} + B^{2} = R^{2}. The angle between is
In the relation :
P is pressure, Z is distance, k is Boltzmann constant and θ is the temperature. The dimensional formula of b will be
A projectile projected at an angle 30º from the horizontal has a range R. If the angle of projection at the same initial velocity be 60º, then the range will be
If sum of angle of projection = 90° for given speed then range for that angle of projection is same.
A block of mass M is pulled along a horizontal frictionless surface by a rope of mass M/2. If a force 2Mg is applied at one end of the rope, the force which the rope exerts on the block is –
A chain of mass M is placed on a smooth table with 1/n of its length L hanging over the edge.The work done in pulling the hanging portion of the chain back to the surface of the table is
W = change in PE of COM of hanging part
A particle of mass 10 kg moving eastwards with a speed 5 ms^{–1} collides with another particle of the same mass moving northwards with the same speed 5 ms^{–1}. The two particles coalesce on collision. The new particle of mass 20 kg will move in the northeast direction with velocity
Here
A uniform cube of side a and mass m rests on a rough horizontal table. A horizontal force F is applied normal to one of the faces at a point that is directly above the centre of the face, at a height 3a/4 above the base. The minimum value of F for which the cube begins to topple an edge is (assume that cube does not slide)
For toppling about edge xx'
At the moment of toppling the normal force pass through axis xx'.
The rotation of the earth having radius R about its axis speeds upto a value such that a man at latitude angle 600 feels weightless. The duration of the day in such case will be :
A metallic rod breaks when strain produced is 0.2%. The Young’s modulus of the material of the rod is 7 × 109 N/m2. What should be its area of crosssection to support a load of 104 N ?
Maximum possible strain = 0.2/100
A liquid is flowing through a nonsectional tube with its axis horizontally. If two points X and Y on the axis of tube has a sectional area 2.0 cm^{3} and 25 mm^{2} respectively then find the flow velocity at Y when the flow velocity at X is 10m/s.
According to principle of continuity
A body of length 1m having crosssectional area 0.75m^{2} has heat flow through it at the rate of 6000 Joule/sec. Then find the temperature difference if K = 200 Jm^{–1}K^{–1}.
Which of the following combinations of properties would be most desirable for a cooking pot?
A particle starts moving rectilinearly at time t = 0 such that its velocity v changes with time t according to the equation v = t^{2} – t where t is in seconds and v is in m/s. Find the time interval for which the particle retards.
Acceleration of the particle a = 2t – 1
The particle retards when acceleration is opposite to velocity.
Now t is always positive
∴ (2t – 1) (t – 1) < 0
This is not possible
or
2t – 1 > 0 & t – 1 < 0 ⇒ 1/2 < t < 1
A sample of gas expands from volume V_{1} to V_{2}.
The amount of work done by the gas is greatest when the expansion is
A cyclic process is shown in the pT diagram. Which of the curves show the same process on a PV diagram ?
Which one the following graphs represents the behaviour of an ideal gas
For an ideal gas PV = constant i.e., PV does not vary with V.
In case of a forced vibration, the resonance wave becomes very sharp when the
A pendulum bob carries a +ve charge +q. A positive charge +q is held at the point of support.
Then the time period of the bob is – [where, L = length of pendulum, g_{eff} = effective value of g]
Two tuning forks A and B sounded together give 6 beats per second. With an air resonance tube closed at one end, the two forks give resonance when the two air columns are 24 cm and 25 cm respectively. Calculate the frequencies of forks.
Let the frequency of the first fork be f_{1} and that of second be f_{2}.
We then have,
We also see that f_{1} > f_{2}
Solving (i) and (ii), we get
f_{1} = 150 Hz and f_{2} = 144 Hz
If an electron has an initial velocity in a direction different from that of an electric field, the path of the electron is
The path is a parabola, because initial velocity can be resolved into two rectangular components, one along and other ⊥ to . The former decreases at a constant rate and latter is unaffected. The resultant path is therefore a parabola.
If on combining two charged bodies, the current does not flow then
Calculate the area of the plates of a one farad parallel plate capacitor if separation between plates is 1 mm and plates are in vacuum
For a parallel plate capacitor
This corresponds to area of square of side 10.6 km which shows that one farad is very large unit of capacitance.
The length of a potentiometer wire is ℓ. A cell of emf E is balanced at a length ℓ/3 from the positive end of the wire. If the length of the wire is increased by ℓ/2. At what distance will be the same cell give a balance point.
