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 20 comprises of 20 questions of one mark each.
(iii) Section B: Q. No. 21 to 26 comprises of 6 questions of two mark each.
(iv) Section C: Q. No. 27 to 34 comprises of 8 questions of three mark each.
(v) Section D: Q. No. 35 to 40 comprises of 6 questions of four mark each.
(vi) Use of Calculators are not permitted.
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From an external point \(P \), tangents \(PA \) and \(PB \) are drawn to a circle with centre O. If \(\angle PAB=50^o \) then find \(\angle AOB \).
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…[Tangents drawn from external point are equal]
…[Angles equal to opp. sides]
In …[Angle sum property of a latex]\triangle [/latex]]
In cyclic quad.
…[Sum of opposite angles of a cyclic quadrilateral is latex]180^o [/latex]]
In Fig. AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m. Find the length of the ladder. (Use \(\sqrt3=1.73 \))
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In rt.,
Length of the ladder,
Find the 9th term from the end (towards the first term) of the A.P. 5, 9,13, …, 185.
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Here First term,
Common difference,
last term,
term from the end
term from the end
Cards marked with number 3,4,5,…., 50 are placed in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the selected card bears a perfect square number.
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Total no. of cards
Perfect square number cards are i.e., cards
P(perfect square number)
If \(x=\dfrac{2}{3} \) and \(x=3 \) are roots of the quadratic equation \(ax^2+7x+b=a \) and \(b \).
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Here
and
Given,
Sum of roots
Product of roots
… From (i)
…(i)
Find the ratio in which yaxis divides the line segment joining the points A(5, 6), and B(1, 4). Also find the coordinates of the point of division.
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Let
Coordinates of
…(i)
Point lies on axis,
Required ratio
From (i), required point ,
How many terms of the A.P. 27, 24, 21, … should be taken so that their sum is zero?
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term,
Common difference,
But , i.e., number of terms cannot be zero.
Number of terms
The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are 60° and 30° respectively. Find the height of the tower.
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Let the height of the tower,
In ,
A solid sphere of radius \(r \) is melted and recast into the shape of a solid cone of height \(r \), then find the radius of the cone.
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If \(x=2 \) and \(x=0 \) are the zeroes of the polynomials \(f(x)=2x^35x^2+ax+b \). Find the values of \(a \) and \(b \)
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The graph of a polynomial \(P(x) \) is shown here. Find the number of zeroes of \(P(x) \)
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Number of zeroes of is 1
Find the value of \(k \) for which one root of the quadratic equation \(kx^214x+8=0 \) is \(6 \) times the other
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…(1)
…(2)
Find the area of \(a \) sector of circle of radius 21 cm and central angle \(90^o \) (take \(\pi=\dfrac{22}{7} \))
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Area
Evaluate \(\sin45^o\cos60^o+\cos45^o\sin60^o \)
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Find angle A in the following figure
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In
Find the discriminant of the quadratic equation \(6x^27x+2=0 \)
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If the length of a tangent to a circle from a point is 0.5 cm and distance of a point from its centre is 1.5 cm. Find its radius
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A standard deck of \(\sqrt2 \) cards is shuffled Ritu draws a single card from the dick. What is the probability that the card is a Jack?
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Number of possible outcomes
Number of favourable outcomes
Two dice are thrown together. Find the probability that the sum of two numbers will be multiple of 4.
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The mean of \(n \) observations is \(\bar x \). If the first term is increased by 1 second by 2 and so on, find the new mean
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In Fig. a circle is inscribed in a \(\triangle \)ABC, such that it touches the sides AB, BC and CA at points D, E and F respectively. If the lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm respectively, find the lengths of AD, BE and CF.
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…[Given]
As we know,
[Tangents drawn from an external point are equal]
Let ,
then,
Similarly, [Given]
The xcoordinate of a point P is twice its ycoordinate. If P is equidistant from Q (2, 5) and R(3, 6), find the coordinates of P.
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Let the point be
… Given
… Squaring both sides
Hence coordinates of point P are (16, 8).
How many terms of the A.P. 18,16,14,…. be taken so that their sum is zero?
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Here term,
Common difference,
…Given
But can not be zero.
Therefore, Number of terms, .
In Fig. AP and BP are tangents to a circle with centre O, such that AP = 5 cm and \(\angle \)APB = 60°. Find the length of chord AB.
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… [Tangents drawn from an external point are equal
Given: latex]\angle APB=60^o [/latex]
…(i)
[Angles opposite to equal sides]
In
Anglesumproperty of a
Hence,
All sides of an equilateral are equal
In Fig. ABCD is a square of side 14 cm. Semicircles are drawn with each side of square as diameter. Find the area of the shaded region.
Use \(\pi=\dfrac{22}{7} \)
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Radius of circle,
Since the radius of all the semicircles is same
Area of 4 semicircles,
Area of Square,
Area of shaded region
(Area of square) (Area of semicircle)
In Fig. is a decorative block, made up of two solids—a cube and a hemisphere. The base of the block is a cube of side 6 cm and the hemisphere fixed on .the top has a diameter of 3.5 cm. Find the total surface area of the block. Use \(\pi=\dfrac{22}{7} \)
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Total surface area of the block
= Total surface area of cube + C.S. Area of hemisphere – Area of circle
Side of cube
Radius of hemisphere
In Fig. ABC is a triangle Coordinates of whose vertex A are (0, 1). D and E respectively are the midpoints of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the midpoint of BC, find the areas of \(\triangle \)ABC and \(\triangle \)DEF.
