CAREER POINT
PRE FOUNDATION DIVISON
IMO Stage-II Exam.-2016
CLASS-10
MATHEMATICS
Date : 14-02-2016 Q.1 If one of the zeros of a quadratic polynomialof the form x2 + ax + b is negative of the other, then it ……… -
(A) Has no linear term and the constant term is negative
(B) Has no linear term and the constant term is positive
(C) Can have a linear term but the constant term is negative
(D) Can have a linear term but the constant term is positive
Q.2 Solve for x and y : b b 1 x + a a 1 y = b – a, b x – a y 4 = 5; ab 0. (A) x = –a, y = b (B) x = b2, y = a2 (C) x = a2, y = b2 (D) x = b, y = – a
Q.3 In the given figure, ABC is a triangle right angled at B and BD AC. If A = 4 cm and CD = 5 cm, find BD and AB respectively –
(A) 2 5 cm, 3 5 cm (B) 3 5 cm, 6 cm (C) 2 5 cm, 6 cm (D) 3 5 cm, 8 cm
Q.4 In ABC right angled at B, BC = 5 cm and AC – AB = 1 cm. Evaluate C cos C sin 1 - (A) 13 5 (B) 13 (C) 13 12 (D) 5
Q.5 Which of the following statement is correct ? (A) sin (A + B) = sin A + sin B
(B) The value of sin increases as decreases (C) The value of cos increases as increases (D) None of these
Q.6 The following table shows the daily pocket allowance given to the children of a multistory building. The mean of the pocket allowance is
18. Find out the missingfrequency – Class interval 11-13 13-15 15-17 17-19 19-21 21-23 23-25 Frequency (in
) 3 6 9 13 ? 5 4 (A) 8 (B) 16 (C) 12 (D) 4 Q.7 Three years ago, the average age of Latika,Garima and Megha was 27 years and that of Garima and Megha 5 years ago was 20 years. Latika’s present age is …….. -
(A) 30 years (B) 36 years (C) 40 years (D) 46 years
Q.8 Find the mode of the following frequency distribution : Marks 10-20 20-30 30-40 40-50 50-60 Number of students 12 35 45 25 13 (A) 20.33 (B) 30.12 (C) 33.33 (D) 60.43
Q.9 A small scale industry produces a certain number of items per day. The cost of production of each item (in rupees) was calculated to be 74 minus twice the number of articles produced in a day. On a particular day, the total cost of production was
540.Which of the following equations represent how to find the number of times produced on that day ?
(A) 74 + 2 x = 540 (B) x2 + 74 x – 540 = 0 (C) 74 – 2x = 540 (D) x2 – 37 x + 270 = 0
Q.10 The sum of first n terms of an A.P. is given by (n2 + 8n). Find the 12th term of the A.P. Also find the nth term of the A.P. -
(A) 31, 2n + 9 (B) 31, 2n + 7 (C) 30, 2n + 6 (D) 30, 2n + 8
Q.11 IN the given figure, PQ is the chord of circle an PT is the tangent at P such that QPT = 60º. Then PRQ is –
(A) 135º (B) 150º
(C) 120º (D) 110º
Q.12 In the given figure, arcs are drawn by taking vertices A, B and C of an equilateral triangle of side 10 cm to intersect the sides BC, CA and AB at their respective mid-points D, E and F. Find the area of the shaded region. [Use = 3.14] –
(A) 39.25 cm2 (B) 48.50 cm2 (C) 78.50 cm2 (D) 28.25 cm2
Q.13 Two cleanliness hoardings are put on two poles of equal heights standing on either side of a roadway 50 m wide between the poles. The elevations of the tops of the poles from a points between them are 60º and 30º. Find the height of the pole -
(A) 50 3 m (B) 3 25 3 m (C) 25 3 m (D) 2 25 3 m
Q.14 Beena gave a simple multiplication question to her students. Bt one student reversed the digits of both numbers and carried out the multiplication and found that the product was exactly the same as the one expected by Beena. Which on e of the following pairs of numbers will fit in the description of the questions ?
