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This exam covers key concepts in mathematics including number theory, geometry, algebra, and the properties of functions. Students should be familiar with solving equations, understanding properties of numbers, and worki…
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Prove that √3 is an irrational number.
Assume √3 is rational, derive a contradiction.
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Find a quadratic polynomial whose sum and product of zeroes are 0 and -9, respectively. Also, find the zeroes of the polynomial so obtained.
Polynomial is x^2 + 9, zeroes are ±3.
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Solve the following system of equations graphically: x + 3y = 6; 2x - 3y = 12.
Graph intersection gives solution.
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x and y are complementary angles such that x:y = 1:2. Express the given information as a system of linear equations in two variables and hence solve it.
x + y = 90°, x = 30°, y = 60°.
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Prove that: 1+√(a)/(1+√(b)) = (1−√(a)/(1−√(b))
Use algebraic manipulation and properties of radicals.
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Find the coordinates of the points of trisection of the line segment joining the points A(−1, 4) and B(−3, −2).
The points of trisection are (-1.67, 2) and (-2.33, 0).
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Prove that: (1+√(a))/(1−√(a)) + (√(b))/(1−√(b)) = 1+√(c)
Use algebraic identities and simplifications.
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Prove that: √(1 - cosA)/(1 + cosA) = tan(A/2).
Use half-angle identities.
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If the sum of first m terms of an AP is 2m^2 + 3m, then find its second term.
Second term is 9.
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The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.
The numbers are 18 and 12.
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Find the value(s) of k for which the equation 2x^2 + kx + 3 = 0 has real and equal roots. Hence, find the roots of the equations so obtained.
k = -6; roots are x = -3.
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State 'Basic Proportionality Theorem' and use it to prove the following: In a quadrilateral ABCD, diagonals AC and BD intersect each other at O such that AO/OC = BO/OD. Prove that ABCD is a trapezium.
Proportional segments imply parallel sides.
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Aarush bought 2 pencils and 3 chocolates for ₹11 and Tanish bought 1 pencil and 2 chocolates for ₹7 from the same shop. Represent this situation in the form of a pair of linear equations. Find the price of 1 pencil and 1 chocolate, graphically.
Prices are ₹2 for a pencil and ₹3 for a chocolate.
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The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that AF × FB = CF × FD.
Use properties of similar triangles.
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In triangle ABC, if AB ⊥ AC and AB^2 = AC × BC, then prove that ∠BCA = 90°.
Use Pythagorean theorem.
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Your elder brother wants to buy a car and plans to take a loan from a bank for his car. He repays his total loan of ₹1,18,000 by paying every month, starting with the first instalment of ₹1,000 and he increases the instalment by ₹100 every month. Based on the information given above, answer the following questions: (i) Find the amount paid by him in the 30th instalment. (1 Mark) (ii) If the total number of instalments is 40, what is the amount paid in the last instalment? (1 Mark) (iii) (a) What amount does he still have to pay after the 30th instalment? (2 Marks) OR (iii) (b) Find the ratio of the tenth instalment to the last instalment. (2 Marks)
30th instalment is ₹3,000; last instalment is ₹4,000.
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In a garden, saplings of rose flowers were planted at equal intervals to form a spiral pattern. The spiral is made up of successive semicircles, with centres alternatively at A and B, starting with centre at A, of radii 50 cm, 100 cm, 150 cm, ... Based on the above information, answer the following questions: (i) What is the radius of the 13th spiral? (1 Mark) (ii) If the radius of the nth spiral is 500 cm, find the value of n. (1 Mark) (iii) (a) Find the total number of saplings till the 11th spiral. (2 Marks) OR (iii) (b) Till which spiral, will there be a total of 450 saplings? (2 Marks)
Radius of 13th spiral is 650 cm; n = 10.
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In a society, there is a circular park having two gates. The gates are placed at points A(10, 20) and B(50, 50). Two fountains are installed at points P and Q on AB such that AP = PQ = QB. Based on the above information, answer the following questions: (i) Find the coordinates of the centre C. (1 Mark) (ii) Find the radius of the circular park. (1 Mark) (iii) (a) Find the coordinates of the point P. (2 Marks) OR (iii) (b) Find the distance of the fountain at Q from gate A. (2 Marks)
Centre C is (30, 35); radius is 28.28.
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Solve the following system of equations for x and y: x^2 + 2y^3 = -1 and x - y^3 = 3.
x = 1, y = -1
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In the given figure, if PQ ∥ RS, then prove that ΔABC ∼ ΔDEF.
Use AA criterion for similarity.
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If p and q are the zeroes of the quadratic polynomial ax^2 + bx + c, then find the value of p^3 + q^3.
p^3 + q^3 = (p + q)(p^2 - pq + q^2)
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If 2sin^2θ + 1 = 2, find the value of sin^2θ.
sin^2θ = 0.
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If the points A(4, 5), B(m, 6), C(4, 3), and D(1, n) taken in this order are the vertices of a parallelogram ABCD, then find the values of m and n.
m = 1, n = 5.
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If the HCF of two positive integers a and b is 1, then their LCM is:
ab
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The number 3+√2 is:
an irrational number
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The discriminant of the quadratic equation x^2 - 3x - 2 = 0 is:
17
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The equation x + 1/x = 3 (x ≠ 0) is expressed as a quadratic equation in the form of ax^2 + bx + c = 0. The value of a - b + c is:
2
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For a point (3, −5), the value of (abscissa − ordinate) is:
8
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The mid-point of a line segment divides the line segment in the ratio:
1 : 1
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Which of the following is not the criterion for similarity of triangles?
RHS
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From the figures given below, which of the following is true about the measure of angle P?
The measure of P cannot be determined
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Which of the following statements is false?
tan 90° = sin 45°
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The value of (sin^2 A - cos^2 A) is:
The graph of f(x) = g(x) is given. The number of distinct zeroes of f(x) = g(x) is:
2
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If p and q are two zeroes of a polynomial g(x) = ax^2 - 2bx + c and p + q = -b/a, then value of p is:
2/3
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If the pair of linear equations: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is consistent and dependent, then:
a1/a2 = b1/b2 = c1/c2
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Which of the following sequence is not an A.P.?
√2, √8, √18,...
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In triangles ABC and PQR, ∠A = ∠P and ∠B = ∠Q, then AB : AC is equal to:
PQ : PR
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If x = 1 and y = 2 is a solution of the pair of linear equations 2x - 3y + a = 0 and 2x + 3y - b = 0, then:
a = 2b
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If the sum of first m terms of an AP is 2m^2 + 3m, then its second term is:
9
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The distance of the point A(4a, 3a) from x-axis is:
3a
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