Potential gradient in the first case
Potential gradient in second case
From equations (i) and (ii),
A conducting circular loop of radius r carries a constant current i. It is placed in a uniform magnetic field such that is perpendicular to the plane of the loop. The magnetic force acting on the loop is
The magnetic field is perpendicular to the plane of the paper. Let us consider two diametrically opposite elements. By Fleming's Left hand rule on element AB the direction of force will be Leftwards and the magnitude will be
dF = Idl B sin 90° = IdlB
On element CD, the direction of force will be towards right on the plane of the papper and the magnitude will be dF = IdlB.
An ammeter reads upto 1 ampere. Its internal resistance is 0.81ohm. To increase the range to 10 A the value of the required shunt is
At the magnetic north pole of the earth, the value of horizontal component of earth’s magnetic field and angle of dip are, respectively
At the magnetic north pole, the magnetic needle will point vertically. There is no component of earth’s magnetic field in the horizontal direction and the angle of dip (the angle that the resultant magnetic field at the place makes with the horizontal) is 90°.
H = 0, δ = 90° (maximum)
Lenz’s law is a consequence of the law of conservation of
The instantaneous current from an a.c. source is I = 6 sin 314 t. What is the rms value of the current?
A coil has resistance 30 ohm and inductive reactance 20 ohm at 50 Hz frequency. If an ac source, of 200 volt, 100 Hz, is connected across the coil, the current in the coil will be
If ω = 50 × 2π then wL = 20Ω If ω' = 100 × 2π then ω'L = 40Ω Current flowing in the coil is
The magnetic field in a travelling electromagnetic wave has a peak value of 20 nT. The peak value of electric field strength is
A planoconvex lens of focal length 30 cm has its plane surface silvered. An object is placed 40 cm from the lens on the convex side. The distance of the image from the lens is
When a mica sheet of thickness 7 microns and μ = 1.6 is placed in the path of one of interfering beams in the biprism experiment then the central fringe gets at the position of seventh bright fringe.The wavelength of light used will be
According to question
n = 7 . m = 1.6, t = 7 x 10^{–6} meter .....(2)
From eqs. (1) and (2), l = 6 x 10^{–7} meter
In Young's double slit experiment, if the slit widths are in the ratio 1 : 2, the ratio of the intensities at minima and maxima will be
In a photoelectric experiment, with light of wavelength l, the fastest electron has speed v. If the exciting wavelength is changed to 3l/4, the speed of the fastest emitted electron will become
Taking Rydberg’s constant R_{H} = 1.097 x 10^{7}m, first and second wavelength of Balmer series in hydrogen spectrum is
For first wavelength, n_{1} = 2, n_{2} = 3
⇒ λ_{1} = 6563 Å. For second wavelength, n_{1} = 2, n_{2} = 4
⇒ λ_{2} = 4861 Å
An Xray tube is operated at 15 kV. Calculate the upper limit of the speed of the electrons striking the target.
The maximum kinetic energy of an electron accelerated through a potential difference of V volt is 1/2mv^{2 }= eV
Nuclear energy is released in fission since binding energy per nucleon is
Nuclear energy is relased in fission because BE/nucleon is larger for fission fragments than for parent nucleus.
Assuming the diodes to be of silicon with forward resistance zero, the current I in the following circuit is
The truth table given below corr espon d to the logic gate
The given gate is a NOR gate. The output is high, when all inputs are low.
Boolean expression:
Given the numbers : 161 cm, 0.161 cm, 0.0161 cm.The number of significant figures for the three numbers are
Each has three significant figures. When zero is used to locate the decimal point, it is not considered as significant figure.
Beryllium resembles with aluminium due to similarity in the size of ions and similarity in electropositive character. This type of resemblance between first element of a group in second period with second element of the next group is termed as diagonal relationship.
Which one of the following ions has the highest value of ionic radius?
The ionic radii follows the order
O^{2–} > F^{–} > Li^{+} > B^{3+}
Which of the following two are isostructural ?
XeF_{2} an d IF^{}_{2} both are lin ear and h ave hybridisation sp^{3}d.
The cooking time in a pressure cooker is less because :
In pressure cooker, pressure is high thus, the boiling point of water increases, resulting cooking time is less than other open pots.