(A) 14, 22 (B) 13, 62 (C) 19, 33 (D) 42, 28
Q.15 For what values of k will the following pairs of linear equations have infinitely many solutions ? 2x + 3y = 4 and (k + 2) x + 6y = 3k + 2
(A) 1 (B) –1
(C) 2 (D) –2
Q.16 The values of for which the quadratic equation x2 + 5/x + 16 = 0 has no real root is - (A) > 8 (B) < –5 (C) – 5 8 < < 58 (D) – 5 8 < 0
Q.17 A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30º, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car to reach the foot of the tower from this point - (A) 3 seconds (B) 6 seconds (C) 9 seconds (D) 5 seconds Q.18 Tow dice are thrown simultaneously. What is
the probability of getting two numbers whose product is odd ? (A) 2 1 (B) 4 3 (C) 8 3 (D) 4 1
Q.19 Find the median of the following data :
Marks 0-10 10-20 20-30 30-40 40-50
No. of Students 10 18 40 20 12
(A) 51.5 (B) 25.5
(C) 28.5 (D) 31
Q.20 A number is selected at random from the numbers :
5, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 24, 25, 25, 27, 30, 30, 36, 37, 37, 39, 40, 40, 46. Find the probability that the selected number is a prime number. (A) 0 (B) 8 1 (C) 6 1 (D) 12 1
Q.21 The value of expression
4 / 1 2 2 / 1 3 / 2 3 / 2 6 / 1 4 / 1 3 / 1 ) 243 .( 3 1 ) 3 .( ) 9 . 0 ( ) 81 . 0 .( ) 9 .( 27 1 . ) 3 . 0 ( + 1 3 1 3 1 0 3 1 2 3 . 2 3 ) 1 . 0 ( ) 6 . 0 ( is - (A) – 0.2 (B) 0.9 (C) 1.27 (D) – 0.06
Q.22 There are two circles intersecting each other. Another smaller circle with centre O, is lying between the common region of two larger circles. Centres of the circle (i.e., A, O and B) are lying on a straight line. AB = 16 cm and the radii of the larger circles are 10 cm each. What is the area of the smaller circle ?
(A) 4 cm2 (B) 2 cm2 (C) 4 cm2 (D) 4 cm2
Q.23 If the points A(–2, 1), B(a, b) and C(4, –1) are collinear and a – b = 1. Find the values of a and b respectively.
(A) 1, 0 (B) 1, – 1 (C) 0, 1 (D) –1, 1
Q.24 A circular paper is folded along its diameter, then again it is folded to form a quadrant. Then it is cut as shown in the figure, after it the paper was reopened in the original circular shape. Find the ratio of the original paper to that of the remaining paper?
(The shaded portion is cut off from the quadrant. The radius of quadrant OAB is 5 cm and radius of each semicircle is 1 cm) (A) 25 : 16 (B) 25 : 9
(C) 20 : 9 (D) None of these Q.25 There are 100 apples in a box. 20 of them are
rotten. At random, two apples are taken one by one consecutively without replacement. What is the probability that both of them are good? (A) 495 316 (B) 495 19 (C) 99 16 (D) 99 32
Q.26 Which of the following statements is INCORRECT ?
(i) In order to divide a line segment internally in the ration m : n, both m and n are real numbers.
(ii) A pair of tangents can be constructed to a circle inclined at an angle of 165º. (A) Only (i) (B) Only (ii)
(C) Both (i) and (ii) (D) Neither (i) nor (ii) Q.27 If and be two zeros of the quadratic polynomial p(x) = 2x2 – 3x + 7, evaluate 3 2 1 + 2 3 1 . (A) – 14 3 (B) 7 3 (C) – 4 5 (D) 14 3 Q.28 5 – 3 – 2 is ... - (A) A rational number (B) A natural number (C) Equal to zero
Q.29 In figure, the line segment LM is parallel to side XZ of XYZ and it divides the triangle into two parts of equal areas. Find the ratio XY XZ . Y X Z M L (A) 2 – 1 : 2 (B) 2 + 1 : 2 (C) 1 – 2 : 2 (D) 2 – 2 : 2 Q.30 Two ships are sailing in the sea on the either
side of the lighthouse, the angles of depression of two ships as observed fro the top of the lighthouse are 60º and 45º respectively. If the distance between the ships
is 200 3 1
3 metres, find the height of the
lighthouse. (A)
3
100m (B) 100 m
(C) 200 m (D) (1 + 3 )m
Q.31 A well of diameter 3 m is dug 14 m deep. The earth taken of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.
(A) 4.50 m (B) 2.125 m (C) 1.125 m (D) 3.25 m
Q.32 In the given figure, if POQ = 130º, then SOR is equal to - S O Q R P (A) 50º (B) 45º (C) 35º (D) 55º
Q.33 In the given figure, XW is a tangent to the circle with centre O at X and YZW is a straight line. Find the value of y.
(A) 30º (B) 35º (C) 40º (D) 50º Q.34 If the common difference of an A.P. is 5, then
a18 – a13 = ………… .
(A) 5 (B) 20 (C) 30 (D) 25 Q.35 If the points A(1, –2), B(2, 3), C(–3, 2) and
D(–4, –3) are the vertices of parallelogram ABCD, then taking AB as the base, find the height of this parallelogram.
(A) 24 26units (B) 13 14 units (C) 26 24 units (D) 14 13 units
Q.36 If the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, –3) in the ratio k : 1, then find the value of x.
(A) 8 (B) 8
67 (C)
3
67 (D) 9
Q.37 ABC is right-angled triangle, right angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 5 cm and 12 cm. Find the radius of the circle. (A) 1/2 cm (B) 13 cm
(C) 2 cm (D) 10 cm
Q.38 In figure, ABCDEF is any regular hexagon with different vertices A, B, C, D, E and F as the centres of circles with same radius 'r' units are drawn. Find the area of the shaded portion –
(A) 2r2 sq. units (B) 4r2 sq. units
Q.39 A cylindrical pipe has inner diameter of 4 cm and water flows through it at the rate of 20 m per minute. How long would it take to fill a conical tank, with diameter of base as 80 cm and depth 72 cm ?