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Let and
Midpoint of AB = Coordinates of D
Midpoint of AC = Coordinate of E
Coordinate of F = Midpoint of BC
Area of ,
As area of is always +ve
In Fig. are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with centre O and radius OP while arc PBQ is a semicircle drawn on PQ as diameter with centre M. If OP = PQ = 10 cm, show that area of shaded region is \(25\left(\sqrt3\dfrac{\pi}{6}\right)\ cm^2 \).
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…[Tangents drawn from an external point are equal]
…[sides are equal]
is an equilatral .
…Angles of an equilateral
Side
Area of the shaded region = Area of + Area of semicircle – Area of sector
If the sum of first 7 terms of an A.P is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.
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Let term , Common difference
Given:
Solving (i) and (ii), we get
Solve for \(x:\dfrac{2x}{x3}+\dfrac{1}{2x+3}+\dfrac{3x+9}{(x3)(2x+3)}=0,x\ne3,\dfrac{3}{2} \)
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[Dividing both sides by 2
latex]\Rightarrow 2 x^{2}+3 x+2 x+3=0\\ \\ \Rightarrow x(2 x+3)+1(2 x+3)=0\\ \\ \Rightarrow(2 x+3)(x+1)=0\\ \\ \Rightarrow 2 x+3=0. \quad \text{or} \quad x+1=0\\ \\ \Rightarrow x=\dfrac{3}{2} \quad \text{ or} \quad x=1\ \\ \\ \text{But}, x \neq \dfrac{3}{2}[/latex]… Given
is the only solution.
A well of diameter 4 m is dug 21 m deep. The earth taken out of it has been spread evenly all abound it in the shape of a circular ring of width 3 m to form an embankment. Find the height of the embankment.
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Radius of well,
Radius of embankment,
Height of the well,
Required height raised
The sum of the radius of base and height of a solid right circular cylinder is 37 cm. If the total surface area of the solid cylinder is 1628 sq. cm, find the volume of the cylinder. Use \(\pi=\dfrac{22}{7} \)
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Let the radius and height of cylinder be and respectively.
…(i)
Total surface area of cylinder
From (i),
Volume of cylinder
In a single throw of a pair of different dice, what is the probability of getting (i) a prime number on each dice? (ii) a total of 9 or 11?
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Two dice can be thrown in ways
(i) “a prime number on each dice” can be obtained as i.e., ways.
P(a prime no. on each dice)
(ii) “a total of 9 or 11” can be obtained as
P(a total of 9 or 11)
A passenger, while boarding the plane, slipped from the stairs and got hurt. The pilot took the passenger in the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passengers could catch the connecting flight, the speed of the plane was increased by 250 km/hour than the usual speed. Find the usual speed of the plane. What value is depicted in this question?
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Let the usual speed of the plane km/hr
Then the increased speed of the plane km/hr
Distance km
According to the question,
Speed of plane km/hr.
Value: By giving utmost importance to the health and wellbeing of the passenger the pilot displays the value of compassion. By making up for the time lost due to this delay he displays concern for the passengers’ time and money and shows dedication towards his airline and job.
Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to smaller circle from a point on the larger circle. Also measure its length.
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We have, and
and are the required tangents.
By measurement
In Fig. O is the centre of a circle of radius 5 cm. T is a point such that OT 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.
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… Tangents is to the radius through the point of contact
We have,
In rt. ,
… Pythagoras’ theorem
… radius of the circle
Let, , then
… Tangents drawn from an external point
In rt. ,
….Pythagoras’ theorem
Find \(x \) in terms of \(a,b \) and \( c\): \(\dfrac{a}{xa}+\dfrac{b}{xb}=\dfrac{2c}{xc},x\ne a,b,c \)
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A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45°. The bird flies away horizontally in such a way that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30°. Find the speed of flying of the bird. (Take \(\sqrt3=1.732 \))
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Let be the tree
In rt.
In rt.
… From (i)
Hence, speed of bird
A thief runs with a uniform speed of 100 m/minute. After one minute a policeman runs after the thief to catch him. He goes with a speed of 100 m/minute in the first minute and increases his speed by 10 m/ minute every succeeding minute. After how many minutes the policeman will catch the thief.
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Let total time be minutes
Total distance covered by thief in minutes
= Speed Time
= metres
Total distance covered by policeman
Thief runs = mins
Policeman runs = mins
Here,
[Dividing both sides by 10
latex]\Rightarrow n^{2}6 n+3 n18=0\\ \\ \Rightarrow n(n6)+3(n6)=0\\ \\ \Rightarrow(n+3)(n6)=0\\ \\ \Rightarrow n+3=0 \quad\ \text{or}\ \quad n6=0\\ \\ \Rightarrow n=3 \quad\ \text{or}\ \quad n=6 [/latex]
But (time) can not be negative.
Time taken by policeman to catch the thief minutes.
Prove that the area of a triangle with vertices \((f, t. – 2),\left(\dfrac{f+2}{f+2}\right) \) and \((f + 3, f) \) is independent of \(f \).
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Let
Area of
Area of is always positive.
Area of sq. units, which is independent of .