For the reaction : N_{2} + 3H_{2} ⇌ 2NH_{3}
Which one of the following is correct regarding DH :
According to thermodynamics’s Ist law ΔH = ΔE + nRT
Where ΔH = enthalpy of reaction at constant pressure
ΔE = heat of reaction at constant volume
R = molar gas constant
T = temperature of the reaction
n = (no. of moles of product) – (no. of moles of reactant.)
From reaction, n = n_{P} – n_{R} = 2 – 4 = – 2 Hence, ΔH = ΔE  2RT .
One mole of an ideal gas at 300 K is expanded isothermally from an initial volume of 1 litre to 10 litres. The ΔE for this process is (R = 2 cal mol^{–1} K^{–1})
For isothermal process, ΔE = 0
At 25°C and 1 bar which of the following has a nonzero ΔH°f ?
Ozone is allotropic form of oxygen and is of higher energy (by 68 K Cal mol^{–1}) than O_{2}.
Hence it can not be taken as the reference in standard state.
If the equilibrium constant of the reaction is 0.25, then the equilibrium constant for the reaction would be
When the reaction is reversed,
The oxidation states of sulphur in the anions follow the order
According to chemical bond method:
Because of smaller size, Mg^{2+} ions are extensively hydrated.
For making good quality mirrors, plates of float glass are used. These are obtained by floating molten glass over a liquid metal which does not solidify before glass. The metal used can be
It is mercury, because mercury exists as liquid at room temperature.
The intermediate formed during the addition of HCl to propene in the presence of peroxide is
The addition of HCl to propene proceeds by ionic mechanism and not by free radical mechanism. Hence it forms intermediate carbonium ion.
Which of the following has zero dipole moment?
Which one of the following contain isopropyl group?
It contains isopropyl group.
The statement which is not correct about control of particulate pollution is:
Par ticulates acquire n egative charge and are attracted by the positive electrode.
The average residence time of NO is 4 days.
The atomic radius of atom is r.Total volume of atoms present in a fcc unit cell of an element is
4 atom are present in fcc.
Which one of the following statements is false?
ΔT_{f} = K_{f} x m x i. Since Kf has different values for different solvents, hence even if the m is the same ΔT_{f} will be different.
The degree of dissociation of Ca(NO_{3})^{2} in dilute aq. solution containing 7.0 g of salt per 100 g of water at 100° C is 70%. If vapour pressure of water at 100° C is 760 mm Hg. The vapour pressure of solution is
When the sample of copper with zinc impurity is to be purified by electrolysis, the appropriate electrodes are
In electrolytic purification cathode is of pure metal and anode is of impure metal.
The pure metal is thus deposited at cathode.
Th e con ductivity of a saturated solution of BaSO_{4} is 3.06 x 10^{–6} ohm^{–1} cm^{–1} and its equivalent conductance is 1.53 ohm^{–1} cm^{2} equiv^{–1}. The Ksp for BaSO_{4} will be
In a cell that utilises the reaction 2
Zn (s) + 2H^{+} (aq) → Zn^{+} (aq) + H_{2} (g)
addition of H_{2}SO_{4} to cathode compartment, will
Addition of H_{2}SO_{4} will increase [H^{+}] and E_{cell} will also increase and the equilibrium will shift towards RHS.
The chemical reaction 2O_{3} → 3O_{2} proceeds as follows:
the rate law expression should be
Note: Intermediates are never represented in rate law equation.
Among th e following statements the incorr ect one is :
Cuprite is Cu_{2}O and Argentite is Ag_{2}S.
Cinnabar (HgS) is an ore of Hg.
Which of the following is used in the preparation of chlorine?
Both MnO_{2} an d KMnO_{4} used for the preparation of chlorine by the action of conc. HCl.
Which of the following elements does not belong to first transition series?
The more is the stability of intermediate carbonium ion, the more is the chance of S_{N}1 mechanism. The intermediates obtained will be
The stabilty is of the order iv > iii > ii > i.
When esters are hydrolysed the product gives hydrogen ions. The product which gives hydrogen ion is
When esters are hydrolysed , then acid and alcohol are formed, where acid gives hydrogen ion.
Which of the following compound can not be used in preparation of iodoform?
Formaldehyde can not produce iodoform, as only those compound which contains either group or
group on reaction with potassium iodide and sod. hypochlorite yield iodoform.
Which of the following compound is obtained by heating ammonium cyanate?
Urea is obtained by heating ammonium cyanate
Which of the following statements about vitamin B12 is incorrect?
Vitamin B_{12} does not occur in plants.
Ammonia forms the complex ion [Cu(NH_{3})_{4}]^{2+} with copper ions in alkaline solutions but not in acidic solutions. What is the reason for it ?
