General Instructions:
(i) This question paper comprises four sections – A, B, C and D. This question paper carries 40 questions. All questions are compulsory.
(ii) Section A: Q. No.1 to 11 comprises of 11 questions of one mark each.
(iii) Section B: Q. No. 12 to 20 comprises of 9 questions of two mark each.
(iv) Section C: Q. No. 21 to 30 comprises of 10 questions of three mark each.
(v) Section D: Q. No. 31 to 35 comprises of 5 questions of four mark each.
(vi) Use of Calculators are not permitted.
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Find the common difference of the Ap \(\dfrac{1}{p},\dfrac{1p}{p},\dfrac{12p}{p},…. \)
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The common difference
In Fig., Calculate the area of triangle ABC (in sq.units).
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In Fig., PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA PB, then find the length of each tangent.
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Construction: Join AC and BC.
Proof: [ latex]\because [/latex] Tangent is to the radius through the point of contact
APBC is a square.
Length of each tangent
= AC = radius = 4 cm
If the difference between the circumference and the radius of a circle is 37 cm, then using \(\pi=\dfrac{22}{7} \), calculate the circumference (in cm) of the circle.
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Circumference of the circle
Solve the following quadratic equation for \(x:4\sqrt3x^2+5x2\sqrt3=0 \)
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Find the common difference of the A.P. \(\dfrac{1}{2b},\dfrac{16b}{2b},\dfrac{112b}{2b},…. \)
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Common difference
A die is tossed once. Find the probability of getting an even number or a multiple of 3.
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‘an even number or multiple of 3’ are 2, 3, 4, 6,
i.e., 4 numbers
Required probability
Find the common difference of the A.P. \(\dfrac{1}{3q},\dfrac{16q}{3q},\dfrac{112q}{3q},…. \)
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Common difference
A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability that the drawn card is neither a jack nor an ace.
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Total number of cards = 52
Numbers of jacks = 4
Numbers of aces = 4
Card is neither a jack nor an ace = 52 – 4 – 4 = 44
Required probability
The sum of first \(n \) terms of an AP is \(3n^2 + 4n \). Find the 25th term of this AP.
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How many threedigit natural numbers are divisible by 7?
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“3 digits nos. ” are 100, 101, 102,….,999
3 digits nos. “divisible by 7” are:
105, 112, 119, 126,….,994
As
In Fig., a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then find the lengths of AD, BE and CF.
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Let
and …. [Two tangents drawn from an external point are equal
latex] AB=12\ cm[/latex]…. [Given
latex]\therefore\ x+y=12\ cm [/latex]….(i)
Similarly, …. (ii) and ….(iii)
By adding (i), (ii) and (iii)
in (ii) and (iii),
A card is drawn at random from a well shuffled pack of 52 playing cards. Find the probability that the drawn card is neither a king nor a queen.
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P(neither a king nor a queen)
(king or queen)
Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular card board of dimensions \(14\ cm\times7\ cm \). Find the area of the remaining card board. [Use latex]\pi=\dfrac{22}{7} [/latex] ]
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Here
Area of the remaining card board
For what value of \(k \), are the roots of the quadratic equation \(kx(x – 2) + 6 =0\) equal?
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Given:
Two equal roots …[Given
latex]\therefore b^24ac=0\\ \\ a=k,b=2k,c=6\\ \\ \therefore(2k)^24(k)(6)=0\\ \\ 4k^224k=0\ \Rightarrow\ 4k(k6)=0\\ \\ 4k=0\qquad or\qquad k6=0\\ \\ k=0\qquad or\qquad k=6 [/latex]…. [Standard form of a quadratic equation latex] ax^2+bx+c=0,a\ne0[/latex]
The nth term of an A.P. is given by (4n + 15). Find the sum of first 20 terms of this A.P.
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For what value of \(k \), the roots of the quadratic equation \(kx(x – 2\sqrt5) + 10 = 0\), are equal?
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….[ latex]\because [/latex] Roots are equal
As
Find the value of \(x \) for which the points \((x, 1), (2,1)\) and \((4, 5)\) are collinear.
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Given: Paints
As [ latex]\because [/latex] Given 3 pts. are collinear
From a point P on the ground, the angle of elevation of the top of a 10m tall building is 30°. A flagstaff is fixed at the top of the building and the angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from the point P. (Take \(\sqrt3=1.73 \))
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Let be the building and be the flagstaff.
In right ,
(Distance)
In right
Length of the flagstaff,
The 24th term of an AP is twice its 10th term. Show that its 72nd term is 4 times its 15th term.
