Class 12

The bisectors of angles A and B of a scalene triangle ABC meet at O. (i) What is the point O called? (ii) OR and OQ is drawn a perpendicular to AB and CA respectively. What is the relation between OR and OQ? (iii) What is the relation between ∠ACO and ∠BCO?

Solution: (i) The point O where the angle bisectors meet is called the incenter of the triangle. (ii) The perpendicular drawn from point O to AB and CA are equal. i.e., OR and OQ. (iii) ∠ACO = ∠BCO....

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(a) In the figure (i) given below, triangle ABC is equilateral. Find ∠BDC and ∠BEC. (b) In the figure (ii) given below, AB is a diameter of a circle with center O. OD is perpendicular to AB and C is a point on the arc DB. Find ∠BAD and ∠ACD

Solution: (a) triangle ABC is an equilateral triangle Each angle = 60o ∠A = 60o But ∠A = ∠D (Angles in the same segment) ∠D = 600 Now ABEC is a cyclic quadrilateral, ∠A = ∠E = 180o 60o + ∠E = 180o...

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(a) If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by r = /frac (a + b – c) – (2) (b) In the given figure, PB is a tangent to a circle with center O at B. AB is a chord of length 24 cm at a distance of 5 cm from the center. If the length of the tangent is 20 cm, find the length of OP.

Solution: (a) Let the circle touch the sides BC, CA and AB of the right triangle ABC at points D, E and F respectively, where BC = a, CA = b and AB = c (as showing in the given figure). As the...

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(a) In the figure (i) given below, O is the center of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, find the radius of the circle. (b) In the figure (ii) given below, from an external point P, tangents PA and PB are drawn to a circle. CE is a tangent to the circle at D. If AP = 15 cm, find the perimeter of the triangle PEC.

Solution: (i) Join OB ∠OBA = 90° (Radius through the point of contact is perpendicular to the tangent) OB2 = OA2 – AB2 r2 = (r + 7.5)2 – 152 r2 = r2 + 56.25 + 15r – 225 15r = 168.75 r = 11.25 Hence,...

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(a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB. (b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values of x, y and z.

Solution: (a) Construction: Join BC, and AC then ABCD is a cyclic quadrilateral. Now in ∆DCF Ext. ∠2 = x + z and in ∆CBE Ext. ∠1 = x + y Adding (i) and (ii) x + y + x + z = ∠1 + ∠2 2 x + y + z =...

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(a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC. (b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find: (i)∠BAD (ii) DBCD.

Solution: (a) ADFE is a cyclic quadrilateral Ext. ∠FEB = ∠ADF ⇒ ∠ADF = 80° ABCD is a parallelogram ∠B = ∠D = ∠ADF = 80° or ∠ABC = 80° (b)In trapezium ABCD, AD || BC (i) ∠B + ∠A = 180° ⇒ 70° + ∠A =...

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(a) In the figure (i) given below, M, A, B, N are points on a circle having centre O. AN and MB cut at Y. If ∠NYB = 50° and ∠YNB = 20°, find ∠MAN and the reflex angle MON. (b) In the figure (ii) given below, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°, find (i) ∠ACB (ii) ∠OBC (iii) ∠OAB (iv) ∠CBA

Solution (a) ∠NYB = 50°, ∠YNB = 20°. In ∆YNB, ∠NYB + ∠YNB + ∠YBN = 180o 50o + 20o + ∠YBN = 180o ∠YBN + 70o = 180o ∠YBN = 180o – 70o = 110o But ∠MAN = ∠YBN (Angles in the same segment) ∠MAN = 110o...

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0.0210\text{ }M\text{ }solution\text{ }of\text{ }{{N}_{2}}{{O}_{5}}\,is\text{ }allowed\text{ }to\text{ }decompose\text{ }at\text{ }43{}^\circ C.\text{ }How\text{ }long\text{ } will\text{ }it\text{ }take\text{ }to\text{ }reduce\text{ }to\text{ }0.0150M\text{ }?\text{ }\left( Given\text{ }k\text{ }=\text{ }6.0\text{ }\times \text{ }{{10}^{-4}}se{{c}^{-1}} \right)

  5600 sec 360.0 sec 560.0 sec  3364 sec Solution: 560 sec $ Given: $ $ {{[R]}_{0}}=0.0210\,M~ $ $ [R]=0.0150\,M~ $ $ k=6\times {{10}^{-4}}\,{{\sec }^{-1}} $ $ For\text{ }a\text{ }first\text{...

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Given that ∆s ABC and PQR are similar. Find: (i) The ratio of the area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1 : 3. (ii) the ratio of their corresponding sides if area of ∆ABC : area of ∆PQR = 25 : 36.

Solution:- From the question it is given that, (i) The area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1 : 3 Then, ∆ABC ~ ∆PQR area of ∆ABC/area of ∆PQR = BC2/QR2 So,...

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An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is 85 m and the height of the cylindrical part is 50 m. If the diameter of the base is 168 m, find the quantity of canvas required to make the tent. Allow 20% extra for folds and stitching. Give your answer to the nearest m².

solution; Given height of the tent above the ground = 85 m Height of the cylindrical part, H = 50 m height of the cone, h = 85-50 h = 35 m Diameter of the base, d = 168 m Radius of the base of...

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Lifetimes of the molecules in the excited states are often measured by using pulsed radiation source of duration nearly in the nanosecond range. If the radiation source has the duration of 2 \mathrm{~ns} and the number of photons emitted during the pulse source is 2.5 \times 10^{15} J, calculate the energy of the source.

