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 29 comprises of 9 questions of three mark each.
(v) Section D: Q. No. 30 to 35 comprises of 6 questions of four mark each.
(vi) Use of Calculators are not permitted.
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Find the roots of the equation \(x^2 – 3x – m (m + 3) = 0\), where \(m \) is a constant.
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In Figure , O is the centre of a circle, PQ is a chord and PT is the tangent at P. If \( \angle\)POQ = 70°, then calculate \(\angle \)TPQ.
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In Figure , AB and AC are tangents to the circle with centre O such that \( \angle\)BAC = 40°. Then calculate \(\angle \)BOC.
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and are tangents
In
Find the perimeter (in cm) of a square circumscribing a circle of radius a cm.
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radius
Perimeter
A card is drawn at random from a pack of 52 playing cards. Find the probability that the card drawn is neither an ace nor a king.
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P(neither an ace nor a king)
= 1 – P (either an ace or a king)
= 1 – [P (an ace) + P (a king)]
Find the area of the triangle whose vertices are (1, 2), (3, 7) and (5, 3).
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Let and be the vertices of
Since area is a measure, which cannot be negative
Area of the triangle = 9 sq. units
Find the value of \(m \) so that the quadratic equation \(mx\ (x 7) + 49 = 0\) has two equal roots.
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Which term of the A.P. 3,14, 25, 36, … will be 99 more than its 25th term?
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Let the required term be term, i.e.,
According to the question,
Here
34th term is 99 more than its 25th term
Find 10th term from end of the A.P. 4, 9,14, …, 254.
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Common difference
Last term
term from the end
term from the end
If the perimeter of a semicircular protractor is 36 cm, find the diameter of the protractor.
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Perimeter of semicircular protractor
Hence diameter of protractor
Find how many twodigit numbers are divisible by 6?
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There are 15 twodigit numbers divisible by 6.
In Figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of side AD.
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Draw a line segment \(AB \) of length \(7\ cm \). Using ruler and compasses, find a point \(P \) on \(AB \) such that \(\dfrac{AP}{AB}=\dfrac{3}{5} \).
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\(\)AB=7\ cm,\dfrac{AP}{AB}=\dfrac{3}{5}\qquad….[Given\\ \\ \therefore AP:PB=3:2\\ \\ AP:AB=3:5\ \text{or}\ \dfrac{AP}{AB}=\dfrac{3}{5} /latex]
Find the perimeter of the shaded region in Figure , if ABCD is a square of side 14 cm arid APB and CPD are semicircles. [Use \(\pi=\dfrac{22}{7} \) ]
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Perimeter of the shaded region
= Circumference of circle + AD + BC
\(\)=2\pi r+14+14\\ \\ =2\times\dfrac{22}{7}\times7+28\qquad….\bigg[\because r=\dfrac{14}{2}=7\ cm\\ \\ =44+28=72\ cm /latex]
Two cubes each of volume \(27\ cm^3 \) are joined end to end to form a solid. Find the surface area of the resulting cuboid.
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Surface area
A cone of height 20 cm and radius of base 5 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the diameter of the sphere.
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Find the value of \(y \) for which the distance between the points \(A\ (3, 1)\) and \(B\ (11, y)\) is \(10 \) units.
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units ….[Given
latex]AB^2=10^2=100 [/latex] ….[Squaring both sides
latex](113)^2+(y+1)^2=100\\ \\ 8^2+(y+1)^2=100\\ \\ (y+1)^2=10064=36 [/latex]
Taking square root on both sides
A ticket is drawn at random from a bag containing tickets numbered from 1 to 40. Find the probability that the selected ticket has a number which is a multiple of 5.
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Total number of tickets = 40
A multiple of 5′ are
5 , 10, 15, 20,…40, i.e., 8 tickets
P (A multiple of 5)
Find the roots of the following quadratic equation : \(x^23\sqrt5x+10=0 \)
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Find an A.P. whose fourth term is 9 and the sum of its sixth term and thirteenth term is 40.
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\(\)\begin{aligned} &\begin{array}{ll} a_{n}=a+(n1) d & \\ a_{4}=9 & a_{6}+a_{13}=40 \\ a^{+} 3 d=9 & a+5 d+a+12 d=40 \\ a=93 d \quad \ldots .(i) & 2 a+17 d=40 \\ & 2(93 d)+17 d=40\text { …[From (i) }\\&186 d+17 d=40\\&11 d=4018=22 \\ &d=\dfrac{22}{11}=2 \end{array}\\ &\begin{array}{l} \\ \text { From }(i), a=93(2)=96=3 \end{array}\\ &\therefore \text { A.P. is }\ a,\ a+d,\ a+2 d,\ a+3 d, \ldots\\ &\qquad\quad\qquad3,\ 3+2,\ 3+4,\quad3+6, \ldots\\ &\begin{array}{llll} \qquad\qquad\quad3, & 5, &\quad 7, & \qquad9,\ldots \end{array} \end{aligned} /latex]
In Figure, a triangle \(PQR \) is drawn to circumscribe a circle of radius \(6\ cm \) such that the segments \(QT \) and \(TR \) into which \(QR \) is divided by the point of contact \(T \), are the lengths \(12\ cm \) and \(9\ cm \) respectively. If the area of \(\triangle PQR=189\ cm^2 \), then find the lengths of sides \(PQ \) and \(PR \).