(A) 5 minutes
(B) 3 minutes 56 seconds (C) 4 minutes 20 seconds (D) 4 minutes 48 seconds
Q.40 If sum of the squares of zeros of the polynomial 6x2 + x + k is
36
25, find k. (A) 2 (B) –2 (C) 1 (D) –1 Q.41 In the Maths Olympiad in a school, two
representatives from two teams, while solving a quadratic equation, committed the following mistakes :
(i) One of them made a mistake in the constant term and got the roots as 5 and 9.
(ii) The other committed an error in the coefficient of x and he got the roots as 12 and 4.
In the meantime, they realized that they were wrong and together they managed to get it right. Find the right quadratic equation. (A) x2 + 4x + 14 = 0
(B) 2x2 + 7x – 24 = 0 (C) x2 – 14x + 48 = 0
(D) 3x2 – 17x + 52 = 0
Q.42 What must be added to the polynomial 34 + 53 – 72 + 5 + 3 so that the resulting polynomial is exactly divisible by 2 + 3 + 1 ? (A) –3 + 1 (B) –3 – 1
(C) 3 + 1 (D) 3 –1
Q.43 In the given figure, DE || BC. If DE : BC = 3 : 5, find the ratio of the area of ADE to the area of trapezium BCED.
(A) 9 : 16 (B) 25 : 9 (C) 16 : 9 (D) 9 : 25 Q.44 If sin cos sin cos = 3 1 3 1 and 0º < < 90º, then find the angle .
(A) 30º (B) 60º (C) 90º (D) 45º Q.45 Evaluate : 4 (sin4 30º + cos4 60º) – 3(cos2 45º
– sin2 90º) + (sin2 60º + sin2 45º) (A) 3 4 1 (B) 4 1 (C) 4 3 (D) 16 7 ACHIEVERS SECTION
Q.46 Which of the following options hold ? Statement-1 : If p, q, r and s are real numbers such that pr = 2 (q + s), then atleast one of the equations x2 + px + q = 0 and x2 + rx + s = 0 has real roots.
Statement-2 : If a, b, c are distinct real numbers, then the equation.
(x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 Has real and distinct roots.
(A) Statement-1 is true, Statement-2 is false. (B) Statement-1 is false, Statement-2 is true. (C) Both Statement-1 and Statement-2 are
true.
(D) Both Statement-1 and Statement-2 are false.
Q.47 Rakesh has to buy a TV. He can buy TV either making cash down payment of ` 14000 at once or by making 12 monthly installments as below :
` 1500 (1st month), ` 1450 (2nd month), ` 1400 (3rnd), ` 1350 (4th month), ………. .
Each installment except the first is ` 50 less than the previous one.
Find :
(i) Amount of the installment paid in the 9th
month.
(ii) Total amount paid in 12 installments. (iii) How much extra he has to pay in addition
to the amount of cash down payment ?
(i) (ii) (iii)
(A) ` 1100 ` 16700 ` 900 (B) ` 1200 ` 14700 ` 600 (C) ` 1100 ` 14700 ` 700 (D) ` 1100 ` 14700 ` 900
Q.48 Find the mode of the distribution from the following table : Marks Less then 20 Less than 40 Less than 60 Less than 80 Less than 100 No. of Students 4 10 28 36 50 (A) 48,904 (B) 50.909 (C) 62.804 (D) 64.324
Q.49 Arrange the given steps in correct order while constructing a triangle similar to given triangle ABC with its sides equal to
4 3 of the corresponding sides of the triangle ABC (i.e, of scale factor
4 3
).
(1) Locate 4 points B1, B2, B3 and B4 on BX
so that BB1 = B1B2 = B2B3 = B3B4.
(2) Draw a line through C' parallel to the line CA to intersect BA at A'.
(3) Draw any ray BX making an acute angle with BC on the side opposite to the vertex A.
(4) Join B4C and draw a line through B3
parallel to B4C to intersect BC at C'.
Then, A'BC' is the required triangle. (A) 3, 4, 2, 1 (B) 2, 4, 3, 1 (C) 3, 1, 4, 2 (D) 2, 4, 1, 3
Q.50 Match the columns :
Column-I Column-II (i) In figure, a circle touches
all four sides of quadrilateral ABCD with AB = 18 cm, BC = 21 cm and CD = 12 cm, AD = ………
(a) 10 cm
(ii) In figure, value of x is
………. (b) 9 cm
(iii) Perimeter of DPST with PQ = 10 cm is ……. (c) 3 cm (iv) In figure, PQ = 6cm, QR = 7cm, RS = 4 cm PS = ………. (d) 20 cm
(A) (i) (b), (ii) (a), (iii) (c), (iv) (d) (B) (i) (a), (ii) (b), (iii) (d), (iv) (c) (C) (i) (b), (ii) (d), (iii) (a), (iv) (c) (D) (i) (b), (ii) (a), (iii) (d), (iv) (c)