Frequency of radiation $(\nu)$, $\nu=\frac{1}{2.0 \times 10^{-9} s}$ $\nu=5.0 \times 10^{8} s^{-1}$ Energy $(E)$ of source $=$ Nhv Where, $N$ is the no. photons emitted $\mathrm{h}$ is Planck's...

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The work function for the caesium atom is 1.9 eV. Calculate
(a) the threshold wavelength and
(b) the threshold frequency of the radiation. If the caesium element is irradiated with a wavelength of 500 nm,(c) calculate the kinetic energy and the velocity of the ejected photoelectron.

Given, the work function of caesium $\left(W_{0}\right)=1.9 \mathrm{eV}$ (a)From the $W_{0}=\frac{h c}{\lambda_{0}}$ expression, we get: $\lambda_{0}=\frac{h c}{W_{0}}$ Where, $\lambda_{0}$ is the...

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In Rutherford’s experiment, generally the thin foil of heavy atoms, like gold, platinum etc. have been used to be bombarded by the α-particles. If the thin foil of light atoms like Aluminium etc. is used, what difference would be observed from the above results?

The findings obtained with a foil made up of heavy atoms will differ from those obtained with a foil made up of comparatively light atoms. The magnitude of positive charge in the nucleus of a...

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What is the energy in joules, required to shift the electron of the hydrogen atom from the first Bohr orbit to the fifth Bohr orbit and what is the wavelength of the light emitted when the electron returns to the ground state? The ground state electron energy is -2.18\times10^{-11} ergs. The ground-state electron energy is -2.18\times10^{-11}ergs.

$ E_{5}=\frac{-\left(2.18 \times 10^{-18}\right) Z^{2}}{(n)^{2}} $ Where, $Z$ denotes the atom's atomic number Ground state energy $=-2.18 \times 10^{-11}$ ergs $=-2.18 \times 10^{-11} \times...

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Glycogen is a branched-chain polymer of α-D-glucose units in which chain is formed by C1—C4 glycosidic linkage whereas branching occurs by the formation of C1-C6 glycosidic linkage. Structure of glycogen is similar to __________.
(i) Amylose
(ii) Amylopectin
(iii) Cellulose
(iv) Glucose

Option (ii) is the answer. Glycogen has a similar structure to amylopeptin. It's an a-D glucose unit branched chain polymer with C1-C4 glycosidic linkage for chain formation and C1-C6 glycosidic...

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Proteins are found to have two different types of secondary structures viz. α-helix and β-pleated sheet structure. α-helix structure of the protein is stabilised by : (i) Peptide bonds (ii) van der Waals forces (iii) Hydrogen bonds (iv) Dipole-dipole interactions

Option ( iii) is the answer. Hydrogen bonding help to keep the -helix structure of proteins stable. By twisting into a right-handed helix and hydrogen bonding the -NH group of each amino acid...

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Which of the following reactions of glucose can be explained only by its cyclic structure?
(i) Glucose forms pentaacetate.
(ii) Glucose reacts with hydroxylamine to form an oxime.
(iii) Pentaacetate of glucose does not react with hydroxylamine.
(iv) Glucose is oxidised by nitric acid to gluconic acid

Option (iii) is the answer. The absence of a free -CHO group is indicated by the fact that glucose pentaacetate does not react with hydroxylamine. Only the cyclic nature of glucose may explain this...

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Structure of a disaccharide formed by glucose and fructose is given below. Identify anomeric carbon atoms in monosaccharide units.
(i) ‘a’ carbon of glucose and ‘a’ carbon of fructose.
(ii) ‘a’ carbon of glucose and ‘e’ carbon of fructose.
(iii) ‘a’ carbon of glucose and ‘b’ carbon of fructose.
(iv) ‘f’ carbon of glucose and ‘f ’ carbon of fructose.

Option (iii) is the answer. Anomeric carbon is carbon that is next to an oxygen atom in the cyclic structure of glucose or fructose. 'a' and 'b' are next to the oxygen atom, as illustrated in the...

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Three structures are given below in which two glucose units are linked. Which of these linkages between glucose, units are between C1 and C4 and which linkages are between C1 and C6?
(i) (A) is between C1 and C4, (B) and (C) is between C1 and C6
(ii) (A) and (B) are between C1 and C4, (C) is between C1 and C6
(iii) (A) and (C) is between C1 and C4, (B) is between C1 and C6
(iv) (A) and (C) is between C1 and C6, (B) is between C1 and C4

Option (iii) is the answer (A) and (C) are in the Cl-C4 range, while (B) is in the Cl-C6 range.

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In nucleoside, a base is attached at 1C position of the sugar moiety. A nucleotide is formed by linking the phosphoric acid unit to the sugar unit of a nucleoside. At which position of sugar unit is the phosphoric acid linked in a nucleoside to give a nucleotide?

When a nitrogenous base is connected to the 1' position of a five-carbon sugar, a nucleoside is produced. The 5' carbon of the sugar in a nucleoside molecule is bonded to the 5' carbon of the sugar...

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Monosaccharides contain carbonyl group hence are classified, as aldose or ketose. The number of carbon atoms present in the monosaccharide molecule is also considered for classification. In which class of monosaccharide will you place fructose?

Carbonyl groups can be found in monosaccharides. As a result, they're categorised as either aldose or ketose. Aldose refers to monosaccharides that contain an aldehyde group. Ketose refers to...

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The letters ‘D’ or ‘L’ before the name of a stereoisomer of a compound indicates the correlation of configuration of that particular stereoisomer. This refers to their relationship with one of the isomers of glyceraldehyde. Predict whether the following compound has ‘D’ or ‘L’ configuration.

On the left side of the C5 carbon atom, the –OH group is connected. As a result, the provided compound is in the 'L' configuration.

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