Q1
mcq
1 mark
The value of k for which the equation kx^2 – 6x – 4 = 0 has real and equal roots, is
- A. 9/4
- B. –4
- C. –9/4
- D. –2
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Q2
mcq
1 mark
The line segment joining the points P(–4, –2) and Q(10, 4) is divided by y-axis in the ratio
- A. 2 : 5
- B. 1 : 2
- C. 2 : 1
- D. 5 : 2
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Q3
mcq
1 mark
If the zeroes of a polynomial p(x) are –3 and 8, then p(x) equals
- A. x^2 + 5x – 4
- B. ( x + 3) (–x + 8)
- C. a( x^2 + 5x – 24)
- D. x^2 – 24
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Q4
mcq
1 mark
In the given figure, PQ is tangent to the circle with centre O. S is a point on the circle such that SQT = 55. The m QPS is
- A. 55
- B. 20
- C. 35
- D. 70
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Q5
mcq
1 mark
Devansh proved that ABC ~ PQR using SAS similarity criteria. If he found C = R, then which of the following was proved true ?
- A. AC/AB = PR/PQ
- B. BC/AC = PR/QR
- C. AC/BC = PR/PQ
- D. AC/BC = PR/QR
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Q6
mcq
1 mark
While calculating mean of a grouped frequency distribution, step deviation method was used (x – a)/h = u. It was found that x = 64, h = 5 and a = 62.5. The value of u is
- A. 0.5
- B. 1.5
- C. 0.3
- D. 7.5
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Q7
mcq
1 mark
A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (use = 3.14)
- A. 314 2 cm^2
- B. 314 cm^2
- C. 3140/3 cm^2
- D. 3140 2 cm^2
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Q8
mcq
1 mark
If a_n represents n-th term of the A.P.– 15/4, – 10/4, – 5/4, …… then value of a_16 – a_12 is
- A. 4
- B. 5/4
- C. 5
- D. 25/4
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Q9
mcq
1 mark
The value of p for which roots of the quadratic equation x^2 – px + 6 = 0 are rational, is
- A. 1
- B. –5
- C. 25
- D. 5
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Q10
mcq
1 mark
Which of the following can not be the probability of an event ?
- A. 39/100
- B. 0.001/20
- C. 10/0.2
- D. 10%
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Q11
mcq
1 mark
A card is drawn at random from a well shuffled deck of 52 playing cards. The probability that it is either a ten or a king is
- A. 1/26
- B. 2/13
- C. 1/13
- D. 8/26
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Q12
mcq
1 mark
A camping tent in hemispherical shape of radius 1.4 m, has a door opening of area 0.50 m^2. Outer surface area of the tent is
- A. 11.78 m^2
- B. 12.32 m^2
- C. 11.82 m^2
- D. 12.86 m^2
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Q13
mcq
An arc of length 2.2 cm subtends an angle $\theta$ at the centre of the circle with radius 2.8 cm. The value of $\theta$ is
- A. 50°
- B. 60°
- C. 45°
- D. 30°
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Q14
mcq
For an acute angle $\theta$, if $\sin \theta = 1/9$, then value of $\frac{9 \csc \theta + 1}{9 \csc \theta - 1}$ is
- B. 80/81
- C. 1
- D. 82/80
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Q15
mcq
In the given figure, PQ || YZ such that XP : PY = 2 : 3. If PQ = 5 cm, then YZ equals
- A. 12.5 cm
- B. 10 cm
- C. 15 cm
- D. 7.5 cm
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Q16
mcq
Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 tickets were sold, the number of tickets bought by her, is
- A. 64
- B. 640
- C. 100
- D. 10
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Q18
mcq
Simplest form of $\frac{\sec A}{\sec^2 A - 1}$ is
- A. $\sin A$
- B. $\tan A$
- C. $\csc A$
- D. $\cos A$
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Q19
mcq
Assertion (A) : The system of linear equations 3x – 5y + 7 = 0 and –6x + 10y + 14 = 0 is inconsistent. Reason (R) : When two linear equations don’t have unique solution, they always represent parallel lines.
- A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- C. Assertion (A) is true, but Reason (R) is false.
- D. Assertion (A) is false, but Reason (R) is true.
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Q20
mcq
Assertion (A) : H.C.F. ($36m^2$, $18m$) = $18m$, where m is a prime number. Reason (R) : H.C.F. of two numbers is always less than or equal to the smaller number.
- A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
- B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- C. Assertion (A) is true, but Reason (R) is false.
- D. Assertion (A) is false, but Reason (R) is true.
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Q21
short answer
2 marks
Prove that $2 - 5\sqrt{3}$ is an irrational number given that $\sqrt{3}$ is irrational.
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Q21
short answer
2 marks
Vertices of a right triangle ABC with $\angle B = 90^\circ$ are A(3, 4), B(1, 1) and C(–8, 7). Find the value of $\tan A$.
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Q22
short answer
2 marks
Using distance formula, prove that the points A(2, 3), B(–7, 0) and C(–1, 2) are collinear.