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…[Given
latex]a+23a=2(a+9d)\quad [\because a_n=a+(n1)d\\ \\ 23d=2a+18da\\ \\ 23d18d=a\quad\Rightarrow\quad a=5d\ ….(i) /latex]
To prove with the help of
From (ii) and (iii), L.H.S = R.H.S
Construct a triangle with sides 5 cm, 4 cm and 6 cm. Then construct another triangle whose sides are \(\dfrac{2}{3} \) times the corresponding sides of first triangle.
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Steps of Construction:
Draw with .
Draw ray making an acute angle with .
Locate 3 equal points on .
Join .
Join .
From point draw .
is the required triangle.
The horizontal distance between two poles is 15 m. The angle of depression of the top of first pole as seen from the top of second pole is 30°. If the height of the second pole is 24 m, find the height of the first pole. [Use latex]\sqrt3=1.732 [/latex] ]
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Let be the pole and be the pole
In ,
Height of the first pole,
Prove that the points (7,10), (2,5) and (3, 4) are the vertices of an isosceles right triangle.
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Let be the vertices of a triangle
…[By converse of Pythagoras theorem
latex]\triangle ABC [/latex] is an isosceles rt.
From (i) & (ii), Points A, B, C are the vertices of an isosceles right triangle.
Find the ratio in which the yaxis divides the line segment joining the points (4, 6) and (10,12). Also find the coordinates of the point of division.
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Let
Let be point on axis
Let
Coordinates of C = Coordinates of C
Required ratio and Required Point
In Fig., AB and CD are two diameters of a circle with centre O, which are perpendicular to each other. QB is the diameter of the smaller circle. If QA = 7 cm, find the area of the shaded region. [Use \(\pi=\dfrac{22}{7} \) ]
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Reqd. shaded area
A vessel is in the form of a hemispherical 66 bowl surmounted by a hollow cylinder of same diameter. The diameter of the hemispherical bowl is 14 cm and the total height of the vessel is 13 cm. Find the total (inner) suface area of the vessel. [Use latex]\pi=\dfrac{22}{7} [/latex]
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Inner surface area of the vessel = C.S. area of Hemisphere + C.S. area of Cylinder
…[C.S. area = curved surface area
latex]=2\times\dfrac{22}{7}\times7(7+6)\\ \\ =44\times13=572\ cm^2 [/latex]
A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the volume of wood in the toy. [Use latex]\pi=\dfrac{22}{7} [/latex] ]
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Here
Vol. of wood in the toy = Volume of cylinder – 2(Volume of hemisphere)
In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc. [Use latex]\pi=\dfrac{22}{7} [/latex] ]
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(i) Length of the arc:
Length of the arc
(ii) Area of the sector formed by the arc:
Area of minor sector
Solve the following for \(x:\dfrac{1}{2a+b+2x}=\dfrac{1}{2a}+\dfrac{1}{b}+\dfrac{1}{2x} \)
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Sum of the areas of two squares is \(400\ cm^2 \). If the difference of their perimeters is \(16\ cm \), find the sides of the two squares.
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Let the side of Large square
Let the side of small square
According to the question,
…(i) …. area of square = (side)
….[ latex]\because [/latex] Perimeter of square = 4 sides
….[Dividing both sides by 4
latex]x=4+y [/latex]
Putting the value of in equation (i),
Side of small square and side of large square
If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first n terms.
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As
From (i) and (ii), we have
In Fig., l and m are two parallel tangents to a circle with centre O, touching the circle at A and B respectively. .Another tangent at C intersects the line / at D and m at E. Prove that \(\angle \)DOE = 90°.
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Proof: Let be and be
…[Consecutive interior angles
latex]\dfrac{1}{2}\angle XDE+\dfrac{1}{2}\angle X’ED=\dfrac{1}{2}(180^o)\\ \\ \angle 1+\angle2=90^o [/latex]…. [OD is equally inclined to the tangents
In latex]\triangle DOE [/latex],
…[Anglesumproperty of a latex]\triangle [/latex]
… [Proved
The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building.
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Let be the tower and let be the building
In right
In right
Height of the building,
The three vertices of a parallelogram ABCD are A(3, 4), B(l, 3) and C(6, 2). Find the coordinates of vertex D and find the area of ABCD.
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Let
Since diagonals of bisect each other
midpoint BD = midpoint AC
4th point,
Water is flowing through a cylindrical pipe, of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. Determine the rise in level of water in the tank in half an hour.
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Radius of tank,
Internal radius of cylindrical pipe,
Length of water flow in 1 second
Length of water flow in 30 minutes,
Volume of water in cylinder tank
= Volume of water flow from cylindrical pipe in half an hour
As
Level of water in cylinder tank rises in half an hour