An aqueous solution of a substan ce gives a white precipitate on treatment with dil. HCl which dissolves on heating. When H2S is passed through the hot acidic solution, a black precipitate is obtained. The substance is a
PbCl_{2} is insoluble in cold water, soluble in hot water and PbS is black ppt in acidic medium.
More the electron density on N, higher will be the basicity. Density on N is influenced by the (i) nature of the group (+I or –I) present in alkyl group or benzene nucleus and (ii) resonance (delocalisation of the electron present on N). In (C_{6}H_{5})_{3 }N: electron pair is delocalised to the maximum extent due to three benzene rings and hence least available for protonation, thus it will be least basic.
In the following question, choose the alternatives which can be substituted for the given word.
Agnostic
In the following question, choose the alternatives which can be substituted for the given word.
Bohemian
In the following question, choose the alternatives which can be substituted for the given word.
Cacographist
DIRECTION: Which one of the following word is correctly spelt?
Spelling testfind correct spelling :
DIRECTION: Which one of the following word is correctly spelt?
Spelling testfind correct spelling:
DIRECTION: Which one of the following word is correctly spelt?
Spelling testfind correct spelling :
DIRECTIONS : In the following questions, choose the alternative which is opposite in meaning to the word given in capital letters.
REPRIMAND
DIRECTIONS : In the following questions, choose the alternative which is opposite in meaning to the word given in capital letters.
IMPERTINENT
DIRECTIONS : In the following questions, choose the alternative which is opposite in meaning to the word given in capital letters.
EQUIVOCAL
DIRECTION : In the following question, choose the most appropriate altarnative to fill in the blank.
It is difficult to believe what he tells us because his account of any event is always full of ............ of all sorts.
DIRECTION : In the following question, choose the most appropriate altarnative to fill in the blank.
The bank clerk tried to ............... money from his friend’s account.
DIRECTION : In the following question, choose the most appropriate altarnative to fill in the blank.
TEight scientists have ............. the national awards for outstanding contribution and dedication to the profession.
DIRECTION : In the following question, some parts have been jumbled up. You are required to rearrange these parts, which are labelled P, Q, R and S to produce the correct sentence.
Freedom, is the restricted kind in the sense/(P), the rich and poor woman/(Q), that a wide gulf separates/(R), which a modern woman enjoys (S)
DIRECTION : In the following question, some parts have been jumbled up. You are required to rearrange these parts, which are labelled P, Q, R and S to produce the correct sentence.
In life, some rules are/(P), as in business/(Q), they seem almost instinctive/(R), learnt so early that/ (S)
DIRECTION : In the following question, some parts have been jumbled up. You are required to rearrange these parts, which are labelled P, Q, R and S to produce the correct sentence.
Kapil, left in an aeroplane/(P), after reading a sailing magazine/(Q), had decided/(R), to build his own boat nine years earlier/(S)
DIRECTION : In the following question, select the related letter/word/number from the given alternatives.
Distance : Odometer :: ? : Barometer
Distance is measured by Odometer.
Similarly, Pressure is measured by Barometer.
One of the, numbers does not fit into the series.
Find the wrong number 13, 16, 38, 124, 504, 2535
The number should be 123.
× 1 + 3, × 2 + 6, × 3 + 9……
DIRECTION: In the question below is given a statement followed by three assumptions numbered I, II and III. You have to consider the statement and the following assumptions, decide which of the assumptions is implicit in the statement and choose your answer accordingly.
Statement: In order to reduce the gap between income and expenditure, the company has decided to increase the price of its product from next month.
Assumptions: I. The rate will remain more or less same after the increase.
II. The expenditure will more or less remain the same in near future.
III. The rival companies will also increase the price of the similar product.
Clearly, the company intends to reduce the gap between income and expenditure by increasing the price of its product i.e. by keeping the expenditure unaltered and increasing the income only. So, II is implicit while I is not. However, the rival companies may or may not follow the same pursuit.
So, III is not implicit.
DIRECTION: In the following question, select the related letter/word/number from the given alternatives.
FLMO : ? : : BFEN : ARSO
If
A denotes '+'
B denotes ''
C denotes '×'
Then what is the value of (10 C 4) A (4 C 4) B 6?