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Construction: Join and
Proof: Tangents drawn from an external point are equal
Let
\(\)\mathrm{ar(\triangle PQR)=ar(\triangle POQ)+ar(\triangle QOR)+ar(\triangle POR)\\ \\ 189=\dfrac{PQ\times OS}{2}+\dfrac{QR\times OT}{2}+\dfrac{PR\times OU}{2}}\ ….\left[\begin{matrix}\because\ \text{area of }\triangle \\ =\dfrac{\mathrm{base\times corr.\ altitude}}{2}\end{matrix}\right. \\ \\ \begin{array}{l} 189=\dfrac{(x+12) \times 6}{2}+\dfrac{(12+9) \times 6}{2}+\dfrac{(x+9) \times 6}{2} \\ 189=3x+12+21+x+9] \\ 63=[2 x+42] \quad \ldots \text { .[Dividing by } 3 \\ 6342=2 x \quad \Rightarrow \quad 2 x=21 \\ \therefore \quad x=\dfrac{21}{2}=10.5 \mathrm{~cm} \\ \therefore \quad P Q=x+12=10.5+12=225 \mathrm{~cm} \\ \text { and } P R=x+9=10.5+9=19.5 \mathrm{~cm} \end{array}/latex]
Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are \(\dfrac{3}{5} \) times the corresponding sides of the given triangle.
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.
A chord of a circle of radius 14 cm subtends an angle of 120° at the centre. Find the area of the corresponding minor segment of the circle. [Use latex]\pi=\dfrac{22}{7} [/latex] and \(\sqrt3=1.73 \) ]
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Here
Shaded region = ar(minor segment) = ar(minor sector) – ar (AOB)
An open metal bucket is in the shape of a frustum of a cone of height 21 cm with radii of its lower and upper ends as 10 cm and 20 cm respectively. Find the cost of milk which can completely fill the bucket at ?30 per litre. [Use latex]\pi=\dfrac{22}{7} [/latex] ]
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Cost of milk at per litre
Point \(P(x, 4)\) lies on the line segment joining the points \(A( 5,8)\) and \(B(4, – 10)\). Find the ratio in which point \(P\) divides the line segment \(AB\). Also find the value of \(x\).
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Let
Coordinates of P = Coordinates of P
\(\)\begin{array}{ll} \left(\dfrac{4 k5}{k+1}, \dfrac{10 k+8}{k+1}\right)=(x, 4) & \\ \dfrac{4 k5}{k+1}=x, & \dfrac{10 k+8}{k+1}=4 \\ x=\dfrac{4\left(\dfrac{2}{7}\right)5}{\dfrac{2}{7}+1} \quad \ldots \text { [From (i) } & 4 k+4=10 k+8 \\ x=\dfrac{835}{2+7} & 4 k+10 k=84 \\ x=\dfrac{27}{9}=3 & 14 k=4 \\ & k=\dfrac{4}{14}=\dfrac{2}{7} \quad \ldots(i) \\ \therefore \text { Required Ratio }=\dfrac{2}{7}: 1 \\ =2: 7, & x=3 \end{array} /latex]
Find the area of the quadrilateral ABCD, whose vertices are A( 3, – 1), B( 2, – 4), C(4, 1) and D(3, 4).
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Area of
Area of
Area of quad ABCD
Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are A(2, 1), B(4, 3) and C(2, 5).
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Area of
(As area cannot be negative)
From the top of a vertical tower, the angles of depression of two cars, in the same straight line with the base of the tower, at an instant are found to be 45° and 60°. If the cars are 100 m apart and are on the same side of the tower, find the height of the tower. [Use latex]\sqrt3=1.73 [/latex] ]
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Let be the tower
Let , and
In rt. ,
In rt. ,
Height of the tower,
The probability of guessing the correct answer to a certain question is \(\dfrac{x}{12} \). If the probability of guessing the wrong answer is \(\dfrac{3}{4} \), find \( x\). If a student copies the answer, then its probability is \(\dfrac{2}{6} \) . If he doesn’t copy the answer, then the probability is \(\dfrac{2y}{3} \). Find the value of \(y \). If he does not copy the answer, which moral value is depicted by him?
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P (of guessing) + P (guessing wrong) = 1
P (he copies) + P (he doesn’t copy) = 1
If he does not copy, he is an honest student and true to his studies.
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
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Given: is a tangent at point to the cirlce with centre .
To prove:
Const: Take a point on other than and join .
Proof: If point lies inside the circle, then will become a secant and not a tangent to the circle.
This happend with every point on the line except the point .
is the shortest of all the distances of the point to the points of
….[Shortest side is latex]\bot [/latex]
The first and the last terms of an A.P. are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum?
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A train travels 180 km at a uniform speed. If the speed had been 9 km/hour more, it would have taken 1 hour less for the same journey. Find the speed of the train.
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Let the speed of the train
Increased speed
A/Q
Speed of the train
Find the roots of the equation\(\dfrac{1}{2x3}+\dfrac{1}{x5}=1 \)
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Roots are
In Figure , three circles each of radius 3.5 cm are drawn in such a way that each of them touches the other two. Find the area enclosed between these three circles (shaded region). [Use \(\pi=\dfrac{22}{7} \) ]
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The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 30°. Find the height of the tower.
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Let be the tower
Let
Let
In rt. ,
In rt. ,
Height of the tower,