Nusret Balcı TMUA Paper 1
20 questions · 75 min · Difficulty 63 · by Nusret Balcı
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Questions in this paper
- Question 1Evaluate the integral ∫_0^1x^5(x+2)^2(3x+4) dx.
- Question 2A sequence of similar sectors S_1,S_2,S_3,… has a constant central angle θ (in radians). The radius r_(n+1) of each sector is defined by…
- Question 3A geometric sequence has first term a and common ratio r, where a and r are positive integers and r>1. Define the alternating partial sums…
- Question 4Find the number of solutions to the equation 5sec6x=3x, -π≤ x≤π.
- Question 5The function f(x) is defined for all real numbers x by: f(x)=8^x-6·4^x+9·2^x Let S be the set of all integer values of k such that the…
- Question 6Let k∈mathbb R. Consider the equation (1+3cosθ)^2=kcos2θ, 0°≤θ≤360°. For which values of k does the equation have exactly four distinct…
- Question 7It is given that f(x)=x^2-4x. Two curves are defined by: C_1:y=f(kx) C_2:y=f(x-p) where k and p are constants such that k>1 and p>0. The…
- Question 8Point P lies on the circle with equation: (x-1)^2+(y-2)^2=4 Point Q lies on the circle with equation: (x-k)^2+(y-14)^2=9 where k is a…
- Question 9The difference between the maximum and minimum values of the function f(x)=a^(sin x)-a^(-sin x) where a>0 and x is real, is 3. Find the sum…
- Question 10The points A(2,3) and B(9,-1) are two vertices of a triangle. The third vertex lies on the line y=x and has coordinates C(k,k). Given that…
- Question 11A family of circles is defined such that their centers lie on the line y=2x. Every circle in the family is tangent to the line y=x. The…
- Question 12The point P(x_P,y_P) lies on the curve C with equation y^2=4x, such that y_P>0. Normals are drawn to the curve C at three distinct points…
- Question 13Consider the polynomial f(x)=x^3-3ax+b where a and b are real parameters. The parameter a varies in the range 0≤ a≤1. The parameter b…
- Question 14Let m be a real constant such that 0≤ m≤1. We wish to approximate the curve y=x^2 with the line y=mx on the interval [0,1]. Let E(m) be the…
- Question 15Let p,q,r,s be real numbers with p<q<r<s. Consider the function f(x) defined by: f(x)=|x-p|+|x-q|+|x-r|+|x-s| The set of values of x for…
- Question 16The function f(x) is a polynomial that satisfies the equation f'(x)+∫_0^x f(t) dt=x^3+kx^2+x+2 for all real values of x, where k is a real…
- Question 17A sequence of real numbers v_0,v_1,v_2,… is defined by v_0=0, v_1=1, and the linear recurrence relation: v_(n+1)=√2 v_n-v_(n-1) n≥1. What…
- Question 18Let f(x) and g(x) be two polynomials with coefficients in mathbb R, defined as: f(x)=x^2-8x+α g(x)=-x^2+4x+β When both f(x) and g(x) are…
- Question 19Consider the following inequality: m|x|+1≤(x-3)^2+8 where m is a real constant. Which one of the following describes the complete set of…
- Question 20Consider a non-degenerate triangle with angles given by x and 2x (measured in degrees). The side lengths of this triangle form a…