Using correct symbols we have:
(10 × 4) + (4 × 4) – 6 = 40 + 16 – 6 = 50
DIRECTION : In this questions, two figure/ words are given to the left of the sign and one the sign:: with four alternatives under it out. of which one of the alternatives has the same relationship with the figures/words to the right of the sign:: as between the two figures/words to the left of the sign (::). Find the correct alternative.
Identify the missing part of the figure and select it from the given alternatives.
Figure (x) is embedded in anyone of the four alternative figures.
Choose the alternative which contains figure (x).
Which symbol will appear on the opposite surface to the symbol x?
The three figures marked X, Y, Z show the manner in which a paper is folded step by step and then cut. From the answer figures (a), (b), (c), (d), select the one, showing the unfolded position of the Paper after the cut.
If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is
Number of proper subsets of A = 2^{n} –1 Given : A = {1, 2, 3, 4, 5}
Here n = 5
∴ no. of proper subsets = 2^{5} – 1
y – 1 ≠ 0
⇒ y ≠ 1
If y = 1 then original equation gives x = 0, so taking y = 1
Also 3y^{2} – 10y + 3 ≤ 0
⇒ (3y – 1) (y – 3) ≤ 0
Hence period = 2π/3
We have,
⇒ 8 cos x  8 cos^{2} x = (1 + cos x)^{2}
⇒ 9 cos^{2} x  6 cos x + 1 = 0
⇒ (3 cos x  1) ^{2} = 0
⇒ 3 cos x 1 = 0
10^{n} + 3(4^{n+2}) + 5 is divisible by ( n ∈ N)
10^{n} + 3(4^{n+ 2} )+5 Taking n = 2;
10^{2} + 3 x 4^{4}+5 = 100 + 768 + 5 = 873
Therefore this is divisible by 9.
If the expression x^{2} – 11x + a and x^{2} – 14x + 2a must have a common factor and a ≠ 0, then, the common factor is
Here Let x – α is the common factor then x = α is root of the corresponding equation
∴ α^{2} – 11α + α = 0
α^{2} – 14α + 2α = 0
Subtracting 3α – α = 0
⇒ α = α/3
For the equation if the product of roots is zero, then the sum of roots is
or x^{2} + (a + b) x + ab = (b – a) x + (b  a) c
or x^{2} + 2ax + ab + ca – bc = 0
Since product of the roots = 0
The letters of the word TOUGH are written in all possible orders and these words are written out as in a dictionary, then the rank of the word TOUGH is
Rank = (4!x 3) + (3!x 2) + (2!x 2)+ 1 = 72 + 12 + 4 + 1 = 89
If in the expansion of and sum of the binomial coefficients of second and third terms is 36, then the value of x is –
The 100th term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,... is
1st term → 1, 2nd term = 2, 4th term → 3, 7th term → 4, 11th term → 5,...
Series is 1, 2, 4, 7, 11,...
If n = 14, then a_{n} = 92,If n = 15, then a_{n} = 106.
The line 3x – 4y + 7 = 0 is rotated through an angle π/4 in the clockwise direction about thepoint (–1, 1). The equation of the line in its new position is
As (–1, 1) is a point on 3x – 4y + 7 = 0, the rotation is possible. Slope of the given line = 3/4. Slope of the line in its new position
The required equation is
Find the vertex of the parabola x^{2} – 8y – x + 19 = 0.
The given equation of Parabola can be written as
If : p Raju is tall and q: Raju is intelligent, then the symbolic statement ~ p v q means
~ p v q : Raju is not tall or he is intelligent.
Given below is a frequency distribution with median 46. In this distribution, some of the frequencies are missing : Determine the missing frequencies.
∴ Median class = 40 – 50
∴ l = 40 , c.f. = 42 + x, f = 65, h = 10
∴ x = 34 (∵ Number of students cannot be in fraction) Now Sfi = 29
∴ x + y + 150 = 229 x + y = 229 – 150 = 79 .......... (i)
Putting the value of x in (i), we get 34x + y = 79
⇒ y = 79 – 34 = 45
∴ x = 34, y = 45
If the function f : (– ∞ ,∞) → B defined by f (x) = – x^{2} + 6x – 8 is bijective, then B =
Since the function f is bijective, therefore f is onto. Therefore range of f = B.
Let y = – x^{2} + 6x – 8
⇒ x^{2} – 6x + (8 + y ) = 0
If A and B are 2 x 2 matrices, then which of the following is true ?