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Q23
short answer
In the given figure, $AB || DE$ and $AC || DF$. Show that $\Delta ABC \sim \Delta DEF$. If $BC = 10$ cm, $EB = CF = 5$ cm and $AB = 7$ cm, then find the length DE.
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Q24
short answer
Evaluate : $\frac{\sin^3 60^\circ - \tan 30^\circ}{\cos^2 45^\circ}$
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Q25
short answer
For acute angles A and B, if $A + 2B$ and $2A + B$ are acute, $\tan (A + 2B) = \sqrt{3}$ and $\sin (2A + B) = \frac{1}{2}$, then find the measures of angles A and B.
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Q25
short answer
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.
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Q26
short answer
3 marks
A circle of diameter 20 cm is equally divided into five sectors. Find the area and perimeter of one of the sectors.
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Q28
short answer
In an A.P., 15th term exceeds the 8th term by 21. If sum of first 10 terms is 55, then form the A.P.
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Q28
short answer
3 marks
Prove that $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$
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Q29
short answer
The sum of first n terms of an A.P. is $2n^2 + 13n$. Find its nth term and hence 10th term.
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Q30
long answer
Solve the system of linear equations : x = 4 and 3x - 2y = 6 graphically.
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Q31
long answer
A circle centered at (2, 1) passes through the points A(5, 6) and B(-3, K). Find the value(s) of K. Hence find length of chord AB.
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Q31
long answer
The dimensions of a window are 156 cm $\times$ 216 cm. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.
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Q32
long answer
Prove that the point P dividing the line segment joining the points A(-1, 7) and B(4, -3) in the ratio 3 : 2, lies on the line x - 3y = -1. Also find length of PA and PB.
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Q32
long answer
5 marks
Venkat can row a boat in still water at the speed of 12 km/h. He ferries tourists 15 km upstream and 18 km downstream in 3 hours. Find the speed of the stream.
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Q35
long answer
5 marks
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.
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D is the mid-point of side BC of $\Delta ABC$. CE and BF intersect at O, a point on AD. AD is produced to G such that OD = DG.
Q36
long answer
5 marks
Prove that OBGC is a parallelogram.
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Q38
long answer
5 marks
Prove that $\Delta AEF \sim \Delta ABC$.
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Q39
long answer
5 marks
Through the mid-point Q of side CD of a parallelogram ABCD, the line AR is drawn which intersects BD at P and produced BC at R. Prove that AQ = QR
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Q40
long answer
5 marks
The mean of the following frequency distribution is 28. If sum of all frequencies is 100, then find the values of p and q : Class Interval 0-10, 10-20, 20-30, 30-40, 40-50, 50-60; Frequency 12, p, 27, 20, q, 6.
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Q41
long answer
5 marks
Find median and mode of the following distribution : Class Interval 0-15, 15-30, 30-45, 45-60, 60-75, 75-90, 90-105; Frequency 15, 10, 12, 9, 8, 10, 6.
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In the above figure, an arch of a railway bridge over a river is shown. It is a parabolic arch which joins two hills at points P and Q. If the polynomial representing the parabolic arch is $p(x) = -0.0025x^2 - 0.025x + 136$.
Q45
short answer
2 marks
Write the zeroes of the polynomial $p(x)$ with the help of the given figure. Verify the relationship between these zeroes and the coefficients of the polynomial.
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Q46
short answer
2 marks
Find the value of the polynomial $p(x)$ at x = 100 and x = -100. Are these two values equal?
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An arch of a railway bridge, built on Chenab riverbed, is shown in the above diagram. It is a parabolic arch connecting two hills at P and Q. If the parabolic curve is represented by the polynomial p(x) = –0.0025x^2 – 0.025x + 136.
Q47
short answer
2 marks
(a) Write the zeroes of the polynomial using diagram and verify the relationship between sum of zeroes and polynomials.
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Q48
short answer
2 marks
(b) Find the values of p(x) at x = 100 and x = –100. Are they same ?
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A wall mounted lamp, made of fabric, is shown below. Lamp has cuboidal shape, open from top and bottom. A spherical bulb of diameter 7 cm is latched with a very thin rod. (Ignore the rod while making calculations.) Dimensions of the cuboid are 24 cm × 12 cm × 17 cm.
Q49
short answer
1 mark
Find the surface area of the bulb.
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Q50
short answer
1 mark
What could be the maximum diameter of the bulb if at least 1 cm space is left from each side ?
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Q51
short answer
2 marks
(a) Find the area of the fabric used if there is a fold of 2 cm on top and bottom edges.
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Q52
short answer
2 marks
(b) Find the space available inside the lamp.
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Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram given above, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window (W) of the building, the angle of elevation of top of the tank is 45° and angle of depression of its foot is 30°.
Q53
short answer
1 mark
Write a relation between d (the height of window) and y.
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Q55
short answer
2 marks
(a) Determine height of the water tank.
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Q56
short answer
2 marks
(b) Find the value of x and height of the window above ground level.
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