For two 2 x 2 matrices, A & B (A – B) × (A + B)
= A × A + A × B – B × A – B × B
= A^{2} – B^{2} + AB – BA
Hence, (A – B) (A + B) = A^{2} + AB – BA – B^{2}
If a > 0, b > 0, c > 0 are respectively the pth, qth, rth terms of G.P., then the value of the determinant
Let A be the 1st term and R the common ratio of G.P., then
a = Tp = AR^{1}
∴ log a = log A + (p  1) log R
Similarly, log b = log A + (q  1) log R
and log c = log A + (r  1) log R
Split into two determinants and in the first take logA common and in the second take log R common
The digits A, B and C are such that the three digit numbers A88, 6B8, 86C are divisible by 72 then the determinant is divisible by
which is divisible by 72.
If y = e^{–x} cos x and y^{4} + ky = 0, where then k =
The set of points of discontinuity of the function 1/log  x  is –
Let
The points of disconinuity of f (x) are those points where f(x) is undefined or infinite. It is undefined when x = 0 and is infinite when
log  x  = 0,  x  = 1, i.e. x = ± 1.
∴ Set of points of discontinuity = {–1, 0, 1}.
The minimum value of the function y = x^{4} – 2x^{2} + 1 in the interval is
For max. or min,
4x (x^{2} – 1) = 0; either x = 0 or x = ± 1
x = 0 and x = – 1 does not belong to
∴ there is minimum value of function at x = 1
∴ minimum value is
y(1) = 1^{4} – 2(1)^{2} + 1 = 1 – 2 + 1 = 0
The value of a in order that f (x) = sin x – cos x – ax + b decreases for all real values is given by
We have ; f (x) = sin x – cos x – ax + b ⇒ f ' (x) = cos x + sin x – a
⇒ f ' (x) < 0 x ∈ R
⇒ (cos x + sin x) < a ∀ x ∈ R
As the max. value of (cos x + sin x) is √2
The equation of tangent to the curve y = sin x at the point (π, 0) is
Therefore the equation of tangent at ( π, 0) is given by
y – 0 = –1 (x – π)
⇒ x + y = π
If = Aln  cos x + sin x 2  +Bx+C.
Then the ordered triplet A, B, l is –
Area of the triangle formed by the line x + y = 3 and angle bisectors of the pair of straight lines x^{2} – y^{2} + 2y =1 is
x^{2} – y^{2} + 2y = 1
⇒ x = ±( y  1)
Bisectors of above line are x = 0 & y = 1
So area between x = 0, y = 1 & x + y = 3 is shaded Region shown in figure.
Area = ½ × 2 × 2 = 2 sq. units
= – xcosx + sinx + C
⇒ x (y + cosx) = sinx + C
If the line lies in the plane 2x – 4y + z = 7, then the value of k is
The point (4, 2, k) on the line also lies on the plane 2x – 4y + z = 7.
So, 8 – 8 + k = 7
⇒ k = 7
A line segment has length 63 and direction ratios are 3, –2, 6. If the line makes an obtuse angle with x–axis, the components of the line vector are
Let the components of the line vector be a, b, c.
Then a^{2} + b^{2} + c^{2} = (63)^{2} ... (i)
Also
(say), then a = 3λ ,
b = –2λ and c = 6λ and from (i) we have
Since a = 3λ < 0 as the line makes an obtuse angle with xaxis, λ = –9 and the required components are –27, 18, –54.
It is given that the events A and B are such that
Then P(B) is
By conditional probability,
The random variable X has the following probability distribution
Then the value of P(X ≥ 2) is
Since, SP_{i}(X = x) = 1
In a triangle ABC, ∠C = 90°, then is equal to:
A + B = 180° – C = 90°
a = 2R sin A, b = 2R sin B, c = 2R sin C
A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is 60º. when he retreats 20 feet from the bank, he finds the angle to be 30º. The breadth of the river in feet is :
Let h be the height of tree PQ and breadth of river PS be x ft. Angle of elevation subtended by a tree is 60°. Also, when he retreats 20 feet, the angle becomes 30°.
Also, in DPQS,
The minimum value of the function z = 4x + 3y subject to the constraints 3x + 2y ≥ 160, 5x + 2y ≥ 200, x + 2y ≥ 80, x ≥ 0, y ≥ 0 is
Two tangents PQ and PR drawn to the circle x^{2} + y^{2} – 2x – 4y – 20 = 0 from point P (16, 7). If the centre of the circle is C then the area of quadrilateral PQCR is
where L = length of tangent and r = radius of circle.
Hence the required area = 75 sq. units.
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